workshop

Editor

G. R. Liu

ADVANCES IN

ESHFREE AND X - F E M

ETHODS Proceedings of the 1st Asian Workshop

on Meshfree Methods

World Scientific

ADVANCES IN

MESHFREE AND X-FEM METHODS Proceedings of the 1st Asian Workshop

on Meshfree Methods

This page is intentionally left blank

ADVANCES IN

MESHFREE ANDX-FEM

METHODS Proceedings of the 1 st Asian Workshop

on Meshfree Methods

Singapore 16-18 December 2002

Editor

G. R. Liu National University of Singapore

V f e World Scientific w b New Jersey'London'Singapore* New Jersey'London • Singapore • Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

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USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661

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ADVANCES IN MESHFREE AND X-FEM METHODS

Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd.

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Printed in Singapore by Mainland Press

V

PREFACE

This volume collects the Proceedings of the 1st Asian Workshop on Meshfree Methods held as a part of the 2nd International Conference on Structural Stability & Dynamics (ICSSD02) onl6 - 18 December 2002 in Singapore. The workshop proceedings contain 36 papers covering a large number of the aspects on meshfree and the extended finite element methods (X-FEM).

The aim of the workshop is to provide researchers working with meshfree methods an opportunity to exchange freely their new ideas, concepts, and techniques in this rapidly developing area of research. The idea of organizing this workshop was conceived among Professors Yagawa, W. Kanok-Nukulchai, H. Noguchi and G. R. Liu during the first Asian-Pacific Congress on Computational Mechanics held in Sydney in 2001. Profs. W. Kanok-Nukulchai, H. Noguchi and G. R. Liu have then worked together to materialize this idea. Since G. R. Liu was at that time helping to organize the ICSSD02 in Singapore, the workshop is then naturally held as a part of the ICSSD02, so that other ICSSD02 dedicates can participate in the meshfree workshop, and the workshop dedicates can also benefit from all the practical engineering examples presented in the ICSSD02 as well as other workshops held under the umbrella of ICSSD02.

I wish to express my sincere appreciation to Professors Yagawa, W. Kanok-Nukulchai, H. Noguchi for their strong support. Without their guidance, help and support, we will not be able to receive this excellent response to this workshop. I would like to also offer my sincere thanks to all the invited speakers, technical session speakers and participants. They are the ones who made this workshop possible and built up this volume through their hard works in preparing their manuscripts.

My special thanks go to Profs. Ohtsubo and Suzuki who provided very strong support to the workshop. In addition, they have put up a very good session of an interesting topic of the X-FEM methods.

I would also like to mention a very encouraging finding in the process of editing this volume: many young students and researchers have contributed a lot of papers on meshfree methods to this workshop. I am very happy to find out about this, as they are the ones to take the meshfree and other advanced methods to a new height in the future.

I would also like to express my sincere appreciation to all the conference sponsors for all the support provided by them. Last but not least, I would like to thank Suwamo, Ms. Lim Hui Leng and other PAC members at the National University of Singapore for their assistance in the preparation of this proceedings and the management of the workshop.

G. R. Liu

vii

The Editor

Dr. G. R. Liu

Dr. Liu received his PhD from Tohoku University, Japan in 1991. He was a Postdoctoral

Fellow at Northwestern University, USA. He is currently the Director of the Centre for

Advanced Computations in Engineering Science (ACES), National University of

Singapore. He is also an Associate Professor at the Department of Mechanical

Engineering, National University of Singapore. He authored more than 250 technical

publications including more than 150 international journal papers and 5 books. He is the

author of the book entitled "Mesh Free Method: Moving Beyond the Finite Element

Method". He is the recipient of the Outstanding University Researchers Awards

(1998), the Defence Technology Prize (National award, 1999), and the Silver Award at

CrayQuest 2000 Nationwide competition. His research interests include Computational

Mechanics, Element Free Methods, Nano-scale Computation, Micro Bio-system

computation, Vibration and Wave Propagation in Composites, Mechanics of Composites

and Smart Materials, Inverse Problems and Numerical Analysis.

Organization

IX

Conference Organization

Department of Civil Engineering National University of Singapore

Meshfree Workshop Organization

Centre for Advanced Computations in Engineering Science (ACES) Department of Mechanical Engineering National University of Singapore

Sponsors

Institution of Structural Engineers, Singapore Branch Singapore Structural Steel Society

Centre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore

Nanyang Centre for Super Computing and Visualization Nanyang Technological University

Army Research Office-Far East (ARO-FE)

Army Research Office (ARO)

Asian Office of Aerospace Research and Development (AOARD)

Office of Naval Research International Field Office Asia (ONR IFOA)

X

Conference Organizing Committee

Chairman: C. M. Wang, NUS

Co-Chairman: N. E. Shanmugam, NUS

Hon. Secretary: K. K. Ang, NUS

Technical Comm. Chairman: G. R. Liu, ACES, NUS

Members: Y. S. Choo, NUS

C. G. Koh, NUS

J. Y. R. Liew, NUS

K. M. Liew, NTU

Q. Wang, NUS

Advisors: Y. B. Yang, National Taiwan University

J. N. Reddy, Texas A &M University

Meshfree Workshop Organizing Committee

Chairman: G. R. Liu, National University of Singapore, Singapore

Co-Chair men: W. Kanok-Nukulchai Asian Institute of Technology,

Pathumthani,Thailand

H. Noguchi, Keio University, Japan

XI

Contents

Preface v

The Editor vii

Organization ix

Section 1. Keynote Paper 1

1.1 Seamless and Parallel Computing by Using Free Mesh Method: A Kind of Meshless Technique 3 G. Yagawa

Section 2. Meshfree Formulations 5

2.1 Analysis of 3D Solid with Complicated Geometry using Finite Cover Method (Invited Paper) 7

K. Suzuki and H. Ohtsubo

2.2 A Meshless Method Using Radial Basis Functions for Solving Wave Equations (Invited Paper) 15

C.S. Chen, Jichun Li and D.W. Pepper

2.3 Meshless Computational Method By using Radial Basis Functions (Invited Paper) 16 Benny Y.C. Hon

2.4 Recent Advances in the Method of Fundamental Solutions 17 A. Karageorghis, G. Fairweather and P.A. Martin

2.5 A Study on the Patch Test of Point Interpolation Methods 23 Y.TGu

2.6 A Comparison Between Radial Point Interpolation Method

(RPIM) and Kriging Based Meshfree Method 29 G.R. Liu, K.Y. Dai, Y.T. Gu and KM Lint

2.7 Radial Basis Point Interpolation Collocation Method For 2-D

Solid Problem 35 Xin Liu, G.R. Liu, Kang Tai and K. Y. Lam

XII

Section 3. Meshfree Methods for Smart Materials/ Structures 41

3.1 Point Interpolation Mesh Free Method for Static and Frequency

Analysis of Two-dimensional Piezoelectric Structures 43 K.Y Dai, G.R. Liu and KM. Lim

3.2 A Hybrid Meshless-Differential Order Reduction ( H M - D O R )

Method for Deformation Control of Smart Circular Plate by

Sensors/Actuators 49 /. Q. Cheng, Hua Li, K.Y. Lam, T. Y. Ng and Y.K. Yew

Section 4. Meshfree Methods for Fracture Analysis 55

4.1 Application of 3D Free Mesh Method to Fracture Analysis of Concrete 57 Hitoshi Matsubara, Shigeo Iraha, Jun Tomiyama and Genki Yagawa

4.2 Meshless Analysis Integrate System for Structural and Fracture Mechanics Analysis 63 Seiya Hagihara, Mitsuyoshi Tsunori, Torn Ikeda, Noriyuki Miyazaki,

Takayuki Watanabe and Chaunrong Jin

4.3 Application of 2- Dimensional Crack Propagation Problem using

Free Mesh Method 69

/. Imasato and Y. Sakai

Section 5. Meshfree Methods for Membranes, Plates & Shells 75

5.1 Analysis of Membrane Structures with Large Sliding Cable using Mixed Displacement Formulation and EFGM (Invited Paper) 77

Hirohisa Noguchi, Yoshitomo Sato and Tetsuya Kawashima

5.2 The Effects of the Enforcement of Compatibility in the Radial Point Interpolation Method for Analyzing Mindlin Plates 84

X. L. Chen, G.R. Liu and S.P Lim

5.3 A Mesh Free Method for Dynamic Analysis of Thin Shells 90

L. Liu and V.B.C. Tan

5.4 A Conforming Point Interpolation Method for Analyzing Spatial

Thick Shell Structures 96

L. Liu, G.R. Liu, V.B.C. Tan and Gu, Y.T.

Section 6. Meshfree Methods for Soil 107

6.1 Characteristics of Localized Behavior of Saturated Soil with Pore Water via Mesh-Free Method 109 S. Arimoto, A. Murakami

6.2 Radial Point Interpolation Method for Interface Problems 115 /. G. Wang, T. Nogami and Md. Rezaul Karint

Section 7. Meshfree Methods for CFD 121

7.1 Application of Free Mesh Method to Viscoplastic Flow Analysis of Fresh Concrete 123 Jun Tomiyama, Yoshitomo Yamada, Shigeo Iraha and Genki Yagawa

7.2 A Meshless Local Radial Point Interpolation Method (LRPIM) for Fluid Flow Problems 129 y. L. Wu

73 Application of Meshless Point Interpolation Method with Matrix Triangularization Algorithm to Natural Convection 135 G. R. Liu and Y.L Wu

7 A The Solution for Convection-Diffusion Equations using the Quasi-Interpolation Scheme with Local Polynomial Reproduction Based on Moving Least Squares 140 Xin Liu, G.R. Liu, Kang Tai and K.Y Lam

Section 8. Boundary Meshfree Methods 149

8.1 Regular Hybrid Boundary Node Method (Invited Paper) 151 J.M Zhang and Z.H. Yao

8.2 Radial Boundary Node Method for Elastic Problem 161 H. Xie, T. Nogami and J.G. Wang

8.3 A Hybrid Boundary Point Interpolation Method (HBPIM) and its Coupling with EFG Method 167 Y.T. Gu and G.R. Lin

Section 9. Coding, Error Estimation, Parallisation 177

9.1 Error Regulation in EFGM Adaptive Scheme (Invited Paver) 179 W. Kanok-Nukulchai and X.P. Yin

XIV

9.2 Object Oriented Development of FMM3D: Foundation Software for Parallel 3D Free Mesh Method 194

Yutaka Nakama, Akio Shimada, Yasuhiro Kanto, Tomoaki Ando and Genki Yagawa

9.3 An Approach for Nodal Selection in MFree2D® 200 G.R. Liu, Edgar Frijters and Y.T. Gu

Section 10. Meshfree Particle Methods 209

10.1 Coupling Meshfree Particle Method with Molecular Dynamics Novel Approach for Multdscale Simulations 211

M.B. Liu, G.R. Liu and K.Y. Lam

10.2 Adaptive Smoothed Particle Hydrodynamics with Strength of Materials, Part I 217 G.L. Chin, K.Y. Lam and G.R. Liu

10.3 Adaptive Smoothed Particle Hydrodynamics with Strength of Materials, Part II 223 G.L. Chin, K.Y. Lam and G.R. Liu

10.4 Numerical Simulation of Perforation of Concrete Slabs by Steel

Rods using SPH Method 229

H.F. Qiang and S.C. Fan

Section 11. X-FEM 237

11.1 Three Dimensional Crack Growth Analysis using Overlaying Mesh Method and X-FEM (Invited Paper) 239

S. Nakasumi, K. Suzuki and H. Ohtsubo

11.2 Buckling Analysis of Composite Laminates with Delaminations

using X-FEM 245

T. Nagashima and H. Suemasu

11.3 Boundary Condition Enforcement in Voxel-Type FEM 251

T. Nagashima

Author Index 257

SECTION 1

Keynote Paper

3

Advances in Meshfree andX-FEMMethods, G.R. Liu, editor, World Scientific, Singapore 2002

SEAMLESS AND PARALLEL COMPUTING BY USING FREE MESH METHOD: A KIND OF MESHLESS TECHNIQUE

G. Yagawa

School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN

yagawa@q. t. u-tokyo. ac.jp

With the amazing progress of computers, the finite element method (FEM) has become widely used in many practical situations. There exists, however, a large gap between its industrial applications and academic fundamental studies. One of its reasons is due to the difficulty in the pre-processing for FEM, i.e. mesh generation. The generation of mesh is a very difficult task if the degree-of-freedom of the analysis model is extremely large, or if the geometry of the domain is much complex. In addition, mesh generation is usually carried out in a sequential algorithm, although parallel computing is becoming available for large-scale numerical. Therefore, we need to develop a new CAE algorithm, where parallel computing is employed throughout the process. Especially, the sequential procedure of mesh generation becomes a serious bottleneck in the whole computational processing, if frequent mesh refinement is required in the problems, such as moving boundary problem, compressible flow involving shocks, crack propagation, large deformation problem.

On the other hand, in order to avoid the troublesome processing of mesh generation for the FEM analysis, various 'meshless methods' have been proposed, in which analysis domain is discretized without employing any 'mesh' or 'elements'. The smooth particle hydrodynamics (SPH), the diffuse element method (DEM), the element-free Galerkin method (EFGM), the reproducing kernel particle method (RKPM), the moving-particle semi-implicit method (MPS) are among others. However, it seems that these meshless methods have not succeeded in replacing the FEM analysis completely, while they show excellent performance in several special fields. On the other hand, new finite element approaches are proposed in order to overcome the difficulty of mesh generation, such as the manifold method the voxel finite element method, the generalized finite element method (GFEM), the extended finite element method (X-FEM) , the finite cover method (FCM), the free mesh method (FMM) and the node-by-node finite element method (NBN-FEM). Although these methods employ elements or mesh, they are not given a priori, so that they are categorized as 'meshless method' in a wide sense or 'mesh free methods'. The FMM or NBN-FEM aims at seamless finite element computing from GAD models to the final numerical solutions in the parallel environments. In the method, both pre-processing and main-processing of finite element analysis can be parallelized in terms of nodes, where the pre-preprocessing involves the local mesh generation and the construction of system of equations, and the main-processing indicates the solution of system of equations. The method is quite suitable for massively parallel environments, while the commercial parallel mesh generator can use only ten processors at most. Here,

http://ac.jp

4

we discuss the algorithm of the method with several numerical examples.

Reference

G. Yagawa, T. Yamada, Free mesh method: A new meshless finite element method, Comp. Mech. 1996;18:383-386.

G. Yagawa, T. Furukawa, Recent developments of free mesh method, Int. J. Numer. Meth. Engrg. 2000;47:1419-1443.

M. Shirazaki, G Yagawa, Large-Scale parallel flow analysis based on free mesh method: a virtually meshless method, Comput. Methods Appl. Mech. Engrg. 1999;174:419-431.

G. Yagawa, Parallel computing of local mesh finite element method, Proc. The First Asian-Pacific Congress on Computational Mechanics, Sydney, 2001;17-26.

T. Fujisawa, G. Yagawa, Node-based parallel mesh generation and finite element solver for high Speed compressible flows, Proc. the Fifth world Congress on Computational Mechanics (WCCM V), July 7-12, 2002,

Vienna, http://wccm.tuwien.ac.jp

G. Yagawa, Node-by-node parallel finite elements: A virtually meshless method, ibid.

http://wccm.tuwien.ac.jp

SECTION 2

Meshfree Formulations

7

Advances in Meshfree and X-FEM Methods, G.R. Liu,editor World Scientific, Singapore,2002

ANALYSIS OF 3D SOLID WITH COMPLICATED GEOMETRY USING FINITE COVER METHOD

K. Suzuki Department of Environmental Studies, Graduate School of Frontier Sciences,

University of Tokyo, Japan

katsu@k. u-tokyo. ac.jp

H. Ohtsubo Department of Environmental and Ocean Engineering, University of Tokyo, Japan

ohtsubo@nasl. t. u-tokyo.ac.jp

Abstract

The new meshless method based on the cover least square approximation, which utilize cover instead of points that are used in the moving least square approximation, is proposed. For the cover distribution, multi scale voxel data is used for the ease of analysis in 3D solid. Several 3D examples are shown for demonstration. The method is applied to adaptive analysis of seepage flow problem of rock including complicated cracks. The rock model with 3 cracks and 1000 cracks are analyzed.

Introduction

The Finite element method has been widely used in the design process of industrial products as CAE tools. As the design process moves to 3D CAD and as the computational power increase, the FE analysis also have been moving to 3D. In the 3D analysis the most of the time is consumed in the generation of FEM model. Especially for the analysis of 3D solid with complicated geometry the generation of model often takes several weeks to months, or sometimes impossible to make model. Under these circumstances, the meshless analysis method has been emerging which does not require the mesh in the analysis, but still meshless approach has several disadvantage compared to FEM, and has not been widely used in the engineering practice. Especially in the analysis of 3D solid, the integration of 3D domain and increase of computational costs prevents the meshless method from becoming popular. The authors have been proposing the alternative meshless approach, and named Finite Cover Method (FCM). The FCM aims at 3D solid analysis and utilizes voxel concept.

http://ac.jp

http://ac.jp

8

Single scale voxel is used in the PU version of FCM (FCM-PU), which employs voxel cover for the cover of Manifold Method. By changing the polynomial order of cover functions it is possible to control the accuracy locally (p-type adaptive). Also we extended the method to utilize multi-level sized voxel to control the accuracy by mesh size (h-type adaptive) using Cover Least Square Approximation (CLSA, FCM-CLSA). The model generation of the voxel analysis is so simple and effective that it is possible to mesh any complex structure.

Formulation

Voxel Analysis

The voxel analysis was proposed by Kikuchi et. al. (Hollister S and Kikuchi N (1994), Terada K and Kikuchi N (1996)). The model generation of the voxel analysis is so simple and effective that it is possible to mesh any complex structure either from CAD data or real object by CT scanner (Figure 1). They used voxel data as HEXA element in FEM. However, because all the elements are uniform, any local refining requirement causes a global refining. When small voxels are necessary to improve the accuracy in a local area, a numerous number of elements and degrees of freedom are required. How to get reasonable accuracy and avoid the sharp increase of degrees of freedom is a problem of the voxel analysis.

Suzuki et.al. (1998) used boundary shape voxel (Figure 2) that subdivide the boundary shape voxel into smaller voxels for better geometric representation of original shape without increasing the analysis voxel, and used the boundary shape voxel for the domain integration and applying boundary conditions (displacement and foruce).

Figure 1. Voxel Analysis (Kikuchi and Diaz (1996))

9

Figure 2. Voxel analysis with Boundary Shape Voxel

Finite Cover Method

Since voxel analysis divide the domain into same size cube, it is impossible to control the accuracy locally, which is commonly used in the FEM by changing the size of the element. In the Finite Cover Method we have developed 2 methods to control the accuracy. In the FCM-PU, (Figure 3 left) by keeping the size of the analysis voxel same, the degree of polynomial for approximation function is changed. In the FCM-CLSA, the size of the voxel is changed to accept multi scale voxel subdivision (Figure 3 right) and Cover Least Square Approximation is proposed for constructing shape function.

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Sample Mathematical Cover field

Boundary of analysis domain

Sample Mathematical Cover field

Boundary of analysis domain

Figure 3 FCM-PU and FCM-CLSA

Finite Cover Method-PU

In FCM-PU, the concept of cover in the Manifold Method (Shi, (1991)) is used, which separate the definition domain for approximation function (cover) and physical domain where basic equation should be satisfied. Since the cover can be defined arbitrary independent from physical domain, the flexibility of the model generation is increased considerably. FCM utilize voxel shape cover for the mathematical cover in Manifold Method. As shown in Figure 3 left, the each physical domain is covered by 4 voxel

10

covers (in 3D, 8 covers) in FCM-PU. The displacement functions are approximated as follows.

k

u(x) {" f#(x)w,(x) (1) i 1

where f (x) is cover function and w,.(x) is weight function. The weight function needs to satisfy following conditions, where U, is common domain of mathematical cover and physical domain. Equation (3) is called Partition of Unity condition that guarantees the reproducibility of the function in cover functions. Suzuki et. al. (1998) also discuss the weight function that guarantee the linear independency of approximation functions for arbitrary degree of polynomial order of cover functions, which allow accuracy control by changing the order of polynomial. The displacement function (1) is substituted into Galerkin formulation to derive the linear set of equations. Figure 5 is the example of the analysis of gear. The gear is divided into 40x40x4 analysis voxel.

Tw,(x) x

4W ,(X)

f »,C

Lo; I K* W l#

^ .- * A * ' 1 •»*•• m

"*- ** ' ixed

Figure 4 Analysis of Gear

Finite Cover Method-CLSA

When the size of voxel is changed as in Figure 3 right, it is not easy to find the weight function that satisfies PU condition. Jin et. al. (2000) proposed Cover Least Square Approximation (CLSA) that is similar to the Moving Least Square Approximation (MLSA) but instead of evaluating the function by nodes, CLSA evaluate the function by cover. In CLSA, the approximation function "(*) is defined as equation (4), which is same function as the one used in MLSA. In the CLSA, to derive approximation function the functional J in equation (5) is minimized. The evaluation of the functional to be minimized is carried out on each cover, while in MLSA the evaluation is evaluated at

Ox U,

Ox U, ( 2 )

0 1 (3)

11

each node. It has been proved that by CLSA, the linear independency between approximation function can be guaranteed. Most meshless approaches do not have the guaranteed linear independency of functions.

H

u(x) = u'(x,x~) = 2_iaj(x)0J(x-x~) = </>& (4)

J(a(x)) = £ w , ( x ) j"ft>,0)e,2(x,x>ft (5)

where e.(x,x) = O,(x,O,)-u'(x,x) = cp,d, -cpa (6)

and w,(5c) is weight function, a>Xx)is localization factor function

Numerical Example

Constant Stress Cube

The following conditions are imposed on a cube of 1 x 1 x 1.

Displacement conditions

ux=0 on X = 0 uy=0 on y = 0 «2 = 0 on Z = 0

Traction condition

P z =l on z = l

The covers are distributed based on a two-level voxel data as shown in Figure 5. The computed results of displacement is linear and the stress is constant everywhere, and it is proven that the CLSA can give an exact solution for constant stress problem.

V \

A

Figure 6 1/8 plate with a hole Figure 5 A Cube with Constant Stress

12

Plate with a Hole

Consider a 20x20x4 plate with a hole at its center, whose radius is 2. Uniform traction acts on a pair of opposite sides. Figure 6 shows 1/8 of it. The boundary conditions are given as

a, = 0 on X = 0 ^ = 0 on y = 0

uz=0 on z = 0 ffw=lony = 10

Three Models as shown in Figure 7 are calculated. In the Model a, four-level voxel data is employed to create the cover distribution. The sizes of the four-level voxels are respectively 10/2x10/2x2/2, 10/4x10/4x2/4, 10/16x10/16x2/8 and 10/64x10/64x2/8. In the Model b, three-level voxel data is employed to create the cover distribution. The sizes of the three level voxels are respectively 10/8x10/8x2/4, 10/32x10/32x2/8, and 10/128x10/128x2/16. In the Model c, four-level voxel data is employed to create the cover distribution. The sizes of the four level voxels are respectively 10/4x10/4x2/4 10/16x10/16x2/8, 10/64x10/64x2/16, and 10/256x10/256x2/32.

The computed results of ayy are shown in Figure 8. The stress values at point A and B (see Figure 6) are listed in Table 1 together with the ANSYS results and the numbers of DOF are listed in Table 2.

Model a Model b Model c

Figure 7 3D Plate with Circular Hole Model

Figure 8 Computed results of a

13

Table 1 Stress values

Point A

Point B

Model a

Model b

Model c

ANSYS

Model a

Model b

Model c

ANSYS

°"*r

—

—

—

—

-1.287

-1.458

-1.502

-1.488

Cyy

3.269

3.339

3.464

3.578

—

—

—

—

^ z z

0.382

0.405

0.420

0.398

-0.387

-0.416

-0.428

-0.405

Von Mises stress

3.092

3.116

3.189

3.389

—

—

—

—

Seepage Flow Problem of Rock with Cracks

Figure 9 is the model of rock with 1000 circular cracks, which is made for the seepage flow problem for the safety evaluation of nuclear waste disposal. For this kind of model of nature, it is impossible to make FEM mesh. Also to evaluate the size of crack, the mesh size of the small area need to be reasonably small, and it is impossible to generate model by voxel with uniform size. By using multiscale voxel subdivision, the model is generated with about 200,000 elements with minimum voxel size is 1/1000 of one side as shown in Figure 10, while 1000 million elements are required if the domain is divided into uniform voxel of the size.

Figure 9 Rock with 1000 cracks (birds view and sectional view)

14

Figure 10 Multi-scale voxel Model

Conclusions

In this paper, the Cover Least Square Approximation method is implemented for linear structure analysis. The covers are distributed using multi-resolution voxel data. Numeric examples show that multi-resolution voxel data based cover distribution can conveniently guarantee the linear independence of CLSA shape functions and the approximation accuracy can be flexibly controlled by locally justifying the density of cover distribution.

References

Fish J and Markolefas (1993), "Adaptive s-method for linear elastostatics", Computer Methods in Applied Mechanics and Engineering, 104, 363-396.

Hollister S and Kikuchi N (1994), Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue, Biotechnology and Bioengineering, Vol.43, No. 7, pp.586-596

Jin C, Suzuki K, Fujii D, and Ohtsubo H (2000), "Methodology and Property of Cover Least Square Approximation", Transaction of the Japan Society for Computational Engineering and Science Vol. 2 pp 213-218

Jin C, Suzuki K and Ohtsubo H (2000), " Linear Structural Analysis Using Cover Least Square Approximation ", Journal of Applied Mechanics, JSCE Vol.3 pp 167-176

Kikuchi N and Diaz A (1998), "CAD/CAE using Image Base Method", 14th Quint Seminar Texbook

Shi, G H (1991), "Manifold Method of Material Analysis", Transactions of the 9th Army Conference On Applied Mathematics and Computing, Report No. 92-1. U.S. Army Research Office.

Suzuki K etal. (1998) "The Analysis of 3D Solid Using Multi-scale Voxel Data", Computational

Mechanics -New Trends and Applications VII, 2-15, CIMNE

Terada K and Kikuchi N (1996), "Microstructural design of composites by using the homogenization method and digital images, Mat. Sci. Res. Int. , Vol.2, No.2, pp.73-81

15

Advances in Meshfree and X-FEM Methods, C.R. Liu, editor World Scientific, Singapore, 2002

A MESHLESS METHOD USING RADIAL BASIS FUNCTIONS FOR SOLVING WAVE EQUATIONS

C. S. Chen, Jichun Li, and D.W. Pepper

University of Nevada Las Vegas [email protected]

Abstract

Using various time difference schemes or integral transforms, a given wave equation can be reduced to solving a series of inhomogogenous Helmholtz-type equations which can then be further split into evaluating particular solutions and solving the related homogeneous equations. Recent development of deriving closed-form particular solution for Helmholtz-type equations using radial basis functions has made it possible to solve time-dependent problems efficiently. As a result, the domain integration can be avoided in the solution process. The method of fundamental solutions (MFS), a meshless and often spectrally accurate boundary method, will be further developed and adopted as the major numerical method to solve the corresponding homogeneous equations in this paper.

mailto:[email protected]

16

Advances in Meshfree andX-FEMMethods, G.R.Liu, editor, World Scientific, Singapore, 2002

MESHLESS COMPUTATIONAL METHOD BY USING RADIAL BASIS FUNCTIONS

Benny Y. C. Hon

City University of Hong Kong Benny.Hon@cityu. edu.hk

Abstract

The recent development of a meshless method by using radial basis functions will be reported in this talk. Application to both multivariate interpolation and solving partial differential equations have demonstrated the spectral convergence of the method for some particular radial basis functions like multi quadric. This talk will also discuss some of the recent proposed techniques for solving the ill-conditioning problem resulted from solving the full resultant matrix.

Advances in Meshfree andX-FEMMethods, G.R. Liu.editor World ScientiflcSingapore, 2002

17

RECENT ADVANCES IN THE METHOD OF FUNDAMENTAL

SOLUTIONS

A. Karageorghis Department of Mathematics and Statistics, University of Cyprus, P. 0. Box 20537,

1618 Nicosia, Cyprus a n d r e a s k f l u c y . a c . c y

G. Fairweather, P. A. Martin Department of Mathematical and Computer Sciences, Colorado School of Mines,

Golden, Colorado 80401, USA gfa i rweaQmir t e s . edu , pamar t i n f lmines . edu

A b s t r a c t The aim of this paper is to describe recent developments in the method of fundamental solutions (MFS) and related methods for the numerical solution of certain elliptic boundary value problems.

K e y w o r d s : Method of Fundamental Solutions, Nonlinear Least Squares, Boundary Collocation.

I n t r o d u c t i o n

The method of fundamental solutions (MFS) is a meshless technique for the numerical solution of certain elliptic boundary value problems which falls in the class of methods generally called boundary methods. Like the boundary element method (BEM), it is applicable when afundamental solution of the differential equation in question is known, and shares the same advantages of the BEM over domain discretization methods. Moreover, it has certain advantages over the BEM.

In the MFS, the approximate solution is expressed as a linear combination of fundamental solutions with singularities placed outside the domain of the problem. The locations of the singularities are either preassigned and kept fixed or are determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible. This is usually achieved by a least squares fit of the boundary data. Early uses of the MFS were for the solution of various linear potential problems in two and three space variables. It has since been applied to a variety of situations such as plane potential problems involving nonlinear radiation-type boundary conditions, free boundary problems, biharmonic problems, problems in elastostatics and in the analysis of wave scattering in fluids and solids.

T h e M F S for H e l m h o l t z p r o b l e m s

To illustrate the essential features of the MFS, we consider a two-dimensional exterior Helmholtz problem which is closely related to the external scattering problem for acoustic waves by a rigid obstacle. We let Q be an unbounded domain in IR2 and Qc its bounded complement in Ht2 with

http://gfairweaQmirtes.edu

http://pamartinflmines.edu

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boundary dU. We consider the problem

Au(P) + k2u(P) = 0, Pen,,

Bu(P) = o, Peda, where A denotes the Laplacian, u is the dependent variable, k a real constant, and Q is a bounded domain in the plane with boundary dil. The operator B specifies the boundary conditions (BCs). The behaviour of u at infinity must also be specified.

In the MFS, the solution u is approximated by a function of the form

N

UA,(c,P;<?)=£CjG(P,,Q), Q e n , J = l

where c = {ci, c 2 , . . •, CN) G C and P is a 2N-vector containing the coordinates of the singularities Pj, which he outside il. The function G(P,Q) is a fundamental solution of the Helmholtz equation given by G(P,Q) = -\H^2\kR(P,Q)), where H^2) is the Hankel function of the second kind of order zero and R(P,Q) denotes the distance between the points P and Q. A set of observation points {Qt}tL\ is selected on dil. When the locations of the singularities are fixed, the coefficients c are determined by boundary collocation leading to the equations BUN(C, P ; QI) = 0, £ = 1,2, • • •, M. When M = N we have a linear system of JV equations in N unknowns whereas when M > N, this yields a linear least-squares problem.

In the case of moving singularities, the AN unknowns, comprising the coefficients c and the locations of the singularities P , are determined by minimizing the functional F ( c , P ) = YltLi IBu^^c.V-fQejl2, which is nonlinear in the coordinates of the Pj. The minimization of this functional is done using readily available nonlinear least squares software, such as the MINPACK routines LMDIF and LMDER [16], the Harwell subroutine VA07AD [21], and the NAG routine E04UPF [36]. The relative merits of these codes are examined in [25] and [40]. The constrained optimization features of E04TJPF are particularly useful for ensuring that the singularities remain outside the region.

The initial placement of the singularities can be extremely important in the convergence of a least squares routine. Usually the singularities are distributed uniformly around the domain of the problem at a fixed distance from the boundary. More details about the available least squares routines as well as the various algorithmic MFS features developed until the mid 90!s may be found in [13]. More recently, the optional placement of the singularities, in problems with boundary singularities, via a simulated annealing algorithm has been the subject of studies by Cisilino and Sensale [11]. This simulated annealing algorithm involves an iterative random search with adaptive moves which enable one to avoid local minima. Saavedra and Power [45] introduce an adaptive refinement MFS algorithm in the case the singularities are fixed and their number is less than the number of boundary points. This leads to a least squares problem and in this algorithm the distribution of singularities is selectively improved. This is done by taking into account the intensities of the fundamental solutions in the MFS expansion and using them as parameters in a multiple linear regression model.

Appl icat ions

It is unclear who first used the MFS with fixed singularities; see the references in [9, 13]. The MFS with moving singularities was first proposed by Mathon and Johnston [35]. A survey of the MFS

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and related methods for the numerical solution of elliptic boundary value problems is presented in [13]. Also, material on the MFS may be found in the books by Golberg and Chen [17] and Kolodziej [31]. Since the approximate solution in the MFS automatically satisfies the differential equation in question, the method may also be viewed as a Trefftz method. A survey of such methods can be found in [30].

Lately, much attention has been devoted to the application of the MFS to diffusion problems [18, 19, 39], especially in conjunction with the dual reciprocity and radial basis function methods for treating the inhomogeneous terms [33, 44]. The MFS combined with radial basis functions has also been used recently by in [12] and [2, 4], for the solution of linear and nonlinear Poisson problems, respectively, whereas the MFS in conjunction with compactly supported radial basis functions has been used in [6] for the solution of Poisson problems and in [20] for the solution of three-dimensional Helmholtz type-problems. The integrations involved in the evaluation of the particular solutions in Poisson problems were treated by a quasi-Monte Carlo method in [7]. More recently, the MFS has been used for the solution of inhomogeneous problems in combination with the fundamental solutions of the modified Helmholtz equation instead of radial basis functions by Alves, Chen and Saler [1]. A comparison of the performance of the MFS combined with radial basis functions and another meshless method, called Kansa's method, is carried out in [34].

The MFS has also been used in [9, 32] to investigate the dependence of the accuracy of the solution on the position of the auxiliary boundary and the number of boundary points. In [3], the MFS was applied to singular problems governed by the modified Helmholtz equation and in [24] for the calculation of the eigenvalues of the Helmholtz equation. Cisilino and Pardo [10] used an MFS-type approximation with a functional integral method. This approach introduces a regularization parameter which can be adjusted to reduce the error. In [29], the MFS is used for the solution of anisotropic problems in elasticity. Rajamohan and Raamachandran [43], also use the MFS for the solution of anisotropic thin-plate bending problems. Fenner uses the MFS for linear isotropic elasticity problems in [15]. In the same paper, a domain decomposition technique is discussed. The MFS with moving singularities has been used recently for the solution of three-dimensional Signorini problems [41], three-dimensional elasticity problems [42] and anisotropic single material and bimaterial problems in combination with a domain decomposition technique [5].

In recent years, the MFS has also been widely used for problems in electrostatics. In particular. Ismail and Abu-Gammaz [22] apply the MFS to calculate the electric field resulting from high voltage transmission systems and Vlad et al [47] apply the MFS to calculate the electric field in plate-type electrostatic separators. In [37], Nishimura et al., investigate the positioning of the singularities via a genetic algorithm, when the MFS is applied to elastostatics problems. The same authors study the positioning of both the singularities and the boundary points in the MFS for axisymmetric elastostatics problems in [38] using the same technique. Recently, the MFS was also applied to three-dimensional shape recognition problems by Kanali, Murase and Honami [23].

When the MFS with uniformly distributed (fixed) singularities and boundary points is applied to certain problems in circular domains it leads to circulant system matrices or system matrices which may be decomposed into circulant submatrices. Ways of exploiting the properties of such systems for the efficient implementation of the MFS are investigated in [46].

Because of their advantages over domain discretization methods for the solution of scattering and radiation problems in acoustics and elastodynamics, various MFS-type formulations have been suggested for such problems. A comprehensive survey of these can be found in [14].

20

References

[1] C. J. S. Alves, C. S. Chen and B Saler, The method of fundamental solutions for solving Poisson problems, to appear in the Proceedings of BEM 24, 2002.

[2] K. Balakrishnan and Ramachandran, A particular solution Trefftz method for non-linear Poisson problems in heat and mass transfer, J. Comput. Phys. 150 (1999) 239-267.

[3] K. Balakrishnan and Ramachandran, The method of fundamental solutions for linear diffusion-reaction equations, Math. Comput. Modelling 31 (2000) 221-237.

[4] K. Balakrishnan and Ramachandran, Osculatory interpolation in the method of fundamental solutions for nonlinear Poisson problems, J. Comput. Phys. 172 (2001) 1-18.

[5] J. R. Berger and A. Karageorghis, The method of fundamental solutions for layered elastic materials, Engng. Anal. Boundary Elements 25 (2001) 877-886.

[6] C. S. Chen, C. A. Brebbia and H. Power, Dual reciprocity method using compactly supported radial basis functions, Comm. Numer. Methods Engrg. 15 (1999) 137-150.

[7] C. S. Chen, M. A. Golberg and Y. C. Hon, Numerical justification of fundamental solutions and the quasi-Monte Carlo method for Poisson-type equations, Engng. Anal. Boundary Elements 22 (1998) 61-69.

[8] Y. L. Chow and S. K. Chaudhury, Non-linear optimization for field scattering and SEM problems, IEE Conf. Publ. 195 (1981) 378-382.

[9] S. Christiansen, Condition number of matrices derived from two classes of integral equations. Math. Meth. Appl. Sci. 3 (1981) 364-392.

[10] A. P. Cisilino and E. Pardo, A functional integral formulation for the method of fundamental solutions, Paper presented at the International Conference on Boundary Element Techniques, New Jersey, USA, July 2001.

[11] A. P. Cisilino and B. Sensale, Application of a simulated annealing algorithm in the optimal placement of the source points in the method of fundamental solutions, Comput. Mech. 28 (2002) 129-136.

[12] M Elansari, D. Ouazar and A. H.-D. Cheng, Boundary solution of Poisson's equation using radial basis function collocated on Gaussian quadrature nodes, Comm. Numer. Methods Engrg. 17 (2001) 455-464.

[13] G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998) 69-95.

[14] G. Fairweather, A. Karageorghis and P. A. Martin, The method of fundamental solutions for scattering and radiation problems, Technical Report TR/04/2002, Department of Mathematics and Statistics, University of Cyprus, 2002.

[15] R. T. Fennel, A force superposition approach to plane elastic stress and strain analysis, J. Strain Anal., 36 (5) (2001) 517-529.

[16] B. S. Garbow, K. E. Hillstrom and J. J. More, MINPACK Project, Argonne National Labora

tory, 1980.

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M. A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, (Computational Mechanics Publications, Southampton, 1996.

M. A. Golberg and C. S. Chen, The method of fundamental solutions for potential. Helmholtz and diffusion problems, Boundary Integral Methods - Numerical and Mathematical Aspects. M. A. Golberg, ed., Computational Mechanics Publications, Boston, 1999, pp. 103-176.

M. A. Golberg and C. S. Chen, A mesh free method for solving non-linear reaction-diffusion equations, Computer Modeling in Engineering and Science, 2 (2001), 87-95.

M. A. Golberg, C. S. Chen and M. Ganesh, Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions, Engng. Anal. Boundary Elements 24 (2000) 539-547.

M. J. Hopper, Editor, Harwell Subroutine Library Catalogue, Theoretical Physics Division, AERE, Harwell, U.K., 1973.

H. M. Ismail and A. R. Abu-Gammaz, Electric field and right-of-way analysis of Kuwait high-voltage transmission systems, Electr. Power Syst. Res. 50 (1999) 213-218.

C. Kanali, H. Murase and N. Honami, Shape identification using a charge simulation retina model, Math. Comput. Simulation 48 (1998) 103-118.

A. Karageorghis, The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation, Appl. Math. Lett. 14 (2001) 837-842.

A. Karageorghis and G. Fairweather, The method of fundamental solutions for the numerical solution of the biharmonic equation, J. Comput. Phys. 69 (1987) 434-459.

A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems, J. Acoust. Soc. Amer. 104 (1998) 3212-3218.

A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric potential problems, Internat. J. Numer. Methods Engrg. 44 (1999) 1653-1669.

A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric elasticity problems, Comput. Mech. 25 (2000) 524-532.

G. Khatiashvili and G. Silagadze, On a numerical solution of two-dimensional problems of elasticity for anisotropic medium by the method of fundamental solutions, Reports of the Enlarged Sessions of Seminars of the I. Vekua Institute of Applied Mathematics, Tbilisi State University, Vol. XIV, No. 3, 1999.

E. Kita and N. Kamiya, Trefftz method: an overview, Adv. Engineering Software 24 (1995) 3-12.

J. A. Kolodziej, Applications of the Boundary Collocation Method in Applied 'Mechanics, (Wydawnictwo Politechniki Poznanskiej, Poznan, 2001). (In Polish).

L. Lazarashvili and M. Zakradze, On the method of fundamental solutions and some aspects of its application, Proc. A. Razmadze Math. Inst. 123 (2000) 41-51.

J. Li, Mathematical justification for RBF-MFS, Engng. Anal. Boundary Elements 25 (2001) 897-901.

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[34] J. Li, Y. C. Hon and C. S. Chen, Numerical comparisons of two meshless methods using radial basis functions, Engng. Anal. Boundary Elements 26 (2002) 205-225.

[35] R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SLAM J. Numer. Anal. 14 (1977) 638-650.

[36] Numerical Algorithms Group Library, NAG(TJK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, U.K.

[37] R. Nishimura, K. Nishimori and N. Ishihara, Determining the arrangement of fictitious charges in charge simulation method using genetic algorithms, J. Electrostat. 49 (2000) 95-105.

[38] R. Nishimura, K. Nishimori and N. Ishihara, Automatic arrangement of fictitious charges and contour points in charge simulation method for polar coordinate system, J. Electrostat. 51-52 (2001) 618-624.

[39] P. W. Partridge and B. Sensale, The method of fundamental solutions with dual reciprocity for diffusion and diffusion-convection using subdomains, Engng. Anal. Boundary Elements 24 (2000) 633-641.

[40] A. PouUikkas, A. Karageorghis and G. Georgiou, Methods of fundamental solutions for harmonic and biharmonic boundary value problems, Comput. Mech. 21 (1998) 416-423.

[41] A. PouUikkas, A. Karageorghis and G. Georgiou, The numerical solution of three dimensional Signorini problems with the method of fundamental solutions, Engng. Anal. Boundary Elements 25 (2001) 221-227.

[42] A. PouUikkas, A. Karageorghis and G. Georgiou, The method of fundamental solutions for three dimensional elastostatics problems, Comput. & Structures, 80 (2002) 365-370.

[43] C. Rajamohan and J. Raamachandran, Bending of anisotropic plates in charge simulation method, Adv. Engineering Software 30 (1999) 369-373.

[44] P. A. Ramachandran and K. Balakrishnan, Radial basis functions as approximate particular solutions: review of recent progress, Engng. Anal. Boundary Elements 24 (2000) 575-582.

[45] I. Saavedra and H. Power, Adaptive refinement in method of fundamental solutions for bidi-mensional Laplace problems, Boundary Element Communications 12 (2001) 3-11.

[46] Y-S. Smyrlis and A. Karageorghis, Some aspects of the method of fundamental solutions for certain harmonic problems, J. Sci. Comput. 16 (2001) 341-371.

[47] S. Vlad, M. Mihailescu, D. Rafiroiu, A. Iuga and L. Dascalescu, Numerical analysis of the electric field in plate-type electrostatic separators, J. Electrostat. 48 (2000) 217-229.

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Advances in Meshfree and X-FEM Methods, G.R. Liu, editor

World Scientific, Singapore,2002

A STUDY ON THE PATCH TEST OF POINT INTERPOLATION METHODS

Y. T. Gu Centre for Advanced Computations in Engineering Science(ACES)

Department of Mechanical Engineering, National University of Singapore E-mail: [email protected]

Abstract

Two meshfree point interpolation methods (PIMs), which are based on the polynomial basis and the radial

basis functions, have been proposed recently to replace the moving least squares approximation for the

field function approximation. However, PIMs cannot automatically ensure the compatibility of the solution

when it is used together with the global energy principles. The difficulty in passing the patch test was

reported for meshfree methods using the PIM shape functions and the global Galerkin weak form. In this

paper, so-called meshfree conforming and non-conforming PIMs are proposed. They are studied for the

patch tests in great detail. Requirements for passing patch tests are discussed.

Keywords: Meshfree Method, Patch Test, Compatibility, and Computational Mechanics.

Introduction

Meshfree methods have recently become an attractive alternative for problems in computational mechanics. Several meshfree methods, such as the Element Free Galerkin (EFG) method and the Meshless Local Petrov-Galerkin (MLPG) method, have been proposed and achieved remarkable progress in solving a wide range of static and dynamic problems for computational mechanics. The moving least squares (MLS) approximation is currently widely used in meshfree methods to construct shape functions. However, there exist some disadvantages in using MLS.

Two meshfree point interpolation techniques, point interpolation methods (PIMs) that based on the polynomial (Liu and Gu, 2001a) and the radial basis function (Liu and Gu, 2001b), respectively, are proposed to construct shape functions with the Kronecker delta function properties. It has been combined with the global Galerkin weak forms to formulate the meshfree point interpolation method (PIM) and the meshfree radial point interpolation method (RPIM).

However, PIM interpolants cannot automatically ensure the compatibility (Liu, 2002),


24

when they are used together with the global energy principles. The difficulty of passing the patch test was reported for meshfree methods based on the PIM and the global Galerkin weak form. In this paper, issues related to the compatibility of PIMs are studied. A technique of background cell-based nodal selections and a penalty method are proposed to overcome these problems. So-called meshfree conforming and nonconforming meshfree PIMs are proposed. They are studied for the patch tests in great detail. Requirements for passing patch tests are also discussed.

Point Interpolation Method

Polynomial basis PIM

Consider a function w(x) defined in a domain Q discretized by a set of field nodes. The point interpolates w(x) from the surrounding nodes of a point x using the polynomials as basis can be written as

«(x) = 2> , (x )a ,=p T (x ) a (1)

where p,(x) is a monomial in the space coordinates XT=[JC, y], n is the number of nodes in the neighborhood (support domain) of x, a, is the coefficient for p,{x) corresponding to the given point x. Thep,{x) in equation (1) is built utilizing Pascal's triangle, so that the basis is complete. The coefficients a, in equation (1) can be determined by enforcing equation (1) at the n nodes surrounding point x. From equation (1), we have

H(X)= pTWP0"HW ue (2)

It can be found that shape functions <|>(x) possess the delta function property, and the essential boundary conditions can be easily imposed.

The point interpolation using polynomial basis is accurate and easy to use. However, like other methods that use polynomial as basis functions, it is tricky to choose a basis for interpolation. If an inappropriate polynomial basis is chosen, it may result in a badly conditioned matrix, which could be even noninvertible, due to rank deficiency of basis functions. In order to avoid the singularity of the moment matrix P 0 , several strategies have been proposed, such as, the slightly moving node method, the transformation of the local coordinates, and so on. Using radial functions as basis in point interpolation is also a good alternative. The matrix triangularization algorithm (MTA) has recently been proposed (Liu, 2002). It has been found that the MTA is very efficient and works well for most situations.

Radial basis PIM

The point interpolation form and the constraint conditions are written as:

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«to = I,*,toa,+I>,(x>*7=RT(x)a + PT(x)'> md 2 > y ( x > / = ° (3)

where R,{r) is a radial basis function, n is the number of nodes in the influence domain of x, pj{x) is a monomial, m is the number of polynomial basis functions, coefficients a, and bj are interpolation coefficients.

The Multi-quadrics (MQ) radial function, is used in this work due to its simplicity. Two parameters need be determined in the MQ radial function (Liu, 2002).

From equation (3), we have

M(x) = [RT(x)Sa +pT(x)SJue =«»(x)ue (4)

Sa = R-1 [1 - PmS4 ] , S6 = [PmTR^Pm ]"' PJR-' (5)

Mathematicians have proved the existence of R^1 for arbitrary scattered nodes.

Therefore, using radial basis functions can completely solve the singularity problem of polynomial PIM. In addition, it is found that the shape functions O(x) formed through the above procedure also possess Kronecker delta function properties.

Compatibility of PIMs

In meshfree methods, the field function approximated is often based on the moving. The compatibility of field function approximation using meshfree shape functions may or may not be always satisfied. In using PIM shape functions, the compatibility is not ensured automatically (Liu, 2002), and the field function approximated could be discontinuous when nodes enter or leave the moving support domain. Because no weight function is used in PIMs, the nodes in the support domain are updated suddenly, meaning that when the nodes are entering or leaving the support domain, they are actually "jumping" into or out of the support domain. Therefore, the function approximated using the PIMs can "jump". However, the compatibility can be ensured automatically in the MLS approximation by the weight functions appropriately chosen.

For one-dimensional problems, the incompatibility problem of PIMs can be easily avoided by using one-piece PIM shape functions for one entire integration cell. For two-or three-dimensional problems, however, the compatibility problem becomes more complex because two neighboring integration cells are connected by common curves or surfaces. Use of one-piece PIM shape functions cannot totally solve the incompatibility problem because it cannot automatically ensure the compatibility along the interfaces between cells. Stitches using the penalty method are needed to enforce the compatibility. Detailed discussions about it will be presented in the following section.

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Meshfree Methods Based on PIMs

The global Galerkin weak form of the 2-D elastostatics is posed as follows.

J(L<5u)T -(DLu)dQ- {&T b d Q - j<5uT tdT + JS(U+ - iT)a(u+ -u">/r = 0 (6) o n r, r;

where a is a diagonal matrix containing the penalty coefficients. u+ and u~ correspond to the displacement field on the two sides of the incompatible interface Tc of two adjacent background cells. Substituting the polynomial PIM and the radial PIM into the above weak form, respectively, we can obtain the meshfree PIM and the meshfree RPIM.

As above discussed, in using PIM shape functions, the compatibility is not ensured automatically, and the field function approximated could be discontinuous in the whole domain when moving interpolation domains are use. There are two incompatibilities using of PIMs to form meshfree methods that are based on the global weak form. The first is the incompatibility among different numerical quadrature points in a quadrature cell. The second is the incompatibility along interface boundaries among different numerical quadrature cells. The first incompatibility can be overcome by simply using one-piece PIM functions for the entire quadrature domain, and the second incompatibility can be overcome by simply using the penalty method. The last term of equation (6), which is the integration along the interface boundary Tc, is used to overcome the second incompatibility. If the compatibility can be ensured, the method is referred to as conforming PIM (CPIM or CRPIM). Otherwise, it is referred to as the non-conforming PIM(NPIMorNRPIM).

Study on the Standard Patch Test

The standard patch test is often used to check the convergence of a numerical method. Both conforming and non-conforming PIMs are used for the standard patch tests. In these patch tests, the displacements are prescribed on all outside boundaries by a linear function of x axi&y (wx=0.6x and uy=0.6y). Satisfaction of the patch test requires that the displacement of any interior node be given by the same linear function and that the strains and stresses be constant in the patch.

The meshfree PIM that is based on the polynomial PIM is tested firstly. Figure 1 shows a patch with 25 nodes of which nine nodes are interior nodes. Total 32 triangular background cells are used for integration in these patch tests. Three Gauss points are used in each integration cell. "One piece" PIM shape functions are used and 9 nodes are selected in the support domain. It is found that NPIM cannot exactly pass the patch test. The displacements along the interface between cell 11 and cell 12 are plotted in Figure 1. It can be found that the NPIM results are not compatible along the interface. This incompatibility leads to the error in passing patch tests. After stitches are used to tie up centre points (e.g. the A point and the A' point) of interfaces, CPIM exactly passes this patch test.

Figure 2 and Figure 3 show patches of 16 and 25 irregular nodes, respectively. Again,

27

NPIM cannot exactly pass these patch tests and CPIM passes these patch tests exactly.

It should be mentioned here the requirements for the numerical method based on the global Galerkin weak form to pass the patch test are listed as follows (Liu 2002):

1) The shape functions are of at least linear consistency; 2) The field function approximation using the shape functions must be compatible; 3) The essential boundary conditions have to be accurately imposed; 4) Accurate numerical operations are required, such as the numerical integration.

The polynomial PIM shape functions can satisfy the first requirements very easily as long as linear polynomials are included in the basis. The polynomial PIM shape function can also satisfy the third requirement, as it possesses the Kronecker delta function property. To satisfy the second condition, however, the constrained Galerkin form is required in constructing the system equation (6). Hence, CPIM can pass patch tests because it satisfies all requirements of passing patch tests. NPIM cannot pass patch tests because it does not satisfy the second requirement. If, however, proper nodes are used to construct the PIM shape functions and properly arranged cells for the integration to prevent incompatibility to occur, NPIM can also pass the patch test.

<? 9

O

o

displacement profile

Figure 1 A patch with 25 regular nodes Figure 2 A patch with 16 irregular nodes

The meshfree radial PIM is also tested for these patches. Detailed results are not presented here. It is found that NRPIM cannot exactly pass the above discussed patches. The CRPIM passes these patch testes exactly when three polynomial bases (m=3) are added to MQ basis function. When the polynomial term is not included (m=0), even CRPIM failed to pass these standard patches. It is because when w=0 the radial PIM can not reproduce the linear function exactly. In other word, without polynomial terms, the meshfree radial PIMs do not satisfy the first requirement for passing the patch test.

23 (>"

28

Note that passing the standard patch test is a sufficient requirement for a numerical method to be able to converge to the true solution as the nodal spacing approaches to zero. It is not a necessary requirement for a numerical method to converge. The ultimate test of a numerical method should be the convergence test. It has been found that NPIM and NRPIM also converge to the true solution (Liu, 2002).

O Q Q _

\ Background cells 1

(1 >v

o o

o

o

O ©

o

o

o o

i O C

1

i ° (

o <

9 a

Figure 3 A patch with 25 irregular nodes

Conclusion

The compatibility of PIMs is discussed in this paper. The PIMs cannot automatically ensure the compatibility. A technique of background cell-based nodal selections and a penalty method are proposed to overcome these problems. Requirements for passing the standard patch tests are discussed in detail. From studies in patch tests and the numerical example, it has been found that meshfree conforming PIMs can pass the standard patch test exactly, when meshfree non-conforming PIMs usually cannot pass the standard patch. However, meshfree non-conforming PIMs are also convergent.

References

Liu G.R.(2002). MeshFree Methods-Moving beyond the Finite Element Method, CRC Press LLC, USA.

Liu G. R. and Gu Y. T. (2001a). "A Point Interpolation Method for two-dimensional solids," Int. J. Num. Meth.Eng.,50, 937-951.

Liu G. R. and Gu Y. T. (2001b). "A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids," Journal of Sound and Vibration, 246(1), 29-46.

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Advances in Meshless and X-FEM Methods, G.R. Liu.editor World Scientific, Singapore, 2002

A COMPARISON BETWEEN RADIAL POINT INTERPOLATION METHOD (RPIM) AND KRIGING BASED MESHFREE METHOD

G. R. Liu, K. Y. Dai, Y. T. Gu, and K. M. Lim

Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering

The National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260 [email protected], [email protected]

Abstract

Point interpolation method based on radial basis function, or RPIM, has been successfully developed and applied in meshfree method. Recently, a new meshfree method is proposed based on the general moving Kriging method. In this paper, both the two methods are formulated in detail to construct their shape functions. It can be found that their shape functions are identical if the same basis function or semivariogram is adopted. Despite of this, the theorems used in Kriging may provide an alternative theoretical support for RPIM that is the simplest in formulation ensuring the minimum error of approximation.

Keywords: Meshfree method, Point interpolation method, Kriging method, radial function.

Introduction

In recent years, meshfree or meshless methods have been developed and achieved remarkable progress in computational mechanics and related fields because no element connectivity is needed in the formulation.

The point interpolation method (PIM) with weak form formulation was proposed by Liu and his coworkers (Liu and Gu, 2001; Wang and Liu, 2001; Wang et al., 2001; Liu, 2002), which uses the nodal values in the local support domain to interpolate the shape functions exactly. Hence the shape functions so constructed possess the Kronecker delta function property. When radial basis functions are employed for interpolations, the method is termed as radial point interpolation method (RPIM).

Recently some good properties have been found in the Kriging method and used in computational mechanics (Gu, 2002). Kriging is a form of generalized linear regression, named after a South African mining engineer D. G. Krige (Cressie, 1993). It was originally developed for mapping in the fields of geology and geophysics, mining, etc. Apart from being an exact interpolator, when employed to meshless method, its shape functions also have delta property. As the RPIM, Kriging also has the property of the partition of unity, and it can reproduce any function in its basis exactly.


30

In this paper, the formulations of RPIM and the Kriging based method are compared in detail. The properties of the constructed shape functions are discussed. It can be found that, although they are developed based on different mathematical approaches, the resultant shape functions are identical.

Formulations of Radial Point Interpolation Method (RPIM)

Consider a field function «(x) defined by a set of arbitrary distributed nodes

x, (/ = 1, 2, • • •, N) in a domain Q with boundary T. N is the total number of the nodes in

the whole domain. It is assumed that only the surrounding nodes of a point xg have

effect on w(x). The domain that includes these surrounding nodes is called influence

domain, or support domain.

Using n nodes in the support domain of a point x g , the radial PIM combining with

polynomial basis function approximates the field variable w(x) in form of (Liu, 2002) n m

a*(x,xe) = !> , (*)« , +y£pJ(x)bJ = r r ( x ) a + p7(x)b (1) .•=i j=\

where a, is the coefficient for the radial basis rt (x), which is in the form of

r,(xj) = exp[-br,2(xJ.)] = expHf lx , - x,)2 + {y} -yf)} (2)

and bj the coefficient for the polynomial basis Pj (x). For example, a basis in two-

dimensional problems is provided by pT = {l,x, y,x2, xy, y2}, m = 6 (3)

The number of the radial basis n and the polynomial basis m is chosen based on the reproduction requirement. Minimum terms of polynomial basis are often adopted for better stability. The coefficients a, and b} are determined in this way such that Eq. (1) pass through n

data nodes in the support domain, hence n equations can be obtained and written in matrix form

u e = R e a + Pmb (4)

where the moment matrix R e is given as follows

R o

r,(x,) /-2(x,) ••• r„(x,)

/-,(x,) r2(x2) ••• r„(x2)

r , K ) r 2 ( x j ••• r , (x.)

(5)

and Pm is a n x m matrix given by

31

P_ =

/>i(xi) PI(X\)

p,(x2) p2(x2) Pm(*l)

Pm(*2) (6)

(nxm) _M*J /> 2 ( X J ••• />m(xJ_ There are altogether (n + m) coefficients to be determined whereas only n equations are available above. To guarantee unique solutions, the following constraints are imposed:

Pla = 0 (7) Combination of Eqs. (4) and (6) yields

Pi 0 Eq. (8) is solved and a and b are written as

a = S„u

= S6u<

(8)

(9) (10)

where

01) (12)

ue =["(*,) "(x2) - K ( X „ ) ] 7 (13) Here I is an identity matrix of sizes n by n. Substituting of a and b into Eq. (1), the interpolation function q>(x) can be obtained

S6 - [ P m R e P < J PmR£ S a =R e ' [ I -P m S 6 ]

i A x , x e ) = [r(x)rSa + p ( x ) r S > £ =<p(x)u* (14)

Formulations of Kriging Based Method

Kriging method was initially formulated for the estimation of a continuous, special attribute at an unsample site. Ordinary Kriging method is widely used to estimate field value at a point of a problem domain for which the variogram is known, without the prior knowledge about the mean. In case that the drift of the function is not a constant mean, the universal Kriging concept may be an alternative in which a polynomial drift model is commonly applied. In this paper, we discuss the interpolation method based on universal Kriging. Consider a field value u(x) defined in the problem domain Q with boundary T. The domain is represented by a set of properly scattered nodes x,(i = 1, 2, •••, N). N is the total number of the nodes in the whole domain. Given n field values M(X,),---,U(X„) in the support domain of a point x0 , we want to obtain an estimate value uh of u at x0 . Supposing that uh is linear in w(x,),---,«(x„), it can be written as (Olea, 1999) '

U"(x,x0) = jA,M(x,) (15)

where ^'s are termed as weights. Suppose that uh is unbiased and the drift w(x) is assumed as a polynomial model, the above equation is subjected to

32

m 00 = Y,MiPiOO, with />0(x) = l (16)

where //, is the coefficient for the polynomial basis />r(x), which has the same form as Eq. (3). The condition that uh minimizes the mean-square prediction error E[u(x0)-u(x0)]

2

requires a constrained linear optimization involving -tj,•••/!,„ and Lagrange multipliers Mi • The constrained linear optimization can be expressed in terms of the Lagrange function for the universal Kriging

( n \ 2 ( n \ k ( « "l

L^,Ml)=E «(x 0)-^2 1«(x,) + 2 ^ YjXi~X + 2Z<"' J^^iPiM-Pi&o) (17)

The data are assumed to be part of a realization of an intrinsic random function «(x) with the semivariogram /(h) , which is in the form of

y(h) = - Var[u(x) -u(x + h)] (18)

where Var[.] is the variance of the random function, h is a vectorial distance in the sampling space. Then Eq. (16) is expanded to

Z,(^,/U;) = 2 £ ^ r ( x , , x 0 ) - J £ l , ^ ( x , ; x > ) + 1=1

2^o

*=i j=\ (19)

5 > , - l + 2^>, !>,/>,(x,)-/>;(Xo) V'=i J /=i v«=i J

Let Aj'sbe the optimal weights. The minimum mean square error is given by those weights that make all first derivatives of the Lagrangian function diminish with respect to the unknowns, which leads to

Gw = g (20) where

>(x,,x,) ••• r(x,,x„) 1 />,(x,) - /? t(x,)'

G = R P

PT O

r(x„,x,) i

Pl(*l)

r(x2 )x„) l ^,(x„) 1 0 0

A (x „ ) 0 0

Pk(*J 0 0

K Mo Mi ••• Mkf

Pkfrn) 0 0

(21)

(22)

(23)

Pk&\)

w = [A, X2

g r=[r(x0) p(x0)] = [x(x0,x,) r(x0.x2) •••y(x0,x„) 1 />,(x0) p2(x0) ••• pk(x0)]

The Eq. (19) is solved for the weights and the estimated field variable computed using Eq (15). For simplicity, the estimated value can be written in an alternative form of (Stein, 1999)

w"(x,Xo) = r ( X ) r s y +P(x)rstu* =a>ue (24) where 4»(x) is a matrix of shape functions and

33

Sh = ( P r R P) P r R ' (25)

S a = R - * a - P S 4 ) (26)

From the above formulations, it can be seen semivariogram plays an important role in the formulations. Let h be the lag. Here only the Gaussian model is given in one-dimensional-space form as follows:

y(h) = c0 -exp hl

, 2 •*0 J

(27)

It should be mentioned here, although the semivariogram is used for the formulations, they are still valid if it is replaced by the covariance of a second stationary random function. The relation between them is proved and given by the literature (Olea, 1999)

cov(h) = C-7(h ) (28)

Here C is a constant sill.

Comparison between the Two Methods

From the above formulations, it can be found that

1) RPIM is derived from the idea of exact interpolation to pass through every node in the support domain. Once the data nodes are given in the support domain and the basis function is selected, matrices R e and Pm will be determined. The

interpolation coefficients in Eq. (4) are then uniquely determined.

2) In Kriging, the weights and polynomial coefficients are determined by minimizing the mean-square prediction error. Through the optimization analysis, when a semivariogram is chosen, matrix G is Eq. (21) is also a constant matrix and hence vector w is also uniquely determined.

3) The RPIM uses the Gauss basis function to construct the shape functions while Kriging uses the semivariogram. Eq. (28) shows that the Gauss basis function in Eq. (2) is just the mathematical expression of the covariance according to the semivariagram given in Eq. (26).

Hence, although they are derived from different mathematical approaches, RPIM and Kriging yield exactly the same interpolation functions. This important fact can be understood simply by intuition: if the same basis or semivariogram and the same data nodes are used, the shape functions using passing-node interpolation will be unique, as long as the basis functions are independent. Therefore, regardless of the mathematical path used to arrive at the shape functions, they will be exactly the same.

The formulation, algorithm and procedure of RPIM are much simpler than those of Kriging. Krging provides an alternative mathematical background. The theory given in

34

Krging may be beneficial to find some efficient basis functions for RPIM. In the practical coding, RPIM should be used.

Let's examine the properties of the shape functions obtained by these two methods. One properly is that the Kronecker delta function is satisfied, i.e.,

[ 1 (/'= /, i =1, 2,•••,n) J J [0 <i*j, i,j = 1,2,-,n)

Furthermore, its shape functions will reproduce any function in the basis exactly (Reproduction property). In particular, if all constants and linear terms are included, it will reproduce a general linear polynomial exactly (Linear consistence), i.e.,

2 > = 1 °»d | > x , = x , (30)

Conclusions

Point interpolation method based on radial basis function has been successfully developed and applied in meshfree method. Recently, a new meshfree method is proposed based on the general moving Kriging method and also used in computational mechanics. In this paper, both the two methods are formulated in detail to construct the shape functions. We found that their shape functions are in the same forms if the same basis function or semivariogram is used for them. The finding is very useful because kriging provides some mathematical background for the RPIM and RPIM can be used in practical coding in computational mechanics. The shape functions obtained by the two methods possess the delta function property, reproduction property and linear consistency property, which are the most important properties for meshfree methods (Liu, 2002).

References

Cressie N.A.C. (1993). Statistics for Spatial Data, John Wiley & Sons, Inc. New York.

Gu L. (2002). Some properties of Kriging Based Meshfree Method (extended abstract), in Proc. Program and Abstracts for the 14th US National Congress of Theoretical and Applied Mechanics, June, Blacksburg, VA, 309-310.

Liu G.R. (2002). Mesh Free Methods: Moving Beyond the Finite Element Method, CRC press, USA.

Liu G.R. and Gu Y.T. (2001). A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids. J. of Sound and Vibration, 246(1), 29-46.

Olea R. A. (1999). Geostatistics for Engineers and Earth Scientists, Kluwer Academic Publishers, Boston

Stein M.L. (1999). Interpolation of Spatial Data - Some Theory for Kriging, Springer, New York.

Wang J. G., Liu G. R. (2001) A Point Interpolation Meshless Method Based on Radial Basis Functions, Int. J Numer. Meth. Eng., (revised).

Wang J. G., Liu G. R., Wu Y. G. (2001), A Point Interpolation Method for Simulating Dissipation Process of Consolidation, Comp. Meth. Appl. Mech. Eng., 190, 5907-5922.

35

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor World Scientific, Singapore, 2002

RADIAL BASIS POINT INTERPOLATION COLLOCATION METHOD FOR 2-D SOLID PROBLEM

Xin Liu, *G. R. Liu, Kang Tai and K.Y. Lam

SMA-Research Fellow, Singgpore-MITAlliance smalx&.nus.edn.ss

Centre for Advanced Computations in Engineering Science (ACES), Dept. OfMech. Eng., National University ofSingpore

srliu(a),nus. edu.sz

SMA-Fellow, Singpore-MITAlliance mktai(cbntu.edu.sg and lamkv&ihpc.nus.edu.ss

Abstract

In this paper, a new local RBF collocation approach based on point interpolation method, which can assure that its coefficient matrices are of bandwidth, has been presented. The main feature of this approach is to use a Hermite-type interpolation scheme, namely making use of the normal gradient at Neumann boundary so that this computational accuracy has greatly improved. The accuracy and simplicity of this presented approach will be shown numerically by well-known 2-D linear elastic Cook's membrane benchmark.

Keywords: Meshless, Radial Basis, Collocation, Hermite Interpolation, and PIM.

Introduction

The primary disadvantage of traditional RBF approach is that its coefficient matrices obtained from this discretization scheme are full. At present, there are mainly two approaches to improve this disadvantage. One is to improve the conditioning of the coefficient matrix and the solution accuracy using some special techniques. The other is to obtain coefficient matrices with bandwidth using compacted radial basis function. It has been shown that these two approaches can not obtain very satisfactory results. So more effective approaches need to be explored.

Point Interpolation Method (PIM) was originally proposed by Liu and Gu (1999, 2001), and has been improved and applied with Galerkin-based formulations. Here, an efficient radial basis point interpolation collocation-based formulation has been presented and applied to solve 2-D linear elastic problem. In contrast to Galerkin-based approaches, collocation method is simple and efficient to solve partial differential equations without the needs of numerical integrations. Another attractive feature is that its formulation is very simple. However, the study shows that the accuracy of results obtained by using direct collocation scheme is very poor especially on boundary. Therefore, collocation scheme is very different from the Galerkin scheme, which deals with Neumann boundary condition easily, due to its difficulties

36

to deal with Neumann boundary condition. Liszka et al. 1996 proposed a Hermite-type interpolation scheme in Generalized Finite Difference Method (GFDM) to improve the accuracy of collocation-based approach for solving solid problems. Zhang et al. (2000) applied Hermite-type interpolation in compacted support radial basis function method successfully. In this paper, Hermite-type interpolation was adopted in PIM in order to improve the accuracy. Approximate displacement field functions are carried out not only with the nodal values of its but also with the normal gradient at the force boundary by taking advantage of a radial point interpolation method (RPIM).

In this paper, the formulation of constructing shape functions with RPIM based on Hermite-type interpolation is presented in section 2. Collocation schemes for general 2-D linear elastic mechanics equations are demonstrated in section 3. In section 4, the accuracy of this presented approach is shown numerically by Cook's membrane problem. We conclude with a summary in section 5.

Hermite Radial Basis Point Interpolation

The approximation of displacement function vector u(x) in 2-D solid mechanics problems may be written as a linear combination of radial basis functions at the n nodes within influent domain near x and its normal derivatives at the rib nodes on force boundaries, viz.

u(x) = «(x) = u(x)

v(x) = 1 >, 0

0 <t>, dn

0

0

9#

dn\ H-

"c0 0"

0 J0

A =<%-*,!), 4 =«K||*-*,1) ^ = / ; ^ + / i dy

(1)

(la)

a/, bj are coefficients which corresponds to radial basis $ of displacements in x and y directions respectively; c,-, dj are coefficients which corresponds to normal derivatives of radial basis $ of displacements in x and y directions at the points on force boundaries respectively; and Co, do are the additional unknown constant terms. ^ is the radial basis. / *, lj are the elements of normal vector at jth point on force boundaries.

In this paper, high order thin-plate spline radial basis has been adopted in order to avoid the adjustment of sensitive shape parameters.

For 2-D problems:

^ = ^(|JC - xJ) = r6 ln(r) + constant

r = -j{x-x,f +{y-y,f

(lb)

(lc)

The coefficients au bh cy, dJt c0, d0 in Equations (1) can be determined by enforcing that the displacements' interpolations pass through all n nodes within the influent domain and the normal derivatives' interpolations of displacements pass through «* nodes on force boundaries.

37

The interpolations of the displacements in x and y directions at the kth point have the form:

= 1 : + l y=«

dn 0

dn J ( : : ) •

"c0 0"

0 d0_

<t>u = 0(\\xk - */|) ,$ = <t>\xk -xbj

k = l — n (2)

(2a)

The interpolations of the normal derivatives of displacements in x and y directions at the mth point on the force boundaries have the form:

m

dn

. dn .

• = •

\dub(xm)}

dn dO\xm)

. dn ,

> = /. \ui

»>\$b

dn 0

dj,

L =

dn

d_

dx

7 +/.

=1'.

c0 0

0 rfn

,m = l,---nb

b,

11- + 11-8y

0

0

(3)

(3b)

In addition, the constant term has to satisfy an extra requirement that guarantees unique approximation of the function, and the following constraints are usually imposed:

i>t=o,$>t=o (4) *=i t=i

They can be expressed by matrix formulations as follows:

(5)

where tie is the vector that collects all variables of the nodal displacements at the n nodes in the influent domain and all variables of normal derivatives of the nodal displacements at the nt nodes on the force boundaries in the influent domain .

(5a) ue = * . « « 3M*

M , V , • • • M „ V „

dn

The coefficients vector a is defined as

a = [a, 6, ••• a , 2>„ c,

aw* 5n

rf, ••

5M„*

dn

• c„ d„

dvbn

dn

co

0 0

d0]T (5b)

38

Thus the unknown coefficients vector

a = Vxue

Finally, the approximation form of function can be obtained as follows:

M(X) = 4>a = 0V~lue = y/ue

(6)

(7)

The matrix of radial basis, its normal derivatives and additional constant terms is defined by

<P = A o

o *

in o

o <P„

8n

0

0

dn

dn

0

0 1 0

d^_ 0 1

dn

(7a)

The matrix of shape functions can be expressed as follows

V, o - Wn o tf o ••• y/=W =

0 y,x 0 </„ 0 y," (7b)

Here y/t (z'=l, ..., w) and y/f (j=\, ..., nt) are shape functions.

Finally, the displacement vector can be expressed as follows:

0 yrk +

> i

< 0 0 ^ "

dM*;

3«

i - / / 2

9« , d \-u d , d \-u d I— + m——— lju — + m———

dx 2 dy dy 2 dx 8 A-u d 8 A-ju 8

mju— + / — — — m — + /—— — dx 2 dy dy 2 dx

(8)

(9)

The size of the nodal influence domain, that is Ro, can be determined by using four quadrants criterion proposed by X. Liu (2000). In order to get better accuracy and stability, R=aoRo has been adopted, here ceo generally is chosen to be 3.0-4.0.

Collocation Schemes

Let us consider a 2D problem of solid mechanics in domain Q bounded by the boundary r = r u +r t , which Tu is the displacement boundary and Tt is the force boundary. The governing equations and boundary conditions can be given as follows:

Lu = \-f

d2 I-ft d

dx2 2 dy

\ + ju 82

2 dxdy

l + fi d2

2 dxdy

d2 l-fi d2

T- + — r dy2 2 dx1

v(*)J = / ( * ) , x e Q (10a)

39

L.u E

l-M2

,d \-n d I — + m———

dx 2 dy d A~M d mju— + / — — — dx 2 dy

, d \-u d lH— + m———

dy 2 dx d A-M d

m — + /——— dy 2 dx

u(x)\

) '\ = t(x),xzrb (iob)

u = u, xeTu (10c)

where t(x) = \tx (x) ty (x)f is the traction vector on force boundaries.

Assuming that there are Nj internal (domain) points and N-Nd=N,+Nu boundary points, which N, are the numbers of force boundary points and Nu are the numbers of displacement boundary points.

In general, the location of the collocation points can be different from the location of nodes in discretization model. However, for the sake of simplicity, collocation points are the same as the nodes of model in our computational procedure.

The following N-Nu=Nd+ Nt equations are satisfied in internal domain nodes:

2*«, ) - / ( * , ) = <> (11a)

The following N, equations are satisfied on force boundary r t :

L,(u,) = t(xl) ( l ib )

The following Nu equations are satisfied on displacement boundary ru :

« ; - « = o ( l ie )

Numerical Simulation

In this section, Cook membrane's problem was numerically analysed. The results were obtained by high order thin plate spline with constant term under Hermite-type collocation (HTC) schemes. Figure 1(a) shows this problem under a uniformly distributed end shear load. An analytical solution does not exist for this model problem. Therefore, a reference solution can be obtained in Hueck and Wriggers' paper (1995), which used a FEM mesh of 128 x 128 elements to obtain vc =23.96, tfAmax=0.2369, aBmax=-0.2035 corresponding to the vertical displacement of node C and the maximum and minimum principal stresses at the nodes A and B respectively. The model parameters is E=\, ju=l/3, F=\, l=l.Two kind of models are used with 9 x 9 and 17 x 17. For 9 x 9 model, N=81, Nt=23, Nu=9, the total number of freedom degrees is (2 x 81)+(2 x 23)=208.For 17 x 17 model, N=289, N,=47, Nu=19, the total number of freedom degrees is (2 x 289)+(2 x 47)=672.The numerical results are shown in figure 2.

Conclusions

A Radial Point Interpolation Collocation Method (RPICM) based on Hermite-type interpolation is presented for the solution of 2-D solid mechanics in this paper. As is known, the accuracy issues are boresome for directed collocation schemes. As our computational experience, right results for the solution of solid mechanics problems can hardly be obtained by using directed collocation. However, the computational accuracy may be greatly improved with the utilization of the interpolation information

40

of normal derivatives at the nodes on the force boundary. This feature makes the truly meshless approach RPICM potential. Of course, further research works need to be done so that the numerical stability and convergence can be developed for this available collocation approach.

(a) Cook's membrane (b) 9x9 Model (c) 17x17 Model problem

Figure 1. Cook's membrane problem and its discrete models

* reference solution

(c) The minimum principal stress of B: aBmax

Figure 2. The computational results show

References

G. R.'Liu, Y. T. Gu, (1999), "A point interpolation method", in Proc. 4* Asia-Pacific Conference on Computational Mechanics, December, Singapore, 1009-1014.

G. R. Liu, Y. T. Gu, (2001), "A point interpolation method for two-dimension solids", Int. J. Numer. Methods Eng., 50, 937-951.

Liu Xin, Zhu Demao, Lu Mingwan and Zhang Xiong, (2000), "h, p, hp adaptive meshless method for plane crack problem", Acta Mechanica Sinica, 3, 308-318, 2000 (Chinese).

Liszka T J, Duarte C A, Tworzydlo W W. (1996), "Hp-Meshless cloud method", Comput. Methods Appl. Mech. Engrg., 139, 263-288.

X. Zhang, K. Z. Song, M. W. Lu and X. Liu, (2000), "Meshless Methods based on collocation with Radial Basis Function", Computational mechanics, Vol. 26, 4, 333-343.

Hueck U., Wriggers P., (1995), A formulation for the 4-node quadrilateral element, Int. J. Numer. Methods Engrg., 38, 3007-3037.

:

reference solution „. reference solution

forBt-node model for 289-node model

(a) The vertical displacement (b) The maximum principal of C: vc stress of A: aAmax

SECTION 3

Meshfree Methods for Smart Materials / Structures

43

Advances in Meshless andX-FEMMethods, G.R. Liu.editor World Scientific, Singapore, 2002

POINT INTERPOLATION M E S H F R E E M E T H O D FOR STATIC AND FREQUENCY ANALYSIS OF TWO-DIMENSIONAL PIEZOELECTRIC STRUCTURES

K. Y. Dai, G. R. Liu and K. M. Lim Center for Advanced Computations in Engineering Science (ACES),

Department of Mechanical Engineering

The National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260

[email protected], [email protected]

Abstract

A point interpolation mesh free method using radial basis with polynomial reproduction is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures. In the present method, the problem domain and its boundaries are represented by a set of properly scattered nodes. The displacements and the electric potential of a point are interpolated by the values of nodes in its local support domain using shape functions derived by the PIM scheme. Variational principle is used to establish a set of system equations for arbitrary-shaped piezoelectric structures. These equations are assembled for all quadrature points and solved for displacements, electric potentials or natural frequencies. A program is coded in MATLAB based on the formulations. Numerical examples are also presented to demonstrate the validity and convergence of the present method and their results compare well with the conventional FEM results from ABAQUS as well as the experimental ones.

Keywords: Mesh free method, Piezoelectric structure, Radial function, static analysis, frequency.

Introduction

Finite element method (FEM) for modeling multi-field problems has been successfully adopted to model and analyze the electroelastic coupling existing in piezoelectric structures (Ostergaard, et al, 1986). It allows for modeling static and eletroelastic vibration problems commonly encountered in the design of transducers, resonators, and other electromechanical devices. The validity of the simulation method has been confirmed by data report as well as experimental results in some literatures (Reinhard, 1990). In recent years, element-free, or meshless method has been developed and received remarkable progress in solid mechanics and related fields due to its flexibility and high convergence rate. Different types of mesh-free methods have been raised and developed so far. Point interpolation method (PIM) was proposed by Liu and his coworkers (Liu and Gu, 2001; Wang and Liu, 2001; Liu, 2002). It uses the nodal values in the local support domain to interpolate the field variable. The shape functions so constructed possess the Kronecker delta function property. According to the basis function employed, it can be mainly classified into two types, i.e., polynomial PIM and radial PIM.



44

Although mesh free methods has made great progress and been regarded as a gradually powerful numerical technique in computational mechanics, most applications are only limited to analyze basic solid problems in elastic mechanics so far. The aim of this paper is to extend the radial point interpolation method to the analysis of piezoelectric structures. In radial PIM, the problem domain is discretized by a set of arbitrarily distributed nodes. A radial basis function is used to construct the shape functions, which possess delta function properties. Piezoelectric structure with arbitrary shape is formulated using radial PIM combined with variational principle and linear constitutive piezoelectric equations. The major advantages of the present method over FEM method are that the predefined mesh or element is not required and it is very robust for irregularly distributed nodes. The selection of nodes does not affect greatly the accuracy of the results.

Numerical examples are presented to investigate the deflection of a piezoelectric bending motor, and the natural frequencies and vibration modes of a transducer. The results obtained by radial PIM are compared with those obtained by finite element method using commercial software ABAQUS. Some important parameters in the implementation of radial PIM are also investigated and discussed.

Radial Point Interpolation Method (PIM)

Consider a field function w(x) defined by a set of distributed nodes x ,(z = 1, 2, • • •, N) in a domain Q with boundary T. N is the total number of the nodes in the whole domain. It is assumed that only the surrounding nodes of a point x 0 have effects on w(x) . The domain is called influence domain, or support domain.

Using n nodes in the support domain of a point xQ, the radial PIM with polynomial

basis function approximates the field variable w(x) in form of n m

K*(x,x0) = £r,(x)a, + 2 > , ( x ) 6 , = rr(x)a + pr(x)b (1) f-i j=\

where a: is the coefficient for the radial basis r, (x), which is in form of

r, (*, y) = Kx-x,)2+(y-y,)2+ cd2min y ( 2 )

for Multi-quadrics (MQ) basis; and bj the coefficient for the polynomial basis Pj (x). For example, a linear basis in two-dimensional problems is provided by

pT={l,x,y}, m = 3 ^

which is also used in the paper. The coefficients a, and b} are so determined that Eq. (1) passes through n data nodes

in the influence domain, hence n equations can be obtained and written in matrix form

u e =R e a + Pmb f 4 )

where ( R e , Pm) are nxn and nxm matrices, respectively. To guarantee unique solutions, the following constraints are imposed:

45

Pla = 0 (5)

Combination of Eqs. (4) and (5) yields a = S0ue (6) b = S6ue (7)

Substituting of a and b into Eq. (1), the interpolation function *F(x) can be obtained

«*(x,x f i)=[r rS ( I+p rS4]u'=¥<x)u' (8)

Note that the shape function obtained through the above procedure possesses delta function and reproducing property. Some literatures prove that Rg1 always exists (Powell, 1992). When only linear or quadratic polynomial terms are included in Eq. (1), i.e., m=3 or 6, the solutions of shape functions can be obtained for general cases.

Variational Form for Piezoelectric Problem of Radial PIM

In this section, we consider a two-dimensional problem with piezoelectricity. The general functional L is determined by a summation of kinetic energy, strain energy, dielectric energy and potential energy of external work:

_ p u r u — S r T + - D r E + urf, -cpq, dA + T,uT Fp +2<pQp (9)

where T, S, D and E are vectors of mechanical stresses, mechanical strains, dielectric displacements and electric fields, respectively. They are coupled by the following linear constitutive piezoelectric equations.

T = c S - e r E

D = eS + eE

The electric field E is related to electric potential § by

E = -grad§

and the mechanical strain S to the mechanical displacement u by

S = V H

(10a)

(10b)

(11)

(12)

where Vs is differential operator in the Cartesian coordinate system.

Invoking Hamilton's variational principle, the following equation holds

bJLdt = 0 (13)

Substituting Eq. (9) to (12) into the above equation and using the approximation method depicted by Eq. (8) in section 2, a set of linear algebraic equations can be obtained to describe one integration point, rather than one single element in finite element method.

46

mii + kuuu + ku4,^ = F (14a)

k^u + k M ^ = Q (14b)

In order to obtain the integrals of the mass and stiffness matrices, a background cell structure is needed, which may be independent on the field nodes for interpolation. In each cell, Gauss quadrature is employed.

Numerical Examples

A Bimorph Beam

To demonstrate the validity of the derived formulations, the first example is conducted about a piezoelectric bending motor, or bimorph beam that converts electrical energy to mechanical energy. It has a 2-layer element in parallel connection. When an external voltage is applied across the beam thickness, the induced internal stresses produce a resultant bending moment which forces the beam bend. In this case, a unit voltage (1 volt) is applied across its thickness. Material properties of the motor are adopted from the book of Tzou (1993) and the motor is assumed to be in a state of plane stress.

In FEM analysis using ABAQUS, 4-node element CPS4E and 8-node element CPS8E are employed for simulation and the beam is discretized into different elements. Their static deflections are listed in Table 1 for tip node.

In the analysis using radial PIM method, multi-quadrics (MQ) radial function is employed as basis function. Two parameters (C and q) will influence the performance of this function. In our program, C is defined as C = c0dmm , where c0 is the coefficient chosen with the range from 0.2 to 5. It is recommended that c0 = 1 leads to a better result. dmm is the shortest distance between Gauss point s.Q and neighbor nodes in its influence domain but no less than 10% of the average nodal space. g=1.03 and g=0.98 are the recommended parameters in the static analysis.

A simple circle influence domain is used here with radius r, which is defined as r =a0dt, where a 0 is a coefficient chosen as 2 .0<a 0 <3 .5 in this paper and di is

FEM solution (ABAQUS) q=0.98, a o B 3 - 2

q=0.9B

q=1.03

... q«1.03

a 0 =3.5

« o - " a 0 - 3 . 5

Coordinates in lanth (m)

Fig. 1. Nodal deflection of a bimorph beam. Fig. 2. Eigenmodes of a piezoelectric transducer.

47

average nodal space. Two types of field nodes are arranged for analysis, i.e. regularly and irregularly

distributed 101 x5 nodes. The results are listed in Table 2. Some of them are also plotted in Fig. 1 together with FEM results. From Table 2, it

can be seen that the tip deflections agree well with ABAQUS results. Irregularly scattered nodes can also give good results. q=0.95~\ .08(g * 1.0) can assure good results. Inclusion of linear polynomial terms gives little larger but reasonable results. The method is very stable and the selection of nodes dose not affect greatly the accuracy of the final results.

Table 1. Static tip deflections of the piezoelectric bending motor by ABAQUS (x 10"8 m) Element type CPS4E CPS4E CPS4E CPS4E CPS8E Mesh 50x2 100x4 100x8 200x16 50x4 Node density 51x3 101x5 101x9 201x17 Node 5 -1.029 -1.062 -1.067 -1.073 -1.074

Table 2. Static tip deflections of the piezoelectric bending motor by radial PIM (x 10 8 m) , . , ^ Background „ Tip deflection Node pattern 5.. q n -

101x5 (Regular)

101x5 (Inregular)

100x4 0.98

1.03

100x4 1.03

" 0

3.2 3.5 3.2 3.5 3.2 3.5

m=0 -1.058 -1.082 -1.021 -1.058 -1.128 -1.162

m=3 -1.141 -1.142 -1.146 -1.147

— —

A Piezoelectric Transducer

This example performs an eigenspectmm analysis of a cylindrical transducer consisting of a piezoelectric material with a brass end caps. This is a typical example described in section 3.1.17 in ABAQUS manual book (1998). The transducer is modeled with a variety of elements in ABAQUS. The solutions obtained are shown in Table 3. When using radial PIM method, two arrangements of regularly distributed nodes are employed. 69 and 175 field nodes are for the coarse and fine nodal distributions, respectively. The results obtained are listed in Table 4 and the first 6 eigenmodes are plotted in Fig. 2, which is identical to those shown in ABAQUS manual.

Table 3. Piezoelectric transducer eigenvalue estimates by ABAQUS (kHz) Element type # of Elements

CAX4E 320 CAX8RE 80 C3D8E 16

Experimental (Mercer, 1987)

Mode 1 18.6 18.6 19.8 18.6

Mode 2 40.3 40.3 41.8 35.4

Mode 3 57.8 57.6 62.0 54.2

Mode 4 64.2 64.2 68.7 63.3

Mode 5 88.1 87.6 95.2 88.8

From Table 4, it can be seen that all the models give reasonable results and all these results match well with the numerical results from ABAQUS and the experimental results reported in Mercer et al. (1987). As the number of field nodes increases, the results

48

become closer to experimental results. In our analysis, q in the range of 0.95-1.08( q* 1.0 ) anda0 between 2.0-3.5 can give good solutions of the natural frequencies. The inclusion of linear polynomial terms makes mode 3rd to 5th a little smaller but still satisfactory.

Conclusions

The radial point interpolation method (PIM) is proposed to study the numerical solution of coupled piezoelectric structures. A radial PIM technique is applied and the shape function constructed by this technique possesses the delta function property. Variational principle is employed to derive the system equations. Numerical examples are given for determining deflections of a piezoelectric bimorph beam under an external voltage and mode-frequency analysis of a cylindrical transducer, which demonstrate the efficiency, convergence and stability of the present radial PIM method.

Table 4. Piezoelectric transducer eigenvalue estimates by radial PIM (kHz)

#. of Nodes P o l y n o m i a l q a Model Mode 2 Mode 3 Mode 4 M o d e 5

terms ^ u

69 1.03

1.03

2.0 19.9 42.7 60.8 65.6 91.9 2.5 19.8 42.6 60.7 65.5 91.5 2.0 19.8 42.5 54.9 59.6 88.4 2.5 19.8 42.6 54.9 59.8 88.3

175 m=0 1.03

m=0 0.98

2.0 19.7 42.2 59.3 65.0 89.9 2.5 19.6 42.2 59.3 64.9 89.9 2.0 19.8 42.3 59.5 65.2 90.3 2.5 19.7 42.3 59.4 65.0 90.1

References

ABAQUS/Standard (1998) Example Problems Manual, Volume I Version 5.8 Hibbitt, Karlsson & Sorensen Inc. 3.1.17.

Lerch R. (1990), "Simulation of Piezoelectric Devices by Two- and Three- Dimensional Finite Elements", IEEE Trans. Ultra. Ferro. Freq. Contr. 37(2) 233-247.

Liu G. R. (2002), Mesh Free Methods: Moving Beyond the Finite Element Method, CRC press, USA.

Liu G. R. and Gu Y. T. (2001), A Local Radial Point Interpolation Method (LRPIM) for Free Vibration Analysis of 2-D Solids, 246(1), 29-46.

Mercer C. D., Reddy B. D. and Eve R. A. (1987), "Finite Element Method for Piezoelectric Media", University of Cape Town/CSIR Applied Mechanics Research Unit Technical Report No. 92, April.

Ostergaard D. F. and Pawlak T. P. (1986), "Three -Dimensional Finite Elements for Analyzing Piezoelectric Structures", IEEE Ultrasonics Symposium, 639-644.

Powell M. J. D. (1992), "The Theory of Radial Basis Function Approximation in 1990", Advances in Numerical Analysis (F. W. Light, editors), 303-322.

Tzou H. S. (1993), Piezoelectric Shells: Distributed Sensing and Control of Continua, Kluwer Academic Publishers, Boston.

Wang J. G., Liu G. R. (2001), "A Point Interpolation Meshless Method Based on Radial Basis Functions", Int. JNumer. Meth. Eng., (revised).

49

Advances in Meshfree and X-P'EMMethods, G.R. Liu, editor,

World Scientific, Singapore, 2002

A HYBRID MESHLESS-DIFFERENTIAL ORDER REDUCTION (//M-DOR) METHOD FOR DEFORMATION CONTROL OF SMART CIRCULAR PLATE BY

SENSORS/ACTUATORS

J. Q. Cheng, Hua Li, K. Y. Lam, T. Y. Ng and Y. K. Yew Institute of High Performance Computing, National University of Singapore

1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, Singapore 117528

[email protected], [email protected], [email protected]

Abstract

A new hybrid meshless-differential order reduction method called the AM-DOR method is proposed in this paper. It is based on an order-reduction technique for partial differential equations and combines a point collocation technique with a fixed reproducing kernel estimate. The method is able to directly impose overlapping boundary conditions, which is a task difficult to achieve by existing collocation-based meshless approaches. In order to examine the efficiency and accuracy of the method, numerical comparisons with exact solutions are made for the bending of thin simply-supported circular and semicircular plates, and they show very good agreement. Further, the method is applied to simulate the deformation shape control of uniformly loaded smart circular plates integrated with piezoelectric sensors/actuators.

Keywords: Meshless method, Differential order reduction, Piezoelectric deformation control.

Introduction

Meshless approaches have attracted world-wide attention in both academia and industry (Belytschko et al., 1996) over the last few years. Generally they may be classified roughly into two groups. One requires a background mesh such as Galerkin-based techniques (Liu et al., 1995; Gosz, et al., 1996). The other does not require a background mesh such as the finite cloud method (Alum et al., 2001). However, they have some drawbacks, for example, it is difficult to implement overlapping boundary conditions, such as the simply-supported boundary condition, where there are more than one boundary conditions involving the same variable. In order to implement the overlapping boundary conditions, a new hybrid meshless-differential order reduction (/iM-DOR) method is developed. It combines the collocation technique with a fixed reproducing kernel approximation, based on a differential order-reduction technique, for partial differential equations (PDE). The present method first introduces variables that are suitable to reduce the differential order of PDEs. This is followed by reconstruction of the PDEs and boundary conditions in coherence with the selected variables. By scattering points in the domain, the collocation technique with fixed reproducing kernel is then imposed on the order-reduced PDEs and restructured boundary conditions, for discretization. Finally, the set of discrete algebraic equations with respect to point values is solved. For the examination of the convergence and accuracy of the developed /iM-DOR method, numerical comparisons with exact solutions are made for the bending of both thin simply-supported circular and semi-




50

circular plates. Further, the method is applied to smart structures for simulation of deformation shape control of circular plates integrated with distributed piezoelectric sensors/actuators. This enables the optimisation of both the dimension and location of the sensors and actuators in structural control systems.

AM-DOR Method - A True Meshless Method

For a domain Q in polar co-ordinate system, based on the classical reproducing kernel particle method (RKPM) (Liu et ai, 1995), the fixed reproducing kernel defines the approximation of an unknown real function f(r, 8) as

fir,6) = \C{r,0,p,cp)K{rk -p,dk-(p)f(p,<p)dpd<p (1) Q

in which K{rk -p ,6 k -cp) is kernel function and (rk,0k) the center point of the kernel. C{r,0,p,(p) = B(p,<p)C'(r,0) is a correction function, C*(r,0) unknown mth-order column coefficient vector, B(p,q>) wth-order basis-function row vector. Usually B{p,qj) is defined according to PDE problems. For example, for bending of circular plate with overlapping boundary conditions, if the cubature Serendipity-based interpolation polynomial is introduced as the basis functions where m=%, we have B(p,<p) = [l,p,(p,p2,pp,(p2,p2<p,p<p2]. Also, depending on the PDE problem, the fixed kernel function K(rk - p,0k -cp) is constructed by different forms of window functions. In this paper, a cubic spline function is used to construct the fixed kernel as,

K{rk -p,0k -<p) = W\{rk-p)IAr)W\(0k - <p) / A0)/(ArA0) (2)

where W*{z) is the cubic-spline form of the window function, z = (rk - p)/Ar or z = (0k -<p)lA0. Ar and A0 represent the cloud size in the r- and ^-directions and they need be adjusted according to the coordinates and accuracy requirement due to the consistency conditions of the reproducing kernel. The unknown coefficient vector C*(r,0) of the correction function is determined by the consistency conditions in the form of a set of linear algebraic equations, C*{r,0) = A\rk,0k)B

T(r,8), in which A{rk,0k) is a symmetric constant matrix. Thus, the approximation of the unknown f(r,0) in Eq. (1) is obtained in the discretized form as

f(r,0) = fj(B(pn,<p„)A~i('-k,0k)BT(r,0)K(rk - p„,0k -<p„)Asn)f„ = f > > , 0 ) / „ (3)

n=l n=l

where /„ is the unknown point values for the point n and Nn{r,0) is defined as the shape function. The approximation in Eq. (3) can be used to solve PDE problems, e.g.

Lf(r, 0) = P(r, 0) in CI, f{r, 0) = Q(r, 0) on TD, df(r, 0)ldr = R{r, 0) on TN (4)

where L is differential operator, Q PDE domain. TD and TN are the Dirichlet and

Neumann boundary conditions. Using the point collocation technique and taking

J(r,9) in Eq. (3) as the approximation of unknown f(r,0), the problem Eq. (4) is

discretized and rewritten with respect to the unknown point value / . (z=l,2,...,

•WT=(./VQ+./VD+NN)) into matrix form

51

[LNÂ,)}^

\Ji >NTx\ MnA,)^ [R(rtA)hK«

(5)

Numerically solving above-mentioned set of linear algebraic equations, the JVY-order

point-value vector f = {f,}NrXl is obtained and the approximation J(rA) is then

computed through Eq. (3). The formulations require that the intersection between different boundary conditions must be an empty set. This results in difficulty if it is applied directly to the PDEs with overlapping boundary conditions. In order to overcome the difficulty, the order-reduction technique is employed here.

The overlapping boundary conditions mean that there are more than one boundary conditions involve the same variable imposed at a scattered edge points. Current meshless approaches require an empty intersection set between two adjacent boundary conditions. For example, the Galerkin-based meshless methods simply use two sets of functions to formulate a variational form. One set consists of trial functions (<D = {f(r, 9)\f(r, 9)&HX, f(r, 9) = Q(r, 9) on TEssential}) and the other of test functions

(V = {g(r,0)\g(rA)zH\g(rA) = 0onrEssentiaI}) (Gosz, et al, 1996), Q(rA) is the given essential boundary condition. The domain boundary is defined by ÊssenM ^^Natural = r w h e n empty set rEssential n ^ , , = (j) is required. The similar empty-set is also required for the collocation-based meshless methods. Therefore, it is difficult to directly solve PDEs with overlapping boundary conditions whereby

® = {w(r,9)\w(r,0)eH\T]E(w) = 7fE(w) and TJN(W) = TJM(W) on TEssential nTNatural}(6)

with rfff(w) and r\N(w) being the essential and natural boundary conditions

respectively. The present boundary conditions are overlapping, i.e., TEssentia, ^>TNatural

= r = raw** =TNaturai ° r Tsienna, ^YNa,ura, " • « « non-empty set. In order to overcome this difficulty, suitable variables F(r, 9) are introduced not only to reduce higher-order PDEs into lower-order PDEs, but also to separate the overlapping boundary conditions respectively for F{rA) and w(x,y), such that they correspond to empty sets

® = {w(rA)\»>(r,9)eH1 and F{r,9)\F{rA)zH\

TJE(W,F) = J7E(W^F) a I l d Ô'-OÔ'^) 0 n rESsentic,l ^Natural)

(7)

By reconstructing the governing PDEs and overlapping boundary conditions to correspond with F{rA) and w(rA), a reduced set of PDEs is developed. Then, scattering points in the domain and using the collocation technique with fixed reproducing kernel, the system is discretized into a set of linear algebraic equations. Finally, approximations are obtained by solving this set of algebraic equations.

Numerical Validation of the AM-DOR Method for Circular Plates

For an isotropic thin circular plate subjected to a static transverse load, the governing equation in polar co-ordinates is written in terms of the transverse deflection w = w(r,6) as(Timoshenkoe/a/., 1959),

52

D0 V2 (wrr +Wr/r + wag/r

2) = q(r, 9) (8)

in which D0 is the bending stiffness and q(r,G) the distributed load in the transverse z direction. The simply-supported boundary conditions given by w = 0 (on TEssential) and Mr = 0 (on r ^ ^ ^ ) are overlapping boundary conditions since both the boundary conditions involve the variable w{r,$) which are required to be imposed on each edge point. As described above, it is not possible to directly impose the overlapping boundary conditions by the point collocation technique. By the present ftM-DOR method however, the problem can be solved by introducing a new variable F(r, 6), selected as

F ( ^ 0 ) = V2W(r,0) = ( w r r + W r / r + w w / r 2 ) (r,0)eQ. (9)

Then the governing PDE (8) and simply-supported boundary conditions can be rewritten in the following reduced form,

V2F(r,0) = (wrr+wr/r+wge/r2) = q(r,0)/Do in Q (10)

(w\=a =0 on TD, [F -(1 -v)(w„ +wr/r + wee /r2)]r=a = 0 on TN (11)

By the point collocation with fixed reproducing kernel technique, the approximate solutions are constructed as

VfrA^f, N*(r»0>n, ^ , 0 , ) = Z Nn(r,A)FH i = l,2,...,NT (12)

Substituting Eq. (12) into Eqs. (9) to (11), the reduced PDEs and boundary conditions are discretized and consequently a set of linear algebraic equations with respect to the unknown point values wn and Fn is obtained as [M]2NTX1NT X2NTXX -d2N xl, where

XT = [wN xl,FN x l ] . By any generic solving technique for linear algebraic equations,

the point values wn and Fn are computed and the approximation w(xr,,_y(.)

(i=l,2,...,NT ) of plate deflection is obtained through Eq. (12).

In order to validate the accuracy of the present method, the transverse deflections of both thin circular and semi-circular plates are investigated numerically. These plates are simply-supported and subjected to uniformly distributed loads q0 . Numerical comparisons are made with the exact solutions for the non-dimensional deflection W -w(r,0)Do/(qoa

4) . For the circular plate, 5 variations of point distributions, 10x10, 20x10, 40x10, 50x10 and 80x10, are considered. As the points in the r-

direction increase while the points in the ^-direction remain fixed, the relative errors \

decrease rapidly in a monotonic manner, for example, £ from 1.40% to 0.126% at r/a=0, l from 1.26% to 0.016% at r/a=0.l, t, from 1.21% to 0.033% at r/a=0.2, and £ from 1.20% to 0.067% at r/a=0.5. For the semi-circular thin plates, 4 variations of point distributions, 20x11, 20x17, 20x21 and 20x23, are considered. As the points in the ^-direction increase while the points in the r-direction remain fixed, the relative errors t, decrease rapidly, for example, % from 122% to 2.38% at r/a=0.25, t, from 85% to 2.84% at r/a=0.5, and £ from 69.6% to 2.86% at AVO=0.75. These show the numerical stability and good convergence of the developed /2M-DOR method.

53

Shape Control of Smart Circular Plates via Distributed Sensors/Actuators

In order to demonstrate the efficiency of the present ftM-DOR method for analyzing smart structures, the bending-deformation shape control of smart structures is simulated for integrated piezoelectric sensor and actuator patches. The uniformly loaded symmetric laminated circular thin plates with simply-supported boundary conditions are considered, as shown in Figures 1 (a, b), whereby the smart layers are partially covered by circular electrodes acting as the sensors/actuators. The uncoupled governing equation for thin bending plate subjected to the mechanical loading and electric field has been obtained by Lee (1990). If each layer of laminated plate is isotropic, the bending stiffnesses of the laminated plate are Du = D22 = (Dn +2D66) = D0. The governing equation derived in polar co-ordinate system is

4 V 2 K + w, / r + w„ I r2) = q{r, 0) = p(r, 6) -YjLtyJifjZftJr, # U (13) m=l

The variables and parameters in Eq (13) are given by Lee (1990). For die deformation-shape control of circular plates, by utilizing AM-DOR method and introducing a suitable variable F(r,0) in the same form as Eq. (9), the governing equation, Eq. (13), is reduced to a lower-order PDE. The approximate solutions w(x,y) and F(r,&) are constructed in the same form as Eq. (12), for PDE discretization. Similarly, the simply-supported boundary conditions are discretized. Thus, a complete set of discretized linear algebraic equations is constructed with respect to the unknown point values wn and Fn, which are solved for the simulation of the deformation shape control of the smart circular plate.

In the present numerical simulation of deformation control, we consider a sandwich-type laminated circular plate with central circular sensor/actuator patches, which are simply-supported and subjected to a uniformly distributed load. It consists of an aluminium-alloy core layer (E, = E2 = £3 =69GPa, G12 = G13 = G23 = 25.94GPa, and //12 = ̂ 13 = ̂ 23=0.33) with two surface layers of the piezoelectric material PXE-52 (£, = £2 = £3=62.5GPa, G12 = G13 = G23=24GPa, //12 = //13 = //23=0.3, dn =700x10-|2m/V, d31 = rf32=-280xl0~12m/V, /t33=3.45xl0~8F/m, with other zero parameters), as shown in Figure 1 (b). The influences of the electro-elastic coupling parameter Q defined by Ng et al. (2002) and central actuator dimension S on the deformation shape of the smart circular plates subjected to the uniformly distributed mechanical load p0 are shown Fig. 1. The maximum deflection of the plate can be controlled easily by the distributed sensors/actuators and the deflections decrease with increasing Q and S values. The dimensional effect of the electrode profile is also examined. By comparing Fig. 1(e) with Fig. 1(f), it is evident that, with increasing electrode profile surface, the deformation shape of the plate is qualitatively changed for the same electro-elastic coupling parameter Q. Further, the deflection mode profiles of the plates show that the simply-supported boundary conditions have been properly enforced by the AM-DOR method.

Conclusion

Due to difficulty in the direct imposition of overlapping boundary conditions encountered by existing collocation-based meshless methods, a new hybrid meshless-differential order reduction (AM-DOR) method is developed here such that these

54

overlapping boundary conditions can be imposed directly. Based on the order-reduction technique for partial differential equations, the AM-DOR method combines the collocation technique with a fixed reproducing kernel approximation. The method is validated for the bending analysis of thin circular plates. The /2M-DOR is found to be very accurate and also possesses high numerical stability. Further, an application of the /M-DOR method is demonstrated for the simulation of deformation shape control in circular plates under uniformly distributed loading and integrated with piezoelectric sensors/actuators. The numerical results all point to the newly developed / J M - D O R

method being elegant, accurate and numerically stable.

f*""V„"*** 4 ' ^ ( I^WV.

(a) Geometry of simply supported smart circular plate (c) S=n(0. la)2, G=0.0 (e) S=JI(0. lo)2, 0=22.0

Electrode 7TTTTT-, rrrrrrrr,

. Smart layer / Jl n.

.NElectrodc

Electrode

W " I C ^ S C S , ^ <* Wltf. 2=5.0 (f) S=.(0,fl)

2, 2=22.0 Figure 1. Control effects of central actuator dimension S and electro-elastic coupling

parameter Q on the deformation shape of simply-supported sandwich-type laminated circular plate subjected a uniformly distributed load.

References

Aluru, N.R. and Li, G. (2001), "Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation," International Journal for Numerical Methods in Engineering, 50, 2373-2410.

Belytschko, T., Krongauz, Y., Organ, D. and Fleming, M. (1996), "Meshless methods: an overview and recent developments," Computer Methods in Applied Mechanics and Engineering, 119, 3-47.

Gosz, J. and Liu, W.K. (1996), "Admissible approximations for essential boundary conditions in the reproducing kernel method," Computational Mechanics, 19,120-135.

Liu, W.K., Jun, S. and Zhang, Y.F. (1995), "Reproducing kernel particle methods," International Journal for Numerical Methods in Engineering, 20, 1081-1106.

Ng, T.Y., Li, H., Cheng, J.Q., Lam, K.Y. (2002), "A new hybrid meshless-differential order reduction (AM-DOR) method with applications to shape control of smart structures via distributed sensors/actuators," Engineering Structure (in press).

Timoshenko S. and Woinowsky-krieger S. (1959). Theory of Plates and Shells, McGraw-Hill Inc. Lee, C.K.(1990), "Theory of laminated piezoelectric plates for the design of distributed

sensors/actuators, Part I: Governing equations and reciprocal relationships," Journal of the Acoustical Society of America, 87, 1144-1158.

SECTION 4

Meshfree Methods for Fracture Analysis

57

Advances in Meshfree andX-FEM Methods, G.R. Liu, editor, World Scientific, Singapore, 2002

APPLICATION OF 3D F R E E M E S H M E T H O D TO FRACTURE ANALYSIS OF CONCRETE

Hitoshi Matsubara, Shigeo Iraha, Jun Tomiyama Dept. of Civil Engineering and Architecture University of the Ryukyus, 1, Senbaru,

Nishihara-chou, Okinawa, 903-0129, JAPAN k018424&.eve.u-rvitk\ni.ac.ip.iun-i&.tec.u-iyukyu.ac,ip, iraha&tec.u-rvuh'u.ac.ip

Genki Yagawa Dept. of Quantum Engineering and Systems Science University of Tokyo, 7-3-1, Hongo,

Bunkyo-ku, Tokyo, 113, JAPAN vaaawa(aj.q.tM-tokvo.ac.ip

Abstract

The numerical analysis technique came to be used for elucidation to various problems of the engineering field thanks to rapid advancement of the computing technology and popularization of computer. To simulate today's problems accurately, some large-scale analysis must be carried out. However, the problem of the input data making is pointed out in Finite Element Method (FEM) etc. Such a situation is received, meshless method is researched actively in a variety of engineering fields. There is Free Mesh Method (FMM) in a kind of these techniques. FMM does not require any connectivity between nodes and elements for an input data and is able to obtain accuracy which is almost equal to FEM. Authors applied the tensile fracture analysis of plane concrete using FMM in 2 dimension, and obtained excellent results. However, actual concrete structures have 3 dimensional behaviors, and the fracture behavior is extremely complex. In addition, the fracture behaviors of concrete are not analyzed enough in 3 dimension. This paper describes an application of FMM in 3 dimension to complex fracture of concrete, and some examples of the numerical analysis are shown and excellent results are obtained.

Keywords: FMM, Digital Image, Concrete, Fracture Analysis, 3 Dimension, and Splitting Test.

Introduction

Recently, element-free methods take much attention according to increase of the calculation scale. It is because that preparing the input data, especially providing the element data, is the most expensive part in those calculations. Omission of the element data can make much efficient processes. Several methods have been proposed in this point of view. The FMM is one of these methods and it has a great advantage that the basic idea is common to the usual FEM, and then the same techniques can be used. On the other hands, concrete structures in recent are large and they have complicated shapes. That's why it is very effective to use FMM which does not need mesh information to analyze them. In addition, the failure behaviors of them in 3 dimension are not analyzed enough, then the elucidation of destruction behaviors in 3 dimension is needed. This

58

paper presents the simple analysis method of tension crack problem of concrete using digital image, and simulates the tensile strength test considered the influences of the coarse aggregates.

Free Mesh Method

It is not necessary for FMM to prepare the element data in preprocessing. In usual FEM, the global stiffness matrix is constructed element by element. On the other hand, it is constructed node by node in FMM. The local meshes around node are generated as shown in Figure 1, and the matrix components for the corresponding node are calculated by using the stiffness matrixes of the local elements.

First, the one node is chosen as the center node. The local mesh is generated by using the satellite nodes which are selected from the candidate nodes within the appropriate distance from the center node. In 3 dimensional problems, several tetrahedron elements are generated as the local mesh as shown in Figure 1. We can calculate only the part of the global stiffness matrix corresponding to the degrees of freedom of the center node. It is 3 row components of the global stiffness matrix. In the next step, we get another pair of lines of the global stiffness matrix. When the all steps are done, we get the same global stiffness matrix to that from usual FEM. In this research, we used the CG law for the method of the equation, and used the Delaunay division for local mesh generation.

@: Current Central Node

# : Current Satellite Nodes

O : Current Candidate Nodes

• : Other Nodes

Figure 1. Local radial element

Simple Method of Fracture Analysis of Concrete

Usually, when we analyze the fracture of concrete, we use the method of increment analysis and use the stress-strain matrix (D-matrix) of cracked material. However in this paper, plasticity in compressive area, re-contact in cracked plane and tension softening of concrete has not been considered. Therefore, this analysis present only elastic behavior of concrete, and the calculation does not require any increment methods. Under cracked condition, in this method, the ratio of a load and displacement necessary to cause the following another cracks is obtained. Then, the node representing the maximum-principal-tensile-stress is defined as the cracked node, and the stress in a vertical direction (ri) in the direction of it is opened; we can express it by using D-matrix of a local element

59

which contributes to the crack node like the equation (1). In this paper, the smeared crack model was used as a crack model.

0

0

0

0

0

0

0

Ec

0

0

0

0

0

0

Ec

0

0

0

0

0

0

p-G 0

0

0

0

0

0

P-G 0

0 0 0 0 0

P-G

En

£s

7 ns

Yst

7 m.

(1)

where, Ec is the Young's modulus, p is Decreasing parameter of the elastic shear modulus (=0.01), G is the Shear modulus of elasticity, n is direction of maximum principal stress and 5, fare vertical direction of n.

For example, under an arbitrary load, the <r, is calculated at each node, and this value is substituted into the equation (2) to find out the biggest value of R,

fl = ^ f, (2)

where, /,is the tensile strength of concrete.

In case of symmetric configuration and symmetric applied load, it is considered that the cracks occur at 2 places at the same time. After Ris determined, all physical amounts (P) just before cracking are given by the equation as follow.

P=p, \ + R (3)

where, P is an arbitrary physical amounts.

Method of Analysis using Digital Image

Concrete is composite material, with coarse aggregates and mortar as the main components. But, in an existing study, the analysis by which these were taken into consideration was difficult. Then, this paper presents the method of analysis considered influence the coarse aggregates using digital image. It is necessary to prepare the material data besides the node data in this method. This data is made by digital image. The method of making the material data is shown below.

(1) Cut concrete, take a picture with the digital camera, and classify the coarse aggregate and mortar by a simple color as shown in Figure 2.

60

(2) RGB information on the pixel of the digital image data made by (1) is read, and the coarse aggregate and mortar are distinguished with different RGB. This is done on the entire cutting surface.

(3) Pixel information is converted into the identification number (mortar: 1, aggrigate: 2) that distinguishes the material.

(4) 3 dimensions are expressed by overlapping each layer. Each layer has the data of the layer number, the pixel coordinates, coordinates in direction of thickness, and the material number.

Cutting

«=^>

(a) Cutting of concrete (b) Digital images

Figure 2. Method of making the material data

N+l-Layer

Coarse Aggregate node

Mortar node

N-Layer

Figure 3. Interpolation of aggregates

In this method, two kinds of data are necessary; they are node and material information. And, the node does not necessarily exist on its digital layer because digital image data is made in 2 dimension, and it is only arranged. Therefore, we solved this problem by the linear interpolation, which showed by Figure 3; material property of node placed between certain two image layers is interpolated from the two layers in the straight line and decided.

Numerical Simulation of the Splitting Test

Here, the situation for tension fracture of concrete is simulated. The model of splitting test is shown in Figure 4. This test has often used to determine the concrete tension characteristics instead of the direct tension test that is very defect sensitive. In test,

61

material behavior is generally assumed to be elastic up to the maximum load (pwaJ,), and so the critical stress (ft) can be deduced by the well-known formula

f, = pmjxdl (4)

where, d is the cylinder radius and / the specimen thickness. The distribution of the interpolated the coarse aggregate is shown in Figure 5. In this analysis, the number of nodes is 63556, and the material properties and tensile strength are shown Table 1. From to Figure 5, we can see the appearance where the coarse aggregates are distributed complexly in concrete, and the authors think that it can be excellently simulated by this method.

Table 1. Material property of concrete

Mortar

Coarse Aggregate

£(JV/W)

20000.00

56122.45

f^N/mm2)

2.214

5.524

V

0.21

0.15

Displacement-Control

Figure 4. Splitting test Figure 5. Distribution of coarse aggregate

Figure 6 shows the analytical result that is the crack the situation of crack propagation progress situation and its load. The crack starts from (a), and has progressed like (b), (c), and (d). Where, in this Figure, only the crack nodes are displayed as seen easily, red spheres are cracking mortar nodes and black ones are cracking the coarse aggregate nodes. According to formula (4), the maximum load is 37.7kN and in this analysis, that is 35.0kN.

The crack occurred in center parts, extended from one surface of the model to center parts, and progressed toward another surface. At that time, we could confirm most coarse aggregates did not crack, and a lot of mortar cracked (it was 10.7% that the crack nodes were coarse aggregate of all crack ones), which is similar in the experiment; almost crack progresses avoiding the coarse aggregate. And, in this result, the crack is not symmetry. It is thought that the crack has generated in a part where the coarse aggregate is comparatively little, and extends from here. Actually, the coarse aggregate in the place where the first crack occurred was less than other places. In 2 dimension, the crack

62

cannot be expressible such as these situation, but we get the result of that situation because we do analyzed in 3 dimension. Therefore it can be said that this method is an effective way to simulate a microscopic crack of concrete.

*— f^Tpt^^vmp^^S^gr!^

fM * If . . 5 '

(a) P=35.0kN (b) p=21.3kN

(c) P=20.0kN ( d ) P=i6.6kN P: Load (kN)

Figure 6. Crack distribution Conclusions

We present an application of 3d FMM to the fracture analysis of concrete, this method is very simple and practicable, and we can get some excellent results. As the previous numerical example, in this method, the crack of concrete can be splendidly simulated the situation which progresses while avoiding the coarse aggregate. This is confirmed in a lot of expenments. Therefore, authors think that this method will be used as an effective fractures simulation technique of concrete. This research is scheduled just to have still started, and we plan to research detailed concrete destruction behavior in 3 dimension.

References

G. Yagawa, and T. Hosokawa. (1997), "Application of free mesh method with delaunay tessellation in a 3-dimensional problem", Transactions of Japan Society for Mechanical Engineers A, 60(614), 1997 2251-2256, (In Japanese) ' '

J. Tomiyama, S. Iraha, G. Yagawa, T. Yamada, T. Yabuki (1999), "Fracture analysis of concrete using free mesh method", Proceedings of International Conference, June, 1999.

S. Iraha, Y. Gushi, and H. Waniya (1982), "Finite element analysis of bond strength between steel and concrete I - splitting bond failure of deformed bars -", Bulletin of Faculty of Engineering, University of theRyukyus, Vol.24, 1982, (In Japanese) y

H. Matsumoto, S. Iraha, J. Tomiyama, G. Yagawa (1999) "Application of free mesh method using easy image processing to two dimension problem", Proceedings of the Japan Concrete Institute Vol 21 No 3 1999, (In Japanese) '

63

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor, World Scientific, Singapore, 2002

MESHLESS ANALYSIS INTEGRATE SYSTEM

FOR STRUCTURAL AND FRACTURE MECHANICS ANALYSIS

Seiya Hagihara Department of Mechanical Engineering, Saga University

1 Honjo, Saga 840-8502, JAPAN hagihara@me,saga-u. ac.jp

Mitsuyoshi Tsunori Ishikawajima-Harima Heavy Industries Co., Ltd., JAPAN

Toru Ikeda and Noriyuki Miyazaki Deparment of Chemical Engineering, Kyushu University, JAPAN

Takayuki Watanabe Department of Control and Information Engineering

Tsuruoka National College of Technology, JAPAN

Chaunrong Jin CRC Solutions, JAPAN

Abstract The Finite Element Method (FEM) is used for a lot of CAE (Computer Aided Engineering) systems. The FEM analysis requires connectivity information between nodes and elements. Although the pre and post processors for the FEM system have been developed, we have still consumed skill and time of the engineers for creating FEM mesh and remesh. Furthermore, it is difficult to generate models of analyses for except experienced engineer^

The element-free Galerkin method (EFGM) is one of the meshless methods. This method is a new numerical method which is expected to be utilized ior many problems of the continuum mechanics and for a main tool of the seamless system between the CAD and the CAE instead of the finite element method. The EFGM is desired as the CAE system reducing time and costs of the designing structures. The EFGM is studied by a lot of researchers. It is tried to be applied to geometrically nonlinear and three dimensional crack propagated problem etc. We also applied the EFGM to creep, elastic-plastic, dynamic and their fracture mechanics problem.

The EFGM has an another feature which is different from the meshless. The EFGM has the continuity of the first derivative i.e. strain and stress for a structural analysis by selecting the weight function. Then we can obtain displacement, strain and stress anywhere. Calculating the fracture mechanics parameter, we can calculate more accurate fracture mechanics parameter for nonlinear fracture mechanics problems.

In the present paper, we developed a 2-D integrate analysis system with the GUI (Graphical User Interface) system. The EFGM is applied to a system of material nonlinear analyses. We can develop the system to calculate elastic-plastic mechanics parameters J and T* integral. In this meshless sytem, we can generate models for the EFGM analysis by this system easily, analyze elastic-plastic problem, calculate fracture mechanics parameters and display all results of analysis graphically.

Keywords: Meshless Method, Elastic-Plastic, Structural Analysis, Fracture Mechanics, Analysis System, Graphical User Inteface.

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64

Introduction

The meshless methods are studied by a lot of researchers. The element-free Galerkin method (EFGM) (Belytschko et ah, 1994) is widely noticed as one of the meshless methods. It was developed from the diffuse element method (DEM) proposed by Nyroles et al. (1992) Since the node-element connectivities for FEM is not required to the EFGM, it is easier to create analysis model of the EFGM than the FEM. The EFGM is expected to be applied to many problems of the continuum mechanics and to be utilized for a tool in a CAE system through CAD instead of the finite element method. Nagashima (1999) proposed node-by-node method for the EFGM. Noguchi et al. (1997) applied the geometrically nonlinear the EFGM to membrane structures. Krysl, et al. (1999) developed the EFGM to apply three dimensional crack propagated problem. We successfully applied the EFGM to elastic-plastic problem which is one of the material nonlinear problems (Hagihara etal. 1998),. If we can develop the meshless analysis system using the EFGM, we can reduce time and costs of the designing structures.

In the present study, we apply the EFGM to elastic problems and calculate the fracture mechanics parameters. We developed a 2D analysis system including the GUI (Graphic User Interface) system using the element-free Galerkin method for material nonlinear analysis and calculating fracture mechanics parameters J and T*. We aimed to develop a user-friendly prototype system for the EFGM analysis.

Element-free Galerkin method

The weak form of the equilibrium equation is solved by the element-free Galerkin method. We consider weak form of the equilibrium equation given as:

f SteifpK-l + Ao^dV - f {Tf~l + Af;) 8Aut - f ( / f~ ' +AF,) 8AutdV = 0 (1)

where o$~' , Acs^ , ASjj , Tl• ~ , ATt ,Ft~ , AF( , and Aut are stress, incremental stress, incremental strain, external traction, incremental external traction, body force, incremental body force, and incremental displacement respectively. Superscript N-l denotes the previous N-lth time step. By using the nodal displacement vector Aqt, we can get the incremental form of stiffness equation of a total system in the form:

KijAqJ = F?N-'+AF<'-Ri (2)

where, Ky = stiffness matrix (elastic-plastic), Aqj = incremental nodal displacement vector, F"N~ = incremental external force vector, AFf = prior to the current time steps and/?, = residual force vector

The incremental nodal displacements within each load increment is obtained by solving equation (2) for Aqj . The total nodal displacements are calculated from accumulating the incremental ones in each step. If the strain is in an equilibrium with the external force, Ff ~ - Rt will become zero, otherwise the imbalance force is corrected in each increment by Newton-Raphson scheme. Background cells are needed to perform integration over analyzed region. Triangles generated by the Delaunay triangulation are used as background cells. Either the Lagrange multiplier method or the penalty method is usually utilized for the treatment of the essential boundary conditions in the EFGM. In the present EFG system, the essential boundary conditions are imposed by the

65

penalty coefficients.

The shape function is created by the moving least square method (MLSM) using the nodes in the domain of influence. For the MLSM, the linear basis function p(x) and the approximate displacement function uh can be written as follows.

p(x)T = [l,x,y]

uh(x)=p(xfa(x)

a(x) are determined so as to minimize the following function.

(3)

(4)

J= Z, w(x - x^ipixj) a(x) -u,] (5)

The exponential weight function w{x) is employed in this analysis (Belytschko et al.,1994).

For fracture mechanics problems, the path independent parameter J-integral proposed by Rice (1968) is well known . The J-integral can be calculated accurately in the EFGM, because the displacements, strains, and stresses can be obtained at the arbitrary point due to using the MLSM. The mathematical representation for J with the contour as shown in Fig.l is given as follows:

J=J [Wnx - ttuiA]ds , W = j * CTijdEij (6)

where, W is the strain energy density defined by the following equation, r is an arbitrary contour enclosing the crack tip counterclockwise in Fig. 1, ds is an infinitesimal arc-length along r, and Tis the traction vector. Tp is a contour enclosing vicinity of the crack-tip. Line integral is implemented on an arbitrary independent circle of nodes and background cells.

To detect the crack-tip severity of the propagating crack ,the path independent integral

USER

EFG system

Input Geometries and B.C.. etc

GUI processor

GUI pre-processor

Input data

Display results graphically

GUI post-processor

* J and T* integral

Displacement, etc. Integral contour, etc.

Material nonlinear element | w [ Fracture mechanics free Galerkin method analysis | ^ | parameter analysis

Displacement, strain, " " " ^ ~ ^ — ^ ~ " stress, etc.

Fig. 1 Crack geometry with Fig. 2 Outline of the meshless analysis contour path system (EFG system)

66

parameter T* proposed by Atluri et al. (1984) can be also calculated in the EFG system. This parameter is able to be applied to the problems of the strain history dependent loading and unloading in the elastic-plastic problems. The mathematical representation for T* with the domain as shown in Fig.l is given as follows:

T*=[ [Wnt - t fr !\ds= f [Wnx -ttuiA]ds- f [WA - a^^yiV (7) Jrp ' Jr Jv-vp

Procedure of creating analysis model

Outline of the meshless analysis system (EFG system) The outline of the mehless analysis system is described in Fig.2. The user can create easily analysis data of model by using graphic user interface (GUI) preprocessor. The information of geometries, boundary conditions and material properties etc. are input into GUI preprocessor. These input data for analysis are informed to the material nonlinear element-free Galerkin method analysis and the fracture mechanics parameter analysis the from the GUI preprocessors. After calculation of the material nonlinear element-free Galerkin method, displacements, strains, stresses, etc. are output to the fracture mechanics parameter analysis to calculate fracture mechanics parameter, J and T*. These output data is also used in the GUI post-processor. The results of the material nonlinear element-free Galerkin method analysis and the fracture mechanics parameter analysis are displayed graphically to the user.

&i%#- 5?55snp^

Fig. 3 Input geometric boudary and nodes on boundary

Fig. 4 Generation of inside nodes in geometric boundaries

W^Z

«

J

^^^rn^^^r^-^

i i i i i i i i

mtmrnqmim-^ •VVi, <"r

* • - . . -

as

-v™**'"'

1

*f "

US* ^®H&P&!&*''Lm.

Fig. 5 Pop-up window to input boundary conditions

Fig. 6 Pop-up window to input material properties

67

Input data of the analysis into the GUI preprocessor The input window of geometric boundaries and boundary nodes of a quarter model of a center-holed plate is shown in Fig.3. We can input geometric boundaries numerically into pop-up window from keyboard and do them to pick the location by a mouse device. If we input these information, geometric boundaries and nodes located on the boundaries are displayed by a personal computer shown in Fig.3. The nodes on the boundaries are generated at even intervals automatically. The nodes in inside of the boundaries are generate by giving intervals of nodes shown in Fig.4. The outside nodes of the geometric boundaries are eliminated by numerical process. In Fig.5, the boundary conditions can be input into pop-up window by mouse and keyboard devices. We can also input numerical information of Young's modulus, Poisson's ratio, yield stress, the rate of strain hardening into pop-up windows shown in Fig.6. The analysis conditions are also input into other pop-up windows.

The input window of a quarter model of a center-cracked plate is shown in Fig.7. In the fracture mechanics problems of cracked plate, we can give the information to the GUI preprocessor to calculate J and T* integrals. The input data are crack-tip information, contour path radius of integral and number of contour path. The graphic window of fracture mechanics analysis of cracked plate is shown in Fig.7. Geometric boundaries, nodes distribution, boundary conditions and contour path are displayed in window shown

&t*mmt&i&®%F%®**x " v ^

Si3i»J MjZMfyXfS

Fig. 7 Center-cracked plate for fracture mechanics problems

Fig. 8 Deformation and contour of von Mises stress

blB?5?****.'?. .7. " .' .

"*'"" i t t s

LIEE'QQl

ITO'DiB

miuftot&ifrefr

•

33 41

33 45

„ _

/

99 \Zl 16b ivd s i . ?

/

/ » 13? IAS m 'SNip

_,.

-~&3«

i

"*"• ss w w s n ^ i P ! "

l,x*&Mtrte. .. .fiSSS-SSS ~ .S*a«»!t«Jt* fc»-V

Fig. 9 Variation of J and T* integral versus loading steps

Fig. 10 Window of path independency of J and T* integral

68

in Fig.7. The all data about the analysis can be input by the GUI preprocessor graphically.

Calculation of the analysis by the EFGM The seamless EFGM analyses of elastic-plastic problems can be performed on the GUI preprocessor. The results of these analyses calculated by the material nonlinear EFGM analysis are output into the GUI post-processor.

Display results of the analysis by the GUI post-processor We can show results of analysis on the window of the GUI post-processor. The deformation of the analysis model can be described in a window shown in Figs. 8. When we show the deformation, we can indicate the displacements, strains and stresses by color contour line shown in Fig.8. The strains are three component of x, y, xy. The stresses are three components of stress, the maximum and the minimum principal stress and von Mises stress.

Figure 10 shows a window of variation of J and T* integral versus loading steps. We can confirm the variation of the fracture mechanics parameters J and T* by this window. The path independency of J and T* can be also confirmed on each step show in Fig. 11.

Conclusions

In the present paper, we developed a 2D analysis system including the GUI (Graphical User Interface) system using the element-free Galerkin method for material nonlinear analysis and calculating fracture mechanics parameters J and T*. This system could generate models for the EFGM analysis easily, analyze elastic-plastic problem, calculate fracture mechanics parameters and display all results of analysis graphically. If we use this developed system, we will reduce time and costs of the designing structures.

Acknowledgment

This software was developed by RISE (Research Institute of Software Engineering) in Japan under support from IPA's(Information-technology Promotion Agency) "Support program for young software researchers" in Japan.

References Belytschko, T., Lu, Y. Y. and Gu, L., (1994), "Element-free Galerkin method", IntemationalJoumalfor

Numerical Methods in Engineering, 37, 229-256. Nyroles, B., Touzot, G. and Villon, P., (1992), "Generalizing the finite element method": Diffuse

approximation and diffuse elements, Computational Mechanics, 10, 307-318. Nagashima, T., (1999), Node-by-node meshless approach and its applications to structural analyses,

International Journal for Numerical Methods in Engineering, 46, 341-385. Noguchi, H., (1997), "Applications of Element Free Galerkin Method to Analysis of Mindlin Type Plate/

Shell Problems", Advances in Computational Engineering Science, Eds. S.N.Atluri and G.Yagawa, 918-911.

Krysl, P. and Belytschko, T., (1999), "The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks", International Journal for Numerical Methods in Engineering, 44, 767-800.

Hagihara, S., et al., (1998), "Analysis of fracture mechanics parameters for stable crack growth problems using elastic-plastic element-free Galerkin method", Modeling and simulation based engineering, Edrs. S. N. Atluri and P. E. O'Donoghue 1, ,65-70.

Rice, J. R., (1968), "A path independent integral and the approximate analysis of strain concentration by notches and cracks", J. Appl. Meek, 35 , 379-386.

Atluri, S.N., Nishioka, T. and Nakagaki, M., (1984), "Incremental path-independent integrals in inelastic and dynamic fracture mechanics", Engineering Fracture Mechanics, 20- 2, 209-244.

69

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor World Scientific, Singapore, 2002.

APPLICATION OF 2-DEMENSIONAL CRACK PROPAGATION PROBLEM USING FREE MESH METHOD

J. Imasato Grtaduate School of Engineering, Yokohama National University, Japan 240-8501

imasato@dolphin. eng.ynu. ac.jp

Y. Sakai Faculty of Environment and Information Sciences, Yokohama National University

[email protected]

Abstract

Recent advances in computer technology have enabled one to solve a number of complicated natural phenomena. Out of computer simulation techniques, the finite element method(FEM) has been most widely used since it is capable to analyze the domains with arbitrary shape. However, to application on moving boundary problem, Ihe work of remeshing needs to carry out frequently with progress of analysis process.

To overcome fhese difficulties, meshfree methods have been developed. They require only nodal information as an input data and can process all calculations including model generation wifliout commissure. Out of these meshfree methods, the free mesh method(FMM) developed by Yagawa and Yamada. This meshless method, does not require any connectivity local element at each node without considering global meshing. A total stiffness matrix is obtained by adding these temporary element matrices.

This paper presents an application of 2-dimensional crack propagation problem using free mesh method. The feature of this method is an analysis model which modeled along with growth of crack surface can be defined easily by nodal control ( node position change and addition) since the free mesh method is the analysis technique by nodal base. Moreover, since free mesh method is the analysis technique on the basis of conventional finite element method, the criterion of fracture in a crack tip can be applied to the knowledge of finite element method.

Keywords: Free Mesh Method, Meshfree Analysis, Finite element mefliod, crack propagation,

Introduction

The crack propagation problem is important in fracture of material, and computational methods for its numerical simulation are essential to failure prediction. Out of computer simulation techniques, the finite element method is a technique have been widely used in various fields since it is capable to the analysis domain of arbitrary shape. Especially, It is effective to apply this finite element method to the crack propagation problem which

http://ac.jp


70

follows a complicated crack path. However, to application on moving boundary problem such that arbitrary crack path propagation problem, the work of remeshing needs to carry out frequently with progress of analysis process.

For this background, many efforts have been performed in developing the methods, not requiring any element or grid, and several meshfree methods have been completed. These researchs have been developed in the reference, showing many research fields in fluid analysis and crack propagation problem.

Out of meshfree methods, the method called as the free mesh method(FMM) proposed be Yagawa and Yamada becomes applicable. This is a kind of meshfree methods based on theory of FEM. In application of this method, the total stiffness matrix is obtained be adding temporary element matrices and this method does not require any connectivity between nodes and elements for an input information. So, this method is expected to be an effective procedure for realizing "CAD/CAE seamless system" in making models and computations.

This paper presents an application of 2-dimensional crack propagation problem which grows in the arbitrary direction using free mesh method. The feature of this method is an analysis model which modeled along with growth of crack surface can be defined easily by nodal control ( changing the node position and addition) since the free mesh method is the analysis technique by nodal base. Moreover, since free mesh method is the analysis technique on the basis of the theory of conventional FEM, the Criterion of fracture in a crack tip is evaluated using the knowledge of finite element method.

Free Mesh Method

The calculation flows are compared in Fig. 1 for the usual FEM and the FMM. We do not need to prepare the element data in preprocessing for FMM. In usual FEM, the global stiffness matrix is constructed element by element. On the other hand, it is constructed node by node in FMM. The local mesh is generated for each node, and the matrix components for the corresponding node are calculated by using the stiffness matrix of the local elements.

First, we choose one node as the center node. The local mesh is generated by using the satellite nodes, which are selected from the nodes within the appropriate distance from the center node. In two dimensional problem, several triangular elements are generated ass the local mesh as shown in Fig. 2. We can calculate only the part of the global stiffness matrix corresponding to the degree of freedom of the center node. In two dimensional problem, we can get only 2 lines of the global stiffness matrix by using the local mesh. In the next step, we get another pair of lines of the global stiffness matrix. When the all steps are done, we get the same global stiffness matrix to that from usual FEM.

71

In the FMM, although the local element is generated around center node, it is necessary to keep the consistency of local element when a center node moves to other nodes. Therefore, generation of local element needs to introduce the treatment which are satisfied of these restrictions. According to Delaunay's triangulation, the same connection is obtained by using the node which is on the same circumscribed circle each other as shown in Fig.3.

Definition of Analysis Model

When applying a moving boundary problem like a crack propagation problem to a meshfree method, it is expected that it is controllable by change, an addition, etc. of node coordinates. Although the conventional finite element method defines an analysis model as an aggregate of an element, as described above, in FMM, it needs to devise the definition of a new analysis model since an element is generated inside an analysis process.

Thus, in this study, the form modeling method of the boundary representation method used in the field of the form modeling introduces, and the data structure which enable it to refer to this information from the node placed on a boundary is proposed. Namely, the analysis model is expressed with the "topology" and the "geometry" which expresses actual shape. The topology is a data structure which expresses a boundary of solid model using object, shell, face, loop, edge, and vertex, and expresses the connection relation. The geometry displays actual shape with surface, curve and point. On one hand, as information of the distributed nodal point, coordinate information of each nodal point and information of topology and the information of geometry that correspond to boundary as shown in Fig. 4, furthermore the normal vector for expressing the inner side or outside of an analysis model is obtained from geometric information using each node coordinates, and the information is also stored. By using these data structures, it becomes possible to generate a local element, referring to the information on a geometric model.

Discrete Equations using Mixed Variable Principle

As mentioned above, in the FMM, it is necessary to hold the consistency of local element when a center node moves to other nodes. Since these restrictions, the element which can be used serves as the minimum degree of freedom, and it is necessary to use three nodes in two dimensions or four nodes element in three dimensions. Therefore, the accuracy of an analysis solution becomes a thing in a primary element. Kanto has proposed that introduces the iterative solution method for the mixed method of Hu-Washizu by Zienkiewicz at el., and can carry out improvement of the accuracy of solution as the Solver of FMM.

The mixed method which is adopted here is based on treating the displacement u, the strain e, and the stress o as a variable of the problem approximated independently. The Hu-Washizu variable principle is expressed using approximated independently as the following equation

72

nHW= J-eTDzdQ.- ju bdSi- j a T ( e - S u ) d Q - juTtdT n ^ a n r,

(1)

whereb, 1 , £1 and Tdenote body force, surface force , domain, boundary where surface force works, respectively. And D, S is the matrices which defining stress-strain relations, appropriate differential operator defining strain-displacement relations, respectively. Approximating independently the three variables by appropriate shape function sets

a = N „ c e = N,e w = N„w (2)

in which o , e and u are sets of nodal parameters, we can write the approximation equations with NCT', Ne ' and N u ' , respectively. Here, each independent variable is approximated using the same C0 continuous interpolation function N and the node values u, e , and a . , we have

[A - C O ]

\-CT 0

[ 0 ET o\[u (3)

where

A=\ NeTDNtdn, E=\ N/BdQ, C= J N/N0dn, / = J N,rWQ+ J NJidT (4)

a n n n r,

Zienkiewicz and others, this kind of element shows the conditions which give a stable solution, and it are shown that this condition is satisfied 2-dimensional and 3-dimensional both cases, if the linear element which assumes each variable on each node is used.

Criterion of Fracture

The analysis of a crack problem must consider the singularity to which the stress and strain become infinite at near a crack tip. For this reason, in order to obtain the stress intensity factor which is fracture mechanics parameter, there is a method using the element (the singular element) which possesses singularity in the element at the tip of a crack. On one hand, using the normal element instead of use of the singular element method , the J integral which is equivalent to energy release ratio is given as follows:

J, 4[Wnk°-Ty^ds (5)

where the upper subscript expresses component of x,° of system of coordinates at crack tip as shown in Fig. 5, W is the density of strain energy, u surface forces respectively, nk is direction cosine.

T is the displacement,

The relation between J integration and a stress intensity factor is given as follows:

73

Because with FMM the local element which is generated from the nodal data becomes the usual element, the stress intensity factor is estimated from value of J integral. Clearly from eq (5), J integration is path integration, Although calculation of J integration by the finite element method uses the connective information on an element for the path information on integration, it is necessary to set up a path by other method in FMM. In generally, it is known that the value of J integration has path independency at a distant place from a crack tip. Then, It considers setting up the circle of radii arbitrary as a center for a crack tip. And the judging of intersection with this circle and the element which generated locally is performed, stress and strain are interpolated on a path using eq. (2) within the crossing element, and J value is obtained.

In the crack propagation problem with crack curving, it is necessary to clarify the criterion in which a crack carries out the extend direction rule with fracture of material. Although several criterion were proposed as a crack curving fracture criterion, in this study, the criterion of local symmetry (Kn = 0 criterion) is used.

Numerical Example

The proposed technique was applied to the problem in which a crack exists aslant to the direction of the principal stress shown in the Fig. 7. And, the crack propagation which advanced until 8 steps in the Fig. 8 was shown. Material constants of Young's modulus and Poisson ratio were 2100MPa and 0.33 respectively. As for result in Fig. 8, it is seen that it is similar to the result due to G. C.Sih and others.

Conclusions

The proposed method was applied to the 2-dimensional crack propagation problem which grows in the arbitrary direction as a moving boundary problem, and showed validity. The feature of this method is the analysis model of a crack propagation problem because the free mesh method is the analysis technique by the nodal base. Node control (node position change and addition) can perform easily.

References

Belytschko T., Lu Y. Y., Gu L. and Tabbara M. (1995), "Element-free Galerkin methods for static and dynamic fracture", International J'ournal of Solids and Structures, 32, 2547-2570

Yagawa G, and Yamada T. (1996), Free Mesh Method: A new Meshless Finite Elemnet Mehthod, Computational Mechanics,!^, 2741-2746.

KantD Y. (2000). "Accurate Free Mesh Method by using mixed Element," Transactions of Japan Sosiety for Computational Engineering and Science, 3, 13-17

74

Zienkiewicz O.C, Taylor R. L.(1996), "TheFinite Element Nethod,(}^aness Edition)" CADTECHS,

Goldstein R.V. and Salganik R.L.(1974),"Brittle fracture of solids with arbitrary cracks'ynternational Jounal ofFracture,10(4),507-523

Erdogan F.,Sih G. C.(1963),"On the crack extension in plates under plane loading and transverse shear" Journal of Basic Engineering ASME,$5,5\9-527

f s t a r t ^

Dlstrfcute Nodes!

Generate golbaJ mesh

Solve system equations I

Croo Mceh Method

^D Distribute Nfodas I

Otatainv*o» strtrness mark

Generate local temporary leteiment

| Obtain local slllThBSsmiatrtx | node

I adcltQ\*ole stiffness rniatrk |

1 t

Solve system equations I

-*-»,

({^Collection nodes tor neighbourhood of ce nter node nodes

(cJSellect local mesh

Figure 1. Algorithm compare FEM and FMM Figure 2. Free Mesh Method

Circum circle for l-r-m

Circum circle for l-m-n

Triangles for center node* I

Triangles for center node-tn

Figure 3. Deluanay tri angulation

H E rui •> modes on vertex © :node* on Edge O modes In interior

!_~. _~.i i Definition of Edges for Analysis Model

(a) Analysis Model

R ] A H E D [vTTA H E

^ ^ i • • • is r ia ?sri-P _„__o _ i"T"^=tlr::-v ° s O < i O O

| - _ ^ » i © ® 0 4

(b) Node Disribution (c) Shape Toporogy

Figure 4. Definition of analysis model

Figure 5. J integral

Figure 6. J path for FMM Figure 7. Analysis for crack Figure 8. result of crack model propagation

SECTION 5

Meshfree Methods for Membranes, Plates & Shells

77

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor, World Scientific, Singapore, 2002.

ANALYSIS OF M E M B R A N E STRUCTURES WITH L A R G E SLIDING C A B L E

USING M I X E D DISPLACEMENT FORMULATION AND EFGM

Hirohisa Noguchi Department of System Design Engineering Keio University 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

noguchiCcb.sd. keio.ac. ip

Yoshitomo Sato School of Sciences for Open and Environmental Systems Keio University

tomodonoguehi.sd.keio.ac.jp

Tetsuya Kawashima Graduate School of Mechanical Engineering Keio University, Japan

kawa'Mnoguchi.sd.keio. ac.jp

Abstract

Many large membrane structures have been constructed in these days and large membrane structures are often stiffened by cables for reinforcement of the strength. In the analysis of cable-reinforced membrane structure, there are several complicated problems, such as the fold of membrane by cable, sliding of cable on membrane surface and so on. It is difficult to analyze these problems by the finite element method, authors have applied mesh-free method based on the element free Galerkin method to the analyses of membrane structures with cable reinforcement. In the conventional element free Galerkin method, the problem that contains discontinuous slope of displacement cannot be analyzed because of the CI continuity condition by the moving least squares approximation. Additionally, large sliding between cable and membrane surface must be considered. In this paper, the mixed displacement formulation and the technique of patch are incorporated in EFGM, and the proposed method is applied to the simple example of membrane with rigid cable and its validity is demonstrated.

Keywords: mesh-free, EFGM, mixed displacement formation, membrane structures, large cable sliding

Introduction

For design of membrane structure, several kinds of analysis have to be considered. They are form-finding analysis, stress analysis and cutting analysis. In the finite element method (FEM), a different model with appropriate mesh in each analysis is utilized, while in the mesh-free method, a set of analyses can be conducted by using only one model, because it has no elements. Bearing this in mind, authors have been developing a meshfree analysis system by using the enhanced element free Galerkin method (EFGM) [Kawashima 1999], [Noguchi 2000].

http://sd.keio.ac.jp

http://ac.jp

78

On the other hand, many large membrane structures are often stiffened by cables to reinforce their strength. In the analysis of cable-reinforced membrane structure, there are several complicated problems, such as the fold of membrane by cable, sliding of cable on membrane surface and so on. In order to overcome these problems, authors combined the technique of patches and the mixed displacement formulation with the enhanced EFGM and analyzed a cable reinforced membrane structures with discontinuous slope [Noguchi 2002]. However, in these analyses, cable slide is limited to be small because only nodes on the cable have the Eulerian degrees of Freedom. In this study, therefore, the formulation has been extended to treat large sliding cable by rearranging of nodal position at each loading step. Simple examples are demonstrated to show the potential of this method.

Consideration of Discontinuous Slope of Displacement

In this section, the treatment of discontinuous slope of displacement in the EFGM is briefly described. In the analysis by EFGM, continuous strain field is obtained by using the moving least squares approximation (MLSA) for the approximation of the field function. In the analysis of structure with discontinuous gradient of displacement due to such as material discontinuity or membrane with cable, however, strain field also becomes discontinuous at the interface. Therefore the conventional EFGM cannot be applied [Cordes 1996].

In order to analyze the structure with discontinuous gradient of displacement, analysis model is divided into patches. Fig. 1 illustrates the domains of influence near the interface of patches for analysis model of membrane with cable reinforcement. The domain of influence for the nodes close to the interface in either patch is truncated. Therefore, nodes in the same patch can only influence points included in each patch. The stiffness matrix is constructed in each patch, respectively. The constraint condition to impose continuous displacement at the interface is given as follows:

^ / '(xrf)u ;-^ /2(x(i)uJ =0 (Summation convention is utilized for J and I) (1)

where <j>i' and §\2 are approximation functions obtained by MLSA from patch 1 and 2 and Xd is a position vector at the interface. The whole stiffness matrix K is obtained by the assemblage of each stiffness with the penalty term, as

K ; = Ki, +KJ, +aI ( ^ ( x j ^ x j - ^ x ^ x j ) dTs (2)

where a is a penalty number, K1, K2 are the stiffness matrices obtained from each patch and Ts is the interface of patch.

79

domain of influence

Figure 1. Domain of influence close to the interface of patch

Mixed Displacement Formulation in EFGM

For analyzing moving discontinuity caused by a sliding cable, the boundary of patch must be coincided with the discontinuous line of slope. In the proposed mixed displacement formulation, the displacement of membrane structure and the sliding of cable on membrane surface are treated as different variable [Noguchi 2002], [Haber 1983],

In the formulation, both the initial configuration lX and the current configuration lx are treated as unknown variables. Each coordinate is described by using an arbitrary spatial reference configuration jcr as follows:

and its displacement 'u is

'X = X(jcr, t), 'x = x(xr, t) (3), (4)

'u = u(:t', t ) = ' x - ' X ( 5 )

The total displacement is separated into the Eulerian displacement 'u and the Lagrangian displacement ' u .

' u = ' u + ' u (6)

' X = ° X - ' u , ' i = ° i + ' u (7), (8)

In the analysis, the Eulerian displacement shows the flow of material point in the initial configuration and the Lagrangian displacement shows the displacements of nodes. In the formulation, the deformation gradient tensor of total displacement lF is decomposed into the Eularian 'F and Lagrangian 'Fas illustrated in Fig. 2. The Eularian 'F describes the mapping from the initial to the reference configuration and the Lagrangian 'F describes the mapping from the reference to the current configuration. The inverse Eularian 'F describes the mapping from the reference to the initial configuration the concept of mixed displacement formulation is illustrated in Fig. 2 [Haber 1984].

80

'F='F'F

Figure 2. Concept of mixed displacement formation

Fhe total potential energy in the reference configuration is written as:

TI = - [r'S:'E'Jdvr- f ('u+'u)-'b'Jdvr+-a[ ('u-'u)2dTL +-a [ ( ' u - ' u ) 2 a f r . 2 * *' 2 •*"* 2 •*"«

(9)

where S and E are the 2nd Piola-Kirchhoff stress tensor and the Green-Lagrange Strain tensor, respectively. vr is the volume in reference configuration, TL and TE are boundaries for Lagrangian and Eulerian displacement, 'u and 'u are prescribed displacements at each boundary and 'J is volume ratio from reference configuration to initial configuration calculated from the determinant of the inverse Eulerian deformation gradient as,

f rf<V= f ^dvr= f >Jdv' (10)

where, 'V is the initial volume.

From the variation of Eq.(10) with respect to the Lagrangian and the Eulerian displacement, Eq.(l 1) is obtained which is equivalent to the virtual work principle.

U,S:S'E'J+-'S:'ES'j\dVr-l(Su+Su)-'b'J+('u + 'u)-'bSJdvr

+a£ Su-('u-'u)drL+a[ SvL-('u-'u)dTE=0 (11)

After the incremental decomposition of Eq.(l 1), the following equation is obtained.

81

l{CS:SEj+'S:6Em'J + S:6EL,J) + ±CS:'VSJm+'8:ESJL + &:'E5JL)}(V

- l{(5b+Su)-''bJ + ('u + 'vi)-''bSJNL + (<a + a)-l'bSJL}dvr

+ a[ SaudFL+a[ SuudTE

= {, {(Ju + £u) • ' V J + (' u + ' u) • %SJL}dvr - 1 1 ' S: SEL'J + - ' S : ' E5JL W

-a[ Su-('u-''u)drL-a[ Su('u-'u)drE

(12)

Each displacement fields are discretized using shape function obtained by MLSA.

u(x) = ̂ (x )u / ; u(x) = ̂ (x)u, (13), (14)

By substituting Eqs.(13) and (14) into Eq.(12) and solving Eq.(12) by using such as the Newton-Raphson method, the analysis considering sliding can be conducted.

Folded Membrane by Rigid Cable

A simple problem is analyzed to verify the proposed formulation. Figure 3 shows a schematic view of analysis model of membrane with rigid cable. Membrane is fixed at the both ends and folded at rigid cable that can move only x and z direction and keep parallel to y-axis. This structure is subjected to a load by the cable and in the analysis the load is equally distributed at the nodes on the cable where the discontinuity of slope exists. Friction between membrane and cable is ignored. The following two cases are considered.

First, only the horizontal load is applied to the cable and the cable displacement in z-direction is fixed. In this case, all the nodes on the membrane are set to have the Eulerian and the Lagrangian degrees of freedom and the Lagrangian displacement is prescribed so that the height of nodes keeps constant. The deformed configuration is shown in Fig. 4. The black circle shows the same material point on the initial and the deformed configuration. Compared with our previous result [Noguchi, 2002], cable slides very largely. Figure 5 shows the load and the Lagrangian displacement curve of node at the cable. The obtained result perfectly agrees with the exact solution that is available in this particular problem. Figure 6 shows the relation between the Eulerian and the Lagrangian displacement at the discontinuous slope line. There is a peak around x=2, where the cable is located outside of the right fixed nodes. It is physically interpreted that after material point moves toward the cable and then moves slowly back to the opposite direction.

Second, the Lagrangian displacements at the cable are prescribed so that the total length of membrane in x-z plane keeps constant. Apparently the cable draws an elliptic

82

locus as shown in Fig. 7. In this case, the nodal rearrangement is made at each deformed configuration and only the nodes on the cable have the Eulerian degrees of freedom. As the length of membrane does not change, the both reaction force and the section membrane force should remain unchanged at the initial value which is zero in this case. As results, not shown in the figure, all the section forces at any points on the membrane are computed to be zero and the present analyses are validated.

rigid cable

Figure 3. Analysis model

1.25

1

0.75

0.5

0.25

0 m A nodes on initial configuration

X nodes on deformed shape

_ identical material point on initial and deformed configuration

- 2 0 2 4 X

Figure 4. Initial configuration and deformed shape along line y = 0

t>0.4 Si>0.2

0 0.5 1 1.5 2 2.5 3

Lagrangian displacement

Figure 5. x-Reaction force and Lagrangian displacement curve

at the discontinuous line

-*—*-

0 0.5 1 1.5 2 2.5 3

Lagrangian displacement

Figure 6. Relation of Lagrangian and Eularian displacement at discontinuous line

83

1.2 1

0.8 N0.6

0.4 0.2

0 - 3 - 2 - 1 0 1 2 3

x

Figure 7. Deformed shape along l i n e ^ O

Discussion and Conclusion

In this paper, the analysis of membrane structures with large cable sliding is carried out by combining the mixed displacement formulation and the enhanced EFGM. In order to enable large cable sliding, first, the Eulerian displacements are given to all nodes, this may be valid when the Lagrangian displacements are known in advance. Second, the rearrangement of nodal position is made at each deformed configuration and the Eulerian displacements are given to only the nodes on the discontinuous slope line. This procedure seems rather complicated, however, it may be suitable for practical use. A simple example is analyzed to demonstrate the potential of the present method and the validity of the method is clarified.

Acknowledgements

This study is partly supported by a grant from Nohmura Foundation for Membrane Structure's Technology. We gratefully acknowledge for this support.

Reference

Kawashima, T. and Noguchi, H. (1999): "The Analyses of Membrane Structures with Cable Reinforcement by Element Free Method," Proceedings of Fourth Asia-Pacific Conference on Computational Mechanics, Vol.2, pp. 1003-1008.

Noguchi, H., Miyamura, T. and Kawashima, T. (2000): "Element Free Analysis of Shell and Spatial Structures," Int. J. Num. Meth. Engrg., Vol.47 , 1215-1240.

Noguchi, H. and Kawashima, T. (2002): "Meshfree Analysis of Cable Reinforced Membrane Structures by ALE-EFG Method," to appear.

Cordes, W. and Moran, B. (1996), "Treatment of Material Discontinuity in the Element-Free Galerkin Method,"Comp«r. Meth. Appl. Mech. Engrg., Vol.139, 75-89.

Haber, R. B., (1984), "A Mixed Eulerian-Lagrangian Displacement Model for Large-Deformation Analysis in Solid Mechanics," Comput. Meth. Appl. Mech. Engrg., Vol.43, 277-292.

Haber, R. B. and Abel, J. F. (1983), "Contact-Slip Analysis using Mixed Displacements," J. Eng. Mech., VoL109,411-429.

' '' /̂ s*s' / j n s V

**/ IJ^J^*^^

iM&^*~ • wp •

* • v

• • .

* " • • * . w • •-,

^ • O .Z^-^ssk. • :

84

Advances in Meshfree andX-FEM Methods, G.R. Liu, editor. World Scientific, Singapore, 2002.

THE EFFECTS OF THE ENFORCEMENT OF COMPATIBILITY IN THE RADIAL POINT INTERPOLATION METHOD FOR ANALYZING MINDLIN PLATES

X. L. Chen, G. R. Liu and S. P. Lim Department of Mechanical Engineering, National University of Singapore, Singapore

119260 [email protected], [email protected], [email protected]

Abstract

The effects of the enforcement of compatibility in the radial point interpolation method for bending analyses of Mindlin plates have been investigated in this paper. The conformability of the radial point interpolation method (RPIM) is enforced using the constrained weak form of static system equation based on the Mindlin plate assumption. Multi-quadrics (MQ) radial basis functions are used in this investigation. Deflections of Mindlin plates are calculated using both conforming RPIM (CRPIM) and non-conforming RPIM (NRPIM). The examples showed that the results obtained using CRPIM and NRPIM are very close, and the NRPIM can ease the shear locking occurring in thin Mindlin plates.

Keywords: Meshfree, RPIM, Radial basis function, Mindlin plate, Bending, Numerical analysis.

Introduction

Recently, a radial point interpolation method (RPIM) has been developed for mechanics problems for solids, structures and fluids (Liu, 2002; Wang and Liu, 2002). The RPIM has some advantages: Its shape function has delta function property due to that its approximation function passes through all the nodes in the influence domain; This property makes the RPIM enforce essential boundary conditions as easy as in the conventional finite element method (FEM). In addition, the RPIM shape functions and their derivatives can be easily obtained. However, the RPIM shape function does not automatically provide the compatibility for displacement interpolation throughout the problem domain, and the formulation based on so-called unconstrained weak form produces a non-conforming RPIM (NRPIM). A conforming radial point interpolation method (CRPIM) needs to be formulated using the constrained weak form. However, the CRPIM may not be necessary always over performing NRPIM, especially for problems of beams, plates and shells.

This paper formulates firstly an MQ-CRPIM for bending analyses of Mindlin plates. The weak form of static system equation is established based on Mindlin plate assumption. The influence domain is defined based on cells of integration instead of interpolation points. The deflections of Mindlin plates are calculated using both MQ-CRPIM and MQ-NRPIM. The shear locking of Mindlin plates is studied using both formulations.




85

Briefing on Point Interpolation Using Radial Basis Function

A displacement w(x) (\e(x,y)) may be approximated using a set of scattered nodes in a small and local influence domain of x in the form of

n m

u(x) = YJR1(x)a,+YPj^)bJ (1)

where the known radial basis R, (x), in this paper, we use the Multi-quadrics (MQ):

R,(x) = [rj +(c0Ar)2]' , the known polynomial basis P,(x) which is chosen from

Pascal's triangle, unknown coefficients a, and b}, the number of nodes n in a influence

domain and the number of polynomial terms m . The shape parameters are c, c0 and q .

r, is the distance between point x and node x,, Ar is the characteristic distance related

to nodal spacing, which is usually the average nodal spacing in the influence domain.

The polynomial bases need to satisfy an extra-requirement to guarantee unique approximation as (Golberg et al, 1999)

Y.PAx^^O, J = \~m (2) i=\

For any interpolation point, the interpolation must pass through all the n nodes within the influence domain. The combination of Eq. (1) and Eq. (2) gives approximated displacement as

w(x) = 0(x)ue (3)

where the shape function vector O(x) has n components (Liu, 2002).

Governing Equations

Consider a plate shown in Fig. 1. Based on Mindlin plate assumption, the displacements, the linear strains and the stresses for plane stress problems may be expressed as (Wang et al, 2000; Chen et al, 2002):

u = LQ, e = LQ, o = De (4)

where Q = (w0,<fix,</> )T for Mindlin plate. L and L are operator matrixes. D is elastic

rigidity matrix.

Different order polynomial bases can be independently chosen for the displacement interpolations. The three displacement variables are independently approximated as

86

V0 = 2>1/W0/ > & = E*2/#rf , <Py = Z 0 3 / ^ / (5) w„ i=\ 1=1 7=i

where <3>u,<$>2/ and <J>3/ are the shape functions.

In formulating CRPIM and NRPIM, there is a difference in choosing nodes for displacement interpolations. For the NRPIM, the local influence domain is defined usually for an interpolation point (which is usually a quadrature point in a background integration cell), as shown in Fig. 2(a). For the CRPIM, however, the local influence domain is usually formed for the geometrical center of an integration cell as shown in Fig. 2(b), and the so-called one-piece shape functions are used for the entire integration cell (all the interpolation points in a cell share the same influence domain and the same set of shape functions), so as to ensure the compatibility of the field function approximation with the cells. The enforcement of the compatibility of field function approximation is achieved using the following constrained weak form (Liu, 2002):

lS(LQ)TD(LQ)dV- j / ( L Q ) r b r f F - | S(LQ)TtdS-

r ° ( 6 )

| (<5Q+-<5Q-) ra(Q+-CT>/r = 0

where b is the vector of body forces, t is the vector of prescribed surface forces. Q+

and Q" are the displacements of any point of common line between two neighbor cells, which are interpolated based on nodes of influence domains of these two cells respectively, a is the matrix of penalty coefficients. Substituting the approximated displacements Eq. (5) into Eq. (6), the weak form of static system equation can be discretized as

(K + K')U = f (7)

where U is the vector of all the nodal displacements. K is the global stiffness matrix and f is the global force vector. K" is derived from the term with penalty coefficients in Eq. (6) and defined by

K " = | [ 0 + - 0 - ] r a [ 0 + - 0 - ] ^ r (g)

where <b+ and <J> are the array matrixes of shape functions obtained based on the influence domains of two neighbor cells respectively.

Numerical Examples

The deflections of Mindlin plates with simply supported and clamped boundaries and subjected to uniform loads are calculated using both conforming and non-conforming

87

MQ-RPIM. The displacements w0 , <f>n and </>s of line between neighbor cells are constrained for the MQ-CRPIM and are not constrained for the MQ-NRPIM. The geometrical and material property parameters of the plates are: a = b = 10.0m; h = 1.0m; £ = 1.0xl09 Af/m2 ; v = 0.3. c0 =2.0 and # = 1.03 for MQ radial basis function are

used and no polynomial terms are included. The deflection coefficient is £ = ^— for qb

uniform load, where D — and q is the uniform load. The size of influence 12(1-v2)

domain is chosen to be 3.9 times the average nodal distance. Regularly distributed nodes 11x11 in the whole plates are used.

The results are tabulated in Table 1. The deflections calculated using the MQ-CRPIM agree very well with those obtained using the MQ-NRPIM and EFG. In using MQ-NRPIM, we tested also two different ways in constructing shape functions: cell-centre based one-piece shape function and quadrature-point-based moving interpolation shape function. The results obtained using both sets of shape functions in the MQ-NRPIM are also very close, and slightly larger than those for the MQ-CRPIM.

Table 1 Deflections of center of uniformly loaded Mindlin plates

Boundaries

Simply supported

Clamped

MQ-

One-shape function

0.004575

0.001499

•NRPIM

Moving interpolation

0.004575

0.001499

MQ-CRPIM

0.004574

0.001499

EFG

0.004619

0.001505

In meshfree methods, desired higher-order approximate displacement fields can be easily obtained without any complication in the algorithm. Therefore, in this paper, higher-order approximate displacement fields for transverse deflection and rotations are constructed to eliminate shear-locking of thin Mindlin plate in the MQ-CRPIM. The same numbers of polynomial terms from Eq. (1) are used for displacements approximation. c0 = 2.0 and q = 1.03 and 15x15 regularly distributed nodes are used. The deflections of the Mindlin plates with different aspect ratios are calculated using different numbers of polynomial terms. It is observed from Fig. 3 that shear-locking occurs when aspect ratio h/a<1.0xlQr2 for the MQ-CRPIM and h/a<1.0x\0-3 for the MQ-NRPIM. The MQ-CRPIM can more easily leads to a shear locking than the MQ-NRPIM. For both MQ-CRPIM and MQ-NRPIM, shear locking can be reduced by choosing higher-order polynomial terms.

Deflections of Mindlin plates with different aspect ratios subjected to uniform load are studied using the MQ-CRPIM. 15 polynomial terms, c0 = 2.0 and q = 1.03 and 15x15

88

regularly distributed nodes are used. It can be observed from the results in Table 2 that the deflections of the plates obtained using the MQ-NRPIM are very close to those obtained using the MQ-CRPIM, but the MQ-NRPIM can reduce shear locking for thin plates.

Table 2 Deflections of center of uniformly loaded simply supported Mindlin plates

Aspect ratio

hi a = 1CT1

h/a = 10"2

h/a = 1(T3

h/a = 10-4

Conforming

0.004609

0.004073

0.003804

0.003551

MQ-RPIM

Partial . conforming5

0.004610

0.004074

0.003862

0.003552

Non- & conforming

0.004610

0.004074

0.004057

0.003830

Other solutions

0.004619"

0.004062*

Compatibility is enforced for all w0, <j>n and (/>s; Compatibility is enforced only for w0;

No compatibility is enforced for w0, <pn and </>s, and one-piece shape functions are used;

a: EFG result (Liu, 2002); b: Timoshenko's solution.

Conclusion

The deflections calculated using the MQ-RPIM agree very well with those of EFG. Shear locking can be reduced by choosing higher-order polynomial terms. The deflections calculated using the MQ-NRPIM agree very well with those obtained using the MQ-CRPIM, but the MQ-NRPIM can reduce shear locking for Mindlin thin plates.

References

Chen, X.L., Liu, G.R. and Lim, S.P. (2002). "An element free Galerkin formulation for analyzing thick plates," Proceedings of the Ninth International Conference on Computing in Civil and Buckling Engineering, April 3-5, Taipei, Taiwan, 419-424.

Golberg, M.A., Chen, C.S. and Bowman, H. (1999). "Some recent results and proposals for the use of radial basis functions in the BEM," Engineering Analysis with Boundary Elements, 23,285-296.

Liu, G.R. (2002). Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, USA.

Wang, CM., Reddy, J.N. and Lee, K.H. (2000). Shear Deformable Beams and Plates (Relationships with

Classical Solutions), Elsevier, Singapore.

Wang, J.G., Liu, G.R. (2002). "A point interpolation meshless method based on radial basis functions," InternationalJournal for Numerical Methods in Engineering, 54,1623-1648.

Figure 1. A Mindlin plate and its notation

Influence domain

Point

/ Cell

Node

Figure 2(a). Influence domain for point-based interpolation

, Influence domain Line between cells

Cell

Node

' Point Figure 2(b). Influence domain for cell-based interpolation

NRPM,m=0 - A - NRPlM.m=6 - » - NRPIM.m=1G -*- NRPIM.m=15 - * - CRPIM.m=0 --0- CRPIM.m=6 - B - CRPIM.m=10 - « - CRPIM.m=15

Figure 3. Shear locking in a simply supported plate with different aspect ratios (£0 is Timoshenko's solution)

90

Advances in Meshfree andX-FEM Methods, G.R. Liu, editor, World Scientific, Singapore, 2002.

A M E S H F R E E M E T H O D FOR D Y N A M I C ANALYSIS OF T H I N S H E L L S

L. Liu, V.B.C. Tan

Centre for Advanced Computations in Engineering Science (ACES) c/o: Department of Mechanical Engineering, National University of Singapore

Singapore 119260. E-mail: mpeliuli(a),nus.edu.sg

Abstract

The implementation of the element free Galerkin method for dynamic analysis of thin shells is presented in this paper. The formulation of the governing equations based on the geometrically exact theory of shear flexible shells is used. The present method uses the moving least squares approximation to construct both shape functions based on a set of scattered nodes arbitrarily distributed in the analysis domain and the surface approximation of general shells. Discrete system equations are derived from the variational form of system equations. A subdivision similar to finite element method is used to provide a background mesh for numerical integration in domain. The penalty method is used to enforce essential boundary conditions. The Newmark method is applied for the time integration. Several numerical examples are presented to show the validity of the proposed method. Results are compared with theoretical and finite element results to demonstrate the efficiency and accuracy of the presented method.

Keywords: Mesh Free Method, Element Free Galerkin Method, Dynamics, and Shell Structures.

Introduction

Dynamic analysis of thin shells involving complex geometries, loading and boundary conditions can only been done by numerical methods. The most widely used numerical method for this type of problems is Finite Element Method. FEM has achieved remarkable success in the static and dynamic analysis of thin shells (Yang et al.,2000 ). However, it might be noted that mesh generation is a far more time-consuming and expensive task even than the solution of the finite element equations in FEM. On the other hand, mesh free methods in the area of computational mechanics have attached much attention in recent decades, requiring only nodal data and no element connectivity and are more flexible than the conventional finite element method. Mesh free methods have been reported in several varieties in prior literature. The EFG method, one of the well-known Mesh free methods, has been applied to problems in static and dynamic fracture mechanics, static analysis of plates and shells, free vibration of thin plates and shells, and many other problems. (Belytschko et al., 1995; Liu 2002; Noguchi, 2000)

The goal of the present paper is to develop and study the EFG method for dynamic problems of thin shells-usually denoted as Kirchhoff shells. The outline of the paper is as

91

follows: In section 2, the governing equations for the analyses of general thin shells and membrane structures are introduced: first, the numerical formulation based on geometrically exact theory accounting of the Kirchhoff hypothesis is given; then, the definition of curved surfaces, kinematics of shells, stress and strain measures, and the constitutive equation adopted in the formulation are described; the essential boundary conditions are enforced by penalty method. The Newmark method is then applied for time integration. Finally, the present method has been applied to several numerical examples on shells of different geometries to illustrate its efficiency and accuracy.

Governing Equation for Thin Shell

The shells considered in the present work are assumed to be thin so that the Kirchhoff-Love theory can be considered appropriate. The governing equations used in this paper is based on geometrically exact theory of shells proposed by Simo et al (1989) and modifications are made to account for the Kirchhoff hypothesis. The Gauss intrinsic coordinates are used to describe the configuration of the shell. The shell in the 3D space is described in a global Cartesian coordinate system. The pair (p.t) defines the position of an arbitrary point of the shell, q>(^,^2) gives the position of a point on the shell midsurface, and t(£',<f;2) is a director unit vector normal normal to the shell surface both in the reference and deformed states according to the Kirchhoff-Love hypothesis. The configuration can be expressed as

^ = {xeR3 |x = <K£,,<f2) + <ft(£,,<?2) with £ ' , £ 2 e K and ^e(h~,h+)} (1)

Here 5R denotes the two-dimensional parametric space, (/T,//+)are the distances of the

"lower" and "upper" surfaces of the shell from the shell midsurface.The midsurface is defined by the differential two-form

dSR=^,x?>2d^d^2 (2)

The linear membrane and bending strain measures can be denoted as follows

^to-^+f i -"- ) (3)

PaP =\{<P°a • *j, +P,V Ata + u 0 -t°, + u , -t0o) (4)

The Kirchhoff-Love hypothesis needs to be introduced explicitly to obtain the definite forms for the strain measures. The mathematical form of this hypothesis is expressed as

t = (7)~>.,**2). 11*11 = 1- (5)

92

Hence the derivatives of the normal vector in the reference configuration t° and partial derivatives of the increment At can be derived from equation (5) as follows

W7T'fcxf;+f>fS.)- (6)

M* = (7°)"' (u i. x <P\ + <i x 9%, + <P% x u,2 + 9° x u 2* )• (7)

The membrane strain measures of equation (3) are not affected by the introduction of the Kirchhoff-Love hypothesis. Considering the symmetry with respect to partial differentiation g>°l2 = <p°2l and u 12 = u 21, the bending strain measures can be rewritten as

A. =-».,, •t°+(70)"'[u1-(^ l X^) + u 2 - ( ^ x ^ 1 ) ]

p22 = -u 22 • t° + (J° )"' [u , • (9% x <pl) + u 2 • {q>\ x q>\2)]

Pn = - « , 2 ' t° + (7°)"' [u,, • {<P°n x <P\ ) + u 2 • {<pl X p°2)] (8)

The stress resultants and stress resultant couples are defined by normalizing the force and torque with the surface Jacobian j = U», x q>A as follows

n"=(7)"'£<igad<?, a = 1,2 (9)

m a = ( 7 ) " ' £ (x-<p)x<,gajdt, a = 1,2 (10)

The effective membrane and bending forces are defined to describe the weak formulation of the shells

n = H^"ap <8> aa

m = xa^aap®aa (11)

By making use of the basic kinematic assumption (1), the dynamic weak form is expressed in the form by the effective resultant as

W>» (Sx)=JL[fi'a • 5 £ ^^ •d K^+ \&w ' ^ ^ _ w « (*x) (12)

Here W , is the virtual work of the external loading given by

93

Wal = \^n-S(p + mSt]ASR+ J^n-^yds+j^ Ja6tjd& (13)

Here n is the apglied resultant force per unit length and rii is the applied direct couple per unit length. iT and m are the prescribed resultant force and the prescribed director couple on the <3„5R and dJU, respectively. However, penalty method is used to enforce essential boundary conditions by adding an additional boundary condition term in the argument of equation (12)

WDy„ {Sx) - I X • (u - u) <5ud3l = 0 (14)

where u , u are the nodal vector and prescribed displacement vector on the surface 5RU. A is the matrix of penalty coefficients which are usually very large numbers .

Discrete Equation

Substituting the equations (3), (4) and (11) into the variational weak form (12), the final dynamic discrete equation can be obtained as follows:

M u „ + C u „ + K u „ = Q f (15)

Here K and M are the global stifmess and global mass matrices, respectively; subscript n denotes time nAt and At is the size of time increment or time step. u„, u„, ii„ and

Q"' are the displacement, velocity, acceleration and force vectors at time nAt. The Newmark method is applied in the paper for time integration. Hence variations for the displacement u„and velocity ii„ in the time interval At to be such that the values at beginning and end of the time step are related by equations of the form

= u„ + u A/ + — a u „ + a u „ j

At1 (16)

u „ + , = u n + [ ( l - ^ K + < f t » „ + . ] A ' (17)

where a and 5 are parameters that can be determined to control integration stability and accuracy. Solving equation (16) for ii„+, in terms of un+l and then substituting ii„+I into equation (17), the equation for ii„+1 and u„+1 in terms of the unknown displacements u„+1

are obtained as follows

aAt u^, -8

aAt •^(f-'Ms-'H <18)

94

aAt2 1 1 .

- u , + — u . aAt2 ' aAt ' \2a 4»m (19)

Substituting equations (18) and (19) into the governing equation of motion (15) at time (n +1) At, the equation is obtained

1 5 \ K + — — M + — C

^ aAt2 aAt J

+C 8 ? u ,+ | 1

aAt \a

" n + i = Q n + 1 + M 1 1 . ( 1 , ...

—— u , + - — u , + I n , aAt2 aAt 12a (20)

The above equation is an equation with unknown variables un+1 only and the dynamic analysis of the structure can be treated at each time step as a static problem by the right part of the equation to be a generalized external forcing acting on the structure.

Numerical Examples for Forced Vibration of Thin Shells

The forced dynamic response of a clamped cylindrical shell as shown in Figure 1 is presented. The following geometry and material properties are used: length L = 600mm, thickness h = 3.0mm and radius R= 300.0mm . The Young's modulus is E = 2.1 x 10" N/m2, the Poisson's ratio v - 0.3 and the mass density is p = 7868kg/m1. Figure 2 and Figure 3 show the history of load and the dynamic response of clamped cylinder subjected to the sine curve excitation at the center of the meridian, in which the force could be expressed as F = F0 sin (1000/) , F„ =1000.0Af . Figure 4 shows the dynamic response of same cylinder subjected to sine curve excitation, while the force could be expressed as F = F0 sin(2000r), F0 = lOOO.ON. In the numerical calculation, the time step At = 2.5e-5s is used, which is almost the 1/35 of the fundamental period of the cylinder. The (12x16) regular nodes are arranged in the axial and circumferential directions. As shown in Figure 3 and Figure 4, close agreements exist between the results obtained by FEM and the EFG method.

Conclusions

The element-free Galerkin method has been developed for the dynamic analysis of thin elastic shells in this paper. In EFG method, MLS technique is used to construct the shape function and the surface geometry of shell. The present method offers distinct computational advantages over classical finite element method; no element is required in this approach eliminating arduous and time-consuming mesh generation in FEM. The essential boundary conditions are forced by the penalty method. The Newmark method is applied for time integration. Numerical examples of thin shells under step and sine curve load are analyzed to demonstrate the efficiency, convergence of the EFG method. It's

95

found that the results compare favorably with other solution methods and the EFG method is easy to implement for dynamic analysis of thin shells and spatial structures.

z^^te*.

<^\^y \ \

\ • - ' " ' **

\ i

^ • ^ i

1200

1000

\ ,-'' 800

' \ F(Af) eoo

\ 400

^ j 200

0

0.0005

Figure 1. The clamped cylinder with concentrated sine curve load

Figure 2. History of load

0.001 0.002 0.003

- B=G FB/I t(s) 0.0005 0.001 0.0015

FB/I — H=G '(*)

Figure 3. Displacement response of Figure 4. Displacement response of the central point in the meridian the central point in the meridian

References

Belytschko T., Gu L., and Lu Y.Y. (1994). "Fracture and crack growth by Element-free Galerkin method," Modeling Simulations and Material Science Engineering, 2, 519-534..

Liu G.R.(2002). Mesh Free Method: Moving Beyond the Finite Element Method, CRC press, USA.

Noguchi R , Kawashima T., and Miyamura T. (2000). "Element free analyses of shell and spatial structures", International Journal for Numerical Methods in Engineering, 47,1215-1240.

Simo J., and Fox D.D. (1989). "On a stress resultant geometrically exact shell model, Part I: formulation and optimal parameterization," Computer Methods in Applied Mechanics and Engineering, 72, 267-304

Yang H.T.Y., Saigal S., Masud A., and .Kapania., R.K. (2000). "A survey of recent shell finite elements," InternationalJournal for Numerical Methods in Engineering, 47, 101-127

96

Advances in Meshless and X-FEM Methods, G.R. Liu, editor,

World Scientific, Singapore, 2002.

A CONFORMING POINT INTERPOLATION METHOD FOR ANALYZING SPATIAL THICK SHELL STRUCTURES

L. Liu, G. R. Liu*, V.B.C. Tan, Gu. Y.T

* SMA Fellow, Singapore-MIT Alliance (SMA) Centre for Advanced Computations in Engineering Science (ACES)

c/o: Department of Mechanical Engineering, National University of Singapore Singapore 119260. E-mail: mpeliugr(a),nus. edu.ss

Abstract

The conforming point interpolation method (CPIM) is presented for the static analysis of spatial thick shell structures. The formulation of the discrete system equations is derived from stress resultant geometrically exact shell theory based on the Cosserat surface. The PIM technique constructs its interpolation functions through a set of arbitrarily distributed points in the problem domain and its shape function has the delta function property. Hence, the implementation of essential boundary conditions can be imposed with ease as in conventional finite element method. To obtain compatible shape functions, the enforcement of compatibility is needed along the common edges of the background cells in the problem domain, which lead to conforming PIM. Several benchmark problems for shells are analyzed to demonstrate the validity of the present method.

Keywords: Shell Structures, Mesh Free Method, and Conforming Point Interpolation Method.

Introduction

Mesh free methods have become interesting and promising methods in computational mechanics due to their flexibility in practical applications. The objective of mesh free methods is to construct an approximate solution entirely in terms of nodes to solve classes of problems which are very awkward with mesh-based methods, such as finite element method (FEM) and finite difference method (FDM). The PIM is an effective mesh free method which does not require special treatment for imposing boundary conditions even for unstructured distribution of nodes. The most attractive characteristic of the method is that its shape functions possess the Kronecker Delta property, and thus, essential boundary conditions can be easily enforced in a straightforward manner as in the classical finite element method. The key for the PIM with polynomial basis functions is to guarantee the existence of shape functions or the inverse of the moment matrix. Non-existence of the inverse or instability of PIM method occurs from improper enclosure of nodes in the influence domain and improper selection of monomials of the polynomial basis. Some numerical techniques have been developed to alleviate this problem, Liu and Gu (2001a) proposed a moving node method to slightly change the coordinates of nodes, Wang et al (2001) developed a local coordinate transformation to

97

change the basis function. Recently, Liu and Gu (2001b) presented a matrix triangularization algorithm (MTA) to overcome the singularity problem.

The objective of the present paper is to develop and study the PIM method for problems of the three-dimensional general thick shells and to conduct element free analyses of them. The point interpolation method is used in both the construction of shape functions for arbitrarily distributed nodes as well as the geometrical surface approximation of general spatial shells. An improved matrix triangularization algorithm (MTA) is used to obtain proper nodes enclosure and basis selection in order to overcome the singularity problem in the PIM using a polynomial basis. From experience, it is necessary to have a certain number of monomials, say k, to achieve the continuity of the shape function. The method is based on the theorem that the dimension of the column space is equal to the dimension of the row space, each being equal to the rank of the matrix.

The outline of this paper is as follows. In section 2, the PIM and its shape functions are briefly reviewed: the working of the method, and the construction of the shape functions. Then in section 3, the necessary governing equations for the analyses of general shell and the corresponding numerical discretization for CPIM are given. In section 4, several numerical benchmarks are presented to demonstrate the convergence, validity and efficiency of the present method. Finally, conclusions are given in section 5.

PIM Interpolation and Shape Function

The point interpolation method interpolates a set of scattered data points to construct the shape functions for surface shape approximation and numerical discretization. If a set of arbitrarily distributed nodes x,(z' = l,2,...,«) and their function values ut are given surrounding point x0, the PIM interpolates a continuous surface u(x) using a polynomial basis with n terms as (Liu and Gu,2001a; Liu 2002)

«(*) = I A«a , («o) = PT(x)a(x0) (1)

where pt(\) is a monomial and xT =(x,y) for two-dimensional problems, n is the

number of nodes in the influence domain of x0 and ^(x,,) are the coefficient of #(x)

corresponding to the given point, i.e.

aT=[a, a2 ••• an] (2)

The coefficients a, (x0) are determined by enforcing the function in equation (1) to pass through all scattered points surrounding point x0. _p,(x) in equation (1) is built utilizing the Pascal's triangle to guarantee completeness of the basis. A basis in two-dimensional domain is provided by

98

pT (x) = [1, x, y, xy, x2, y2, x2y, xy2,---] (3)

Interpolation at the z'th point is expressed as

",=PT(x,)a(x0), i = l,2,...«. (4)

It is noted that u, is the nodal value of u at x = x,. Equation (4) can be rewritten in the following matrix form

u'=P0a (5)

where ue and the moment matrix P0 are given by

ue=[w„ u2, ..., w„]T (6)

P0=[p(x,), p(x2), ...,p(x„)f (7)

If the inverse of moment matrix P0 exists, a unique solution for a(x0) is obtained

a = P0-'ue (8)

Hence, the function w(x) can be written as

W(x) = 0(x)u< (9)

where the shape function ^(x) is defined by

0(x) = PT(x)Po-' =[p,(s), <P2(*)> ••-, <PM] (10)

The shape function ^(x) depends only on the distribution of scattered nodes. It has the

following properties:

(i) #>.(x) has the delta function property which can be expressed as

*w=lJ *:*; c/^o (ID

(ii) <pt(x) has the property of partition of unity

99

2 > , « = 1 (12)

(iii)^.(x) i s a polynomial of order similar to the rank of the moment matrix P0.

Since the interpolation functions constructed have delta function property, essential boundary conditions can be easily imposed in the PIM. The interpolation function has the unity partition property, which means that the interpolation function has the ability to represent the rigid body motion. It can also be seen that accurate integrations of shape functions and their derivatives can be easily obtained because the shape functions also have polynomials as the basis functions.

Governing Equations for Shells

The formulation of governing equations used in this paper is based on the geometrically exact theory of flexible shells proposed by Simo et al (1989a). The Gauss intrinsic coordinates are used to describe the configuration of the shell. The pair (<p,t) defines the position of an arbitrary point of the shell; where the map q> defines the position of the middle surface of the shell; the map t defines a unit vector field at each point of the surface and is referred to the director field. The configuration of the shell is defined as

Vr = {xeR3 |x = ̂ 1 , ^ ) + ̂ ^ 2 ) with £ ' ,<feA and £e(/T,/z+)} (13)

Here (/f ,A+)are the distances of the "lower" and "upper" surfaces of the shell measured

from the shell midsurface.

Making use of the definition of spatial tensors, the corresponding linearized strain measures are defined relative to the dual spatial surface basis as

Here, At is the incremental spatial rotation.

For isotropic elastic shell structures, the constitutive relations for the effective membrane stress n , and the stress couple resultant m can be written as

100

n = <

ft"

h22 Eh

l-v2

' S\\

£22

2en

•,m = -

' « " '

in22

mn

Eh'

12(l -v 2 )

Pu

Pu 2Pn.

(15)

Here E is the Young's modulus, v is Poisson's ratio, h is the thickness of the shell. The shear stress resultants q are given by

•-$—(P (16)

Here, K is the shear reduction coefficient, G is the shear modulus.

By making use of the basic kinematic assumption equation (13), the weak form of the governing equation for shells under static load is

Wsla{S*)= l[*P" •fcl)a+mPa -SK^ +?Sya]m-Wexl{Sx) (17)

Here, dSR = jd^d^2 is the current surface measure and W^ is the virtual work of the

external loading given by

we* = f [fi • S<P + ™ • s t ] d 9 { + [ s ' Smjds+ f mStjds (18)

where n is the applied resultant force per unit length and m is the applied direct couple per unit length, n and m are the prescribed resultant force and the prescribed director couple on the boundaries 9„A and dmA, respectively.

The enforcement of compatibility is needed along the common edges of the background cells in the problem domain, to produce the CPIM. Hence, when the penalty method is utilized to impose the compatibility, the modified variational form for CPIM can be written as

Wm (Sx) -Slr(u+-u)T-a- (u+ - u")dS = 0 (19)

Here, u+ and IT are the displacements on the two sides of the incompatible interface A r .

a are the matrix of penalty coefficients, which are usually very large numbers.

The displacement vector u and incremental rotation vector At can be expressed in the

global Cartesian basis E^- as

101

m

(20)

and

At(C) = v-AT = v-£0I(t)[(AT1)1E1+(AT2)iE2] = y.±<f,,(C)AT1 (21)

where m is the number of points in the neighborhood of Q; [/,, F, and W, are the components of the displacement vector of the / th point in E , , E2 and E3 direction, respectively. The vector U ,T and AT are defined as

V = (U V W)\ T = (7; T2)T and AT = (A7; Ar2)T

Wsta (*x) = J \[HmmSV]T n+[Him^U + HH<TT]T in + [Hjm5U + HS^T]T qldM (22)

- F ^ (<?U,<5T)- J^ ( ^ U - ^ U ) T -a -(<p, -p , )dS

Here the matrices are defined as

H „ „ —

a •' a^1

a •2 a^2

a a ^ iTTT + 9'2 •̂' a<« '2 a^1 _

. H s m -

3x3

r t ^ a£«

tJ-d^

. Hrf =

2x3

V,i

. * * . 2 . T,.

H i m ~

t , A

• 2 a ^ 2

t , — + t , — •' d? ,2 a^1

H 4 4 -

3x3

a /71 ^—^— *V

a •2 a^2

a a •' a£2 • a^1 _

T , (23)

Substituting the displacement field from equations (20) and (21) and the enforcement of compatibility from equation (19) into the variational form (22), the static discrete system equations for CPIM can be obtained as

102

(K + K)U = F (24)

where K is the global stiffness. The contribution of the membrane stress, bending stress resultant and shear stress resultant to the stiffness matrix associated with node {I, J) are denoted by Kj}, KJ, and UL'U , which are given by

K£ = i H l m D . H ^ j d t t (25)

"•bm(I)

•mn)

H T

D2(HtoU) HM(y) )d<R (26)

A(H„ W ) H j i U ) )d<R (27)

F7 = j A ( 0 / n +(y )_I <D/Trii)dSR+ J ^ O / n d s + j ^ / m d s (28)

*" = JA, ( ° ' - ° ^ ) T -a- ( 0 ' " ° ^ ) d S (29)

where U is written as

V = (U V W AT{ AT2)T )

The essential boundary conditions are easily enforced in the same way as in the FEM because PIM provides shape functions that possess Kronecker delta function property.

Numerical Examples

Both CPIM are tested on several well-known benchmark problems to demonstrate their validity and efficiency. The phenomena of membrane locking and shear locking are also discussed.

BARREL VAULT ROOF

The performance of the present method is evaluated on a standard test problem of a barrel vault roof shown in Figure 1. The shell roof is loaded by its own uniform vertical gravity load. It is supported by rigid diaphragms along the curved ends, which allow displacement in the axial direction and rotation about the tangent to the shell boundary, but is free along the straight edges. The following parameters are used: length L = 600, radius R = 300, thickness h = 3.0 and the semi-span-angle of the section is 8 - 40°. The

103

Young's modulus is E •• p = 0.20833.

3.0xl06; the Poisson's ratio is v = 0 and the mass density is

Figure 1. Barrel vault roof

This problem is extremely useful for evaluating the ability of the shell formulation to accurately solve complex states of membrane strain. A substantial part of the strain energy is membrane strain energy. Using symmetry, only one-quarter of panel needs to be modeled. There is a convergent numerical solution of magnitude of-3.618 for the vertical deflection at the point A, which is used to normalize the results in Figure 2.

CRM Simo et al

A — ffG-Krysl et al ffG( Quadrature)

6 10 14

Number of background elements/side

Figure2. Convergence of vertical displacement at A

104

The convergence by the CPIM is very good in comparison with the result from FEM and other numerical methods. It can be seen from Figure 3 that errors between the present numerical solutions using CPIM and the FEM convergent numerical results are less than 1.0% when the value of k is greater than 17. The increase of k leads to improved accuracy as it can be seen that the displacement approaches to the accurate result from below with the increase of k. Figure 4 shows the variation of shear, membrane and bending energies with respect to the value of k.

CI

e

T3 <L> N

O

Z

Figure 3. Variation of displacement with k for CPIM

&

60%

50%

ene

tota

l

o

actio

n

40%

30%

20%

10%

0% —•— —f— —f—

—A— Bending

o Membrane

— • • • • _

A

\

10 12 14 16

k 18 20 22

Figure 4. Variation of membrane, shear and bending energies with k for CPIM

105

PINCHED CYLINDRICAL SHELL

The second test problem involves a thin cylindrical shell loaded by two centrally located and diametrically opposing concentrated forces as shown in Figure 5. The ends of the cylinder are supported by rigid diaphragms. The length of the cylinder is L = 600, the radius is /J = 300, and the thickness is h = 3. The material properties are is=3.0xl06

and v = 0.3.

F=1.0

-* »>-« »-

sym »A

SL2\ sym B /

F=1.0

Figure 5. Pinched cylindrical shell problem

This problem poses one of the most critical tests for both inextensional bending and complex membrane states of stress. Only one octant of the cylinder needs to be modeled because of symmetry conditions. There is a convergent numerical solution of 1.8248e-5 for the radial displacement at the loaded points, which was used to normalize the results in Figure 6. However it can be seen that the convergence rate of the present method is slower in comparison with the results from FEM (Simo et al, 1989b), the EFG method for thin shells (Krysl and Belytschko, 1996) and the RPIM method for thick shells (Liu and Liu, 2002).

Conclusions

In this paper the CPIM is utilized to analyze spatial thick shells based on the stress-resultant shell theory proposed by Simo et al. Because the PIM shape function are not compatible, when energy principles are utilized to formulate the approximation, it can be both conforming and non-conforming. When the constrained energy principles are used to enforce the compatibility, it will be conforming as long as the numerical implementation is accurate.

A CPIM can provide the upper bound of the solution, and the displacement can approach to the exact solution from below with the increase of nodes in the problem domain. The CPIM can always pass the standard patch test. The phenomena of membrane locking and

V

106

shear locking are highlighted by determining the membrane and shear energies of the shell structures. It can be seen that membrane locking is alleviated by enlarging the domain of influence of the scattered nodes and correspondingly increasing the number of monomials in the basis functions. Numerical examples are also presented to demonstrate the convergence and validity of the PIM with MTA for selection of nodes in the influence domain and the corresponding basis functions.

110% -, ,

30% I I 4 6 8 10 12 14 16

Number of background elements/side

Figure 6. Convergence of vertical displacement at A

References

Krysl P. and Belytschko T. (1996) "Analysis of thin shells by element-free Galerkin method," International Journal of Solids and Structures, 33, 3057-3080.

Liu G.R.(2002). Mesh Free Method: Moving Beyond the Finite Element Method, CRC press, USA.

Liu G.R. and Gu Y.T.(2001a). "A point interpolation method for two-dimensional solids," International Journal for Numerical Methods in Engineering, 50, 937-951

Liu G.R. and Gu Y.T. (2001b). "A matrix triangularization algorithm for point interpolation method," Proceedings of the Asia-Pacific Vibration Conference, Nov. 20-23, 2001, 1151-1154.

Liu L., Liu G. R., Tan V. B. C , and Y. T. Gu. (submitted). Radial point interpolation method for spatial thick shell structures. Computational Mechanics 2002.

Simo J., and Fox D.D. (1989a). "On a stress resultant geometrically exact shell model, Part I: formulation and optimal parameterization," Computer Methods in Applied Mechanics and Engineering, 72, 267-304

Simo J., Fox D.D., and Rifai M.S. (1989b). "On a stress resultant geometrically exact shell model, Part II: the Linear Theory," Computer Methods in Applied Mechanics and Engineering, 73, 53-92.

Wang J.G., Liu G.R., and Wu Y.G.(2001). "A point interpolation method for simulating dissipation process of consolidation," Computer Methods in Applied Mechanics and Engineering, 190, 5907-5922.

SECTION 6

Meshfree Methods for Soil

109

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor, World Scientific, Singapore, 2002.

CHARACTERISTICS O F LOCALIZED BEHAVIOR O F SATURATED SOIL WITH PORE WATER VIA MESH-FREE METHOD

S. Arimoto, A. Murakami Graduate School of Natural Science & Technology, Okayama University, Japan

sauzar(cp,cc.okavama-u.ac.ip. [email protected]

Abstract

Characteristics of localized behavior of saturated soil with pore water is examined via Element-Free Galerkin (EFG) method while comparing with the FEM solutions under elasto-plastic constitutive equation. A set of weak forms of the nominal stress rate for soil skeleton and the continuity of pore water is derived for the field of finite deformation. An example problem concerning the triaxial test is dealt with where numerical difficulties are appeared in the computation. The mesh-free method provides the wellposed solution for this type of problem while overcoming the mesh dependency of the localization of the test specimen.

Keywords: Localized Behavior, Saturated Soil, Shear Band, Finite Deformation, and Updated Lagrangian.

Introduction

Finite Element Method (FEM) is a numerical method that most widely used in structural deformational analyses of foundations. When dealing with strain localization problems in FEM, there is a problem that the numerical solutions depend on mesh type and size. Using elements in analysis causes this problem. On the other hand, it is expected that the mesh free method, represented by Element-Free Galerkin Method (EFG), will become a leading means to avoid this problem. To examine the validity of the EFG, a hypothetical element test on saturated soil is analyzed where the usual FEM provides inconsistent solutions depending on the different mesh arrangements.

Numerical scheme within the framework of finite deformation

Governing equations

A set of governing equations for saturated soil is listed below:

- Equilibrium of the nominal stress rate divS, =0 (1)

where S, = f + (trD)T - (DT + TD) + LT is called the nominal stress rate.


110

T = T - W T + TW is the Jaumann stress rate, L is the velocity gradient, D = l/2(L + L r ) is the stretching tensor, and W = l / 2 ( L - l / ) is the spin tensor. The above equation is derived by taking the material derivative of the equilibrium of the Cauchy stress.

- Effective stress concept T = T-pwl (2)

- Constitutive equation f '=CD (3)

Cam-clay model is used as the constitutive equation.

- Continuity of pore water trD + d iw i v =0 (4)

- Darcy's law vw = -klgradh (5)

-Head h = pjrw+a (6)

- Boundary conditions s, =S,n (on rCT), v = v (on r„)

q=vw-n (onT,), h=h (onr„) (7)

The discretization of the weak forms of the above equations is presented based on the EFG strategy, e.g., the moving least square method (MLSM), in the following subsection.

Discretization of the equilibrium of nominal stress within the EFG strategy

The equation for the nominal stress equilibrium changes the following equation by applying dv e {SL = graddv, Sv = 0 on T„} and penalty method for essential boundary conditions, namely,

J S, :<5LrfK-[. s dvdS+lr p(\-v) SvdS = 0 (8)

Here, the displacement rate and the head of the pore water are expressed as follows using the shape function of EFG.

v = N] 0

0 Nl

Na 0

0 Na = [N]{v'} (9)

111

'SP = [Nh]{h } (10)

Finally, the following form is obtained as the stiffness equation:

([K] + [K']){Au'}-[Kj{h'\l+J = {AF}-[Kj{h'\,} + {Ap} (11)

Where

[K] = jv([B]T [C][B] + [B]T {T'}[BV]~ 2[B]T [T"][B] + [M]T [T'"][M]

-{Bjpw[Bv] + [M]T[P][M] + [Bv]Tyw[N'])dV,

[K'] = pjr[N]T[NVS, [Kv] = l[Nj[N„W,

{AF} = At I [Nf {J, }dS, {Ap} = pjr [N]T {Au}dS,

IT"]-

r „ o rn/2

0 T'22 T\J2

T'J2 TJ2 (T'u+T'22)/4_

, [T'"] =

n 0 T

0 T\2 0

0 T\.

12

o r o T\2 o

0 T\.

[P] = {T'} =

~PW 0 0 0

0 pw 0 0

0 0 0 Pw

, 0 0 pw 0

{v'} = {Au'}/At, {v}-{Aw*}/A/, [N'] = [o N]

Discretization of the continuity of pore water within an EFG strategy

T'z

r,

o Na]

The weak form of the continuity equation for pore water is obtained by multiplying an arbitrary function, namely, 5h e {Sh = 0on Th},

- f (trD)ShdV + f vw • grad<5WF- f qShdS- f 0(pw -pJShdS = 0 (12)

The above weak form of the continuity for pore water is also discretized by approximating the pore water pressure through the nodal point quantity, {h'}, as found in Eq. (10), in other words,

h = [Nh]{h'} (13)

112

The discretization is done in the same manner as that of the stress equilibrium; the resultant stiffness equation is described as

-[Kv]{Au'}-(l-0)AtaKh] + [K'h]){h' U} = {AQ} + 0At[Kh]{h' |,}-{A/?} (14)

[Kv] = \v[Nhf[BvW, [Kh] = l[Bj[k][B„W

[K\ ] = Ph[N„ f [NhVS, {AQ} = Atl[Nh]TqdS

{Ap} = pAt\[Nh]ThdS, [k] = k/rw o

. o k/K

Numerical simulation of a soil test

To examine the validity of the EFG, the simulation of a triaxial compression test is performed, as shown in Fig. 1. The triaxial compression test is performed in order to pursue the material parameters of the soil. The experimental contents show that the specimen for the cylinder form is compressed in the direction of length. Fig. 1 shows the central cross section of the specimen. The test is analyzed with a 1/4 cross section by using the symmetry of the cross section. The boundary conditions are shown in Fig. 2, because the symmetry of the cross section is used. As for the upper surface, the transverse direction is restrained in consideration of friction with a loading board. This restraint causes localization. And all boundaries are under undrained conditions.

S o i l

Figure 1. Triaxial compression test Figure 2. Numerical model of the triaxial compression test

Two models, which differ in their pattern of element division, are prepared. One is type A (221 nodes, 400 elements) and the other is type B (841 nodes, 1600 elements). These models are calculated by FEM, using the material parameters shown in Table 1. Consequently, the discovery of a shear band may differ, like types A and B, as shown in Fig. 3.

113

Table 1. Material parameters

X K

P V'

P'o (kPa) k (cm/sec.)

0.0899 0.0198

1.5 0.2 196

0.778* 10-5

I ' • 111 ( I t l l l . l 1 . i . l t l U i - n l i i

» - i . • » • > • * i l i r r . i i " l l l ' i , ! * ' V \t*\i\ 1 •

1. I . I I ' M K I ' ' '

• • '< • 1 I . I | » H > • > ) *

. . 1 1 1 1 * , I , • 1 - 1 ^

i • V • •

i n * J l

, J « » - 1

i . i u • .

» •» " . . . 111! * , , |

. ' " . 1 I I . '

' f | -« M i l l i

Figure 3. Example of the dependency of mesh size by FEM

The same models are analyzed by EFG. As shown in Fig. 4, the initial node arrangement has been set so that it will be in agreement with FEM. The values in Table 2 are used as the material parameters.

Table 2. Material parameters

* • • « # « • - • •

•AVAV.JV.VWAV-.V/ .VAV.V.VAWWW,

*»>A*MA»>J» .*« !»A»JMJ* .« '

X K

M v'

v0=l+e0

P'o (kPa) k (cm/sec.)

0.108 0.025 1.53

0.344 1.847 294.3

1.39x 10"'

Figure 4. Initial collocation of nodal points for EFG

Fig.5 shows the deformation profile at an axial strain of 5%, and Fig. 6 shows the load-axial strain curve. Fig.5 shows the contour for shear strain at a linear strain of 2.5% and 4.5%. The shear band has been discovered from the upper right angle of the domain toward the angle at the lower left of the domain for both types A and B. The generating sites and the thicknesses of the shear bands differ for a while. The cause of the difference in thickness is thought to be due to the support radius of type B being made small in accordance with the difference in node density. Therefore, the slope of the approximation function becomes easily changeable.

Conclusions

In this paper, EFG analysis was performed for a mesh dependability problem, which arises in FEM. In EFG, a phenomenon for which only an element of a specific sequence deforms in FEM was not seen. And, irrespective of different arrangement of nodal points,

114

resultant profile of shear strain within a specimen and the load-axial strain curve are found to be consistent with each other. Generally, approximation function becomes discontinuity between elements so that the shape function is independent every elements in FEM. In EFG, approximation function becomes consecutive in the analysis domain. The nodal points layout does not have influence on the direction of shear band in EFG.

Figure 5. Deformation profiles by EFG Figure 6. Load-axial strain curve

axial strain 2.5% axial strain 4.5% axial strain 2.5% axial strain 4.5% (a) Type A (a) Type B

Figure 7. Contour of shear strain

References

Belytschko T., Lu Y. Y. and Gu L. (1994), "Element-Free Galerkin Methods", Int. J. Num. Meth. Eng., 37, 229-256.

Lambe T. and Whitman R. (1969), "Soil mechanics", Wiley: New York.

Yatomi C, Yashima A., Iizuka A. and Sano I. (1989), "Shear bands formation numerically simulated by a non-coaxial Cam-clay model", Soils and Foundations, 29(4), 1-13.

Oka F., Yashima A. and Sawada K. (1998), "Static and dynamic characteristics of strain gradient dependent elastic and elasto-viscoplastic models", Localization and Bifurcation Theory for Soils and Rocks (Adachi, Oka Yashima, eds.), 71-79.

Kobayashi I. (1998), "Mechanical stability of saturated soil and its failure phenomena", Doctoral Dissertation, Kanazawa University, 34-39, (in Japanese).

Nakai T. (2002), Private communication.

115

Advances in Meshfree andX-FEMMethods, G.R. Liu, editor, World Scientific, Singapore, 2002,

RADIAL POINT INTERPOLATION METHOD FOR INTERFACE PROBLEMS

J. G. Wang\ T. Nogami" and Md. Rezaul Karima

"Tropical Marine Science Institute, bDepartment of Civil Engineering National University of Singapore, 10 Kent Ridge Crescent, SI 19260

[email protected], [email protected], [email protected]

Abstract

Interface is an important element for the interaction problem of soil masses and object. This paper proposes an interface layer method to treat the interface of saturated soil medium and rigid object (solid). In the domain, the radial point interpolation method (radial PIM) with compact support is applied. An interface layer is proposed to treat the compression, opening and shear friction of the interface. This method has following advantages over traditional interface element (Goodman, 1968). First, the node distribution on both sides of interface is not required to be the same. Such a scheme is special helpful to the meshless methods for both side domains. Second, water flow in the interface can be considered if the interface is open. Third, the accuracy for interface interpolation is adjustable according to the node distribution. Numerical examples are studied to demonstrate the capability of the current interface layer method.

Keywords: Biot's Consolidation Theory, Interface, Pore Water Pressure, Meshless Method

Introduction

Interface is an important component in the soil-structure, multi-domain and solid-fluid interaction problems. In geotechnical engineering, Goodman interface element with zero-thickness (Goodman et al., 1968) is well developed. Later, thin-layer interface element was proposed to treat the thin-layer mechanical properties (Desai et al , 1984). Wang et al. (2002) theoretically developed a constitutive law of zero-thickness material from a thin-layer material with a limit concept. An interface has its own characteristics: First, the displacement on both sides of the interface can jump and slide. The displacement generally is discontinuous across the interface. Second, the interface cannot penetrate each other. This insures that an interface can open, close and slide. Third, frictional effect can be considered for suitable constitutive laws of interface. Herault and Marechal (1999) discussed the possible forms for the interface condition in meshless methods: Lagrange multiplier which treats the discontinuity of its normal derivative in weak sense instead of shape functions; jump function in approximation to treat the displacement or derivatives jump at element level. Because the material properties of interface are usually different from those on both side domains, interface in soil-structure problem should be treated as a special material in numerical algorithm.




116

Goodman interface element is successful in the numerical analysis although some numerical problems such as stress oscillation have to be solved (Day and Potts, 1994). For the interaction problem of structure-saturated porous media, Goodman interface element should be extended to consolidation or liquefaction problems where pore water pressure is an important factor. Some extensions were only coupled with finite element method in both side domains (Dluzewski, 2001; Yoshida and Finn, 2001). If the domain problems are solved through meshless method, node distributions on the interface have two cases: matching and non-matching node distributions. Matching node distribution has the corresponding nodes on each side of an interface. Non-matching node distribution does not match the node on the each side of the interface.

This paper studies the coupling of interface element with radial point interpolation meshless method (radial PIM) for consolidation problems. The interface element including consolidation properties was developed based on a limit concept of motion and continuity equations of a thin-layer. That is, governing equation for consolidation problem in thin-layer is first presented based on the Biot's consolidation theory. The governing equation for interface is then developed as the limit case where the thickness is approaching to zero. Weak form is developed from Galerkin principle and discretized through compactly supported radial PIM. Finally, examples are studied to check the effectiveness of the current methods.

Governing equations for thin-layer soil mass

The motion equation for fluid-filled porous medium can be expressed as follows:

do' dP -JL + ^ + bi=0 (1) OXj OXt

P is the pore water pressure. M; is the displacement of soil skeletons, b, is the body force. The effective stress principle gives au = a'j+SyP. For linear soil skeleton, the Hooke's law is in following form:

The continuity equation for fluid flow is

4ir (3)

Qot where Q is the compressibility of soil masses. \IQ = riIKf + (\-n')lKs. Kf,Ks are compression moduli for fluid and soil grains, respectively, ri is the porosity of soil skeleton. kv is the permeability of the thin-layer soil. X,G are Lame constants of soil skeleton. In thin-layer soil, a convenient coordinates is taken as the n-s system, n is the normal direction and the s the tangential direction of thin-layer. In such a coordinate system, above governing equations are still true.

dt

3M,

13*J r. BP)

ljdx

117

Governing equations for zero-thickness layer

1. Motion equation for fluid-filled interface

The thickness of a thin-layer is denoted by b as shown in Fig. 1(a), where the dashed lines indicate the thin-layer and the displacement in the thin-layer is continuous. When the thickness approaches to zero, a zero-thickness layer is formed where the displacement has a jump as shown in Fig. 1(b).

TangentiaJ direction

Displacement

Normal direction

(a) Continuous zone (b) Discontinuous zone

Fig. 1 Thin-layer to interface

Following formulation expresses this limit concept:

[ [<HHK1 [dM]r=iim[fe>rJr (4)

where \du„\ and [di/,] denote the increments of normal and shear displacement jumps, respectively. Taking the variational Sut of displacement w, as weight function, the energy in thin-layer soil can be expressed as

J--L Su,dV (5)

Where the integration domain Q, refers to the thin-layer soil. When the thickness approaches to zero, the force equilibrium should satisfy:

ffljnjLp=^tjnJ Loom

After integration by parts, above energy can be finally written as

J.=-ls{«}lSu\dS-lsP{m}[Su]dS

(6)

(7)

This energy is the contribution of interface to the whole energy system of the porous medium domain. The weak form for the porous medium domain with interface can be

118

l{Sz}T{a}dV + l{a}lSnjdS + lP{m}lSn]dS = l{Su}T{b}dV+jjSu}T{t}dS(S)

2. Continuity equation for fluid-filled interface

The volume strain of a thin-layer is expressed as

£v=-^- = £„+e, (9)

where en is the normal strain along normal direction of the interface and es the normal strain along interface direction. The normal stress along interface is usually small and negligible (crs = 0 ) . When the thickness approaches to zero, the relationship of normal strain and displacement jump can be developed as

i™ t e-=Kl» u$te,=ldu,] (10)

It is noted that | ^ M , J *• [^w,] because the later is from shear strain. Taking the variational SP as weight function, the weak form on the interface domain £2, is:

J<=k dt dx.

_d_

dx,

1_3P_

Qdt SPdV (11)

After integration by parts, one finally gets the weak form:

where ktj = l i m ( M ? ) , Q = \im(Q/b) and # is the flux along upper and lower sides of

interfaces. Eq.(12) is the contribution of interface to the continuity equation.

Discretization

Following discretizations are introduced into the weak form:

[u] = CSv{x,y,z)7b„ P = NP(x,y,z)RPe (13)

where Tue is the relative displacement, RPe is the relative pore pressure, and C is the matrix for coordinates transformation. The weak form for equilibrium is discretized as

Ki„Au + Li„AP = AFln (14)

The weak form for continuity is discretized into:

v " " ' * '" dt '" q )

The matrix form for above equations is obtained as

119

K „ L „

L,„ + Lins -S,„

I *

.dt.

.= "0 0 "

{PM rfF'

dt

q .

(16)

where the coefficients for an interface element are as

K = L TTNTuC

TDCNuTdS, Ifin = J TTNTuC

TnNpRdS, F;„ = AF/ + AF/

1 5 : = ^RTNT

pNpRdS, Hl=\sJlTpKBpdS,

Le.=\ RTNTITCB^—dS ins is p ^ D

D = K o o k.

(17)

Numerical example

A simple example is used to study the effectiveness of the current method. In the interface element, the Iflris is omitted in the computation because it does not have obvious physical meanings. Domains and interface are discretized through compactly support radial point interpolation method and interface element, respectively. Fig 2 is the model. Table 1 gives the material parameters. The loads are Pa = 10 kPa and Pb = 15 kPa.

Table 1 Interface properties for computation Property E (kPa) lkn{kPalm) Poisson ratio / ks (kPa/m) Permeability kxlks(mld) Permeability kvlk„ (mid)

p

Soil A 1000

0.3 0.001728 0.001728

SoilB 1000 0.3 0.001728 0.001728

p,

Interface 1000 100 Variable Variable

, t < ! , i , . ! . . t . , » . • . . » . . . , . * . . . • • .

5ai*

Fig. 2 Interface element at the middle line (8m long) and domain discretization

120

Interface does not affects the initial pore water pressure, however, it will heavily affect the dissipation of pore water pressure as shown in Fig. 3.

? *

50 100 150 2 0 0 ^ 2 5 0 ^ 3 0 0 ^ 3 5 0 400 *50 Consolidation time (day)

50 100 150 200 250 300 Consolidation time (day)

(a) High permeability (b) Low permeability

Fig. 3 Dissipation of excess pore water pressure in interface points

Conclusions

An interface element is developed for the consolidation problem based on the limit concept proposed by Wang et al.(2002). This interface element is incorporated with compactly supported radial PIM. Numerical example shows that the current development can describe the behavior of fluid-filled interface in porous medium.

References

Goodman EL, Taylor RL, and Brekke AM(1968), "A model for the mechanics of jointed rock," J of Soil Mech. And Found. Div.. ASCE, 94, 637-659

Desai CS, Zaman MM, Lightner, and Siriwardane HJ(1984), "Thin-layer element for interfaces and joints," Int. JforNumer. &Analy. Methods in Geomechanics, 8,19-43

Wang JG, Ichikawa Y, and Leung CF(2002), "A constitutive model for rock interfaces and joints," Int. J of Rock Mech. And Mining Sci, in press

Herault C and Marechal(1999), "Boundary and interface conditions in meshless methods," IEEE Transactions on Magnetics, 35(3), 1450-1453

Day RA, and Potts DM(1994), "Zero thickness interface elements - numerical stability and application," Int. JforNumer. &Analy. Methods in Geomechanics, 18,689-708

Dluzewski JM(2001), "Nonlinear problems during consolidation process," Adv Numer Appl and Plast in Geomechanics, eds by DV Griffiths and G Gioda, Springer Verlag, pp.81-158

Yoshida N and Finn WDL(2000), "Simulation of liquefaction beneath an impermeable surface layer," Soil Dynamics and Earthquake Engineering, 19,333-338

SECTION 7

Meshfree Methods for CFD

123

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor, World Scientific, Singapore 2002

APPLICATION OF FREE MESH METHOD TO VISCOPLASTIC FLOW ANALYSIS OF FRESH CONCRETE

Jun Tomiyama, Yoshitomo Yamada, Shigeo Iraha Dept. of Civil Engineering and Architecture, University of the Ryukyus, 1, Senbaru,

Nishihara-chou, Okinawa, 903-0129, JAPAN iun-tdiitec. u-rvukvu. ac. ip. b985553(a),tec. u-rvukvu. ac.jp, iraha(d),tec. u-rvukvu. ac. ip

Genki Yagawa Dept. of Quantum Engineering and Systems Science, The University of Tokyo, 7-3-1

Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN yagawa(o),q.t.u-tokyo.ac.ip

Abstract

This paper presents an application of Free Mesh Method (FMM) to the flow analysis of fresh concrete. FMM, which is a kind of meshless method, is suitable for distributed parallel processing, adaptive analysis and moving boundary problems because it is performing the calculation from the local mesh generation to the construction of the global matrix on a node-by-node basis. In this study, flow behavior of fresh concrete has been assumed viscoplastic fluid, and the constitutive law of fresh concrete is given as the Bingham-model. As the numerical examples, the L-type flow test has been simulated by this method, and the analytical results are in good agreement with the experimental ones.

Keywords: FMM, Fresh Concrete, Flow Analysis, Bingham Model, FEM and L-type Flow Test.

Introduction

Recently advance in computer technology has enabled a number of complicated phenomena to be simulated, which were observed by experiments. Out of computer simulation techniques, the finite element method (FEM) and the finite difference method (FDM) have been widely used in various fields. But, there are many problems in generating meshes for large scale and complex models. Then, automatic mesh generation techniques for the finite element method have been used to decrease the labors. However, the techniques mentioned above are not prepared by simple procedure, requiring more experiences in the works.

For this background, several meshless method which do not require element or grid has been developed, and the research on the meshless method has been applied in many engineering fields. Within these meshless methods, there is Free Mesh Method (FMM) proposed by Yagawa (Y. Yamada, 1997). FMM does not require any connectivity

124

between nodes and elements for the input information and the calculation has been performed by node-by-node basis, so this is an excellent feature of local data reference. Therefore, this is suitable for distributed parallel processing, adaptive analysis and moving boundary problems.

On the other hand, the new types of concrete developed lately such as high-fluidity concrete or high-strength concrete. Within these concrete, there are the cases that the flow behavior of fresh concrete cannot estimate with conventional experiments. From this reason, a study on developing the analytical method for estimating the flow behavior of them has been carried out actively in recent years (H. Mori, 1994).

This paper describes an application of FMM to flow analysis of fresh concrete. In this analysis, fresh concrete is assumed to be a viscoplastic fluid, and the constitutive model of fresh concrete is given as the Bingham model. As numerical examples, the flow behavior in L-type flow test was simulated by proposed method.

Free mesh method

FMM does not require the global element, so that the total stifmess matrix is assembled by the temporary local element matrices around each node. Therefore, it needs the algorithms for a construction method of local elements. The following algorithms achieve the creation of them.

First, lets us put nodes, appropriately in the domain to be analyzed. Then, a node, say /, is chosen among the nodes, around which temporary triangular elements are created using several surrounding nodes, say m.n.o.p, etc. which are chosen as nodes based on a rule given in the following section (see Figure 1). The node / is called here as a current central node, whereas the nodes m,n,o,p, etc. as current satellite nodes. Third, the contributions from the temporary elements to the total stiffness matrix are assembled. For example, an element matrix [K\m is calculated from the triangular element l-m-n, and only the node /-related components are added to the pertinent components of total stiffness matrix. The same process is repeated for the other elements l-n-o, etc. The above procedure is performed on each node in the domain.

It is noted that, in this algorithm, the calculation has been performed on a node-by-node basis. Therefore, it is considered to be suitable for distributed parallel processing, adaptive analysis and moving boundary problems.

/ O m n \ 8 / » • O \ o

/ \ ^ / \ ^ o \ A : Central Node • J J K 7 • : Satellite Nodes

"^ J \ J J O : Candidate Nodes \ 1 P / ® : Other Nodes

O N . o /

Figure 1. Local radial elements around node /

125

Constitutive equation of fresh concrete

In this study, it is assumed that the constitutive law of fresh concrete is Bingham modal as shown in Figure 2. Therefore, the flow behavior model of one is constructed by viscosity element and plasticity element as shown in Figure 3(a). But, in Bingham model, the flow does not occur when the shear stress due to gravity is smaller than yield stress. That is, the fresh concrete behaves as a rigid body and does not deform. In this case, this analytical method can't implement simulation of flow. Therefore, we model this situation as high viscosity fluid which is very slow speed fluid (see Figure 3(b)).

Figure 2. Behavior of fresh concrete by the Bingham model

\r7

e?

T e? t :__: :—I

PE: Plasticity element

VE: Viscosity element

HVE: High viscosity element

(a) The case of flow (b) The case of stop

Figure 3. Constitutive models for fresh concrete (T. Yamada, 2001)

As the yield criterion, we assumed the shear strain rate is governed by the associative flow rule and a material obeys von Mises yield criterion. Also, the shear stress and shear strain rate relation of viscosity element is expressed with Newton viscosity law. Thus, the constitutive equation when it starts flow is given by the following equation,

^ = - ^ + 2 | " + V n - j (1)

where r / , EJ is the stress component and strain rate component of visco-plastic element,

respectively. P is the hydrostatic pressure, S,t is Kronecker delta, n is the plastic

viscosity and xy is the yield stress. It is notice that this equation is material nonlinear

because of n = 2eJsJ . Therefore, the calculation requires the solution of nonlinear

equation. In this analysis, the direct iteration method is used.

126

Also, we assumed the unmoving state of fresh concrete is modeled as high viscosity fluid, so that the constitutive equation until it starts flow is given by the following equation,

T" =-PS +2 n + (2)

where r j , e] is the stress component and strain rate component of viscosity element,

respectively, n = (in J 2 , nc is strain rate of the flow limit. Here it is defined as follows,

.P*, (3)

where p =0.1. This value was obtained by the preparative analysis on L-type flow test.

Formulation of kinematic equation

Concrete and mortar are composite materials, with aggregates, cement, and water as the main components. However, this study assumed isotropic continuum material as to fresh concrete. Also, in this analysis, the kinematic equation is formulated by using the principal of virtual work as the finite element discretization as follows, and the acceleration term of this equation is expressed by using Newmark-B method which is a kind of numerical integration method.

^^H^-^^kKWfw (4)

where [M] is the lumped mass matrix, [K] is the viscoplastic matrix, {u}, {ii} are velocity and acceleration vectors, respectively, {F} is body force vector. At is time step(O.OOlsec). Also, we calculated viscoplastic matrix using the penalty function method in order to satisfy incompressibility of fresh concrete.

Numerical examples

To evaluate the proposed analytical method, this section shows the numerical examples on L-type flow test. Then, we assumed the flow of fresh concrete as the plane strain problem. In this example, the plastic viscosity is constant(50 PaDs), and Table. 1 shows the yield stress of fresh concrete that was used for analysis. Figure 4 shows the L-type flow test machine, and the analytical model.

Table 1. The yield stress of fresh concrete (Pa)

Casel

25

Case2

50

Case3

100

Case4

125

127

In this analysis, the nodes in domain have been moved according to the amount of calculated deformation. Therefore, with progressing deformation, the node distribution becomes distorted in similar to FEM because this method is based on FEM (see Figure 5(a)). In this reason, we proposed to perform the smoothing (Laplacian Smoothing) of nodes in domain, and the remeshing as to local elements around the central nodes. Figure 5(b) show the effective of the smoothing and the remeshing. In FMM, these processing can be easily implemented because of node-by-node analysis method. Also, in this analysis, it has been considered that the slip at contact side between fresh concrete and this test machine occur.

. . . i i

' (a) Test machine 0 20 40 60

(b) Analytical model (70 nodes)

Figure 4. L-type flow test and analytical model (70 nodes)

40

30

20

10

0

0 10 20 30 40 (a) Non processing

40

30

20

10

|

u 0 10 20 30 40

( b)Im plem entpi roces sing

Figure 5. Effective of the smoothing and the remeshing (Case2)

Figure 6 shows L-flow and yield stress relations, and the analytical results were compared with the approximation curve of experimental ones (Miyamoto, 2001). From Figure 5, it is shown the analytical results are in good agreement with experimental ones.

80

~ 70

•£ 60 i 50

t 40

30

20

^Trf ^ - - ^

150 0 50 100

YbH Stress fa)

Figure 6. L-flow and yield stress relations

128

Figure 7 shows flow behavior of Case 1. As can be seen from Figure 7, the flow behavior of fresh concrete with the passage of time can be simulated by the proposed method.

:H

iH i , r

; 1 1 1 h — H

, : I

30

20

10

%

1 1 1 ' r '

1 1 1 I 1

m-rrrr 10 20 30 HO 50 tO 70 10 SO 3D M0 SD tO 70

(a)0.0(s) (b) 0.2 (s)

a

; ; : : ; : ' ':

fe?vvk;f1"t" ID

t»

; : : ; ; :

! ! ! : ; ! , - , , - - , - - - i - - T - -i i i i i i

Afc:ftA-.fo-V r .1 . 10 SO 30 40 50 bO 70 M0 £0 faQ 70

(c) 0.7 (s) (d) 7.0 (s)

Figure 7. Flow behavior of fresh concrete

Conclusions

In this paper, viscoplastic flow analysis using FMM on Bingham model was discussed. As the numerical examples, L-type flow tests were simulated by the proposed method, and the numerical result became excellent. Therefore, it is found that FMM has the great advantage about flow analysis like a fresh concrete, since this method is calculated on a node-by-node basis.

Acknowledgements

The authors would like to express their sincere gratitude to Dr. T.Yamada for his advice on FMM and his preliminary development of its computer program.

References

Tomonori Yamada. (1997), "Free Mesh Method on a Massively Parallel Computer", M.S ThesisfThe University of Tokyo), (In Japanese)

Hiroshi Mori and Yasuo Tanigawa. (1994), "The State of the Art on Flow Analysis of Fresh Concrete", Concrete Journal, Vol.32, No. 12, pp.30-40 (In Japanese)

Yoshitomo Yamada, Atsushi Tobaru and Takeshi Oshiro. (2001), "Viscoplastic Flow Analysis of Fresh Concrete by Finite Element Method", Proceedings of the Japan Concrete Institute, Vol.23, No.2, pp.253-258, (In Japanese)

Yoshiaki Miyamoto and Yasuhiro Yamamoto, (2001), "Study on The Fluidity and The Mix Proportion of High-Fluidity Concrete by Using The J Shaped Flow Test", J. Struct. Constr., AIJ, No.547, pp.9-15, (In Japanese)

129

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor,

World Scientific, Singapore 2002

A MESHLESS LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FLUID FLOW PROBLEMS

Y. L. Wu

Centre for Advanced Computations in Engineering Science (ACES) Department of Mechanical Engineering, National University of Singapore,

10 Kent Ridge Crescent, Singapore 119260 E-mail: [email protected]

Abstract

The Local Radial Point Interpolation Method (LRPIM) is an effective meshless method that is formulated based on the radial point interpolation scheme and local weak form method. It employs no global mesh throughout the process of both interpolation and integration. In this paper, the LRPIM method is adopted and formulated to simulate the incompressible flow in closed domains which is governed by the vorticity-stream function equation. The results agree very well with the available data in the literature, and it is much more accurate than those obtained using Finite Difference method (FD) with the same mesh size. It was found that the LRPIM method has a promising prospect in computational fluid dynamics (CFD) because it is always competent for different geometries of problem domains as well as arbitrary nodal distribution.

Keyword: LRPIM, Meshless method, Navier-Stoke equation, Natural convection

Introduction

The LRPIM method, have been developed based on local weak forms and radial point interpolation scheme (Liu & Gu, 2001). It does not need any "element" or "mesh" for both field interpolation and background integration. The advantages of this method compared with other meshless methods include 1) the shape function has delta function property, so the essential boundary conditions can be implemented as easy as in the FE; 2) stability for randomly scattered nodal distribution, which demonstrates that radial point interpolation scheme is effective for interpolation of arbitrarily scattered data points.

LRPIM method is so far only used in solid mechanics analysis. In this paper, LRPIM method is applied to solve computational fluid mechanics problems. Because efficiency plays a key role in fluid problem, some important modifications targeted to improve the efficiency are presented here.


130

Point interpolation using radial functions basis

Suppose there are n data points, x,,---,x„, with the data vector u = (ux,---,un) in the

support domain of point x, an approximation to u by radial point interpolation scheme is a function of the form

M*(x,xe) = £ t f , . (xH(x e ) = R r(x)a(x e) 0 )

1=1

Rt is a radial basis function, and a, (xe) is the coefficient for the Rt corresponding to the given point xg. The coefficient vector a in Eq.(l) is determined by the collocation method.

a = V ' U s (2)

where Us is the vector that collects all the field nodal variables at the n nodes in the support domain. Substitute Eq. (2) into (1)

u"(x) = RT(x)RQ'Vs=Q>(x)Vs (3)

For good performance (Liu & Gu,2001), we choose MQ RBFs as the basis function in this paper. The expression of Ri that we consider are

Ri(x,y) = (ri2+C2y=[(x-xi)

2+(y-yi)2+C2} (4)

where the positive constants C, q are called shape parameters, r being the distance between point x and x,.

It is well known that the accuracy of RBFs interpolants depend heavily on the choice of the shape parameters. As shown in Liu (2002), C is defined as

C = acdc (5)

where, ac is a dimensionless shape parameter, dc is a characteristic length which is taken the shortest distance between the node / and neighbor nodes in this paper. It is found ac = 8 . , <7=1.03 gives the better performance (Liu, 2002). The number of nodes in support domain n in LRPIM method does influence the accuracy of interpolation scheme. In this paper, we use n=30 and it gives very good results.

Local residual weak form

Local residual weak form was firstly proposed by Atluri and Zhu (1998) in their MLPG method, in which the equilibrium equations are satisfied at each node in a local weak sense by applying the weighted residual method over a local sub-domain. The local sub-

131

domain is conveniently taken to be any simple geometry like circles, rectangles, ellipses centered at each field node in two-dimensional cases. In LRPIM, local residual weak form method is adopted to discretize the PDEs similarly as in MLPG, except (1) radial point interpolation shape functions are used; (2) terms of integrals related to essential boundary conditions are removed.

Numerical integration

In the LRPIM method, for each node Xi, the Gauss quadrature is employed over a local regular-shaped integration cell (for example circle used in this paper) for the numerical integration in the weak form equation. To get the accuracy we wanted, a lot of Gaussian points are needed in the local numerical integration. How to construct the shape function for each Gaussian point can affect the computational efficiency of simulation of fluid flow problems dramatically. In our computation, for every Gaussian point in the quadrature domain of certain field node, say fth node, we use the same support domain as that of the ith node itself instead of seeking individual support domain for each Gaussian point. By this way, the support domain for each node can be determined firstly and stored throughout the whole iteration process. The computation of the inverse of the moment matrix RQ1 in equation (2) can also be pre-computed for each node before iteration. Therefore, the only work in the each Gaussian point is to calculate the vector of R(x,xk),k = l,...n (recalling that n is number of nodes in support domain) and get the

product of <D(x) = [if, (x), R2 (x) • • • Rn (X.)]RQ The computational cost can then be

saved significantly when iteration goes on.

Application of LRPIM method to natural convection in closed domains

Using the LRPIM method, we solve incompressible Navier-Stokes equations in the vorticity stream function form. The governing equations for natural convection in closed domains in Cartesian coordinate system are as follows:

ay 3V

— + v — = P r ( — + — ) - P r - R a - — ( 6 )

dx dy dx2 dy2 dx

dx dy dx2 dy2

where0),i/f,Tare the vorticity, stream function, temperature, u, v are the components of velocity in the x and j> directions.

To show the LRPIM method is applicable to irregular geometry, the computation is implemented for problems with different geometries of being coincided and not coincided with Cartesian coordinate axis. The geometries and boundary conditions are given in Figure 1. Because LRPIM method is a meshless method, we take different sets of nodal distribution to valid this method.

132

For the natural convection in square cavity, Table 1 lists the numerical results for different sets of nodes for Rayleigh numbers of 103,104,105 respectively. Table 1 also includes the FD results for Ra=l03 with the grid 16x16 (equivalent to 256 regular nodes) for comparison. It is found the results by LRPIM method agree very well with the results of Davis (1983). It is much more accurate than that from FD method if the mesh size is the same. Figure 2-Figure 4 show the nodal distribution together with the streamlines and isotherms of Ra=103,104,105.

For the natural convection between concentric annuli, the radius ratio is defined as rr = R0I Rt, where Rt and Ro are the non-dimensional radii of the inner and outer cylinders. The average equivalent conductivity for the inner and outer cylinder keqi and keqo by LRPIM method with general scattered distributed nodes for the case of Pr = 0.71, rr = 2.6 and Rayleigh numbers of 102,103, 104 are computed. They are compared with the benchmark solution of Shu (1999) in Table 2. It can be observed that the results of the LRPIM method agree very well with the benchmark solution of Shu (1999). Figure 5 shows the streamlines and the isotherms of LRPIM results for i?a=104. The separation of inner- and outer-cylinder thermal boundary layer and the symmetry of flow pattern can be seen very clearly.

We also found in the computation that the accuracy of the results on the randomly scattered nodes set is very good, in most cases, even better than that on uniformly distributed nodes set. It shows radial point interpolation scheme to be particularly well suited to scattered data interpolation problems compared with other approximation methods.

Conclusion

LRPIM method is a very effective meshless method. It can be applied to simulate CFD problems very well. The major advantages of the LRPIM method are: 1) Nodal distribution in the problem domain can be arbitrary; 2) Very high accuracy can be achieved by using few nodes; 3) Excellent performance on randomly scattered node distribution which is a very favourable feature for a meshless method.

References

Atluri SN, Zhu T. (1998), "A new meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics," Comput.Mech.22.Wl-121

G.de Vahl Davis, (1983), "Natural convection of air in a square cavity: a benchmark numerical solution," InU.Numer.Meth.Fluids 3:249-264

Liu GR (2002), Mesh Free Methods: Moving Beyond the Finite Element Method, CRC press.

Liu GR, Gu YT,(2001), "A Local Radial Point Interpolation Method(LRPIM) for free vibration analyses of 2-D solids," Journal of Sound and Vibration 246(1): 29-46

Shu C. (1999), "Application of differential quadrature method to simulate natural convection in a concentric annulus," Int. J. Numer. Methods. Fluids 30: 977-993.

http://Comput.Mech.22.Wl

133

Table 1 Comparison of numerical results forRa=10 , 104,10

Ra

103

104

105

256 nodes

FD(16xl6)

Davis(1983)

257 nodes

Davis(1983)

430 nodes

Davis(1983)

\w I \T max|

1.175

1.268

1.174

5.065

5.071

9.766

9.612

"max

3.634

3.905

3.649

16.148

16.178

35.243

34.730

v max

3.687

3.963

3.697

19.694

19.617

69.447

68.590

^ " m a x

1.507

1.608

1.505

3.547

3.528

9.197

7.717

^ " m i „

0.692

0.656

0.692

0.587

0.586

0.729

0.729

Table 2 Comparison of average equivalent heat conductivity for concentric annuli

Ra

102

103

104

keqi (inner cylinder)

Present

0.998

1.080

1.970

Shu (1999)

1.001

1.082

1.979

Kqo (outer cylinder)

Present

1.001

1.083

1.963

Shu (1999)

1.001

1.082

1.979

dn

T=\

Y dn ty

y dn %

r=o

= a = v = 0,7 = 0

a. Square cavity b. Concentric annuli

Figure 1 Schematic of the problems

134

Figure 2. 256 regular nodes, streamlines and isotherms for cavity flow (Ra=103)

Figure 3. 257 scattered nodes, streamlines and isotherms for cavity flow (Ra=104)

Figure 4.430 scattered nodes, streamlines and isotherms for cavity flow (Ra=10 )

Figure 5 967 scattered nodes, streamlines and isotherms for concentric annuli (Ra=104)

135


APPLICATION OF MESHLESS POINT INTERPOLATION METHOD WITH MATRIX TRIANGULARIZATION ALGORITHM TO NATURAL CONVECTION

G.R.Liu, Y.L.Wu Centre for ACES, Dept. of Mechanical Engineering,

National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Email: [email protected]; [email protected]

Abstract

Paper presents the application of the meshless Point Interpolation Method (PIM) for solving the incompressible Navier-Stokes equations. In order to build a well-conditioned matrix for the computation of weighting coefficients of interpolant, Matrix Triangularization Algorithm (MTA) is introduced. The present method is applicable to arbitrary domains and different scattered sets of nodes. Another remarkable feature of this method is its easier implementation of essential boundary conditions than other mesh-free methods. The present method is validated by its application to simulate natural convection in a closed domain. Numerical experiments showed that this method has a greater flexibility than the traditional finite difference method. Numerical results of present method also agree well with the benchmark solutions.

Keyword: Meshless Method, PIM, MTA, Natural Convection

Introduction

The Point Interpolation Method (PIM) was firstly proposed by Liu and Gu (2001a). Based on a polynomial basis interpolation, the PIM is very easy and flexible to implement. However, this method may encounter a singularity problem in some situations. To solve this problem, Liu and Gu (2001b) presented the Matrix Triangularization Algorithm (MTA) for Selecting a proper node enclosure and polynomial basis automatically without a prior knowledge on the nodal arrangement.

In this paper, the PIM method with MTA is applied to solve computational fluid mechanics problems. It is found that the present method possesses a higher efficiency in the construction of shape functions compared with other meshless method. Due to its high efficiency, this method is a very promising numerical approach in simulation of fluid problems.

Meshless Point Interpolation Method

Point interpolation approximation

In general, a meshless method requires a local interpolation or approximation to represent the trail function. The point interpolation approximation is used in this work.



136

The point interpolation approximation to «(x) from the surrounding nodes of a point x e

can be expressed by

M*(x,xe) = £/Ux)a, .(x e) = P r(x)a(x e) (1)

Pi (x) is a monomial in the space coordinate and in a two-dimensional case can be provided by P r(x) =[l,x,y,xy,x2,y2,x2y,xy2,x2y2,...]; a,(xe) is the coefficient for the Pi corresponding to the given point xg, n is the number of nodes in the neighborhood of 1r -Q-

The coefficient vector a in Eq.(l) is determined by the collocation method.

a = P e V (2)

where u" = (ux ,u2 ,...,«„) is the data vector, and the moment matrix P e are given by

pe =

By substituting Eq. (2) into (1), on

" 1 x , yx •••'

i * 2 y2 •••

. l x» y» •••_

obtains

(3)

M * ( x ) = P 7 ' ( x ) p - 1 U e = « » ( x ) u C (4)

where <D(x) is the shape function.

Local weak form

In the local weak form approach (Atluri et al. 1999), the nodal trial and test functions are

non-zero only over local sub-domains €lJtr and Q.'le (which are centered at the nodes J and

I) respectively, and vanish at the boundaries 3Q,̂ and 3Qfe of Q.Jlr and Q.'u respectively.

The local domains Q^ and Q.'u can be of arbitrary shapes, such as circles, rectangles,

ellipses in two-dimensional cases. The PIM method adopts the local weak form approach so that both interpolation and integration are implemented without a mesh.

Equation (4) is taken as the trial function in the PIM method, and the following spline weight function is used in this paper as the test function:

137

W,(*) =

1-6 'd* 'd*

0

'd* 0<d< K

d^rle

(5)

where rf,-=|x-x,| is the distance from node x, to the sampling point x, rle is the radius of the sub-domain Q.le, in which weight(test) function is non-zero, i.e., ff,(x) •£• 0.

In the PIM, the shape functions 1>(x) obtained through the procedure in last section possess delta function properties, i.e.

*>i(Xj) = S« (6)

Therefore, we can impose the essential boundary conditions as readily as in FEM method.

Matrix Triangularization Algorithm (MTA)

The PIM is an effective meshless method (Liu and Gu, 2001a; Gu and Liu, 2001). However, in some situations, the moment matrix PQ can be badly conditioned, and even invertible. People have to choose the nodes in the supporting domain and polynomial basis appropriately to avoid the singularity of PQ. This process is very complex and

problem-dependent.

To completely overcome the singularity problem of PIM, Liu and Gu (2001b) presented the Matrix Triangularization Algorithm (MTA). The basic idea of MTA is: in equation (3), the rows of the moment matrix PQ correspond to the nodes in the supporting domain of an interpolation point x, and the columns correspond to the monomials in the polynomial basis. The singularity problem of PQ arises from the rank deficiency due to

the improper nodes enclosure and basis functions choosing. Therefore, through the row triangulariztion process, we firstly detect that the rows which are responsible for the rank deficiency and should be removed from nodes enclosure. According to the matrix theory, the row rank and the column rank of certain matrix are exactly same. Thus, P^ is triangularized to find out which rows (i.e. the columns in PQ) are responsible for the rank deficiency and should be removed from the basis functions.

Suppose the rank of PQ or PTQ is r, after being rid of the corresponding rows and

columns, the nxn matrix PQ is reduced to a new r x r matrix PQ that has a full rank

and already in a triangularized form. The shape functions then can be calculated very easily.

138

In the triangularization process, the zero rows (which represents the rank deficiency) are always related to a pair rows with same entries in the original matrix. Therefore, in MTA, the one in original P^ corresponding to the node that is farther from the point x and

higher order monomial in the basis functions are removed firstly to ensure the local characteristic of interpolations and the completivity of the basis.

Application of the PIM Method with MTA to Natural Convection in Square Cavity

In this section, the method described in the preceding sections is used to simulate the natural convection in a square cavity. The governing equations in terms of vorticity-stream function formulation are as follows:

ay ay

dx dy dx dy dx

dT dT__¥T_ d2T

dx dy dx2 dy2

where 0), y/, T are the vorticity, stream function, temperature while u, v are the components of velocity in the x and v directions. The boundary conditions can be written as

1) x = 0 , 0 < j ; < l : ^ = 0 , ^ = 0,r = 0, 2) x = l,0< y< l:y/ = Q,^- = Q,T = \, dx dx

3 ) ^ = 0 , 0 < x < l : ^ = 0 , - ^ = 0 , ^ = 0, 4) v = 1,0<x< 1 : ^ = 0 , - ^ = 0 , ^ = 0. dy dy dy dy

The present method is validated in the case of Ra=103, Pr=0.71 with two types of nodal distributions, as shown in Figure 1.

(a) TYPE 1: 256 regular nodes (b) TYPE II: 257 scattered nodes

Figure 1 Two types of nodal distribution for a square cavity

139

Table 1 lists the numerical results obtained by the present method with the two types of nodal distributions. The table also lists the results from FDM (Finite Difference Method) with the grid 16><16 (equivalent to 256 regular nodes). The benchmark solution of Davis (1983) is included for comparison. It is found the results by present method agree very well with the results of Davis (1983). They are much more accurate than that from FDM if the mesh size is the same. In this computation, there are 9 nodes in the support domain for the individual interpolated point. However, after MTA, the actual number of nodes may be less than 9. Therefore, the computational cost for calculation of the shape functions is very low.

Table 1 Comparison of numerical results for Ra=103,Pr=0.71

Present method with TYPE I nodal distribution

Present method with TYPE n nodal distribution

FDM (uniform mesh size 16x16)

Davis(1983)

kmaxl

1.180

1.079

1.268

1.174

" m x

3.602

3.516

3.905

3.649

v max

3.664

3.656

3.963

3.697

W"max

1.660

1.530

1.608

1.505

W"min

0.629

0.734

0.656

0.692

Conclusions

In this paper, the PIM method with MTA is adopted to simulate CFD problems. The simulation of natural convection in a square cavity is used as an example to demonstrate that the present method works very well with different distributions of nodes. The accuracy it achieved is much higher than that obtained by FDM. The major advantages of the present method are (a) the nodal distribution in the problem domain can be arbitrary and (b) computational cost for construction of shape functions is very low which is very favourable for simulation of fluid problems.

References

Atluri SN, Kim HG and Cho JY (1999), "A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG) and Local Boundary Integral Equation (LBIE) methods," ComputMech. 24:348-372

G.de Vahl Davis, (1983), "Natural convection of air in a square cavity: a benchmark numeical solution," lnt.J.Numer.Meth.Fluids 3:249-264

Gu YT, Liu GR(2001), "A local point interpolation method for static and dynamic analysis of thin beams," Comput.Methods Appl. Mech. £/ig/-g. 190:5515-5528

Liu GR, Gu YT(2001a), "A point interpolation method for two-dimensional solids," Int.J.Numer.Meth.Engng 50: 937-951

Liu GR, Gu YT(2001b), "A Matrix Triangularization Algorithm for Point Interpolation Method," Proceeding of the Asia-pacific vibration conference, Hangzhou, China, Nov. 18-31,2001: 1151 -1154.

140

Advances in Meshfree and X-FEM Methods, G.R. Liu, editor. World Scientific, Singapore 2002

THE SOLUTION FOR CONVECTION-DIFFUSION EQUATIONS USING THE QUASI-LNTERPOLATION SCHEME WITH LOCAL POLYNOMIAL

REPRODUCTION BASED ON MOVING LEAST SQUARES

*Xin Liu, **G. R. Liu, ***Kang Tai and "*K.Y. Lam

SMA-Research Fellow, Singgpore-MITAlliance smalx(d).nus.edu.sg

Centre for Advanced Computations in Engineering Science (ACES), Dept. OfMech. Eng., National University of Singpore, Singapore 119260

grliu((Vniis.edu.s«

SMA -Fellow, Singpore-MIT A lliance mktaiCdtntu.edu.ss! and lamkv(diihpc.niis.edu.sg

Abstract

In this paper, a quasi-interpolation scheme with special emphasis on local polynomial reproduction based on the Moving Least Squares (MLS) was applied to solve convection-diffusion problems. The formulations with local polynomial reproduction of different orders have been deduced and employed to investigate the accuracy and h-convergence of the presented method. The influence on accuracy with different supported domain schemes has also been demonstrated. A 2D steady-state convection-diffusion problem and a 2D transient-state convection-dominated "Rotating Cosine Hill Problem" were numerically analyzed and many available results were obtained.

Keywords: Local Polynomial Reproduction, Moving Least Squares, Meshless, and Convection-Diffusion.

Introduction

In recent years, Meshless methods have increasingly developed to solve various kinds engineering and mathematical problems, and are becoming a hot research topics in the fields of computational mechanics and computational mathematics (G. R. Liu 2002). The key point in meshless research is to provide good interpolation approach to multivariate scattered data approximation. Up to now, there mainly exist two kinds of popular scattered data approximate approaches, one is the Moving Least Squares (MLS) method and other is Radial Basis Functions (RBFs). In this paper, interest is paid to MLS method. MLS method was originally presented by Lancaster and Salkauskas (1981) for multivariate scattered data approximation. It has been applied to meshless method by Belytschko et al. (1994). For several years, it has emerged as the basis of numerous meshless approximation methods like Element free Galerkin method (EFGM), Finite Point Method (FPM), Hp-Clouds and Meshless Local Petrov-Galerkin (MLPG) etc.

141

Local polynomial reproduction plays an important role in numerical approximate analysis and computation. Higher order polynomial reproductions and approximations are very significant for solving engineering and mathematical problems effectively and high accurately. Wendland (2001) presented the error estimates with the local polynomial reproduction based MLS. Fasshauer (2001a) obtained a fast explicit approximate MLS method with high approximation order. The matrix-free formulations for polynomial based MLS was presented by Fasshauer (2001b), and its major advantage is that no linear systems need to be solved. The only task is to evaluate a sum for computing the approximation at a certain point.

In this paper, at first, the quasi-interpolation approximation of function with local polynomial reproduction based on MLS was briefly generalized. Secondly, collocation schemes and time integration schemes for general time dependent convection-diffusion equations are proposed, and then numerical testings are given to solve steady-state and transient-state convection-diffusion equations. As a result, the significant results on accuracy and ^-convergence property of the presented method are obtained in our numerical computations. The influence on accuracy with different supported domain schemes is also investigated. In the end, some concluding remarks are given.

The Quasi-interpolation Procedure

In this section, we briefly generalize some key formulations about quasi-interpolation expression of function using local polynomial reproduction based the MLS as follows. More detailed formulations can be found in Fasshauer (2001b).

For a set of distinct data points Xi and their function value M(XJ)=«J, namely {x{, u\),

i=l, ..., n. n is the number of nodes in supported domain. A quasi-interpolation form of function at x can be defined as follows:

n n

i=i i=i

Here y/\(x) do generally not satisfy the following delta function property

However polynomial reproduction can be enforced in order to obtain good approximation construction, and meanwhile determine the function yA(x).

Minimized the following quadratic form

±Irf(*)*'^L) (3) Subjected to the linear constraints

| > , ( x ) M * , ) = />,(*),* = 0,---m (4)

142

for all

/>(*) = W * ) />.(*) ••• Pk(x) ••• Pm(x)}T (5) Eq. (4) is polynomial reproduction conditions. Eq. (5) shows the (m+l) monomial terms which need to be reproduced.

For ID case:

• Constant reproduction: m=0, po(x}=\; • Linear reproduction: m= 1,p0(x)= l,pt(x)=x; • Quadratic reproduction: m=2,po(x)=l,p\(x)=x,p2(x)=x2;

For 2D case:

• Constant reproduction: m=0, po(x)= 1; • Linear reproduction: m=3,po(x)=l,p\(x)=x,pi(x)=y; • Quadratic reproduction: m=5, po(x)=l, p\(x)=x, p2(x)=y, p^(x)=x2, p4(x)=xy,

Ps(x)=y2;

In Eq. (3), we choose

I I* - x, I </>(- ~) = (6)

A x-xt w( ^ ) Pi

Combining Eq. (4) and Eq. (3) by introducing Lagrange multipliers fa, k=0, ..., m, we obtain

1 n v — X \\ m n

- £ tf (*#(" n '"2) + £ K < £ Yi MP* (*. ) - Pk (*» (7) *• i=\ Pi *=0 j=l

After minimized the eq. (7), following representation can be acquired

\\x — x \\ m

yr, (x) = *<!! ^ ) £ XkPk (*,) (8) Pi k=0

where the fa are the unique solutions of

m n j£ — X \\

£ ^ * £ < %»*(*>,(**) = Pj(x), 0 < j < m (9) k=\ f=i A

Here, weight functions w(xjc„pi) are local compactly supported continuous functions with central point Jtj and the size /?, of supported domain. In generally, radial basis functions with compactly supported domain may be a well choice as weight functions.

In this paper, the following weighting functions are chosen

II II ~<->2 - < - ) 2

wi(x) = w(x,xl,pi) = wCl —) = - 7 — dOa) P i , "< »

l-e c

143

r = | * - * , | 2 , c = 0.5A (10b)

For constant and linear polynomial reproduction, the resulting formulas can be analytically deduced, and no linear systems need to be solved in computations.

• Constant reproduction

YM = ^ & - ( i i)

7=1

It is known as Shepard's method. • Linear reproduction

Ux)=jjWu ~M20M02] (12a)

4 ( * ) = l [ M 1 0 M 0 2 - M 0 1 M n ] (12b)

A2(x) = ±-[M„Mm-M10Mll] (12c)

Q = M^M01 +M20M2

0] -M00M20Mm -2Ml0M0lMu +MmM^ (12d) The moments Mig in above Eqs. (12a-d) can be defined as

Mki =fj(x-xi)k(y-yiywi(x),k,F0,l,2,k+}<2 (12e)

i = l

As a result, y/,(x) can be expressed as follows

^ ( * ) = [A0(X) + A 1 (X) (X-X / ) + A I(JCX>'-J ' ()1W ((*) (13)

Collocation Schemes for Time Dependent Convection-Diffusion Equation Let us consider the following transient-state convection-diffusion equation in Q, which is given in the following standard form (X. Liu 2000).

L(u) = p— + vT Vu-VT{DVu)-q=0 (14a)

dt

together with the general boundary and initial conditions:

Neumann boundary condition on r t l :

Lbi(u) = nTDVu + q„=0 (14b) Dirichlet boundary condition on T i2:

u-u=0 (14c)

Initial condition:

«(x,/) = «°(x),r = 0 (14d)

144

Assuming that there are Nd internal (domain) points and Nb =Nbi + Nb2 boundary points, which Nn are Neumann boundary points and Afo are Dirichlet boundary points.

In general, the location of the collocation points can be different from the location of nodes in discretization model. However, for the sake of simplicity, collocation points are the same as the nodes of model in the computing of this paper.

At time t=f, the following Nd equations are satisfied in internal domain nodes:

P^r-R°=0 ,i = l ,Nd (15) at

R°=-vTVii° + VT(DVii°) + q (16)

The following Nbi equations are satisfied on Neumann boundary TM :

nTDVu°+qn=0 ,i = \, Nbl (17)

The following Nb2 equations are satisfied on Dirichlet boundary r i 2 :

tt('-M=0,/ = l, Nb2 (18)

u' can be obtained by eq. (1). Its derivatives can be obtained by following equations:

v«; (x) = v«; (x) = £ v ¥j (x)Cu°) (19)

Here (suej) is the function value at jth node at time f.

• Explicit time integration schemes:

(20)

(21)

•

•

Implicit time integration

Crank-Nicolson schemes

dt us+] -us

P At

schemes:

t?+1 -us

P At

At

-Rs

-Rs+]

= 0

= 0 (22)

« * + 1 - i ) s 1

p- — - (R s + '+R*) = 0 (23) At 2

I'JS+1 -ii' p- — + 6(-Rs+l) + (l-e)(-Rs) = 0 (24)

At 0 explicit

^ = -! 1 implicit (25)

0.5 Crank - Nicolson

145

Numerical Simulation

In this section, a steady-state 2-D convection-diffusion equation and a transient-state convection-diffusion equation were numerically analysed. The steady-state example was used to test the accuracy and convergence of the proposed method under different order polynomial reproduction, namely linear and quadratic polynomial reproduction approximations. The ^-convergences were also computed with uniform regular models. The results obtained with different supported domains were compared. The transient-state example was solved to demonstrate the accuracy and efficiency of the proposed method for solving time dependent complicated problems.

In the following computation, the L2 error in computational results of Tables and Figures were defined as follows:

\i(ur-utr *= PS (26)

I 2>n2

where K** and u denote respectively the exact and the approximate solution. The rates of convergence of the relative error, R(e), are also computed. It is defined as follows:

R(e) =log(eM+i)/log(hi+1 /h,) (27)

where fa+i and hi are the distance between nodes for uniform model in the current and previous case respectively.

Example 1. 2D Steady-State Convection-Diffusion Equations

This problem is to determine a function u(x,y) satisfying the following 2D steady-state convection-diffusion equation:

-£V-V« + v V « = / , j c e £ 2 = [0,l]x[0,l] (28a)

£ = 10-5,v = [l,l]; (28b)

Subjected to the Dirichlet condition:

"(*,>oL=0 (28c) along the whole boundary of the domain. The exact solution is given by

u"{x,y) = sin(^x) sin(fly) ( 29)

f[x,y) may be obtained by substituting Eq. (29) into Eq. (28a).

This problem was solved using four different uniformly distributed node models, namely 6x6 (A=0.2), 11x11 (A=0.1), 21x21 (A=0.05) and 41x41 (h=0.025) models, in order to investigate ^-convergence of presented method. Here two schemes for choosing supported domain are shown in Figure 1. 5-node scheme shown in Figure 1 (a) is used for linear reproduction case. 9-node scheme shown in Figure 1 (b) is used for quadratic reproduction case. The results obtained with different regular models and different order polynomial reproductions are listed in Table 1 and h-convergences are shown in Figure 2.

146

The convergence rate is about 2.0 for both linear reproduction and quadratic reproduction cases. In addition, a 406-node irregular scattered points model shown in Figure 3 is computed to observe the influence on computational results using different supported domains. For irregular model, the supported domain at certain evaluation node is determined by including the nearest nodes around it with prescribed numbers. The results obtained with irregular model are listed in Table 2-3. From the results, it can be seen that linear reproduction acquired poor accuracy while quadratic reproduction greatly improved the accuracy of numerical analysis.

Table 1. The results for example 1 with regular models

Model

6x6 11x11 21x21 41x41

Linear precision 1,2-error (%)

7.11 1.78 0.45 0.11

R(e)

1.998 1.984 2.030

Quadratic precision Z,2-error (%)

1.13 0.133 0.033 0.016

R(e)

3.087 2.011 1.018

Table 2. The results for example

Supported domain

L2 error (%)

Table 3.

Supported domain

L2 error (%)

5-node

39.35

The results for

8-node

3.27

6-node

35.52

1 with irregular models (Linear precision)

7-node

10.66

example 1 with irregular

9-node

1.77

10-node

12.34

8-node

23.24

9-node

25.20

model (Quadratic precision)

11-node

14.40

12-node

0.88

Example 2. Time Dependent Convective-Diffusion Equation

A known benchmark on time dependent convection-dominated problem, namely Rotating Cosine Hill Problem, was numerically analyzed to test the accuracy of presented method.

This problem satisfies Eq. (14a), and its coefficients in Eq. (14a) are given by

D=0, p=l (30)

The solution domain is x e £2 = [0,l]x[0,l].

The Boundary conditions are

u(x,t)=0,xed£2 (31)

The initial conditions are

M(X,0) = l + cos^( -)

a 0,

l + c o s ^ r ( ^ - ^ ) a

(x-x0)2+(y-y0)

2<<J2

(32) otherwise

The initial position of the center and the radius of the cosine hill are

147

( W o ) = ( | , | ) , < 7 = 0.2 (33)

The advection field is a pure rotation with unit angular velocity given by

v = (-y + j,x~) (34)

A uniformly distributed 61x61 nodes model over the unit square of solution has been employed in the calculations and quasi-interpolation formulation with quadratic reproduction in Eq. (1) has been used for the spatial discretization. In the procedure of time integration, 100 time steps with time interval is 2rc/100 have been adopted. The supported domains are chosen according with 9-node scheme in Figure 1 (b). They give the elevations of the rotating cosine hill after one full revolution. To compare the accuracy of the presented method, the maximum and minimum values of the computed solutions and the error are provided in Figure 4.

Conclusions

In this paper, the quasi-interpolation approximation scheme with local polynomial reproduction based on MLS is employed to solve steady-state and transient-state convection-diffusion problems. The presented formulation is simple to use and no linear systems need to be solved. The computational results demonstrate that the high order polynomial reproduction well improved the accuracy of the numerical solution. In addition, the influence on accuracy using different supported domain has also been investigated. It is apparent that more nodes in supported domains are needed for quadratic reproduction than for linear reproduction, but too much nodes in supported domain will destroy the accuracy. In the steady-state convection-diffusion numerical example, it is also found that the ^-convergence rates achieve about 2.0 even higher for both linear and quadratic reproduction. However, higher accuracy was observed by adopting quadratic precision.

References

G. R. Liu, (2002), "Mesh free Methods, Moving beyond the Finite Element Method", CRC Press.

Lancaster P, Salkauskas K., (1981), "Surfaces generated by moving least squares methods", Math. Comput., 37, 141-158.

Belytschko T, Lu Y Y, Gu L., (1994), "Element-free Galerklin methods", Int. J. Numer. Methods Engrg., 37, 229-256.

H. Wendland, (2001), "Local polynomial reproduction and moving least squares approximation", IMA Journal of Numerical Analysis, 21, 285-300.

G. E. Fasshauer, (2001a), "Approximate Moving Least Squares approximation with compactly supported radial weights", http://amadeus.math.iit.edu/~fass.

G. E. Fasshauer, (2001b), "Matrix-free multilevel Moving Least-Squares methods",

http://amadeus.math.iit.edu/~fass.

Xin Liu, (2000), "Application of Finite Point Method in Reservoir Simulation and Numerical Simulation of Option Pricing Models", Research Report of Post-Doctoral Fellow, Tsinghua University.

http://amadeus.math.iit.edu/~fass

http://amadeus.math.iit.edu/~fass

148

• ° T

l h

o •*-(a) 5-node scheme

O O 0

O • O

O 0 O I* (b) 9-node scheme

Figure 1 .Two supported schemes Figure 2. /i-Convergence for example 1

(a) 406-node scattered points model (b) The solution for irregular model Figure 3. The solution for example 1 with irregular model

(a) max=1.0, min=0.0 (b) max=0.993, min=-0.040 (c) max=1.008, min=-0.055 t=0 t=n/2 t=n

Figure 4. The solution for example 2 at t =0, n/2, n

SECTION 8

Boundary Meshfree Methods

151


REGULAR HYBRID BOUNDARY NODE METHOD

J. M. Zhang, Z. H. Yao Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China

demyzh@tsinghua. edu. en

Abstract

A new meshless method, called regular hybrid boundary node method (RHBNM) is developed by the authors, which combines the MLS interpolation scheme with the hybrid displacement variational formulation, and the source points of the fundamental solutions are located outside the domain. In this paper, the formulation of the RHBNM is presented briefly, taken the 2D potential problem as example. Then the numerical examples, not only for 2D potential problem, but also for 3D potential and 2D, 3D elasticity problems, are given to show the accuracy and applicability of this method.

Keywords: Meshless method, Boundary node method, Hybrid variational formulation, Moving Least Square approximation, Regular approach, RHBNM.

Introduction

The idea of meshless methods was initially introduced by Lucy as the Smooth Particle Hydrodynamics (SPH) method for modeling astrophysical phenomena (1977), although the meshless methods first gained popularity after the publication of the diffuse element method (Nayroles et al., 1992) and the element free Galerkin method (Belytschko et al., 1994). The element free Galerkin (EFG) method uses a global symmetric weak form and the shape functions from the moving least-squires approximation. Since 1998, the Meshless Local Boundary Integral Equation (MLBIE) method (Zhu et al., 1998) and the Meshless Local Petrov-Galerkin (MLPG) approach (Atluri et al., 1998; Kim and Atluri, 2000; Lin and Atluri, 2000) have been developed. Both methods use local weak forms over a sub-domain and shape functions from the MLS approximation, in which no 'finite element or boundary element mesh' is required either for the variable interpolation, or for the 'energy' integration. In 1997, Mukherjee et al. proposed a meshless method, called Boundary Node Method (BNM). They combined the MLS interpolants with Boundary Integral Equations (BIE) in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter. This method only requires a nodal data structure on the bounding surface of a body; but an underlying cell structure is again used for numerical integration. A question arises here — is there possibly a method of solving boundary value problems, that only requires nodes constructed on the surface and requires no cells either for interpolation of the solution variables or for the numerical integration? This method will simplify the input data structure greatly, compared with MLBIE and MLPG; and it does not use any mesh either for interpolation or for integration, compared with BNM.

152

The answer is positive. The new method is called Hybrid Boundary Node Method (Hybrid BNM) (Zhang, Yao et al, 2002), which combines the MLS interpolation scheme with the hybrid displacement variational formulation. However, the Hybrid BNM has a drawback of "boundary layer effect", i.e. the accuracy of results in the vicinity of the boundary is very sensitive to the proximity of the interior points to the boundary. To avoid this pitfall, a new Regular Hybrid Boundary Node Method (RHBNM) (Zhang, Yao, 2001, 2002) has been proposed, in which the source points of the fundamental solutions are located outside the domain rather than at the boundary nodes as in the Hybrid BNM or other hybrid boundary element models. Compared with the Hybrid BNM, the present method does not involve any singular integration and the results are no moVe sensitive to the proximity of the interior points to the boundary, high accuracy can be achieved with a small number of boundary nodes. Recently, another new kind of meshless method related to boundary integral formulation, the boundary point interpolation method is developed by Gu and Liu (2001,2002). In this paper, the formulation of the RHBNM is presented briefly, taken the 2D potential problem as example. Then the numerical examples, not only for 2D potential problem, but also for 3D potential and 2D, 3D elasticity problems, are given to show the accuracy and applicability of this method.

Formulation of the Regular Hybrid Boundary Node Method

The Regular Hybrid Boundary Node method proposed in this paper is based on a modified variational principle. The functions assumed to be independent are: potential field in the domain, u; boundary potential field, u ; and boundary normal flux, q . The corresponding variational functional n^B is defined as follows:

n AB = \a j u„ u„ dQ. - \vq{u - U)dT - j ^ q-udr (1)

where, the boundary potential u satisfies the essential boundary condition, u=u on r„ .

With the vanishing of STl^, one can obtain the following equivalent integral equations:

jr(V ~ q)SudY - j u,„ SudQ. = 0 (2)

jr(u-u)8qdr = 0 (3)

f (q - q)8udT = 0 (4)

If we impose the flux boundary condition, q = q , after the matrices have been computed, the equation (4) will be satisfied. It can be seen that the equations (2) and (3) hold in any sub-domain, for example, in a sub-domain Qs and its boundary r , and Ls (Figure 1). We can use the following weak forms on a sub-domain Qs and its boundary r , and Ls to replace equations (2) and (3):

jr+L(g-q)vdr-jau,liVdQ = 0 (5)

153

f («-S>fr = o (6)

r r, =an,nr

Figure 1. The local domain centered at a node Sj and the source point

of fundamental solution corresponding to a node s,

where v is a test function. It should be noted further that the above equations hold irrespective of the size and the shape of Qs and its boundary 3Q s . We now deliberately choose a simple regular shape forQs. The most regular shape of a sub-domain should be an n-dimensional sphere for a boundary value problem defined on an n-dimensional space. In the present paper, the sub-domain £ls is chosen as the intersection of the domain Q and a circle centered at a boundary node Sj .

Now we approximate u and q on Ys in equation (5) and (6), by the MLS, as:

«(*)=!*/(*)«, ?(s) = X*,(s)«, where

®I(s) = fjPj(s)[A-1(s)B(s)}JI

with the matrices A(s) and B(s) being defined by

N

A(s) = '2Jw!(s)p(sl)pT(sl)

7=1

B(s) = [w, (5)p(5,), w2 (s)p(s2 ),---,wN (s)p(sN )]

Gaussian weight function corresponding to node s, can be written as

exp[-(d, / c, f ] - exp[-(rf; / c, f ] W,{S) = ' 1-expK^/c , ) 2 ]

0 < d, < d,

d, >d,

(7)

(8)

(9)

(10)

(11)

154

However, u and q on Ls has not been defined yet. To solve this problem, we deliberately select v such that all integrals over Ls vanish. This can be easily accomplished by using the weight function in the MLS approximation as v, with the radius d, of the support of the weight function being replaced by the radius r, of the sub-domain Q,,i.e.

Vy(fi) = expKrf,/c,)2]-exp[-(r,/c,)2] 0 < r f < r

l-exp[-(r,/cy)2] ' (12) 0, djlTj

where dj is the distance between a point Q, in the domain Q, and the nodal point Sj . Therefore, v vanishes on Ls. The u and q inside Q and on T are defined as

2k 2k Till u = 2u,x, q ^ ^ x , (13)

where Ut is the fundamental solution with the source at a point P,, which locates at the outside of the domain and is corresponding to a node 57; x, are unknown parameters; NN is the total number of boundary nodes. For 2-D potential problem, the fundamental solution is

U,=~\nr(Q,P,) (14)

where Q and P, are the field point and the source point respectively. And P, is determined by following equations

x(Pr) = x(S[) + hn£ y(PI) = y(SI) + hn^ (15)

where x and y are coordinates; h is the mesh size; nx and ny is the components of the outward normal direction to the boundary at node P,; and E, is a scale factor. As can be imagined, the scale factor, | , plays an important role in the performance of the present method. Too small value for % will lead to nearly-singular integrals and thus inaccurate results; on the contrary, too large one will lead to an ill-posed system of algebraic equations. From our computations, the proper range for | is between 3.0 and 6.0 . As u is expressed by the first equation of (13), the last integral in the left hand in equation (5) vanishes. By substituting equations (7) and (12)-(14) into equation (5) and (6), and omitting the vanished terms, one can obtain

dU,

I Jr u,vJ(Q)xIdr = X j r <f;(s)v;(e)Mr (16)

155

Using the above equations for all nodes, one can obtain the following system of equations:

Ux = Hq Vx = Hu (17)

The evaluation of the matrices U and V is much more simple in this approach than in BEM and BNM. No integrations of singular functions are involved. For a well-posed problem, either u or q are known at each node on the boundary. However, transformations between w7 and u,, qt and q, must be performed due to that the MLS interpolants lack the delta function property of the usual BEM shape functions (Atluri et al. 1999). For u prescribed edges, u, can be obtained by

u, ^RffUj =%RaUj (18)

and for q prescribed edges, qt can be obtained by

<7,=J!>z,5./=|X^ (19)

Therefore, by rearranging the governing equations (17), one obtains the final system in term of x only, and the unknown vector x is obtained by solving it.

Numerical Examples

For the purpose of error estimation and convergence studies, a 'global' L2 norm error,

normalized by \u\ is defined as I max

-±M^-^ where 1 / ^ is the maximum value of/over N sample points, the superscripts (e) and (n) refer to the exact and numerical solutions, respectively.

In all examples, the size of support for weight function, d,, is taken to be 9.5/J , with h being the mesh size, and the parameter c7 is taken to be such that d, /c, is constant and equal to 4.0. The size of the local domain (radius ry) for each node is chosen as 1.0A in all computations and the parameter c, in equation (12) is taken to be such that TJJCJ is constant and equal to 4.0. In all integrations, 5 Gauss points are used on each section of two half-parts of r , .

1. Dirichlet problem on a circle (2D Laplace equation)

The example solved here is the Laplace equation on a circle of radius 3 unit, centered at the origin. The exact solution is

u = x (21)

156

A Dirichlet problem is solved, for which the essential boundary condition is imposed on the whole circle. To study the convergence of the present method, three regular meshes of 10, 20 and 40 nodes have been used. Numerical results of u and q (with normal vector (1,0)) along the radius (form (0,0) to (3,0)) from the RHBNM with | = 5.0 and from the Hybrid BNM, together with the exact solution, are shown in Figure 2.

Figure 2. u and q along the radius (from (0,0) to (3,0)): a) from Hybrid BNM, b) from RHBNM

Results for potentials are in all case accurate. The internal fluxes from the Hybrid BNM, however, show considerable error for points close to the boundary when a small number of nodes are used. The results improved considerably when the RHBNM is used. In the RHBNM, it is very appealing that high accuracy can be achieved with a small number of nodes, and the results is no more sensitive to the proximity of the interior points to the boundary whereas in the Hybrid BNM or other hybrid boundary element methods.

2. Dirichlet, Neumann and mixed problem on a square (2D Laplace equation)

The case of Laplace equation on a 2 x2 domain is presented as the second example. The exact solution is a cubic polynomial

u = -xi-y3+3x2y + 3xy2 (22)

For the mixed problem, the essential boundary condition is imposed on top and bottom edges and the natural boundary condition is prescribed on left and right edges.

Scatotacwr, £ b ScMhctw. | « Scartfwwr. £

Figure 3. Relative errors of normal flux on the edge (from (0,0) to (2,0): a) for Dirichlet problem, b) for Neumann problem, c) for mixed problem

The effect of the selection of the scale factor £ has been studied on four different regular nodes arrangements: (a) 5 nodes on each edge; (b) 10 nodes on each edge; (c) 20 nodes

157

Dihchlet problem Neumann problem Mixed problem

a - -»

3 1 -6 - Exact solution

RHBNM solution for Dirichlet problem RHBNM solution for Neumann problem RHBNM solution for mixed problem

-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 °-5 1 0 1 -5 2 0

L°8,0(n> X

Figure 4. Relative errors and convergence Figure 5- <7« a t y = 0 for Dirichlet, rates for Dirichlet, Neumann Neumann and mixed problems

and mixed problems (5 nodes on each edge are used)

on each edge; (d) 40 nodes on each edge, with E, varying from 0.5 to 10. Figure 3 shows the relative errors of normal flux on the edge (from (0,0) to (2,0), 13 uniformly distributed sample points) with different meshes. It is noted that the results for all meshes are accurate enough when E, >3.0. However, as E, grows beyond 8.5, results become unstable for the meshes (b), (c) and (d). This implies that the equations (17) are approaching nearly ill-posed. The convergence of the method has also been studied on the four nodes arrangements with E, = 5.0. The results of relative errors (equation (20)) and convergence of potential on the diagonal (from (0,0) to (2,2), 19 uniformly distributed sample points) are shown in Figure 4, and numerical results of normal flux on the edge (from (0,0) to (2,0)) from the RHBNM when 5 nodes are used on each edge, together with the exact solution are shown in Figure 5. It can be seen that the present RHBNM has high rates of convergence.

3. Circular hole in an infinite elastic plate (2D Navier equation)

This is the well-known Kirsch problem, a portion of an infinite plate with a central circular hole subjected to unidirectional tension S as shown in Figure 6.

~J D —

I L i

Figure 6. Model for the Kirsch problem

The problem is solved here for the plane stress case with S = \, a=\, £ = 60, iT = 2.5, v =0.3. Twenty uniformly spaced nodes are used on the inner circle and the outside square boundary, respectively. The computed stresses along AB and CD, together with

158

the exact solution, are shown in Figure 7. The numerical results agree excellently with the exact solutions again.

° , 2.0-

Exact solution • RHBNM solution

0 0

-0.2

3 .0.4 s

f ~ -0.8

-10-1

Exact solution • RHBNM solution

-30 -25 -20 -15

Figure 7. Stress distribution for Kirsch problem a. along CD; b. along AB

This problem can also be dealt with as an "external" domain without introducing the

artificial boundaries. In this case, the solutions are decomposed into two parts, i.e. au = au + al' w n e r e m e superscript u refers to the uniform solution, c", = S, and cr," = 0

for i * 1 and j T* 1; and c to a complementary solution, respectively. Therefore, the

complementary tractions on the circular hole can be computed by ft = ti -t" = -t". The

solutions a'j are firstly obtained with twenty uniformly spaced nodes on the hole, then,

we get av by adding the uniform solution <x,". The results we finally get are exactly the

same as shown Figure 7a and Figure 7b. It can be seen that the RHBNM retains the advantage of solving infinite domains of the traditional BEM.

4. Dirichlet problem on a sphere (3D Laplace equation)

The example solved here is the Laplace equation on a sphere of radius 2 unit, centered at the origin. The usual spherical polar coordinates 6 and (j) are used. On the surface, 800 MLS points and 136 uniformly spaced nodes are used. The exact solution is a cubic polynomial

u = xi+y3 + zi -3yx2 -3xz2 -3zy2 (23)

0.32

_ 0-28-c :

§ 0.24 4> CL

2 0.20 o

a> £ 0.16-

iS £ 0.12-

0.08-

c

-DM-U -DM-q -SF-U

SF-q

-0.8-1

-1.2

-1.6

> p--2.4-

-2 . 8 l

-3.2-

*\

• \

N\ /\\ ' u 1 k.

V. ' "

— ^ - DM-u —•— DM-q —»-SF-u — I — SF-q

0.8 1.2 1.6

Sub-domain radius, ^ , (* i

2 4 6

Scale factor. £

Figure 8. Relative errors for various sub-domain radius, rj

Figure 9. Relative errors for various scale factor, ^

159

The Dirichlet boundary conditions corresponding to the exact solutions have been imposed on the surface of the sphere. The relative errors of u and its x-derivative inside the sphere, denoted by DM-u and DM-q in the figure, are evaluated over 11 sample points uniformly distributed from (0,0,0) to (2,0,0); and the relative errors of u and q(=du/dn) on the surface, denoted by SF-u and SF-q in the figures, are evaluated over 11 sample points uniformly distributed along the half equator of the sphere (0 < 9 < n). Results for various sub-domain radius, rj, are shown in Figure 8. It should be noted that the U F, do not cover the whole bounding surface when ry < 0.5/i (where h is the mesh size), and the r , will be overlapped when r, > h. Figure 8 shows that results are in all case accurate no matter whether Ts are overlapped, or even uncover the body's boundary. The scale factor % in equation (15) is also studied in this example. The relative errors for various £, are shown in Figure 9. It can be seen that results are all accurate when t, > 2.0. As mentioned before, too large value of Z, will lead to ill-conditioned equations. Actually, further computations of this example show that the biggest values of t, that ensure the RHBNM non-degenerate is 26.0, and this value is independent of boundary conditions while dependent on the domain geometry and meshing.

5.3D Kirsch problem (Navier equation)

This problem is a portion of an infinite cube with a small spherical cavity subjected to an unidirectional tensile load of c 0 in the z-axis direction as shown Figure 10. The exact solution for the normal stress (<rK) in the plane z = 0, in polar coordinates is given by

o„=an 1 + 4-5v "1 __9 U

2(l-5v){r ) ' 2(7-5v)(r (24)

The problem is solved here for a0 = 1, a = 0.1 and b = 1. 72 uniformly spaced nodes are used on the inner sphere and 96 nodes on the outer cube boundary. Figure 11 shows a comparison between the RHBNM solution and the exact solution for the normal stress along the x-axis ahead of the cavity. Again, it can be clearly seen that the RHBNM solution is in excellent agreement with the analytical solution.

2 2

2.0

18

= 1.6

1.4

1.2

1.0

\ \

\

I • Numerical solution

Exact solution

•

0.4 0.6 0.8 X

Figure 10. 3-D Kirsch problem Figure 11. Normal stress distribution along the x-axis ahead of the cavity

160

Conclusions

A type of regular hybrid boundary node method has been presented. It is based on a hybrid model that involves three types of independent variables, and coupled with the MLS interpolation scheme over the boundary variables. Compared with the MLBIE and MLPG, the new approach has the well-known dimensionality of the BEM, compared with the conventional BEM, it is a meshless method, only requires a nodal data structure on the bounding surface; compared with the BNM, no cells are needed either for interpolation purposes or for integration purposes. Numerical examples of 2D, 3D potential and elasticity problems have shown high accuracy and high convergence rate. Though some drawbacks exist, e.g. many constant parameters have to be determined by experience, the advantages of the RHBNM, such as meshless nature, high accuracy, high convergence rates and no singularities etc., are so attractive that this method is certainly worthy of attention.

Acknowledgements: Financial support for the project from the National Natural Science Foundation of China, under grant No. 10172053 is gratefully acknowledged.

References

Atluri, S.N., Zhu, T. (1998). "A new meshless local Petrov-Galerkin approach in computational mechanics". Computational Mechanics, 22,117-127

Belytchko, T., Lu, Y.Y., Gu, L. (1994). "Element free Galerkin methods". International Journal for Numerical Methods in Engineering, 37,229-256

Gu, Y.T., Liu, G.R.. (2001). "A coupled element free Galerkin/boundary element method for stress analysis of two-dimension solid". Comput. Meth. Appl. Mech. Eng., 190,4405-4419

Gu, Y.T., Liu, G.R. (2002). "A boundary point interpolation method for stress analysis of solids". Comput. Mechanics, 28(1), 47-54

Kim, H.G., Atluri, S.N. (2000). "Arbitrary Placement of Secondary Nodes and Error Control in the Meshless Local Petrov-Galerkin (MLPG) Method". Computer Modeling in Engineering & Sciences, 1(3), 11-32

Lucy, L.B. (1977). "A numerical approach to the testing of the fission hypothesis". The Astronomy Journal, 8,1013-1024

Mukherjee, Y.X., Mukherjee, S. (1997). "The boundary node method for potential problems". International Journal for Numerical Methods in Engineering, 40, 797-815

Nayroles, B., Touzot, G., Villon, P. (1992). "Generalizing the finite element method: Diffuse approximation and diffuse element", Computational Mechanics, 10,307-318

Zhang, J.M., Yao, Z.H. (2001). "Meshless regular hybrid boundary node method". Computer Modeling in Engineering & Sciences, 2, 307-318

Zhang, J.M., Yao, Z.H. (2002). "Analysis of 2-D thin structures by the meshless regular hybrid boundary node method". Acta Mechanica Solida Sinica, 15(1), 36-44

Zhang, J.M., Yao, Z.H., Li, H. (2002). "A hybrid boundary node method". International Journal for Numerical Methods in Engineering, 53,751-763

Zhu, T., Zhang, J., Atluri, S.N. (1998). "A local boundary integral equation (LBIE) method in computation mechanics, and a meshless discretization approach". Computational Mechanics, 21,223-235

161


RADIAL BOUNDARY NODE METHOD FOR ELASTIC PROBLEM

H. Xie, T. Nogami Department of Civil Engineering, National University of Singapore, Singapore 119260

engp9365@nus. edu.sg, cvetn@nus. edu.sg,

J.G. Wang Tropical Marine Science Institute, National University of Singapore, Singapore 119260

tmswjg@nus. edu.sg

Abstract

In this paper, using an improved point interpolation technique, a radial boundary node method (RBNM) is proposed. The RBNM retains the dimensional reduction of the BEM and the BNM, and its shape functions possess the delta function property. Using normalized radial basis functions, the shape parameters are studied in detail. Numerical results show that the shape parameters should be carefully selected and the RBNM has good accuracy.

Keywords: Boundary Integral Equation, Point Interpolation Method, and Radial Basis function.

Introduction

The finite element method (FEM) and the boundary element method (BEM) are widely used to solve partial differential equations. However, defining elements in the FEM is time consuming for especially large domain and three-dimensional problems. The BEM discretizes only boundaries with elements. This largely reduces the workload for meshing. Nevertheless, the BEM still has some deficiencies for the interpolation is confined to an element. In large deformation or moving boundary problems, the shape functions over the heavily distorted elements are of poor properties, and hence the numerical results may be not acceptable.

An alternative called the meshless method can overcome the difficulties associated with 'elements'. Most of the meshless methods are based on the moving least-square (MLS) (Lancaster and Salkauskas, 1981) method and applied to domain problems. Mukherjee and his colleagues (1997) proposed a boundary node method (BNM) that combines the MLS with the boundary integral equation (BIE). However, the BNM cannot implement boundary conditions accurately because its shape functions constructed by the MLS lack the delta function property.

This paper formulates a radial boundary node method (RBNM). Firstly, this RBNM proposes a normalized radial basis function for boundary nodes to improve the radial point interpolation method (radial PIM) proposed by Wang and Liu (2002a) for domain problems. In the RBNM, the interpolation over nodes surpasses over elements, and thus could trace large strains. Moreover, the interpolation is of the delta function property. This makes the implementation of boundary conditions much easier. Secondly, the shape parameters in the radial basis functions are studied in

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detail for the RBNM. The appropriate ranges of the shape parameters are found out through numerical examples. These shape parameters can be used in other cases.

Boundary Integral Equation for 2-D Elasticity

A weak form for an elastic problem is expressed as follows:

where Q. is the domain of the problem bounded by boundary T; bk is the component of the body forces; j,k = 1,2 for two-dimensional problems;, and u'k is a weight.

After integrating twice by parts, Eq.(l) can be rewritten as

j ^ V Q + [bkukdn = -[PkukdT + [pkukdY (2)

If the weight u'k is taken as the fundamental solution of following problem,

alj+S(X-Z)e,=0 (3)

The integral equation for any load point is obtained as follows:

«/(£)+ [plk{X-4)uk{X)dT = [ulk{X-^)pk{X)dT + lbk{X)uk{X)dn

(4) In which, £ and X denote the positions where a unit force is located and the any

field point respectively; e, is a unit vector; u]k and p'k are kth components of

displacements and tractions due to a unit point load in the / -direction.

If the loading point £ is moved to boundary, and there is no body force, the boundary integral equation of Eq.(4) is rewritten as

clk^)uk^)+[p;k(X-^)uk{X)dT=[ulk{X-^)pk{X)dT (5)

or in matrix form:

^)^)+[V-{X-^{X)dT=[u{X-^V{X)dT (6)

where clk (£) is a coefficient related to the boundary smoothness.

Numerical Implementation by RBNM

The unknowns in Eq.(6) are boundary displacement uk and traction pk. If the NE

boundary T is divided into NE cells for integration, that is, T = ^ r ; , the Eq.(6) is

divided into:

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NE NE

c\ku[ + X J PnPkdT = Z J u'kPkdr (7)

In each element, the integration can be made through Gaussian quadrature. An approximation of displacement and traction at each Gaussian point can be obtained through radial PIM (Wang and Liu, 2002a):

ND ND

«*=£&«»> Pk=TithPx (8) x=l x=\

where ND is the number of nodes in the influence domain of an interpolation point. ux and px denote the nodal value at the xl node. ^ is the shape function of the x node to the li Gaussian point.

The discretized equation for the Eq. (7) is obtained as following matrix form:

NE NE

£ H , U ; = £ G , P ; (9) y=i j=\

Reordering the matrices in terms of the known u and p on the boundary, a system equation is obtained as follows:

Ax = F (10)

where x is the unknown; F is the load; and A is the system stiffness matrix.

The radial PIM is used to obtain the shape functions in Eq.(8). As an example, the displacement u is studied here (the traction uses the same interpolation):

ND

u(X) = £ 5 / ( Z ) a , = B r ( X ) a (11)

where Bt(X) is the radial basis function which has the following general form:

B,(X) = 5,(r() = B,(x,y), r^^x-x)1+(y-yi)2 (12)

Applying Eq.(l 1) to all ND scattered points within the influence domain, we have

ND

"(w^ZMWtK k = \,2,-,ND (13)

or in matrix form:

u = B0a or a = Bo'u (14) where

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u = [u„u2 ,---,unJ r

B0 =

A ( ^ J M ) ) £ 2 ( * M » J V D ) ••• ^ K C J V D )

(15)

The interpolation is finally expressed as

u(X) = BT(X)B~lu = i?(X)u (16)

<p(X) is the shape function and is obtained according to the following equation

ip(X) = BT(X)B-0l=[h(X),t2(X),..;4>,(X),-<t>ND(X)] 07)

where (/>k (X) = ]Tfl( ( ^ ) 5 , t , in which 2?a is the element of matrixB;;1.

Two particular forms of the radial basis functions introduced by Wang and Liu (2002a), the exponential (EXP) radial basis and the multi-quadric (MQ) radial basis, are written with following normalized forms:

B>(x,y) = exp v 2 \

(EXP), Bl(x,y) = (f Y

+ R V/max J

(MQ) (18)

where b, q and R are shape parameters; and rmax is the maximum distance of neighborhood node within an influence domain.

Study of the Shape Parameters of the RBNM

In the radial PIM, Wang and Liu (2002b) obtained the suitable shape parameters b = 0.002 ~ 0.03 and q = 0.98 -1.03 for the EXP and the MQ radial basis functions respectively. This paper uses the normalized EXP and MQ radial basis functions, thus theoretically the b value in this paper should be retimes that in Wang and Liu's paper. A cantilever beam referred in Wang and Liu (2002b) is studied here as an example. An Li error norm is defined as follows to evaluate the RNBM performance:

-\t^P'-r)'^(%) (19)

where e is the percentage Li error over Af nodes; / " and fe refer to the numerical and exact values of solutions, respectively; and | / | is the maximum value of / over N nodes. Here / may be a component of displacement or traction.

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Figure 1 L± error with different b for EXP radial basis

Figure 1 illustrates the L^ errors for displacements and shear stress along the boundary and the middle section. It is observed that the L2 error fluctuates when b < 0.03, and a steady range with low 1^ is from 0.03 to 0.1. This range is a little different from that obtained in the radial PIM (Wang and Liu, 2002b). Possible reasons have two: the interpolation of the current RBNM is only along boundary; and the BIE is used in the current RBNM.

Parameter q (R>0.05) Parameter q (R«5.00)

Figure 2 L2 error with different q and R for MQ radial basis

Figure 3 L^ -error at q varying from 0.9 to 1.05

Figure 2 shows the L^ error with two shape parameters for the MQ radial basis function. The curves in the figure also show that, when the R is small, there are two

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obvious points at which the L2 error would approach zero. When the R increases, the point around q = 1 still exists, while the other point vanishes. Figure 3 shows that at the exact location q = 1, the error is very big. This may be due to the singularity of the matrix B0 at q = 1. From the above study on the MQ radial basis function, it is suggested that the shape parameter R should be small, and a suitable range of q could be from 0.96 to 0.99 and 1.01 to 1.05.

Figure 4 compares the results with both the EXP and the MQ basis functions. Compared to the BEM, the results computed by the current RNBM is in as good agreement with the analytical solutions as those produced by the BEM

x-coordinate (m) x-coordinate (m)

(a) With EXP basis (b) With MQ basis Figure 4 Displacements of the lower boundary with different radial basis

Conclusions

This paper combines the radial PIM with the BIE to form the RBNM. The shape parameters for the EXP and MQ basis functions are investigated in detail through case studies. From these studies, it is suggested that the RBNM could obtain results comparable to the BEM. The shape parameters should be selected with much cautious since the shape parameters have great influence on the accuracy of the RBNM.

References

Lancaster P and Salkauskas K. (1981), "Surfaces generated by moving least squares methods," Math. Comput,37, 141-158

Mukherjee YX and Mukherjee S, (1997), "The boundary node method for potential problems," Int. J.

for Numerical Methods in Engineering, 40(5), 797-815.

Wang JG and Liu GR (2002a), "A point interpolation meshless method based on radial basis functions," Int. J for Numerical Methods in Engineering, 54,1623-1648

Wang JG and Liu GR (2002b), "On the optimal shape parameters of radial basis functions used for 2-D

meshless methods," Computer Methods in Applied Mechanics and Engineering, 191(23-24),2611-

2630

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Advances in Meshfree andX-FEM Methods, G.R. Liu, editor, World Scientific, Singapore 2002

A HYBRID BOUNDARY POINT INTERPOLATION METHOD (HBPIM) AND ITS COUPLING WITH EFG METHOD

Y. T. Gu1, G. R. Liu1'2

Centre for Advanced Computations in Engineering Science(ACES) Department of Mechanical Engineering, National University of Singapore

E-mail: [email protected]; [email protected] 2 SMA Fellow, Singapore-MIT alliance

Abstract

A hybrid boundary point interpolation method (HBPIM) is presented for solving boundary value problems of two-dimensional solid mechanics. In the HBPIM, the boundary of a problem domain is represented by properly scattered nodes. The point interpolation method (PIM) is used to construct shape functions with Kronecker delta function properties based on arbitrary distributed nodes. In HBPIM, the 'stiffness' matrix so obtained is symmetric. This property of symmetry can be an added advantage in coupling the HBPIM with other established meshfree methods. A novel coupled EFG/HBPIM method for 2-D solids is then presented.

Keywords: Boundary Integral Equation, Meshfree method, Boundary Element Method, Coupled method.

Introduction

Meshless methods have become recently attractive alternatives for problems in computational mechanics, as they do not require a mesh to discretize the problem domain, and the approximate solution is constructed entirely in terms of a set of scattered nodes. Meshless methods may be largely divided into two categories: domain type methods and boundary type methods. In these two types of meshless methods, the problem domain or only the boundary of the problem domain is discretized by properly scattered nodes. The boundary type meshless methods proposed include Boundary Node Method (BNM) (Mukherjee,1997), Boundary Point Interpolation Method (BPIM)(Gu and Liu, 2002a), and Boundary Radial Point Interpolation Method (BRPIM) (Gu and Liu, 2002b).

In the late eighties, alternative BE formulations have been developed based on generalized variational principles. DeFigueiredo and Brebbia (1989) proposed and developed a Hybrid displacement Boundary Element (HBE) formulation. The HBE formulation leads to a symmetric stiffness matrix. In this paper, a hybrid boundary point interpolation method (HBPIM) is proposed for solving boundary value problems of two-dimensional solid mechanics. The HBPIM is formed by the combination the polynomial PIM with the hybrid



168

displacement BIE. In HBPIM, the 'stiffness' matrix so obtained is symmetric. This property of symmetry can be an added advantage in coupling the HBPIM with other established meshfree methods. A novel coupled EFG/HBPIM method for 2-D solids is also proposed. The compatibility condition on the interface boundary is introduced into the variational formulations of HBPIM and EFG using Lagrange multiplier method. The validity and efficiency of the present HBPIM and EFG/HBPIM are demonstrated through numerical examples.

Point interpolation method

Consider a function w(x) defined in domain Q discretized by a set of field nodes. The point interpolates w(x) from the surrounding nodes of a point xe using the polynomials as basis can be written as

M(x) = J A (x ) a / =,p T (x )a (1)

where /?,(x) is a monomial in the space coordinates XT=[JC, y], n is the number of nodes in the neighborhood of x, a* is the coefficient for p,{x) corresponding to the given point xe. The/>,(x) in equation (1) is built utilizing Pascal's triangle, so that the basis is complete. From equation (1), we have

w(x)= (|)(x) ue (2)

where the shape function (|>(x) is defined by

$(x)=pT(x)P0-l=[ HA HA HA , Hx)] (3)

It can be found that shape functions (3) possess the delta function properties. However, like other methods that use polynomial as basis functions, there is possible singular problem of the moment matrix P0. In order to avoid the interpolation singularity, several strategies have been proposed, such as, using radial basis function and the moment matrix triangularization algorithm (MTA) (Liu, 2002).

The domain displacement and traction vectors are approximated as a series of products of fundamental solutions U , T and unknown parameters s. As in the conventional BEM formulation, the boundary displacement u and traction t of boundary nodes can be constructed independently using PIM, equation (2). The interpolation formulations for displacement and traction at a boundary point on the boundary T from the surrounding boundary nodes uses PIM to get

. u=U*s ( 4 a )

t=T*s (4b>

u = O T u„ (4c)

169

t=0>Tt„ (4d)

where the shape function <&(s) is defined

a>T=p-Po~' (5)

Hybrid boundary point interpolation method (HBPIM)

Consider the following two-dimensional problem of solid mechanics in domain Q bounded byT:

Va+b = 0 in Q (6)

where cris the stress tensor, which corresponds to the displacement field u={w, v}T, b is the body force vector, and V is the divergence operator. The boundary conditions are given as follows:

u = u on r„ (7a)

<r-n = t on Tt (7b)

in which the superposed bar denotes prescribed boundary values and n is the unit outward normal to domain Q. The compatibility condition should also be satisfied

u = u on T (8)

where u is the displacement field on the boundary, and u is the displacement in the domain but very close to the boundary. Now subsidiary condition (8) is introduced into the principle of minimum potential energy by introducing a set of Lagrange multipliers A. It can be found from the Euler equations that the Lagrange multipliers X represent the traction on the boundary, t . Thus the modified variational principle can be written as

n = J - e T o d Q - J u T b d Q - J u T t d r + | 7 T ( u - u ) d T (9) n ^ a r, r

The first term on the right hand side can be integrated by parts. The starting integral relationship (9), which is an integral in the domain, can be reduced to an integral on the boundary by using the fundamental solution. The displacement and traction vectors are approximated as a series of products of fundamental solutions (DeFigueiredo and Brebbia, 1989) U*, T* and unknown parameters s. The boundary displacement and traction vectors are written as the product of known interpolation functions by unknown parameters (displacement and traction of boundary nodes). The interpolation procedure in the HBPIM, as shown in Figure 1, is based only on a group of arbitrary distributed nodes. The

170

interpolation at a sampling point in the HBPIM is performed over the support domain of the point, which may overlap with the support domains of other sampling points.

The support domain for the sampling point

Boundary nodes

Background quadrature cell

A sampling point

Figure 1. the interpolation in HBPIM

We can obtain

where

n=-l/2sTAs-tTGTs+tTLu-uTf-sTb

A= JiTT'dr, r

G= JYU'dT, r

L= JT€»TdT, r

f = jotdr, r

b = jlJ 'bdQ

(10)

(11a)

(lib)

(lie)

(12a)

(12b)

The stationary conditions for n can be found by setting its first variation to zero. As this must be true for any arbitrary values of 8s, 8u and 8t, one obtains:

Ku=f+d

K=R'AR

R=(GVL

d=RTb

(13)

(14a)

where

171

R=(GT)_1L

d=RTb

(14b)

(14c)

It can be proved that matrix A is symmetric, and hence the matrix K. It is possible to conclude from equation (14) that this hybrid displacement boundary formulation leads to an equivalent stiffness approach. The matrix K may be viewed as a symmetric stiffness matrix, but the above integrals are only needed to perform on boundaries, and the domain needs not to be discretized.

Coupling of EFG and HBPIM

Consider a problem consisting of two domains Q1 and Q2, as shown in Figure 2, joined by an interface T/. The EFG formulation is used in Q1 and the HBPIM is used in Q2 . Continuity conditions on Tj must be satisfied, i. e.

K<»: t(2)

F/1)+F/2>=0

(15a)

(15b)

where u'1' and uj2) are the displacements on T/ for Q1 and Q2, F/(1) and F/(2) are the forces on T/ for Q ' and Q 2, respectively.

Q(2>

HBPIM region

• HBPIM nodes

i

i

i

r , ' i

i

i

i

i

-*

IS

1 O

1 O

1 O

1 O

1 O

1 O

1 O

1 O

1 O

1 O

, *

e— o o o o

o

o o o o

EFG

— e o o o o

o

O

O

o o O

O

o o o o

region

5 Interface nodes

o o

— e &•

o o o o o o o o o o o o o o o o o o

O EFG nodes

Figure 2. A problem domain divided into the EFG region and the HBPIM region

172

Because the shape functions of EFG are derived using MLS, which lack Kronecker delta function properties. It is impossible to couple EFG and HBE directly along T/.

A sub-functional is introduced to enforce the compatibility condition (15) by means of Lagrange multiplier y on the interface boundary

n,= JY.(u/» -u/2>)dr= Jya/'Mr- jyu/^dr^'-nr (16)

In equation (16), H 1 and III2 are the boundary integration along the EFG side and the HPIM side. Introducing n / and III2 separately into functions of HBPIM and EFG, generalized functional forms can be written as

f l T nEFG = j"-£T • odQ - juT • b d Q - juT • tdT - J£TEFG • (u - u)dr

jYT.u/1)dT

nHBPM = j - £ T o d f i - JuT b d Q - JuT t d r + JVHBPIM - ( u -u )d r

-JyT-u/2)dr

(17)

(18)

In these variational formulations the domains of EFG and HBPIM are connected via Lagrange multipliers y.

In the EFG domain, u is given by the MLS approximation, y is given by interpolation functions A and nodal value y/ of the interface boundary

y=ATy, (19)

For A, PIM interpolants can be used. Using the stationary condition, the following EFG equations can be obtained

"•(EFG)

MT

1T1(EFG) n (EFG)

M(EFG)

0

0

"(EFG)

0

0

U(EFG)

^(EFG)

. "tl

• = •

'(EFG) + " ( E F G )

Q(EFG)

0

(20)

where H(EFG) is defined as

173

H(EFG) = jA*^E F G )dr (21)

Integrating the first term on the right hand side of equation (18) by parts, substituting equations PIM shape functions and using the stationary condition, lead to the following HBPIM (or HBRPIM) system equations

**-(HBPIM) **(HBPIM)

~ "(HBPIM) "

'(HBPIM) I

Y/ J *(HBPIM) + ^*(HBPIM) (22)

where H(HBPIM) is defined as

H = J A * ? W (HBPIM) _ J " • ^ (HBPIM) r,

dr (23)

Because two domains are connected along the interface boundary T/, assembling of equations of EFG and HBPIM yields a linear system of the following form

^•(EFG)

0

"(EFG) R T °(EFG)

K

0

(HBPIM)

0

H (HBPIM)

'(EFG)

0

0

0

(EFG)

(HBPIM)

0

0

U(EFG)

U (HBPIM)

1(EFG)

I T 1 J

. = <

'(EFG) + "(EFG)

'(HBPIM) + "(HBPIM)

1(EFG)

0

(24)

The coupling conditions are satisfied via the above technique.

Numerical example

As shown in Figure 3, a plate with a circular hole subjected to a unidirectional tensile load of 1.0 in the x direction is considered. Due to symmetry, only the upper right quadrant (size 10x10) of the plate is modeled as shown in Figure 1. When the condition bla>5 is satisfied, the solution of the finite plate is very close to that of the infinite plate. Plane strain condition is assumed, and £=1.0xl03, v=0.3. Symmetry conditions are imposed on the left and bottom edges, and the inner boundary of the hole is traction free. The tensile load in the x direction is imposed on the right edge.

The exact solution for the stresses of infinite plate is available Timoshenko and Goodier (1970). The exact solution for the stresses of an infinite plate is

a2 ,3 3a" c r(x, y) = 1—-{—cos26> + cos4#} +—-cos46> r2 2 2r4

(25)

2 i i 4

aJx,y) = —r-{-cos 20-cos 40} -cos 40 y 7 r2 2 ' 2rA

(26)

174

a1 A . 3a4

a O, y) = — j {- sin 20 + sin 40} + —j sin 40 r2"2 • 2r«

(27)

where (r, 0) are the polar coordinates and 0 is measured counter-clockwise from the positive x axis.

• 4 -

j

r k-

>y

">

*W

ttttt

Figure 3. A plate with a central hole subjected to a unidirectional tensile load

The nodal arrangements in HBPIM and EFG/HBPIM analyses are plotted in Figure 4 and Figure 5, respectively. In the EFG/HBPIM analysis, the plate is divided into two domains, where EFG and HBPIM are applied, respectively.

As the stress is most critical, detailed results on stress are presented here. The stress ax at x=0 obtained by the present methods are presented. The results obtained using HBPIM and EFG/HBPIM are shown in Figure 6. It can be observed from this figure that the present HBPIM and EFG/HBPIM yield satisfactory results for this problem considered. Compared with the domain-type methods, fewer nodes are needed in the HBPIM and EFG/HBPIM. The saving is considerable.

Remarks

A hybrid boundary point interpolation method (HBPIM) and the coupled EFG/HBPIM method are proposed and developed for solving boundary value problems of two-dimensional solid mechanics. In the HBPIM, the PIM is combined with the Hybrid displacement Boundary Formulation. The 'stiffness' matrix so obtained is symmetric. In HBPIM/EFG method, the compatibility condition on the interface boundary is introduced into the variational formulations of HBPIM and EFG using Lagrange multiplier method. The validity and efficiency of the present HBPIM and EFG/HBPIM are demonstrated

175

through the numerical example. It is found that the HBPIM and EFG/HBPIM are very efficient for solving problems of computational mechanics.

References

DeFigueiredo T.G.B. & Brebbia C.A. (1989). "A new Hybrid Displacement Variational Formulation of BEM for Elastostatics," In: Brebbia CA (ed) Advances in Boundary Elements vol. 1, 33-42.

Gu Y.T. & Liu G.R.(2002a). "A boundary point interpolation method for stress analysis of solids," Computational Mechanics, 27,47-54.

Gu Y.T. & Liu G.R.(2002b). "A radial basis boundary point interpolation method for stress analysis of solids," (submitted).

Liu GR(2002). MeshFree Methods-Moving beyond the Finite Element Method, CRC Press LLC, USA.

Liu G.R. & Gu Y.T.(2001). "A point interpolation method for two-dimensional solids," Int. J. Numer. Meth. Engng, 50, 937-951.

Mukherjee Y.X. & Mukherjee S.(1997). "Boundary node method for potential problems," Int. J. Num. Methods in Engng. 40: 797-815.

Timoshenko S.P. & Goodier J.N.(1970). Theory of Elasticity. 3rd Edition. McGraw-hill, New York .

176

% w i » i • » * • • » - • — > • • •

HBPIM

Figure 4. Nodal arrangement of HBPIM

Figure 5. Nodal arrangement of EFG/HBPIM

2.9

2.5

Str

ess

1.7

1.3

n Q

S s ^ ^

Analytical solution

. . . . HBPIM

EFG/HBPIM

10

Figure 6. The distribution of ax along the section of x=0

SECTION 9

Coding, Error Estimation, Parallisation

179


ERROR REGULATION IN EFGM ADAPTIVE SCHEME

W. Kanok-Nukulchai andX. P. Yin Asian Institute of Technology, Pathumthani 12120, Thailand

worsak(o),ait.ac,th

Abstract

In order to regulate and control error distribution over domain of the EFGM solution, a scheme for adapting the sizes of the domains of influence (DOI) and the integration cells is proposed in this paper. For 2D problems, the triangles established by Delaunay triangulation using nodes as vertices are adopted as integration cells with fixed number of integration points in each cell. To avoid numerical ill-conditioning and to improve computational efficiency, the size of DOI for any node is varied to cover a fixed number of cells surrounding the node. By adapting the sizes of background cells and DOIs based on nodal density, a higher degree of accuracy can be obtained where it is needed. New nodes can then be added into any region where accuracy is found to be inadequate. With the subsequent addition of nodes, the triangular background cells can be refreshed by a new round of Delaunay triangulation. The proposed scheme performs very effectively especially in conjunction with the proposed error estimator based on the local residual of the Galerkin weak form. The convergence and the effectiveness of the present scheme are confirmed in numerical tests.

Keywords: Element-Free Galerkin, domain of influence, integration cell, adaptive modeling, Delaunay triangulation

Introduction

Finite Element Method (FEM) is characterized by the interpolation of variable fields by a set of shape functions constructed over each element. Researchers find adaptive mesh generation inconvenient in FEM, particularly in problems having sharp sensitive areas and material discontinuities. Recently, various mesh-free and element-free methods have been developed to avoid the same limitations caused by the element structure in FEM. Among them, the Element-Free Galerkin Method (EFGM) proposed by Belytschko et al. (1994) employs the Galerkin weighted-residual formulation to obtain approximate solutions over the problem domain using moving least-squares - MLS (Lancaster et al., 1981). In this way, the requirement of an element mesh structure for constructing shape functions is removed. This is the key difference between EFGM and FEM.

In EFGM, the problem domain may be subdivided into a grid of integration cells in the background so that the weak-form Galerkin equation can be integrated numerically. Unlike FEM, the background cells do not serve as supporting structures for the definition

180

of shape functions. For this reason, nodes can be added, moved, or removed freely within the domain.

In practical engineering problems, field variables may vary sharply over some sensitive area within the domain, e.g., a round a small opening or crack. To achieve an accurate numerical solution under these conditions, a sufficient number of nodes should be allocated in this sensitive area. As EFGM employs MLS as approximate functions, the nodal domain of influence (DOI) plays an important role in determining the accuracy of approximation. As depicted in Fig. 1, if the DOI of a node is too large, MLS approximation may not sufficiently reflect the local characters of the problem as the solution tends to be smeared over a large area. On the other hand, the choice of a too small DOI may result in ill-conditioning of the system. Thus, the size of the DOI should be varied adaptively to cover a sufficient number of coupled nodes. Subsequently, the size of integration cells should also be refined accordingly. In this paper, a scheme is proposed to adapt the size of the triangular integration cells based on nodal density. In addition, the DOI of a node is varied to cover a sufficient number of nodes.

Why not uniform DOI?

- Too small DOI will not meet the visibility requirement, causing ill-conditioning

- Too large DOI covers too many nodes and cannot reflect locality effects.

Fig. 1. Adaptivity of nodal domain of influence (DOI) under consideration

The key aspect of adaptive modeling is a good estimator of discretization errors. An error estimator based on the projected stress has earlier been proposed by Chung et al. (1998). Recently, Gavete et al. (2002) introduced an error approximation by comparing the gradients calculated by the EFGM and by MLS using a Taylor series expansion. In the present paper, an a posteriori error estimator based on the canonical Galerkin weak form is proposed, both locally over individual integration cells and globally over the entire domain. A good estimation of local errors helps identify regions where new nodes should be added to enhance accuracy. In addition, a reliable estimation of global error allows users to gauge the overall accuracy of the current discretization. Several numerical examples are used to test the effectiveness of the proposed adaptive modeling scheme in conjunction with the newly proposed error estimator.

181

The Proposed Adaptive Scheme

Adaptive Integration Cells

Gauss quadrature is commonly used to evaluate the Galerkin residual equations over individual background cells. If solution gradient is typically high in certain part of the problem domain, it is desirable to allocate relatively more nodes in that region. This region of relatively high nodal density also requires highly accurate numerical integration. It has been reported (Dolbow et ah, 1999) that the accuracy of Gauss quadrature for a rational function, such as the MLS approximation function, has an upper bound even if the quadrature rule is increased. However, by dividing the domain into increasingly more cells, integration accuracy can be improved significantly even with a relatively small number of Gauss points in each cell. In this way, the background cells need to be adaptively regenerated when more nodes are added in a region where higher accuracy is desired.

The triangular integration cell is adopted in this study due to the ease with which triangular meshes may be created. Delaunay triangulation (Mern and Eppstein, 1992) is one of the most commonly used techniques for triangular mesh generation. In this work, integration cells are defined as straight-sided triangles using nodal points as vertices. A constant number of integration points are used in each integration cell. A typical triangular mesh of integration cells is shown in Fig. 2 with 13 integration sampling points in each cell.

Fig. 2. Proposed integration cells using nodes as vertices

182

Variable Domain of Influence

The accuracy of EFGM is sensitive to the size of DOI. For a circular DOI, its radius rd

must be large enough to guarantee a non-singular matrix A that is required to obtain the MLS shape functions. Since the rank of matrix A is m, where m is the number of monomials in the set of basis functions, at least m nodes should have an influence over any sampling point at which the shape functions or their derivatives are evaluated. The DOI can be varied in such a way that at least m nodes are visible from all sampling points. Additionally, certain nodal patterns, as discussed in Yin (2001), can result in an ill-conditioned A matrix even when more than m nodes are visible from a Gauss point.

In this paper, only circular DOIs are considered. The radius of the DOI of node i is defined as :

where rdi is the radius of the DOI of node i, fd is an amplifying factor for which the appropriate value is 1.5-2.0 (Yin, 2001), and d, is the longest distance from node i to the vertices of np layers of cells that surround node /, where np is the number of monomials in the set of MLS basis functions, as illustrated in Fig. 3.

1D

i J

: 1 fle

J^*tN^ • "IT't . . WiSS

Fig. 3. For linear basis function, the size of DOI is varied to cover 1 layer of integration cells surrounding the node, amplified by factor^

183

Weak Form Residual Error Estimation and Adaptive Discretization

Consider the following equilibrium equation and natural boundary conditions:

Vcr + b = 0 inQ (2)

a • n = T on Tt (3)

where a denotes stress tensor, bT the body force vector, Q is the problem domain, n is the outward unit vector normal to the boundary, and T is a prescribed traction on the natural boundary

The Galerkin weighted residual method employs the same space for the weight field as the trial functions. For 2D elastostatics, the domain and the boundary residuals are defined, respectively, as:

RD = Vo- + b in Q, and (4)

RB = r - a • a on Tt (5)

For an exact solution, these two residuals should be zero at any point in the problem domain as well as on the boundary. The residual error is minimized over the entire domain and boundary in the Galerkin weak form. In the EFGM, the total residual in the problem domain and its boundary, including RD and Rg, as well as the residual terms corresponding to the Lagrange multipliers, is forced to be zero. However, in an individual integration cell, the cell residual is not necessarily zero. It is evident that the overall accuracy of the numerical solution increases as the residual in individual cells decreases. Therefore, the size of the residual in a cell is a good indicator of the quality of the numerical solution near the cell, and consequently whether more nodes should be added around the cell. Residual in a cell is defined as:

REC = j r(Vo- + b)wfi?Q + J c(T-a-n)wdT (6)

where Qc is the spatial domain of the cell and T , c is the boundary of the cell that intersects the domain boundary. The second term reflects residual error on the natural boundary. Applying integration by parts leads to:

REC = | r<rnwdT-j raVwdQ + j bwdQ+j c (r ~ a • n)wdT (7)

where Tc is the boundary of cell. Eq. (7) can be rewritten as:

184

REC =j.ca- nwdT + j ^ rwdT -jac<J- VwdQ, + j a c bwdQ. (8)

where r\ denotes the part of the cell boundary other than iy. Since the weighted residual method can be enforced with any test function w, the unit weight function w^l is adopted in this paper. Thus, Eq. (8) becomes:

REC = j_e a • ndT + \_c zdT + J bdQ,. (9)

In Eq. (9), the term regarding the stresses within each cell has disappeared and only stresses on the boundary of the cell need to be calculated. The form of Eq. (9) is analogous to checking the force equilibrium of the cell. For the entire problem domain, the sum of all cell residuals defined in Eq. (9) should be exactly zero, except for the error arising from the use of Lagrange Multipliers. However, for individual cells, the cell residual is not zero until the converged state is achieved (each cell approaches a point). One can imagine that RE0 as an unbalanced force in the free body diagram of the cell. For 2D, the residual error of each cell defined in Eq. (9) can be calculated for the x and y directions, respectively. In order to consider the accuracy in both the x and y directions, the residual error in a cell, to be referred as the local residual error, can be written as:

LRE = ^RE:f+(<RE;?l\\F\\2 (10)

where bdQ is the total external force applied in the domain and on the boundary.

The mean residual error in each direction for all cells can be defined as the absolute sum of the residual error in the same direction for all cells. Thus the mean residual error over all Nc cells, or MRE, of the solution is the resultant of both components normalized by the external applied force, i.e.,

MRE = -

Z\"* N.

) 2 + ( ^ RE[

AT (11)

Adaptive EFGM Procedure

Combining the adaptive integration cells, DOIs and error estimation based on the weighted residual, an adaptive EFGM procedure, depicted in Fig. 4, can be performed as follows:

1. Input initial nodal definitions;

185

2. 3. 4. 5.

6. 7.

Establish triangular background cells by Delaunay triangulation; Perform EFGM analysis; Evaluate LRE (Eq. 10) for all cells; For any cell, if LRE > allowable error, then add 3 more nodes at mid-sides of the cell as shown in Fig. 5; If nodes have been added in any cell, refresh the background cell, goto Step 2; Else Stop Program.

1 NO

1 YES

si

f„C:i:j

i

i

Fig. 4. Diagram showing the algorithm of present adaptive procedure

(a) Initial nodal (b) Cells with (c) Adding nodes (d) Improved distribution excessive errors at cell mid-sides nodal distribution

Fig. 5. Nodes added at mid sides of cells, shaded in (b), found to have inadequate accuracy

Numerical Results

In the selected test examples, there is a specific region where stress gradient is sharply higher than the rest in the domain. Thus, it is desirable to have more nodes assigned in this sensitive area. The energy error norm, defined below, shall serve as a benchmark to confirm the effectiveness of the residual error used in the present study.

186

Energy Error Norm - 12 $n(e"-ee)D(ea -ee)dQ

(12)

- f e'Ds'dQ 9 Jn

In the following investigations, the MLS approximation functions are derived based on linear basis functions. Following the previous work [8], the size of DOI follows Eq. 1 where the magnifying factor,/i , is taken as 1.5. Numerical integration in all background triangular cells employs a fixed number of 13 integration points.

Example 1. A bar under high stress gradient due to point load

Stress distribution at 13 nodes

m 0.2 0.4 0.6 0.8 1.0

-EFGM — E x a c t

Stress distribution at 38 nodes

^ 0 . 5

« w » m - t « - « — o

0.5W 0.2 0.4 0.6 0.8 i

x EFGM — E x a c t

tZ5

Stress distribution after reaching 58 nodes

1 -o ®—-«—ooooo

0

-*— EFGM ——Exact

>̂'<H><MM> • • | 0 o — - ^

.0.5 OJO 0.2 0.4 0.6 0.8 1.0

x

Fig. 6. EFGM solutions capture stress jump around the point load

Consider a clamped bar under a patch of uniform axial load at the mid-span. If the width of the patch approaches zero at the limit, a point load situation can be simulated. This situation will allow a very high stress gradient on the boundaries of the patch. Thus, the stress profile is almost a step function. The main idea is to test the performance of the

187

present adaptive procedure. It is expected that the algorithm will add more and more nodes around the high stress gradient region until all cells satisfy the required accuracy.

Shown in Fig. 6 are EFGM solutions of an extreme case (Tin, 2002), where the width of the patch load is 0.01 of the span length. Following on the present algorithm, the nodal distribution pattern is gradually refined around the stress jumps, until a rich nodal concentration can meet the accuracy demand at the high stress gradient zone.

Square Patch Subjected to Quadratic Tractions

A square patch subjected to quadratic tractions, as shown in Fig. 7, is analyzed. The exact solutions for displacements and stresses can be expressed as:

2 3 XV UX

u = ^— + E 3E

(13a)

2 3

x y uy

~E~ ~TE

(13b)

<*u=y

( 7 , , = -X

Tn=0.

(13c)

(13d)

(13e)

Fig. 7. Square patch subjected to quadratic tractions

188

A relatively high gradient of stress is located near the top-right corner of the patch. Therefore, a higher density of nodes should be adaptively placed in this region, as compared with the bottom-left corner of the patch, to assure a uniform distribution of accuracy. Initially 9 nodes are assigned in the domain and eight triangular background cells are generated from Delaunay triangulation as shown in Fig. 8. The local residual error for any cell is limited by the allowable error of 0.003. When local residual errors of all cells are found to be all lower than 0.003, the refinement will be stopped. In this problem, the maximum local error was computed as 0.002957 when the number of nodes reached 759. The final nodal distributions and the 3 intermediate cases are shown in Fig. 8. The plot of maximum local residual errors was presented in Fig. 9 in comparison with the energy error norm. In Fig. 10, the mean residual error over all cells reduces sharply as more nodes are assigned where they are needed. The numerical results for all the 13 iterations are summarized in Table 1.

Based on the proposed adaptive EFGM procedure, the nodal distribution over the domain appears to correspond with the distribution of the stress gradient of the problem. The nodes are densely distributed near top right corner in both the last two meshes in Fig. 8. As more nodes are added, both the maximum local residual error and the mean residual error of cells decrease.

Fig. 8. Square plate under quadratic traction. An adaptive refinement of nodal discretization based on the present algorithm.

189

Max Local Error & Energy Norm

0.25

0.05

0.00

200 400 600 800 Number of Nodes

• Max. Local Error —•— Energy Norm

0.10 -|

irro

i

r\ o

rage

5

C

s.

C

Ave

3 S

0.00 -

Average Error

[

I

— r* * « » n nm

200 400 600 Number of Nodes

800

Fig. 9. Maximum local residual error Fig. 10. Mean residual error

Table 1. Numerical results for the square patch

Iteration

1 2 3 4 5 6 7 8 9 10 11 12 13

No. of Nodes

9 25 81

271 390 470 552 597 654 687 720 740 759

No. of Cells

8 32 128 428 705 862 1022 1111 1225 1291 1357 1397 1435

Max. Local Residual Error

0.159263 0.071205 0.020272 0.006613 0.005095 0.006368 0.004451 0.004875 0.006820 0.004690 0.003972 0.006945 0.002957

with quadratic tractions

Mean Local Residual Error

0.0942261 0.0247658 0.0052441 0.0011981 0.0009653 0.0009086 0.0008604 0.0008397 0.0007730 0.0007489 0.0007431 0.0007436 0.0006852

Energy Norm

0.191394 0.176028 0.070075 0.034004 0.028894 0.029037 0.028869 0.031420 0.029800 0.029964 0.030716 0.030910 0.031043

190

Square Plate with Center Hole

A 10 by 10 square plate with a circular hole of unit radius at its center is analysed to

777^79? TTY

Fig 11. Plate with a center hole

demonstrate the effectiveness of the proposed adaptive EFGM scheme. By taking advantage of symmetry, only one quarter of the plate is modelled, as shown in Fig. 8. The plate is subjected to a unit in-plane traction applied in the x-direction. Since the size of the hole is relatively small, the analytical solutions for stress and displacement of an infinite plate with a hole, given as follows, can be assumed as exact solutions for this problem.

<y„ = 1 — - — cos2# + cos4# +—-cos4# r2yi ) 2r4

(14a)

- %r\ - s in 20 + sin 40) + ̂ s i n 40 r2 {2 J 2rA (14b)

2 / 1 "\ 1 4 •-T- — cos20-cos4# -cos4<9 r2\2 ) 2r4

(14c)

l + v u = •

l + v -rcos0 +

2 a2

cos (9 + —cos 30 cos 30 1 + vr2 2r 2rJ

(14d)

l + v -v . n l-va2 . „ a2 . „„ a4 . „„ rsin6> -s in# +—sin 30 -sin 30

l + v l + v r2 2r 2r3

(14e)

191

where (r,d) are polar coordinates, and 6 is measured counter-clockwise from the positive x-axis. The purpose of this test is to assess the effectiveness of the present scheme in capturing the stress concentration in the vicinity of the hole. The allowable local residual error is specified as 0.001. Initially, 10 nodes are assigned and 9 integration cells are generated by Delaunay triangulation as shown in Fig. 12. More nodes are added after each iteration until all cell residual errors are less than the allowable error. At that state, the number of nodes reaches 603 and the maximum local error is computed as 0.000997.

Prescribed maximum ceil residual < O.OOl

Fig. 12. Square plate with center hole. An adaptive refinement of nodal discretization based on the present algorithm

The nodal distributions and background cells at four intermediate stages are displayed in Fig. 12. Nodes appear to be allocated densely around the hole, where high stress concentration is expected. As shown in Figs. 13 and 14, the maximum local residual error and the mean residual error decrease sharply as more nodes are added. The convergence of the normal stress in the x-direction above the hole, as shown in Fig. 15, illustrates the effectiveness of the proposed adaptive procedure.

192

Max. Local Error & Energy Norm 0.35

S 0.30 w 0.25

- Mxa. Local Error

200 400 600 800

jSTumber^of Nodes

• Energy Normr

Fig. 13. Maximum local error

Average Error

0.05

v. 0.04 f o £ 0.03

0.02

< 0.01

0.00 >» i m —

200 400 600

Number of Nodes

800

Fig. 14. Average error of cells

trn

1 1.5 2.5 3 3.5 4 4.5

y Coordinate (Position)

--<>.-- 9Nodes - - n - - 200Nodes —*—521 Nodes ---©•-- 603Nodes —x—Exact

Fig. 15 (Tu along left boundary (x=0) with increasing number of nodes

193

Conclusions

This study investigates the effectiveness of employing a new error estimator based on the local weak-form residual to identify regions of inadequate accuracy, where nodes should be added. Based on this new error estimator, a completely automated adaptive EFGM procedure can be achieved. In this procedure, integration cells are constructed in the background as a network of triangles that employ nodes as vertices using Delaunay triangulation. The nodal DOIs are allowed to vary to cover the same number of cells around each node. Test examples confirm that with the present scheme, one can achieve an ideal situation where a specified level of accuracy can be maintained rather uniformly over the problem domain.

References

Belytschko, T., Lu., Y.Y. Lu, and L. Gu, L. (1994), "Element-Free Galerkin Methods," Int. J. Num. Meth. Eng., 37, 229-256.

Chung, H..J. and Belytschko, T. (1988) Belytschko, "An error estimate in EFG method," Computational Mechanics, 21, 91-100.

Dolbow, J., and Belytschko, T. (1999), "Numerical integration of Galerkin weak form in meshfree method", Computational Mechanics, 23, 117-127.

Gavete, L. Gavete, Cuesta, J. L. Cuesta, and Ruiz, A. Ruiz,(2002) "A procedure for approximation of the error in the EFG method," Int. J. Num. Meth. Eng., 53, 677-690.

Lancaster, P. and Salkaushas, K. (1981), "Surfaces Generated by Moving Least Squares Method," Mathematics of Computation, 37,141-158.

Mern, M., and Eppstein, D. (1992), "Mesh Generation and Optimal Triangulation", Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, Vol 1,23-90., World Scientific, Singapore.

Tin, S. S. (2002), Error estimator for an adaptive scheme of EFGM, Master Thesis, School of Civil Engineering, Asian Institute of Technology, Thailand.

Yin, X. (2001), An Enhancement of the Element Free Galerkin Method, Master Thesis, School of Civil Engineering, Asian Institute of Technology, Thailand.

194


OBJECT ORIENTED DEVELOPMENT OF F M M 3 D : FOUNDATION S O F T W A R E F O R P A R A L L E L 3 D F R E E M E S H M E T H O D

Yutaka Nakama, Akio Shimada Fuji Research Institute Corporation, Chiyoda-ku, Tokyo, Japan

[email protected]. co.jp, [email protected]. co.jp

Yasuhiro Kanto Toyohashi Univ. of Tech., 1-1 Tempaku-cho, Toyohashi, Japan

kanto@mech. tut. ac.jp

Tomoaki Ando Advanced Simulation Technology of Mechanics R&D Co, Ltd., Japan

andou@astom. co.jp

Genki Yagawa University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan

yagawa@q. t. u-tokyo. ac.jp

Abstract

Free Mesh Method (FMM) is a kind of node-based finite element method, which does not require mesh information. Therefore it has a main merit of meshless method; reduced cost of calculation by omitting mesh creation. Then it seems suitable for an adaptive method, a crack propagation problem, and a large-scale problem. On the other hand, it still has merits of finite element method, i.e., it can use many existent techniques developed for finite element method. Although FMM has been demonstrated to be applied to many fields of research, no general purpose program based on FMM has been developed yet. So we designed " FMM3D " ; Free Mesh Method program for 3D problem. Some of main features of FMM3D are: (1) Large-scale: It can be executed in parallel environment, such as PC cluster with domain decomposition method (DDM). (2) Expandable: It is designed by object-oriented approach and new functions can be added very easily. Here we describe the object-oriented design of FMM3D, generalized recursive bisection method for DDM, and object-oriented parallel matrix solver.

Keywards: Free Mesh Method, FMM, Domain Decomposition Method, OOP, Parallel Computing.

Introduct ion

As the latest technological innovation advances, it is required to analyze complex 3-dimensional

problems and large-scale problems efficiently. The former Finite Element Method (FEM) was

not fully practical, since enormous human working hours were necessary in mesh generation for

complex 3-dimensional problems, and a big amount of computer resources was needed in terms of

memory and calculation time for large-scale problems. As a solution for those problems, research

on Free Mesh Method (FMM) is carried out as one of the methods in which users do not require

mesh generation (G.Yagawa and T.Yamada,1996). Also research on Parallel Computing based

on Domain Decomposition Method (DDM) is carried out, for large-scale problems in order to

reduce memory and calculation time. Then, we have developed a basic software FMM3D based

on Free Mesh Method using DDM. FMM3D has such characteristics; no need of node and mesh

generation, applicable to large-scale for both of distributed and shared memory type parallel

computer as well as cluster environment connecting EWS.

In addition, we have provided a mechanism in the software, which enables users to integrate

various analyses easily, by applying object-oriented approach to above two research results. By


http://co.jp

http://co.jp

http://ac.jp

http://co.jp

http://ac.jp

195

using FMM3D, developers can easily integrate original analysis functions (e.g. fluid analysis

etc.) without changing existent parts of the program. Moreover, it enables developers to build

a parallel program automatically without knowledge of parallel computing.

S u m m a r y o f F r e e M e s h M e t h o d

Free Mesh Method is a kind of meshless methods based on node-based Finite Element Method,

Because the calculation is performed by a node-by-node style, we focus on the current node

denoted as the central node (I) in Fig. 1. A local mesh around the central node is generated by

gathering satellite nodes (m-r), which are selected among the candidate nodes inside the local

region. Element stiffness matrices are made by usual FEM procedure for the local elements,

and only the row components corresponding to the central node are added to the global stiffness

matrix. This process is iterated for entire nodes to obtain the global stiffness matrix.

local element / .candidate node

satellite node

center node

'> local region

Figure 1. Local Mesh

Analysis Condition

remarks

a input file

Program

flow of data

Condition «rttJr>g>" , ' ^ M Q & j i f c

Model . Bdudary o - da tas * Materials

Node > H N o d 8 9 O T S f a i H Visualizlno d e n s i t y ' ' ' ' • I V V " V ' " ; ' ' ' • ' " ' ' ' '•'•'

Rgute Boudary condition

•rials k ^ t d : Ka°^,d,na,.r5a£v55r [ $dlVBf \-—•

How to Visualize

Visualization

Figure 2. Diagram for FMM3D

S u m m a r y o f F M M 3 D

FMM3D can be divided into four parts; the analysis condition setting program, the node gener

ation program, the solver program, and the visualizing program as shown in Fig. 2. Acceptable

input formats are IGES form, VRML form and other forms as solid model (surface model data)

for FMM3D.

The analysis condition setting program

Analysis conditions (boundary condition, material condition) are set up for solid model in the

analysis condition setting program. Since surface model has to be treated as a form (figure) in

FMM3D, if the input model data is IGES format, surface patch is generated by dividing NURBS

surface into triangles using Advancing Front Method (P.J. Frey et al., 1996). And smoothing

processing is done by Laplacian method.

Boundary condition can be set on the plane of surface patches, ridgelines, and apexes (nodes),

and material information can be set up for each surface patch.

The node generation program

FMM3D does not require node information, either. The node generation program automatically

generates nodes on the surface and inside the solid model satisfying the given node density

condition. The nodes are generated on ridgelines at first, on surface patches next, and inside of

196

the solid model defined by surface patches at last. The program supports models with several

closed regions. DThc actual procedure is explained as follows.

1. Cluster of additional candidate nodes should be generated for sufficiently narrow interval

calculated from the given node density.

2. In-out judgement is carried out for all candidate nodes by counting the number of crossing

points of a ray from the candidate node across the surface patches.

3. Calcutatc the intervals between a candidate node and the adopted nodes. When the all

intervals between adjacent nodes are wider than that satisfying the node density conditions,

the candidate node should be adopted.

Besides, by using the bucket method, calculation cost for node generation such as In-out judge

ment can bo reduced.

The solver program

( l ) D D M o f F M M 3 D

In order to realize high efficient parallel processing, it is significant to equalize processing amount

of each P E and to reduce the amount of data transfer. In FMM3D, Recursive Bisection Method

(RBM), one of the DDMs. is used to equalize the number of nodes included in each region,

and to reduce the amount of data transfer between PEs by reducing the area of the domain

cross-sections.

In RBM, a domain is divided into two parts by cutting across the longitudinal direction as the number of nodes in each part becomes equal. This process is iterated until required decomposed region number (equal to the number of PEs) is obtained. Usually number of PEs is restricted to 2 n , however, FMM3D can decompose a domain into any number of PEs by applying the following algorithm itcrativcly. "When each domain needs to be decomposed into m subdomains. dividing should be done retaining the ratio of nodes, m\:m.2 {mi = [m/2],mi +m.2 = m) " Figure 4 shows an example of 7 subdomains of a 3D-cube.

(2) Local Mesh Generation

In Free Mesh Method, it is needed to generate temporary local mesh in local region. In FMM3D,

in order to generate local mesh efficiently, the radius of local region is determined from the node

density, andidatc nodes arc chosen, and Dclaunav tossclation is applied to the central node and

candidate nodes.

However Dclaunay tossclation for local clement generation would cause nonuniqucness of local elements. Figure 5 shows one example in 2-D case. Dclaunay tossclation requires the condition that no other node exists inside the circumcirclc of each clement. But when more than three nodes lay on the same circle (shrinking), the elements composed by those nodes cannot be determined uniquely. For example, four nodes A to D arc the same for Case A Figure 5. nonuniqucness of local and B, but Dclaunay tossclation would generate different clement generation patterns of mesh as shown in Fig. 5.

• . ' ^ • / t

YJb&^

Figure 4. Decomposition of 3D-cubc

197

In mesh generation for FEM, this kind of problem never occurs with Dclaunay tcssclation, as it

is carried out with all nodes in whole domain at the same time.

In the local mesh generation process of FMM3D, the following limitation enables to retain

uniqueness of local elements.

• Give unique global numbers to all nodes. When generating local mesh by Dclaunay tcssc

lation, nodes arc added in the order of the numbers.

• While Dclaunay tcssclation, in case that a new-added node is on the circumcirclc of a

clement, new two elements should be regenerated after deleting the old one.

This local mesh generation method used for FMM is suitable for parallel processing. During

local mesh generation, only the coordinates information of local nodes arc needed, but no data

transfer is required.

(3) Matrix solver and it's Parallelizing

In FMM3D, as the analysis scale becomes enlarged, preconditioned iterative method is employed

due to its smaller memory requirement, less amount of operation, and less error accumulation

than other methods. The matrix is a symmctic matrix, which was obtained by dispersing

static structural analysis problem and heat transfer analysis problem with FMM. Therefore,

preconditioned CG method, an iterative method for symmetric matrix is employed here. There

arc two types of preconditioning to accelerate convergence of CG method. Though the most

popular preconditioning method for symmetric matrix is Incomplete Cholcsky Factorization, it

is hard to apply to parallel computation with commonly used algorithm. Therefore, in FMM3D,

the method approximating the inverse matrix by polynomial approximation is adopted, so that

the forward and backward sweeps arc not required as the preconditioning method.

(4) Matrix solver class

The matrix solver developed here is implemented by object oriented approach, in which the solver function is included as a method. It is also implemented such that the solver class of parallel version inherits form the matrix solver class of single CPU version.

Since the interface is the same to the single and parallel version of the CG Solver class, the modification of analysis part, such as structural analysis or heat transfer analysis, from single version to parallel version would be minimized. The interface, or the set of the methods, is summarized in Table 1.

Object-Oriented D e s i g n

Class diagram

The solver program of FMM3D is designed by object-oriented approach, so that the function

extension of the program is easy. The class diagram of the solver program of FMM3D is shown

in Fig. 7.

Table 1. Methods in CG Solver class

Method minit(N,F) mclearQ maxc(N) resd(R) setk(ij,R) setf(i,R) addk(ij,R) addf(i,R) dirc(i,R) msolve() geta(i) setpe(n.i)

Function Initialize (N:node count, Frdeg. of freedom) zero clear of A and f. Set max iteration count to N. Set residual to R. Set i,j-element of A to R. Set i-th element of f to R. Add R to i,j-element of A. Add R to i-th element of f set Dirichlet condition (modify A and f) solve and calculate x. get i-th element of x Set PE number to i. (for parallel calc.)

198

How to expand the solver program of FMM3D

Since FMM3D uses C + + as its main language, "inheritance" mechanism is applied for the

function expansion. In the first-dcvclopcd version, only static stress analysis and static heat

transfer analysis arc implemented. If wc want to add other analysis functions such as dynamic

analysis or fluid analysis, wc should add a few inheriting methods to some classes; the Analysis

class, the Element Equation class, the Material class and the Boundary condition class denoted

by screened boxes in Fig. 7. Any line is not necessary to be edited in the existing source code

of FMM3D. Table 2 summarizes required modifications for the function expansion.

FMM3D uses MPI for message passing and the source code of the parallel part is completely

independent of any classes related with the analysis function. Therefore, if wc add the inheriting

functions of new analysis according to the expansion method of FMM3D, wc can get both of

the single CPU version and the parallel version of the program.

L h^itogtemflS;::

BSBrtfigll

> - < ^ FMM-Paiallel

^M Analysis

Region Matrix

T Parallel Mit, Solver

Decomposition k

3 Element Matrix

IX \

Local Mesh |-— Patches

^ ^ X ~ Element Mesh Gene.

_L ~L_

. i^K*l t t i^ i r« i ( j j [jjlm: Mafrfftermal);: j

X X Hi

x X |.MMeiiii:<S!?«««);l jMjiaMjiSaBjBlj feJ!*PS§S#;i t ^ ^ S K H g g s g t e

Q30rd,nary Class [X ! Analysis-dependent-subclass.— Reference (pointer;

Table 2. Additional source code

Class Boundary Cond. Material

Element Matrix Analysis

Additional source code the part of reading boundary conditions from file the part of reading material properties from file the part of making the element matrix some parts such as iteration (in case of non-linear analysis and dynamic analysis)

Figure 7. Class diagram

Parallel Efficiency of the Solver Program

Parallel efficiency for static stress analysis with the solver program of FMM3D is shown in Fig.

8 for the analysis model of 9500 nodes. Speeding factor is plotted as a function of the number

of CPUs. The ideal value is shown by a solid line and the results of FMM3D arc shown by

x-marks for 1, 2, 4 and 6 CPUs. Wc can sec the parallel efficiency of the program is very high

from the figure.

Figure 8. Parallel Efficiency

199

Analysis procedure for FMM3D

FFMM3D does not require mesh and node data as input data, but it requires only the solid model and the boundary conditions. The analysis procedure of FMM3D is described as follows.

1. First, the solid model is created by some CAD program and saved by IGES or VRML

format. Figure 9 (1) is an example of a cube model.

2. Secondly, the boundary conditions are set up. The input file of boundary conditions is

created with a text editor. Figure 9 (2) shows an example of the boundary conditions; fix

displacement on the back surface and loads on the front surface

3. Next, the nodes are automatically generated according to the given node density, asn

shown in Fig. 9 (3).

4. At last, FMM3D solves the problem and put the result in VRML format. Figure 9 (4)

shows a displacement diagram as a result.

Figure 9. Procedure for static stress analysis with FMM3D

Conclusion

Object-oriented design and implementation of FMM3D are described briefly. The software does not require mesh data and even node data as input data, and it can deal with arbitrary numbers of PEs for parallel calculation with modified RBM. The main part of the basic software is made with object-oriented approach in order to expand it for various problems easily. In addition, developers does not need to consider parallel processing because the part of parallel calculation is completely independent of function expansion of analysis.

Acknowledgement

FMM3D is based on the software developed under The Development of Computer Aided Engi

neering Software by Free Mesh Method with Domain Decomposition Method in 1998, supported by Information-technology Promotion Agency, Japan (IPA). The authors acknowledge IPA and the researchers who participated in The Research Society of Free Mesh Method.

References

G.Yagawa, T.Yamada, " Free Mesh Method : A new meshless finite element method", Computational Mechanics,18, pp.383-386,(1996).

P.J. Frey, H. Borouchaki, and P.L. George., "Delaunay Tetrahedralization Using an Advancing-Front Approach", 5th International Meshing Roundtable (IMRT'96), Sandia National Labs., Pittsburgh, Pennsylvania, pages 31-43. 1996.

200


A N A P P R O A C H FOR N O D A L SELECTION IN M F R E E 2 D

G. R. Liu1'2, Edgar Frijters1 and Y. T. Gu1

' Centre for Advanced Computations in Engineering Science(ACES) Department of Mechanical Engineering, National University of Singapore

E-mail: mpeliusrffv/tus. edit.sg; [email protected] SMA Fellow, Singapore-MIT Alliance

Abstract

The software package MFree2De is developed by ACES to solve two-dimensional elastostatic problems using meshfree methods. The first version of MFree2Dc is based on the Element Free Galerkin (EFG) method, which uses so called domains of influence to select nodes for interpolation. This paper reports the new inclusion of the radial point interpolation method (RPIM) into this package. At the same time, an approach is proposed to select nodes based upon the triangular background cells, which is used to perform the background integration in MFree2Dc. The new approach is applied to both EFG and RPIM. Numerical examples are shown to demonstrate the efficiency and accuracy of RPIM and the proposed nodal selection procedure. It is found that the present approach can obtain good results. However, although the new selection procedure can work well for most cases, it may lead to less accurate solutions for some cases when the nodal distribution is very irregular.

Keywords: Meshfree method, Meshless Computational mechanics, MFree2D, Stress analyses.

Introduction

MFree2D® is a software package developed by researchers in the centre of Advanced Computations in Engineering Science (ACES), Singapore. This package uses meshfree methods for adaptive elastostatics analyses for two-dimensional (2-D) solids. The method used in the earlier version of MFree2D® is the Element Free Galerkin (EFG) method which was firstly proposed by Belytschko et al. (1994).

The Radial Point Interpolation Method (RPIM) is an efficient meshfree method which originally developed by Liu and his co-workers (Wang and Liu, 2001; Liu and Gu, 2001). The most important feature of RPIM is that RPIM shape functions possess the delta function property. Hence, boundary conditions can be enforced as easy as in conventional FEM. RPIM is now newly added in a test version of MFree2D°. In this paper, the performance of RPIM in MFree2D® is examined.


201

One of the important features of meshfree methods is that the interpolation is based on a group of nodes selected in the local support domain or based on the concept of domain of influence (Liu, 2002). When too few nodes are used for interpolation, the approximation can be inaccurate. If too many nodes are used, the computational cost will be too high. Hence, it is very important to develop an efficient nodal selection method in the development of a meshfree method. In the original version of MFree2D° using EFG, the nodal selection is performed based on domains of influence. Although this method is efficient and easy to use, it cannot always ensure to select a good nodal distribution for the interpolation for some special cases. In this paper, an alternative nodal selection procedure, the background cell based method, is presented. It automatically and efficiently selects nodes for the interpolation making direct use of the background integration cells. Numerical examples demonstrate that this new nodal selection method is efficient and robust.

Basic equations of elastostatics and Standard variational formulation

Consider two-dimensional elastostatics. The governing equation is given by the following standard partial differential equation:

Vo + b = 0 o n n (1)

in which a is called the stress tensor corresponding to the displacement field u = (u, v) r ; b represents the body force vector. The adequate boundary conditions for this problem have the following standard form:

a • n = t on the natural boundary r, (2)

u = u on the essential boundary r„ (3)

where n stands for the outward unit normal and t and u stand for prescribed values on the natural and essential boundaries respectively. The standard Galerkin weak form of this problem can be given in the form of potential energy.

n = J£TD£dQ - juTbdQ - j*uTtdr (4) n « r,

In this formulation, e is the strain vector and u is the displacement vector. The system equation can be obtained by setting the variation of n to zero.

Approximation schemes

In order to approximate the displacement vector u(x), several schemes can be used. The general form of the approximant uh(x) is given as:

m

u*(x) = I / , (x)*,(x) = f » a ( X ) (5)

202

When the EFG method is employed, the vector f (x) consists of monomials (polynomial terms) and the coefficient vector a(x) is determined by using the well-known Moving Least Squares (MLS) approximation (Belytschko, et al.). In the case of RPIM, f (x) consists of radial basis functions. In RPIM, a(x) is determined by forcing the interpolant to pass through all the nodes used for the interpolation.

We can obtain the following expression for the meshfree approximant:

U*(X) = O T ( X ) U (6)

where O(x) is called the shape function. A detailed discussion of meshfree approximation techniques can be found in the book by Liu (2002).

Selection procedures

In this section, two different procedures for nodal selection will be discussed. The first method makes use of the domain of influence based on the influence radius. This method was originally used in EFG of MFree2D°. This nodal selection method indirectly uses the background cells to determine the influence radius. The second nodal selection technique, however, makes direct use of the background cells. This method is suitable for meshfree methods which are based on the triangular background cell for integration.

Nodal selection based on influence domain

Using this technique, each node in the problem domain is assigned a local domain of influence (Liu, 2002). The domain of influence is constructed using a pre-defined influence radius for a given node. When doing interpolation for a point, it is simply checked for all the nodes in the problem domain whether the given point is within the domain of influence. In more popular words, if a node has an influence on this point under consideration, the node is used in the interpolation for this point.

The influence radius for each node is computed by using an average 'element area' (Liu, 2002). In formula, the influence radius for node i can be given as:

r,=ar^2Ai (7a)

where: l " A

in which a, is the factor, A • is the area of y'th triangle where node i is a vertex, and n is

the total number of triangles of which node i is a vertex.

203

Nodal selection based on triangular background cell

We note that the background cells (triangles) are used in MFree2D® for integration. Hence, the information of the background cell can be used to select nodes. This method makes direct use of the triangular background cells and the selection procedure consists of two steps:

Step 1: Consider Figure 1 (a). In the case of a Gauss point, three vertices of the triangle, in which the Gauss point is situated, i.e. nodes 1, 2 and 3, are firstly selected. For a field node, shown in Figure 1(b), this node is also vertices for surrounding triangular cells. Hence, the node itself and all 'outer' vertices of these triangles, of which the node is the vertex, are selected: i.e. nodes 1-7 in Figure 1(b).

Step 2: This selection step is to select all 'outer vertices' of surrounding triangles until a minimum number of nodes has been selected for interpolation. 'Outer' is referred to those nodes that are vertices of triangles one of whose vertices have been selected. For a Gauss point in Figure 1(a), nodes 4-10 are then selected. For the field node in Figure 1(b), nodes 8-19 are selected. When the minimum number of nodes for interpolation has been selecetd, the procedure is ended. Hence, the flnial nodes selected for interpolation of the Gauss point is: 1-10, and for the field node is: 1-19.

(a) Gauss point (b)Node

Figure 1: selection using surrounding triangles

It should be noted here that in order to save computational cost, as less nodes as possible should be used for interpolation as long as the interpolation accuracy can be ensured. Instead of selecting all outer vertices after selecting the initial three nodes, only a maximum less than ten nodes from surrounding triangles are selected. Those nodes,

204

which are to be selected for expansion, are firstly obtained from the 'neighboring' triangles that share the common edge with the initial triangle. For example, in Figure 1(a), nodes 5, 7 and 10 are selected. And then, several other nodes, which have shorter distances, will be selected until the maximum number reaches. In general, it is expected that most of the Gauss points will have 6-10 nodes in the support domain. When the initial triangle is located on a boundary, there are 4-6 nodes will be selected for interpolation.

Numerical examples

Cantilever beam

/

A /

D

Parameters: L = 48 D = \2

-+x £ = 3.0xl07

p v = 0.3 h : H P = 1000

Figure 2: A cantilever beam subjected to a parabolic traction at the free end

RPIM and the nodal selection procedure are tested using a cantilever beam problem, as shown in Figure 2. The beam has length L and height D and is subjected to a parabolic traction P at the free end. The beam has unit thickness. A plane stress problem is considered. The analytical solution can be found in a textbook by Timoshenko and Goodier (1970). The following error indicator has been used:

num exact \\ (8a) \\£exact ||

in which

^ (8b) H=^pw« re is an energy norm. The results of re are obtained and given in Table 1.

It can be seen from Table 1 that RPIM leads to good results using both two nodal selection methods. Comparison between these two nodal selection methods, though the background cell based nodal selection method can obtain good results for both EFG and

205

RPIM, the accuracy for EFG and RPIM is different. In RPIM, this nodal selection method gives better accuracy than the influence domain based nodal selection method. It is because the background cell based nodal selection method can select good distributed nodes for interpolation. However, in EFG, this method leads to worse accuracy than the influence domain based method. It is because the weight function chosen in the former method cannot always be "compatible" with the nodal distribution in the support domain for interpolation.

The average number of nodes used in a support domain for interpolation is also obtained and listed in Table 1. It can be found that the background cell based nodal selection method usually uses fewer nodes for interpolation. Hence, it can save computational cost of interpolation.

Table 1: Relative error for the cantilever beam.

Methods

EFG-influence

EFG-triangle

RPIM-influence

RPIM-triangle

re

0.0268

0.047452

0.037572

0.028155

Average number of nodes in a support domain

15.5

11.4

14.6

12.7

Plate with an infinite hole

In this example, a plate with a central circular hole subjected to a unidirectional tensile load of 1.0 in the x-direction is considered. Due to symmetry only the upper right quadrant of the plate is modeled, as shown in Figure 3. Plane strain condition is assumed. Symmetry conditions are imposed on the left and bottom edges. The inner boundary of the hole is traction free.

Stress <TM at x=0 is investigated and plotted in Figure 4. In this figure, the influence domain based method is used in EFG and the background cell based method is used in RPIM. EFG and RPIM are both compared to the exact solution. It can be found that both methods lead to very good results for this problem.

The background cell based method is also used in EFG. It is found that the accuracy of this method is worse than that of the influence domain based method. It is because that the nodal distribution is very irregular in the plate problem. The weight function chosen in the background cell based method is difficult to be "compatible" with the nodal distribution in the support domain.

206

Discussion and conclusions

In this paper, the radial point interpolation method (RPIM) is added in MFree2Ds>. In the meantime, a background cell based nodal selection method is proposed. The approach is applied to EFG and RPIM. Numerical examples are shown to demonstrate the efficiency and accuracy of RPIM and the proposed selection procedure. It is found that the new approach works quite well and can save some computational cost. However, it leads to less accurate solutions for some cases when the nodal distribution is very irregular. Further investigation still needs to be performed on this new method in order to improve the efficiency and accuracy of this method.

References

Belytschko T., Lu Y.Y. & Gu L. (1994). "Element-free Galerkin methods, " Int. J. Numer. Methods Engrg., 37, 229-256.

Liu G.R. (2002). Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press LLC, USA.

Liu G. R. & Gu Y. T. (2001). "A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids," Journal of Sound andVibration, 246(1), 29-46.

Wang J.G. & Liu G.R. (2001). "A point interpolation meshless method based on radial basis functions," Submitted.

Timoshenko S.P. & Goodier J.N. (1970). heory of Elasticity, 3rd Edition. McGraw-hill, New York.

207

-

^

+-

^

i

r

^

>

J la

^

^

— •

_ •

Figure 3: Plate with an infinite hole

3

y

Figure 4: Stress crxx at JC=0 for the plate

SECTION 10

Meshfree Particle Methods

211


COUPLING MESHFREE PARTICLE METHOD WITH MOLECULAR DYNAMICS NOVEL APPROACH FOR MULTISCALE SIMULATIONS

M. B. Liu, G. R. Liu Center for Advanced Computations in Engineering Science (ACES),

Department of Mechanical Engineering, National University of Singapore, Singapore [email protected], [email protected]

K. Y. Lam Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn

Singapore Science Park II, Singapore 117528 lamky@ihpc. nus. edu.sg

Abstract

One of the major outstanding challenges in the computational science is to provide a systematic frame, which bridges the gap between nano, micro, meso and macro scales for physics on multiple scales. In this paper, a novel approach for coupling length scale (CLS) is presented by combining smoothed particle hydrodynamics (SPH) method with molecular dynamics (MD). The molecular dynamics is applied to the atomic-sized regions for accuracy, whereas the meshfree particle method is applied to other peripheral regions for efficiency. Handshaking MD/SPH is implemented by using transitional SPH particle or virtual molecules, which interacts with both SPH particle and MD molecules. A preliminary numerical example is presented, which shows the validity of the novel CLS approach for simulating multi-scale physics.

Keywords: Coupling Length Scale (CLS); Molecular Dynamics (MD), Smoothed Particle Hydrodynamics (SPH).

Introduction

Coupling length scale is very important for multiple scale physics especially for problems related to nano science and technology. The rational design of nano or micro (fluidic) devices requires theoretical and computational tools that span from nano scale to micro, meso and sometimes even macro scales, accounting for the different phenomena that dominate. How to effectively simulate the physics for different scales related to nanoscale flows in micro devices is one of the toughest tasks and also one of the hottest topics in nano science and technology. Modeling the nano and micro structure and fluidics with molecular dynamics (MD) is prohibitive since atomistic MD simulations are limited to very small length scale (order of 1 angstrom) over very short times (order of 1 femtosecond). Application of macro continuum numerical methods such as FEM and FDM is invalid for atomistic regions due to the invalid continuum assumptions. Coupling



212

atomistic molecular dynamics with continuum methods is a good approach for this multiple scale computation.

Different from the former pioneering work that couples atomistic with FEM and FDM (Broughton, 1999; O'Connell and Thompson, 1995; Hadjiconstantinou, 1999), in this paper, a novel approach is presented by coupling meshfree particle method with molecular dynamics (MD). A preliminary numerical test on the classical Poiseuille flow is presented to show the validity of this CLS approach.

Molecular dynamics

Molecular dynamics is a well-developed atomistic simulation method with various applications (Rapaport, 1995). The molecular dynamics is to be employed in the nano sized regions where there are usually large gradients in field variables. In MD, the time evolution of interacting particles can be determined by using an interaction potential once the initial conditions are set. The evolution of the system in time can be followed by solving a set of classical equations of motion (i.e. Newton's law of equation). The force is to be calculated from the interaction potential, which is a function of the atom positions. Once the initial positions and velocities of all atoms are defined, the equations of motion can be numerically solved as an initial value problem. The thermodynamic and transport properties can be obtained from the direct results of atom position and velocity as a function of time.

In our work, liquid argon (Ar) is employed as the fluid simulated. The interaction between the molecules of liquid argon is described by the Lennard-Jones (LJ) potential

«(r,) = 4« [ ( - ) , 2 - ( - ) 6 ] . rvZrc (1)

where ^ is the distance between two fluid molecules i and/ a denotes the characteristic length describing the range of inter-molecule force; e defines the characteristic energy governing the intensity of the molecular interaction. The LJ potential features a strongly repulsive core arising from the nonbonded overlap between the electron clouds and the attractive tail representing van der Waals interaction due to electron correlations. In the calculation, the interaction potential is truncated at rc =2.5crto reduce computational time. The initial molecular velocities are randomly assigned with the appropriate Maxwell-Boltzmann distribution for the given temperature. The initial molecules are positioned on an FCC lattice with its spacing chosen to obtain the desired density. Integration of the equation of motion gives the positions and velocities at the next time step until the system reaches the equilibrium or stable state.

Smoothed particle hydrodynamics (SPH)

In recent years, the meshfree methods have been highlighted as a class of next generation of computational techniques. Meshfree methods were originally intended either to modify

213

the internal structure of the grid based methods (e.g. FDM and FEM) to become more adaptive, versatile and robust, or to characterize physical circumstances where the main concern of the object is a set of discrete physical particles rather than a continuum. There are many kinds of meshfree and particle methods (Liu, 2001). The smoothed particle hydrodynamics (SPH) (Gingold and Monaghan, 1977) has been widely applied to different areas due to its combination of meshfree, Lagrangian and particle nature. In our work, this meshfree particle method is employed to simulate macro areas where the classical continuum assumptions apply.

The SPH method employs particles to represent fluids. In interpolation, the SPH method uses integral function representation (Liu, 2001). The SPH particles, on the one hand, carry material properties and move with the fluid flow, one the other hand, act as interpolation points, which form the computational frame for solving the partial differential equations (PDEs) governing the continuum fluid dynamics. The integration of the multiplication of a function and the smoothing function gives the kernel approximation of the function, while

Fig. 1 Illustration of SPH approximation summing over the nearest neighbor particles yields the particle approximation of the

function at a certain discrete point or particle. For a function/, its kernel approximation at a certain position x, denoted as <f>, is

< Ax) >= \Ax')W(x -X', h)dx' (2)

where W(x- x',h) is the smoothing function with influencing area of Kh . h is the smoothing length, K is a scale factor depending on the smoothing function. The corresponding particle approximation is

</;>=2>,M)/,^ (3)

where, m; and pt are the mass and density of particle,/' . iVis the total number of particles. Similarly the kernel and particle approximation for the gradient of/are then:

< V • Ax) >= $ Ax') -VxlV(x-x', h)dx' (4)

<VX>=X('";//';)/;'V,^ (5)

214

Approximation of higher order derivatives can be carried out by nested approximations on lower order derivatives. Substituting the SPH approximations for a function and its derivative to the PDEs, Lagrangian SPH formulation can be obtained to calculate the fluid flow as an initial value problem.

Handshaking MD with SPH

The handshaking algorithm is the most important part in the hybrid atomistic continuum combination. In the earliest approach of directly applying atomistic and continuum description in different regions, the material properties are not continuous around the interface region. Later approaches of handshaking need boundary input or field variable averaging from one description to another. As a meshfree Lagrangian particle method, SPH has much in common with the MD in the particle sense and therefore is well suited for coupling with MD to simulate nano systems with multi-scale physics.

In this MD/SPH CLS approach, the domain is divided into two parts (Fig. 2) according to different characteristics, one region for MD simulation with the same potential cutoff distance rc for every molecules, another region for SPH particle simulation. According to the increasing distance from the MD region, the SPH particles gradually change from finer distribution to coarser distribution. For the transitional region where SPH particles neighbor with molecules, the particle separation is of approximately the same length scale as the molecule separations. Each SPH particle has its corresponding smoothing length representing the influencing area. The smoothing length of the SPH particles in the transitional region is taken as the interaction potential cutoff distance of MD. For molecules near the interface, they not only feel the influence from other molecules, but also experience interaction with the SPH particles in the transitional region. Instead, for SPH particles in the transitional region, they may also experience forces from the other SPH particles and neighboring molecules. The interactions between MD molecules and the interactions between SPH particles are treated traditionally. For the interaction between molecules and SPH particles, some kind of potential force with cutoff distance is applied to the interacting pair of molecule and SPH particle. It is convenient to employ the force in Lennard-Jones (LJ) form.

«(',) = ( % ( - ) M 4 ( - ) X , rv<rc (6)

where xtj is the position vector of two molecules. Since the smoothing length of the particles in the transitional region is equal to the cutoff distance of the molecular interaction, the force exerting on a molecule by an SPH particle and the force on the SPH particle by the molecule are equal in magnitude but opposite in direction. This ensures the momentum conservation during the interaction of molecules and SPH particles. Through the interaction of the molecules and the neighboring SPH particles, the momentum and energy are exchanged. Due to the average nature of the SPH method, the thermodynamic and transport properties around the interface should be continuous. Actually since the SPH particles in the transitional region are of the same length scale as

215

the neighboring molecules, these particles can be regarded as virtual molecules extending outside to the continuum region. These virtual molecules or transitional SPH particles, on the one hand, possess the SPH features in interaction with other neighboring virtual molecules and SPH particles, on the other hand, feel the molecular potential force with the real molecules. It is clear that this handshaking technique is carried out on two types of moving particles, i.e. MD molecules and SPH particles. It can be extended to unsteady flows with momentum and energy exchange, and is very suitable to solve the contact line motion.

MD atoms with rc SPH particles with xh

Fig. 2 Schematics for the handshaking

Numerical test

To demonstrate the validity of the MD/SPH CLS algorithm, the planar Poiseuille flow of liquid argon is tested. The geometry of flow system is shown in Fig. 3. The central part around the symmetric line is modeled with MD, while the upper and lower parallel parts close to the wall are modeled with SPH. The system measures 13.6o-x8.5ax80.6a Periodic boundaries are imposed in the x and y direction, while non-slip boundary condition is imposed in z direction. The thickness of the central MD region is 17.0er. Totally 1600 molecules are distributed within the FCC lattice. This thickness of upper and lower SPH region is 31.8<r each. Total 600 SPH particles are employed in each of the upper and lower SPH region. With the increasing distance from the MD region, the SPH particle separation exponentially changing from a to 3.8<r with an exponential factor of 1.1.

The MD simulation is carried out by using a Leapfrog integrator with a time step of 0.005 T0, where r0 =-\ma1je is the characteristic time scale for molecular motion. The parameters used in the MD simulation are as follows. The characteristic length, energy and time scales of liquid argon are a = 3.4 A, e/kB ==120 K, and r0 =2.l61xl0~12 s. The fluid temperature is constant at T = 1.2 e/kB , where kB is the Boltzmann constant. The fluid density is initially given as per3 = 0.80. The flow is driven by the external force of g = 0.1 CT/TQ in the x direction to maintain a low shear rate. The constant temperature T is to be maintained by weakly coupling the y component of the molecular velocity to a thermal reservoir (Grest and Kremer, 1986). For the SPH region, the dynamics of the fluid is modeled using the Navier-Stokes (NS) equation with constant viscosity H = 2.2(me)1,2/cr2 and density p = 0.80/ff3 . For each SPH particle /, the external force

t£ 7?///SS//*~ SPH

Y / • • • A • • • { * • • « • • • • { • • •4a •at ' . • • • { • •

" MD

SPH

Fig. 3 Schematics of the flow geometry

http://13.6o-x8.5ax80.6a

216

g = 0.1(CT/TO )(/»,//») applies to move the flow. The equation is integrated again by using Leapfrog method with a time step of 0.005 T0 . Fig. 4 shows the analytical solution, the full MD solution, and the MD/SPH CLS solution. The results are extracted from 50 layers in z direction. The velocity magnitude and the z coordinate are nondimensionalized by the maximal velocity and the thickness of the entire computational domain in z direction respectively. It is seen that the velocity profile obtained by the averaging of molecular motion coupling with the up and

downstream continuum SPH solution is in good agreement with the analytical solution. The results around the interfacing region are quite continuous.

7 0 . 5 -

| M

8 0.3 S

0.2

1 1 —

: *

---- t-B- i - -

0 i

<> : 1

? i 1 °,. 1

' ! \ - - Total MD aolutlon

Analytical solution c SPHresullsforCLS r MDresultsforCLS c SPHraaultaforCLS

i I

I f l

1 'fe 1 «

_ J - - JB

1 * ' T r i "̂

z (non-dlmanaiona])

Fig. 4 Velocity profiles

Conclusion

This paper presents a novel approach to couple the meshfree particle method of SPH with molecular dynamics. The handshaking interface is treated using transitional SPH particle or virtual molecules, which interacts with both SPH particle and MD molecules. Since MD is applied to atomistic region, while SPH is a continuum approach with meshfree, Lagrangian particle nature, this coupling length scale is very attractive for solving multiple scale physics. Due to the particle nature of MD and SPH, this novel CLS approach will be useful for studying complex flows including studies of convection, coalescence, spreading and wetting, instability in boundary lubrication, and moving contact flows. The numerical test on the planar Poiseuille flow shows the preliminary success of this new atomistic continuum CLS approach.

References

Broughton J. Q. et al. (1999), "Concurrent coupling of length scale: methodology and application", Physical Review B, 60(4): 2391-2403.

Gingold R. A. and Monaghan J. J. (1977), "Kernel estimates as a basis for general particle method in hydrodynamics," Journal of Computational Physics, 46:429-453.

Grest G. S. and Kremer K. (1986) "Molecular dynamics simulation for polymers in the presence of a heat bath", Physics Review A, 33, 3628.

Hadjiconstantinou N. G. (1999), "Combining atomistic and continuum simulations of contact-line motion", Physical Review E, 59(2): 2475-2479.

Liu G.R. (2001), Mesh Free Methods, CRC Press.

O'Connell S. T. and Thompson P. A. (1995), "Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows", Physical Review E, 52 (6) 5792-5795.

Rapaport D. C. (1995), The art of molecular dynamics simulation, Cambridge University Press.

217


ADAPTIVE SMOOTHED PARTICLE HYDRODYNAMICS WITH STRENGTH OF MATERIALS, PART I

G.L. Chin, K.Y. Lam, G.R. Liu Department of Mechanical Engineering, National University of Singapore, 10 Kent

Ridge Crescent, Singapore 119260 [email protected], [email protected], [email protected]

Abstract

The SPH method suffers from two generic problems. The first problem is that the isotropic kernel of SPH is seriously mismatched to the anisotropic volume changes that generally occur in many problems. The second problem is that artificial viscosity causes unphysical heating outside of shocks. The adaptive smoothed particle hydrodynamics (ASPH) method has been developed for astrophysics by other workers to solve the two problems stated. A plain strain adaptive smoothed particle hydrodynamics (ASPH) with strength of materials hydrocode has been developed, by formulating the hydrodynamic ASPH method developed for astrophysics with the stress tensor for problems with strength of materials. The ASPH method gives much higher resolution than the SPH method.

Introduction

Smoothed particle hydrodynamics (SPH) was invented in 1977 by Lucy (1977) and Gin-gold and Monaghan (1977) to simulate nonaxisymmetric phenomena in astrophysics.

Shapiro, Martel, Villumsen, and Owen (1996) have identified the following two generic problems with standard SPH that led to the development of adaptive smoothed particle hydrodynamics (ASPH). In the SPH method with spatially and temporally variable smoothing length, the smoothing length h is usually adjusted in proportion to p~ D , where p is the mass density and D is the number of dimensions; this is adequate only for isotropic volume changes, but is seriously mismatched to the anisotropic volume changes that occur in problems in astrophysics (Shapiro et al., 1996; Owen et al., 1998). In general, the local mean inter-node spacing varies in time, space as well as direction. The second problem that standard SPH has is that due to the use of artificial viscosity, cosmological simulations often show evidence of non-physical pre-heating in shock forming regions (Shapiro et al., 1996; Owen et al., 1998).

The ASPH method was first described by Shapiro, Martel, Villumsen, and Owen (1996) to solve the two stated problems. The ASPH method replaces the isotropic smoothing algorithm of standard SPH, which uses spherical interpolation kernels characterized by a scalar smoothing length that varies spatially and temporally according to the local variations of the density, by an anisotropic smoothing algorithm that uses ellipsoidal kernels characterized by a different smoothing length along each axis of the ellipsoid and varies these three axes so as to follow the value of the local mean separation of




218

nodes surrounding each node, as it changes in time, space and direction. By deforming and rotating the ellipsoidal kernels so as to follow the anisotropy of volume changes local to each node, ASPH adapts its spatial resolution scale in time, space and direction. This significantly improves the spatial resolving power of the method over that of standard SPH for the same number of nodes. An algorithm to restrict the effects of artificial viscous heating to those nodes actually encountering shocks has also been introduced. Owen, Villumsen, Shapiro, and Martel (1998) have described further developments of the ASPH method, further tests, an alternative mathematical prescription for the evolution of the anisotropic interpolation kernels, a method to spatially localize the effects of the artificial viscosity, and the implementation of ASPH in both two and three dimensions.

Just as volume changes in astrophysical problems are in general not isotropic in nature, volume changes in problems with material strength are also, in general, anisotropic. While some grid based methods can follow the deformation and rotation of each cell or element, SPH, whether hydrodynamic or with material strength, artificially constrains the volume changes of interpolation nodes to be isotropic, and has the problem of un-physical artificial viscous heating of nodes not encountering a shock; SPH with material strength also has the same problems as hydrodynamic SPH that led to the development of ASPH.

This project, based on the hydrodynamic ASPH method of ASPH Paper II (Owen et al., 1998), has formulated ASPH with the stress tensor for problems with strength of materials.

The Kernel

Following Owen et al. (1998), the kernel is the commonly used W$ B-spline (Monaghan and Lattanzio, 1985; Monaghan, 1992), which for two dimensions is:

K = G r,

K = | K | , (1)

^ 4 (K,G) = ^ | G | <

3 3 1- -K 2 + -K3 O ^ K ^ l

- ( 2 - K ) 3 1 < K ^ 2 (2) 4 0 K > 2 ,

{- 3 K + - K 2 (KKSC1

- ^ ( 2 - K ) 2 1 < K < 2 0 )

0 K > 2 , where r is the position.

Kernel for Artificial Bond Viscosity

The method of Owen et al. (1998) for suppressing the artificial bond viscosity is followed. A different interpolation kernel Wn is used with artificial bond viscosity. Wn is

219

more spatially compact and has a sharper gradient than W; the influence of the artificial bond viscosity is spatially restricted. The kernel Wn used is a simple variant of the Gaussian kernel (Owen et al., 1998), which for two dimensions is:

^Gauss2 (K, G ) = — - | G | exp (~KK4) , (4) TO-

op-j^.2 V^Gaus^CK, G) - - 2 f i J L |G |G-Kexp ( - * K 4 ) . (5)

Evolving the G Tensor

The G tensor varies both spatially and temporally. Following Owen et al. (1998), to conserve linear momentum, the kernels are symmetrized according to Hernquist and Katz (1989):

K,- = G r ( r , - r , ) ,

Kj = Gj-{ri-rj), (6)

Wij = \[W (Kt, GO + W (xj, Gj)}, (7)

VWij = l- [VW (*, G,) + VW (K, , G ; ) ] , (8)

W$=l- [Wu (Kt, G,) + Wu fa, Gj)], (9)

VWU = \ \VwU (*'•' G ' ) + V r " (*/' G;)] • (10)

The two-dimensional G tensor is evolved by Owen et al. (1998)

Gn = Gn ( 0 -^2 i ) - G n ^ i i ,

G22 — ~G\2 (9 + ^12) - ^22^22,

Gn = G226 - Gll^l2 - Gl2^22,

A <J11^12 ~ <J22^21 +<?12 (^22 ~ £ l l )

G11+G22 ^ = Vv.

(ID

The maximum principal smoothing length hi and minimum principal smoothing length hi for two dimensions are the inverse of the eigenvalues of G.

Smoothing the G Tensor Field

Following Owen et al. (1998), each G is periodically replaced by an averaged G', which

220

for D dimensions is

(cr1) = 1 * J ' J | (12)

G' = \G\® KG-y^f* (G~1)-1

= |G |^ | (G- 1 ) | ° (G- 1 ) - 1 .

Mass Density Equation in ASPH

The continuity equation is

^ + pV-v = 0, (13)

where p is the mass density, t is time and v is the velocity vector. The divergence of velocity V • v, following Libersky and Petschek (1991), is

V-v^S-^vy-vO-Vffy. (14) J pJ

The mass density is evolved using the continuity equation (13) and (14).

Momentum Equation in ASPH

The momentum equation of Libersky and Petschek (1991), with the kernel Wjj for the artificial bond viscosity, is

(15) DM v ——• = > rrij Dt J

Gi °Z 1 .ww..-.n..i.vwn 4 + 4 •v^7-ni7i-v^v

This form of the momentum equation formally conserves linear momentum.

Artificial Bond Viscosity in ASPH

The artificial bond viscosity n,y follows the implementation of Owen et al. (1998), which is based on the formulation of Monaghan and Gingold (1983):

nv = ±(n,+ny),

^ — ^ ( v l - v 7 - ) . ( r , - r y ) < 0

otherwise, (16)

K; • K,- + £art

Kj • Kj + eart

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Ylij is active only for convergent flows within the material, a and P are parameters of order unity (Owen et al., 1998). eart prevents singularities. The symmetry of (16) ensures that it conserves linear momentum.

The Monaghan and Gingold (1983) formulation of artificial bond viscosity produces artificial shear viscosity, which can cause spurious transport of angular momentum in rotating systems (Owen et al., 1998). Following Owen et al. (1998), the multiplicative correction factor / of Balsara (1995) has been implemented,

[v-v̂ l Ji IV-Vfl + IVxv. l+O.OOOl^ ' ( 1 7 )

hui=^{h\i + h2i).

(16) has been implemented both with and without the correction term of (17).

Conclusions

The formulations of a plan strain adaptive smoothed particle hydrodynamics (ASPH) with strength of materials hydrocode have been described.

Acknowledgement

The authors would like to acknowledge the private correspondance with the main author of ASPH Paper II (Owen et al., 1998), J. Michael Owen, which helped to clarify a number of issues in their ASPH paper.

References

D. S. Balsara. Von-Neumann stability analysis of smoothed particle hydrodynamics- suggestions for optimal-algorithms. Journal of Computational Physics, 121(2):357—372, October 1995.

R. A. Gingold and J. J. Monaghan. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181:375-389, November 1977.

L. Hernquist and N. Katz. TREESPH: A unification of SPH with the hierarchical tree method. The AstrophysicalJournal Supplement Series, 70:419-446, June 1989.

L. D. Libersky and A. G. Petschek. Smooth particle hydrodynamics with strength of materials. In Harold E. Trease, Martin J. Fritts, and William Patrick Crowley, editors, Advances in the Free-Lagrange method: including contributions on adaptive gridding and the smooth particle hydrodynamics method: proceedings of the Next Free-Lagrange Conference held at Jackson Lake Lodge, Moran, Wyoming, USA, 3-7 June 1990, number 395 in Lecture notes in physics, pages 248-257, New York, 1991. Springer-Verlag.

L. B. Lucy. A numerical approach to the testing of the fission hypothesis. The Astronomical Journal, 82(12): 1013-1024, December 1977.

222

J. J. Monaghan. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 30:543-574, 1992.

J. J. Monaghan and R. A. Gingold. Shock simulation by the particle method SPH. Journal of Computational Physics, 52:374-389, 1983.

J. J. Monaghan and J. C. Lattanzio. A refined particle method for astrophysical problems. Astronomy and Astrophysics, 149(1): 135—143, August 1985.

J. M. Owen, J. V. Villumsen, P. R. Shapiro, and H. Martel. Adaptive smoothed particle hydrodynamics: Methodology.il. The Astrophysical Journal Supplement Series, 116(2): 155-209, June 1998.

P. R. Shapiro, H. Martel, J. V. Villumsen, and J. M. Owen. Adaptive smoothed particle hydrodynamics, with application to cosmology: Methodology. The Astrophysical Journal Supplement Series, 103:269-330, April 1996.

http://Methodology.il

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A D A P T I V E S M O O T H E D PARTICLE H Y D R O D Y N A M I C S W I T H S T R E N G T H OF M A T E R I A L S , PART II

G.L. Chin, K.Y. Lam, G.R. Liu Department of Mechanical Engineering, National University of Singapore, 10 Kent

Ridge Crescent, Singapore 119260 [email protected], [email protected], [email protected]

Abstract

An ASPH plane strain code based on the equations described in Part I has been developed. The results of test problems using the ASPH code are given in Part II. The details of orientation and anisotropy of volume changes of the nodes are clearly seen, which are not possible using SPH; the ASPH method gives much higher resolution than the SPH method. There are many problems to be solved before SPH with strength of materials can be considered for production use. The ASPH method has been demonstrated to improve the SPH method for problems with material strength. While ASPH with strength of materials also suffers from many of the same problems as SPH with strength of materials, the advantages of ASPH demonstrated here make it a good candidate as the starting basis of other improvements to the SPH method.

Introduction

The formulations of a plane strain adaptive smoothed particle hydrodynamics (ASPH) with strength of materials hydrocode have been described in Part I. An ASPH plane strain code based on the equations described in Part I has been developed. The ASPH code is based on a SPH code developed previously. The ASPH and SPH results are compared here.

Impact of Plate Against Rigid Surface

A cylinder (infinite plate in plane symmetry) is impacted against a rigid surface, using both the SPH and ASPH solvers.The cylinder is of initial length 25.46 mm and diameter 7.6 mm. The particles are initialized as squares of 0.38 mm side dimension, arranged in a rectangular cartesian array. There are 67 particles along the length and 20 particles along the diameter. The rigid surface is simulated using 19 layers of ghost particles reflected across the rigid surface. There are a total of 1340 real particles and 380 ghost particles. The cylinder is initially in contact with the rigid surface. All problems are run to 90 [is. The material is Armco iron with the Johnson-Cook yield model (Johnson and Cook; Johnson and Holmquist, 1988). The Mie-Gruneisen equation of state, of the form in Zukas (1990); Libersky, Petschek, Carney, Hipp, and Allahdadi (1993), is used.




224

SPHResults

Case AV has reference parameters and spatially and temporally variable smoothing length. Figure 1 shows the plot

&. of case AV. There are voids around the centre of the plate.

ASPH Results

Figure 2 shows the plot of case AK5S2. The constant K is set to 1.54, the G tensor field is smoothed every 2 time steps, and the initial h\ and &2 are equal to 1.0 time the initial particle side dimension. The plots of the ASPH cases

Figure 1 • Plot of case clearly show that the particles are flattened along the direc-AV at 90 us ^o n °f ^ irapact- xt is necessary to smooth the G tensor

field; without smoothing, the G tensor field is strongly disordered. The voids are also too big with much smaller frequency of smoothing. Numerical fracture due to tensile instability is not solved by the ASPH method, even though the G tensor adapts according to the anisotropy of the volume changes. Only certain combinations of K and frequency of smoothing G give satisfactory results. There is a trend of turning points with regard to the length, diameter, number of time steps, the fraction of artificial viscosity energy, and energy conservation as K increases. The energy conservations of the ASPH cases are better than the SPH case AV; one of the likely reasons is the smaller time steps due to the anisotropy of the kernel.

Comparing the ASPH cases with the SPH reference case AV, the ASPH cases generally predict less residual kinetic energy. The ASPH results also clearly show the anisotropy of the deformations. If the multiplicative correction factor of Balsara (1995), (17) in Part I, is used for these plate impact problems, the plates will break. The artificial shear viscosity is necessary for this type of plate impact problem. The success of suppressing the artificial bond viscosity using a different interpolation kernel Wu with artificial bond viscosity depends on the value of A" used. The value of K

Figure 2: Plot of case chosen has been demonstrated to have a great effect on the AK5S2 at 90 JUS. results. The solutions for such plain strain impact problems are not known. It is found that the parameters chosen have a great effect on the results.lt is not known the proper set of parameters to use. It may be necessary to use a certain set of parameters for each problem type.

Hypervelocity Impact of Sphere on Plate

Hypervelocity impact problems have great deformations and are very suitable for SPH codes. Hiermaier, Konke, Stilp, and Thoma (1997) have done experimental tests and numerical simulations of the hypervelocity impact of aluminium spheres on thin plates. The numerical simulations are done in plane symmetry using their SPH code, SOPHIA. The shapes of the debris clouds and crater widths at 20 //s from the experimental and numerical results are compared. Their experiment of the impact of aluminium sphere

http://results.lt

225

Table 1: SPH results of hypervelocity impact. Case

HI H1.5

Crater diameter

(mm) 32.5 28.9

Debris cloud length (mm) 107.4 105.1

Debris cloud width (mm) 79.9 86.1

Ratio of length to width

1.34 1.22

Time steps

3773 2702

on aluminium plate has been reproduced numerically here. Hiermaier et al. (1997) have made no mention of failure/fracture modelling in their numerical simulations. There is no failure/fracture model in the developed code.

Model Set-Up

The sphere is of 10 mm diameter, the plate is 4 mm thick. The plate length of 100 mm is used. The particles are all initialized as squares of 0.2 mm side dimension. The particles in the sphere, which is an infinite cylinder in plane symmetry, are arranged in circumferential rings as this gives the most realistic representation of the geometry. There are 50 particles across the diameter. The particles in the plate are arranged in a rectangular cartesian array. There are 500 particles along the length and 20 particles along the thickness of the plate. There are 1956 particles in the sphere and 10000 particles in the plate, for a total of 11956 particles. The sphere is initially in contact with the centre of the plate. The problems are run with the plate free of constraints. The impact speed of the sphere is 6.18 km/s. The problems are run to 20 //s. The room temperature and initial temperature of the sphere and plate are 0 °C. The Johnson-Cook yield model and Tillotson equation of state are used, with the same constants given by Hiermaier etal. (1997).

SPH Cases

Two different cases were run using the SPH code. Spatially and temporally variable

smoothing length is used in both cases. The initial smoothing length h is set to 1.5

times the inter-particle distance in Hiermaier et al. (1997). The code uses the equation,

to update the smoothing length. Case HI uses d=l, case HI.5 uses d = 1.5. a = 2.5 and p = 2.5, which follow Hiermaier et al. (1997), with eart = 0.01 and co = 0.3.

SPH Results

The results are given in Tables 1 and 2. The maximum energy deviation for cases HI and HI.5 are 0.104% and 0.267% respectively. The experimental results given in Hiermaier et al. (1997) for the crater diameter is 27.5 mm including crater lip, 34.5 mm excluding crater lip, and the ratio of length to width of the debris cloud is 1.39. The experimental results of debris cloud length and width are not given. Their simulation results give the crater width as 35 mm, and the ratio of length to width of the debris cloud as 1.11.

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Table 2: SPH energy results of hypervelocity impact. Case

HI H1.5

Case

H1NS6 H1WS6 H1.5WS6 H1.5WS18

ASPH Cases

Total

(J) 4168.4 4175.5

Kinetic Internal (J) (J)

3618.1 550.3 3581.8 593.8

Plastic strain Artificial viscosity

(J) (J) 35.0 35.7

Table 3: ASPH results of hypervelocity impact. Crater

diameter (mm)

35.7 34.4 32.1 32.1

Debris cloud Debris cloud Ratio of length length width to width (mm) (mm) 105.3 105.4 105.2 105.4

72.3 74.7 82.2 79.3

1.46 1.41 1.28 1.33

681.6 727.0

Time steps

15460 15521 21279 25489

Cases H1NS6 and H1WS6 have the initial h\ and hi set to 1.0 time the initial particle side dimension. Cases H1.5WS6 and H1.5WS18 have the initial h\ and fo set to 1.5 times the initial particle side dimension. Case H1NS6, unlike the other cases, does not use the multiplicative correction factor of Balsara (1995), (17) in Part I. Cases H1NS6 and H1WS6 have the time step size safety factors set to 0.3, with the G tensor field smoothed every 6 time steps. Cases H1.5WS6 and H1.5WS18 have the time step size safety factors set to 0.1. Case H1.5WS6 has the G tensor field smoothed every 6 time steps. Case H1.5WS18 has the G tensor field smoothed every 18 time steps. K is set to 1.54, which is the recommendation of Owen et al. (1998) for the problems that they run. a = 2.5 and P = 2.5, which follow Hiermaier et al. (1997), with eart = 0.01.

ASPH Results

Figure3: PlotofcaseH1.5WS18at20//s.

The results are given in Tables 3 and 4. The maximum energy deviation for cases HlNS6,HlWS6,H1.5WS6andH1.5WS18 are 1.55%, 1.64%, 0.402% and 0.432% respectively. Figure 3 shows the plot of case H1.5WS18. Figure 4 shows the close-up plot of case H1.5WS18. Figure 5 shows the plot of the frontal region of the debris cloud of case HI.5WS18. In comparison with case HI.5, the orientation and anisotropy of the deformation of the particles can be clearly seen in the plots. The benefits of using the ASPH method for hypervelocity impact problems have been very clearly demonstrated. With orientation and anisotropy of volume changes clearly shown, the ASPH method gives much higher resolution than

227

Table 4: ASPH energy results of hypervelocity impact. Case

H1NS6 H1WS6 H1.5WS6 H1.5WS18

Total

(J) 4233.3 4233.1 4181.0 4181.5

Kinetic

(J) 3791.4 3807.0 3695.4 3685.8

Internal

(J) 441.9 426.1 485.6 495.7

Plastic strain

(J) 64.5 66.3 51.5 50.3

Artificial viscosity

(J) 546.5 528.7 583.3 597.7

Figure 4: Close-up plot of case H1.5WS18 at 20 /is.

Figure 5: Plot of the frontal region of the debris cloud of case H1.5WS18 at 20 /is.

228

the SPH method. The ASPH results also have lower absolute values of artificial viscosity energy; the artificial viscosity energy is even less with the use of the multiplicative correction factor of Balsara (1995), (17) in Part I. The energy conservations of the ASPH cases are not as good as the SPH cases, although many more smaller time steps are needed. The computational expenses of the ASPH cases are much higher, mainly due to the greater number of smaller time steps necessary, but also due to the effect of the anisotropic kernels on the neighbour search. The increases in memory usage are small. One of the reasons for some loss of symmetry is the inaccuracies of the trigonometric functions over certain ranges, and the consequent effects on the neighbour search. As with the plate impact ASPH results, the choice of parameters has been shown to have a great effect on the results. Again, it is not known the proper set of parameters to use.

Conclusions

There are many problems to be solved before SPH with strength of materials can be considered for production use. The ASPH method has been demonstrated to improve the SPH method for problems with material strength. While ASPH with strength of materials also suffers from many of the same problems as SPH with strength of materials, the advantages of ASPH demonstrated here make it a good candidate as the starting basis of other improvements to the SPH method.

References

D. S. Balsara. Von-Neumann stability analysis of smoothed particle hydrodynamics- suggestions for optimal-algorithms. Journal of Computational Physics, 121(2):357—372, October 1995.

S. Hiermaier, D. Konke, A. J. Stilp, and K. Thoma. Computational simulation of the hyper-velocity impact of Al-spheres on thin plates of different materials. International Journal of Impact Engineering, 20(l-5):363-374, 1997.

G. R. Johnson and W. H. Cook. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In Proceedings of the Seventh International Symposium on Ballistics (The Hague, The Netherlands, 1983), pages 541-547.

G. R. Johnson and T. J. Holmquist. Evaluation of cylinder-impact test data for constitutive model constants. Journal of Applied Physics, 64(8):3901-3910, October 1988.

L. D. Libersky, A. G. Petschek, T. C. Carney, J. R. Hipp, and F. A. Allahdadi. High-strain la-grangian hydrodynamics - a 3-dimensional SPH code for dynamic material response. Journal of Computational Physics, 109(l):67-75, November 1993.

J. M. Owen, J. V. Villumsen, P. R. Shapiro, and H. Martel. Adaptive smoothed particle hydrodynamics: Methodology.il. The Astrophysical Journal Supplement Series, 116(2): 155-209, June 1998.

J. A. Zukas, editor. High Velocity Impact Dynamics. John Wiley & Sons, Inc., New York, 1990.

http://Methodology.il

229

Advances in Meshfree andX-FEM Methods, G.R. Liu, editor World Scientific, Singapore, 2002

NUMERICAL SIMULATION O F PERFORATION O F CONCRETE SLABS BY STEEL RODS USING SPH METHOD

H . F . Qiang, S. C. Fan Protective Technology Research Center, School of CEE, Nanyang Technological University, Singapore 639798

hfqiang(a>,hotmail. com, cfansc&.ntu.edu.sg

Abstract

A numerical simulation of penetration/perforation process of a concrete slab by a cylindrical steel projectile using the Smoothed Particle Hydrodynamics (SPH) method is studied in the paper. In the simulation, the available hydrocode AUTODYN2D is employed with the improved RHT concrete model, in which a Unified Twin-Shear Strength (UTSS) criterion is adopted in defining the material strength effects, and constructed a dynamic multifold limit/failure surfaces including elastic limit surface, failure surface and residual failure surface. The proposed model is incorporated into the AUTODYN hydrocode via the user defined subroutine function. The results obtained from the numerical simulation are compared with available experimental ones. Good agreement is observed. It demonstrates that the proposed model can be used to predict not only the damage areas and velocity reduction of the projectile during the perforation process but also the debris clouds of spalling process.

Keywords: SPH, Concrete, Perforation, UTSS criteria, Multi-limit surface, Damage

Introduction

SPH method was first applied by Lucy (1977) to astrophysical problems and was extended by Gingold and Monaghan (1982). Cloutman (1991) has shown that SPH could be used to model hypervelocity impacts. Libersky and Petschek (1991) have shown the SPH can be used to model material with strength. Liu et al (2003) have studied blasting simulation with explosive in fluid media, and Liu (2002) has reviewed mesh free methods and introduced this method systematically. In fact, SPH is a gridless Lagrangian technique. The main advantage of the method is to bypass the requirement for a numerical grid to calculate spatial derivatives. This avoids the severe problems associated with mesh tangling and distortion which usually occur in Lagrangian analyses involving large deformation impact and explosive loading events. The grid based methods, such as Lagrange and Euler, assume a connectivity between nodes to construct spatial derivatives. SPH uses a kernel approximation, which is based on randomly distributed interpolation points with no assumptions about which points are neighbours, to calculate spatial derivatives.

230

In this paper, a two-dimensional axi-symmetric numerical simulation for the projectile-target model is carried out using the SPH procedure. In the simulation, the available hydrocode AUTODYN2D is employed with the improved RHT concrete model, in which UTSS criterion (Yu, 2002) is adopted in defining the material strength effects, and constructed a dynamic multifold limit/failure surfaces including elastic limit surface, failure surface and residual failure surface. The proposed model is incorporated into the AUTODYN hydrocode via the user defined subroutine function. The results obtained from the numerical simulation are compared with available experimental data.

Material Model For The Concrete Slab

A dynamic plastic damage model is proposed by using UTSS criterion based on RHT concrete model (Riedel et al, 1999). Dynamic multi-limit surface models are employed here, i.e., the elastic limit surface, failure surface and residual strength surface. The failure surface is a bounding surface, no stress state is allowed to exist beyond it. The shapes of the failure surface could be changed in the stress space during impact process. However, the loading surface changes its shape non-uniformly from the initial surface to the failure surface with the development of the effective plastic strain. Once failure surface is reached, residual strengdi surface is determined according to the scalar damage value.

Based on the consideration, the dynamic material model is proposed. The main characteristics of this model are: 1) strain-rate dependent failure surface is considered, 2) UTSS criterion is employed in the failure surface, 3) linear strain hardening is used to impose the plastic flow consideration, 4) isotropic damage is used in this model due to increase strain after the stress in the reached the failure surface. The material model can be split as follows:

The Failure Surface

Amongst the strength models available, UTSS theory has a clear mechanical concept and simple mathematical formula. The advantage of the UTSS theory is that it takes account of the second principal stress on material strength. However, the Von Mises criterion is based on the average principal stresses while the Mohr-Coulomb criterion neglects the intermediate principal stresses. The envelope is then completed by defining a piece-wise linear interpolation function in the deviatoric plane. The beauty of the twin-shear-unified strength criteria is her feasibility in defining the convex shape of the surface. Setting the value of the controllable convex parameter b to 0 or 1 yields the lower and upper limit of the convex shape function. For any arbitrary value of b, the shape function can be written in the following form

r.K sin 60° r, R is (i_i) + ft_L_ when 0°<6<6, (i)

f r,sin<9 + rcsin(6O°-0) cos0

231

R, rtrc sin 60°

r, sin(9 + rcsin(60°-<9) {l-b) + b

cos(60°-6») whenfl„<6><60°

Where 6k - arctan 1

VJ * L - 1 cos 36 = — j = , J2 and / 3 are the second and third 2 JA"

stress invariants respectively, r, and rc are the tensile and compressive meridians respectively. It is worth noting that b reflects the influence of the intermediate principal stress on the material strength. Besides, it encompasses all prevailing yield or failure criteria. When b = 0 , it can represent the Tresca's criterion; when b - 0.5 , it is equivalent to the Von Mises' criterion. The shapes represented by different values of b in deviatoric plane and multi-limit surface in meridian plane are shown in Figure 1. In the present investigation, b is set to 0.6.

UTSS Theory (Xu, 198S)

Uniaxial Tension

Uniaxial Compression Fa{,m Surfm ( D = 0 ) ^

v. Nf V r

]e Sbear s|rengtb T (Mol»j,1900)

Meridian plane Deviatoric plane

Figure 1 Multi-limit surface in meridian and deviatoric projection

The failure surface is defined as a function of pressure (P), the Lode angle (0) and strain rate ( s ).

* fait ~ *TXC(P) " * V ' rSATE(i) (2)

where YTXC - fc\A(P* -P^UFRATE)"]' m which fc is compressive strength.^ is failure

surface constant; N is failure surface exponent; P* is pressure normalized by fc; P* u

is defined as P' ( / / / ) .

232

f • \D

e for P>—fc (compression)

f • V 6'

, in which D is the compressive

for P <—f (tension)

strain rate factor exponent; a is the tensile strain factor exponent.

The Elastic Limit Surface

The elastic limit surface is scaled from the failure surface using

Y =Y F •F 1 elastic 1 fail 1 elastic I CAP{P)

(3)

where Felaslic is the ratio of the elastic strength to failure surface strength. This is derived from two material parameters: tensile elastic strength / , and compressive elastic strength fc • FCAP(P) is a function that limits the elastic deviatoric stresses under hydrostatic compression via

FCAP = <

1

P -P

for P<PU

for PU<P<P0

for P0<P

(4)

Strain Hardening

Linear hardening is used prior to the peak load. During hardening, the current yield surface (T*) is scaled between the elastic limit surface and the failure surface via

V

where e Y —Y

fail elastic

pl{pre-sofiening)

G

<Y, -Y ^ elastic J

failure •*• elastic (5)

/?/(pre-softening ) 3G

elastic

C —C \ elastic plastic J

Geias„c !(Getas,* ~ Gplastic) i s defined by the user.

Residual Failure Surface

A residual (frictional) failure surface is defined as

Y = RP residual

(6)

233

where B is the residual failure surface constant; M is the residual failure surface exponent.

Damage Model

Following on from the hardening phase, additional plastic straining of the material leads to damage and strength reduction, see Figure 1. Damage is accumulated via

h p

6f"-= JD,(p*-/ ' ! ; / /)D 2>6vm t a (8)

where D, and D2 are damage constants; t™n is the minimum strain to failure.

The post-damaged failure surface is then interpolated via

* faraured = U ~~ D)lfaiiure + L>lTesidml (?)

and the post-damaged shear modulus is interpolated via

GfraauKd={\-D)G + DGreMual (10)

where Gresidua, is the G*, the residual shear modulus fraction.

Numerical Example

To verify and calibrate the present model, a numerical simulation was carried out, which is to illustrate the results of the enhancements incorporated using SPH method. It adopts the same configuration and materials used in the tests by Hanchak et al. (1992). The target is a 680mm x 680mm square of 178mm thick reinforced concrete panel. The projectile is an ogival-nose shaped, 143.7mm long steel rods with a diameter of 25.4mm and a 3.0 caliber-radius-head. The impact velocities vary between 300 and 1058m/s. The experimental results are compared with simulating results in the present investigation, and the unconfined compressive strength of concrete is 48MPa, and other parameters for material model refer to Riedel et al (1999).

In the simulation, both the projectile and the target regions are modeled using the SPH. In order to simplify it to a 2D axi-symmetric analysis, the square panel is approximated by a circular one of radius of 303mm The target is discretized into 13528 particles while the projectile is represented by 1678 particles. The panel is lightly reinforced. However, Hanchak's results verify that the small amount of reinforcement does not have a major influence on the penetration resistance. Therefore, the steel bars are ignored in the modeling.

234

The material model for the projectile adopts the linear EOS and the Johnson & Cook strength model. The mechanical properties are based on die AUTODYN's material library for steel 4340: initial density p0 =%.\glcmi, bulk modulus K =\59GPa, shear modulus G = Sl.SGPa, yield stress fy =792MPa etc. The same EOS is adopted for concrete slab, only the bulk modulus K is different. The exit velocities and corresponding penetration depths of the projectile are shown in Table 1. The decrease in the velocity of the projectile is due to the resistance from the target. After perforation, the velocity of the projectile remains constant because the target material can no longer offer any resistances. This constant velocity is defined as the residual (or exit) velocity of the projectile. If perforation did not occur, the projectile would have come to rest and embedded inside the target with zero residual velocity. Meanwhile, the reduction of the projectile velocity is recorded over the penetration depth and compared to the residual velocities measured in the normal strength concrete tests {fc = 48MPa). At high initial velocities the results of the dynamic constitutive law match the experimental values very closely. The ballistic limit of about 300 m/s is also correctly predicted in the simulation. The distribution of the compressive damage for the constitutive theory with or without its dynamic part emphasizes the importance of a realistic consideration of the strain-rate effect. In Figure 2 the contour plots of the compressive damage for time steps during 750 m/s impact are shown, at the same time the dynamic constitutive law exhibits a rather homogeneous damage distribution.

m Table 1 Comparison of exit velocities

and penetration depths

Jlll»-.. Hi

s/n

(1)

(2)

(3)

(4)

Impact velocity

(m/s)

1058

749

360

301

Test (Hanchak

1992)

947

615

67

0

Simulation Exit

velocity (m/s)

950

625

71.5

0

Depths (mm)

178

178

174

163

Figure 2 Damage contour plot at cycle 4800

Conclusions

A multi-limit surface dynamic plastic damage model is developed based on RHT model using UTSS criterion. The present material model was coded and incorporated into AUTODYN. The numerical simulation was carried out for a case of perforation through concrete slab by a steel projectile. Numerical results were compared experimental results

235

by others. They agreed favourably well. It demonstrates that the present model could be used to predict not only the damage areas and the velocity-decrease of the projectile during the perforation process but also the debris clouds of spalling process with an acceptable degree of accuracy.

References

Lucy L.B. (1977). "A numerical approach to the testing of the fission hypothesis," The Astronomical Journal, 82(12), 1013-1024

Gingold R.A. and Monaghan J.J. (1982). "Kernel estimates as a basis for general particle methods in hydrodynamics," Journal of Computational Physics, 46(4), 429-453.

Cloutman L.D. (1991). SPH simulations of hypervelocity impacts, Lawrence Livermore National Laboratory, UCRL-ID-105520.

Libersky L.D. and Petscheck A.G. (1991). "Smoothed particle hydrodynamics with strength of materials," , Proceedings of the Next Free Lagrange Conf, Springer-verlag, NY, 1991, 248-257.

Hanchak S.J., Forrestal M.J., Young E.R. and Ehrgott J.Q. (1992). "Perforation of concrete slabs with 48-MPa and 140-MPa unconfined compressive strength," International Journal of Impact Engineering, 12(1), 1-7.

Liu M.B., Liu G.R., Zong Z. and Lam K.Y. (2003). "Computer simulation of the high explosive explosion using smoothed particle hydrodynamics methodology," Computers & Fluids, 32(3), 305-322.

Liu G.R. (2002). Mesh Free Methods: moving beyond the finite element method, CRC press, Boca Raton.

Yu, M.H. (2002). "Advances in strength theories for materials under complex stress state in the 20lh

Century," Applied Mechanics Reviews, 55(3), 169-218

Riedel W., Thoma K., Hiermaier S. and Schmolinske E. (1999). Penetration of reinforced concrete by BETA-B-500-Numerical analysis using a new macroscopic concrete model for hydrocodes, Proceedings of^ Int. Symp. IEMS, Berlin, 1999, 315-322.

SECTION 11

X-FEM

239

Advances in Meshfree and X-FEMMethods, G.R. Liu, editor, World Scientific, Singapore 2002

THREE DIMENSIONAL CRACK GROWTH ANALYSIS USING OVERLAYING MESH METHOD AND X-FEM

S. Nakasumi Department of Environmental and Ocean Engineering, University of Tokyo, Japan

sumi@nasl. t. u-tokyo. ac.jp

K. Suzuki Department of Environmental Studies, Graduate School of Frontier Sciences, University

of Tokyo, Japan

katsu@k. u-tokyo. ac.jp

H. Ohtsubo Department of Environmental and Ocean Engineering, University of Tokyo, Japan

[email protected]. ac.jp

Abstract

In this paper, a new methodology to analyze three dimensional crack problems with flexible modeling by means of overlaying mesh method and extended finite element method (X-FEM) is presented. The overlaying mesh method increases the accuracy of analysis locally by superimposing additional mesh of higher resolution on the global mesh which represents rough deformation of structures. In this method the boundaries and nodes in the two meshes do not have to coincide with each other. It makes modeling process becomes very flexible. In X-FEM, discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This technique allows the entire crack to be represented independently of the mesh. As numerical example, an inclined semi-circle surface crack under tension is analyzed.

Keywords: Overlaying mesh method, extended finite element method, crack, fracture, partition of unity, and enrichment

Introduction

The finite element method (FEM) has contributed greatly to the progress of the fracture mechanics. However, a serious difficulty in applying the FEM to the analysis of three-dimensional crack problem lies in the mesh generation process. In our laboratory we are trying to resolve this issue by superimposing another mesh of higher resolution and call it overlaying mesh method. This method is intended to improve the quality of the finite element calculations in the regions of unacceptable errors has

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240

occurred. The concept of mesh overlaying analysis has been presented by Fish (Fish, 1992; Fish and Markolefas,1993), and termed as adaptive j-method. He applied this methodology to the analysis of the composite materials (Fish and Guttal, 1996). In this technique, the nodes of the two meshes do not need to agree with each other, so mesh generation becomes very flexible and low cost process. In the case of modeling of crack problems, the crack surface is modeled only on either mesh. However, in the analysis of crack growth, the mesh must be remeshed, and in three-dimensional problem, this process is very cumbersome. Recently, As an effective methodology for the analysis of the crack problem, a technique called extended Finite Element Method (X-FEM) has been proposed by Belytschko et a/.(Belytschko and Black,1999; Mogs, Dolbow and Belytschko,1999). The essential idea of X-FEM is to use a displacement field approximation that can model an arbitrary discontinuity and the near-tip asymptotic crack fields using the notion of partition of unity (Babuska,1997). This enables the domain to be modeled by finite elements with no explicit meshing of the crack surface. Using this technique, the three-dimensional crack problem was first analyzed by Sukumar et o/.(2000). They analyzed several planer crack models. And Mogs et a/.(2002) analyzed non-planer crack models. In this study we propose a new technique which has the merits of both methodology. That is to say, the mesh of the whole structure is represented by a coarse mesh, and the local domain which includes the crack surface is represented by another refined mesh. The two meshes are perfectly independent with each other, and the crack surface is not explicitly expressed on either mesh. But the discontinuity displacement is represented implicitly in the displacement field of the local model. As numerical examples, we analyze semi-circle surface crack problem.

Formulation

Overlaying mesh method

Consider the domain Q bounded by boundary r • The boundary r is composed of the sets P and r", such that r = r" u r ' as shown in Fig. 1. Prescribed displacements are imposed on r", while tractions are imposed on r". Although the model described in Figure. 1 is in two-dimensional model, this concept is easily extended to three-dimensional problems. The domain Q is discretized into a finite element mesh, which we call as global mesh. In local domain nL (QL C Q), another fine mesh is defined, and we call this as local mesh. There two meshes are generated independently and the boundaries or nodes on both meshes do not have to agree with each other. Displacement field u° and uL are defined on global mesh and local mesh respectively. In the domain nL, we define that the true displacement is the sum of them defined on each meshes, namely

u = uG+u i=N°dG+N id i in QL (1)

Where d° and dL are the discretized nodal displacement on each finite element mesh, respectively. Similarly, N° and NJ are the finite element shape functions defined on each

241

mesh. The upper subscript, G or L indicates that the quantity is defined on each model. To satisfy compatibility condition, Eq.(2) is required.

u ' = 0 on rG (2)

Crack surface (segment) Local mesh (uL)

Global mesh (u°)

Figure 1. The concept of mesh overlaying

By partial differentiating Eq.(l), we get the strain field as follows :

£ = E°+e'-=Ba(r+B'-(r in Q. (3)

Where BG and B' are the strain-displacement matrices. Eq.(l)~(3) are substituted to the principle of virtual work, and discrete equilibrium equations can be obtained from arbitrariness of global and local variations as follows:

K 0 KGL

KLa K i dL

Where,

K G = XJL B G r D B G r f f i

e=l e

K G i = S L B G r D B ^ n

KU}=£j&BLTDBadCl=KGLT

fG = |nNGrbrfQ+ {r,N°rWr

(4)

(5)

(6)

(7)

(8)

(9)

(10)

242

Where D is a constitutive matrix, and b and t are the prescribed body force and traction, respectively. In Eq.(5),(6),(9),(10), KG, KL, fG and fL are the ordinary finite element stiffness matrices and load vectors which are defined on the global mesh and local mesh respectively. Whereas KG1 and KiG are the matrices which represent the interaction between both meshes.

Nodal enrichment to local model

Now, we consider the case in which the displacement is discontinuous in domain cil. In this case, we assume that the discontinuity of displacement is represented only by local model displacement. Namely, uL in Eq.(l) is represented as follows,

01)

Where, J is the set of nodes whose support (union of the elements connected to the node) are completely bisected by the crack surface. Whereas K is the set of nodes whose support are partially cut by the crack surface (See Figure 2). H(x) is the Heaviside function to represent the discontinuity of the crack surface (crack segment), i.e.

, 1 y>0 K ' 1-1 v < 0

(12)

This is defined in the local crack co-ordinate system. And, i//,(x) represents the two-dimensional asymptotic displacement field around the crack front (crack tip):

Iv,W = £v"M) \ r • 0 r 9 r • 0 • n r Q • n = i^Jrsm—, yjrcos—, -Jrsm—smd, -jrcos—smff

2 2 2 2

(13)

Where (r ,6) are the local polar co-ordinates in the crack tip. And b and ct are the enriched nodal degree associated with H(x) and y/{x), respectively.

Crack surface (crack segment)

T \

(i—v—p*

\ 5^'( <JJ^£i

^o—i >—o — —

} i

Crack tip (crack front)

Figure 2. Enrichment to the nodes on local mesh. The circled nodes are enriched by the jump function, and the squared nodes are enriched by the crack tip functions

243

Numerical example

An inclined semi-circle surface crack subjected to tensile loading

We analyzed a model with an inclined semi-circle surface crack subjected to unit tensile loading. The meshes used are shown in Figure 3. Young's modulus is 29000000 and Poisson's ratio is 0.32. Hexahedral elements are used. The global model and the local model are discretized with 192 elements and 1280 elements, respectively. The crack is located at the midpoint of the surface of the structure and the crack plane is oriented at angle 30 degree to a cross-sectional plane of the plate.

Local mesh (1280 elemetns) Global mesh (192 elements)

t? 16

1 M i ' A

Cmck ! ,-V L-' -1'-" * - , sd l_ ,_ ,_ ->_,_i

; j

i 1 V

10

Local mesh (front view) Geometry of the crack

Figure 3. Meshes of an inclined semi-circle surface crack subjected to tensile loading

The von Mises stress on the local mesh is shown in Figure 4. The quantitative evaluation such as stress intensity factor will be announced at the oral presentation.

244

Figure 4. von Mises stress of inclined semi-circle surface crack

Conclusions

A new methodology to analyze three dimensional crack problems was presented. In this method, overlaying mesh method and X-FEM are used. The former is the technique in which the accuracy of the analysis in local area is increased by superimposing additional mesh of higher resolution. In X-FEM, the finite element space is enriched by adding special function to the approximation using the notion of partition of unity. This technique allows the entire crack to be represented independently of the mesh. In the case of analyzing crack problems in complex shape structures, the mesh of the whole model needs not to be re-meshed. Only additional mesh around the crack which is not conformed to the crack surface is needed. As numerical example, an inclined semi-circle surface crack under tension is analyzed.

References

Fish J (1992), "The s-version of the finite element method", Computers & Structures, 43(3), 539-547.

Fish J and Markolefas (1993), "Adaptive s-method for linear elastostatics", Computer Methods in Applied Mechanics and Engineering, 104, 363-396.

Fish J and Guttal R (1996), "The s-version of finite element method for laminated composites", IntJ. Numer. Meth. Engng, 39,3641-3662.

Belytschko T and Black T (1999), "Elastic crack growth in finite elements with minimal rerneshing", IntJ. Numer. Meth. Engng, 45(5),601-620

Mogs N, Dolbow J and Belytschko T (1999), "A finite element method for crack growth witiiout rerneshing" Int. J. Numer. Meth. Engng, 46,131-150

Babuska I (1997), "The partition of unity method", Int. J. Numer. Meth. Engng, 40,727-758

Sukumar N, Mogs N, Moran B and Belytschko T (2000), "Extended finite element method for three-dimensional crack modeling", Int. J. Numer. Meth. Engng, 48,1549-1570

N. Mogs N, Gravouil A and Belytschko T (2002), "Non-planar 3D crack growth by the extended finite element and level sets - Part I: Mechanical model", Int. J. Numer. Meth. Engng, 53, 2549-2568

245


BUCKLING ANALYSIS OF COMPOSITE LAMINATES WITH DELAMINATIONS

USING X-FEM

T. Nagashima and H. Suemasu Faculty of Science and Technology, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo

102-8554, JAPAN nagashim@me. sophia. ac.jp, suemasu@sophia. ac.jp

Abstract

Carbon fiber-reinforced plastic (CFRP) composite materials are extensively used in engineering applications. These materials are used in the form of laminates for aerospace structures. Structures constructed of composite laminates may acquire impact damage, for example in the form of delamination. Therefore, clarification of the damage mechanism of composite laminates for structural design using composite materials is very important. In analytical studies, finite element models that consider delaminations in composite laminates have been used. Considerable effort and time are usually required in order to prepare the finite element meshes for creating models. In particular, for structures containing discontinuities such as delaminations the meshes cannot be constructed easily, even if the automatic mesh generation technique is used. Recently, Belytschko et al. proposed the extended finite element method CXFEM) based on the concept of partition of unity. They applied this method to the evaluation of stress intensity factors and performed crack extension simulation. X-FEM can be used to simplify the modeling of continua containing several cracks and hence can be used to perform effective stress analyses related to fracture mechanics. In the present study, X-FEM is applied to buckling analyses of composite laminates with holes and delaminations. The interpolation functions of solid finite elements used in three-dimensional analysis are extended to perform eigenvalue analyses for buckling loads in composite laminates. The numerical results show that X-FEM is an effective method in performing buckling analyses in composite laminates with a hole and with a delamination, respectively.

Keywords: X-FEM, Composite laminate, Crack, Open hole, Delamination.

Introduction

Carbon fiber-reinforced plastic (CFRP) composite materials are used extensively in a number of engineering applications. These materials are used in the form of laminates for aerospace structures. The composite laminate structures may sustain impact damage, including delamination and transverse cracks. Therefore, in the field of structural design, understanding the damage mechanism of composite laminates is very important. Several experimental and analytical studies have been performed (Suemasu et al, 1998). A number of analytical studies have used finite element models that consider cracks and delamination in composite laminates.

The finite element method (FEM) is widely used in industrial design applications and several different software packages have been developed based on FEM techniques. However, considerable effort and time are usually required to prepare the finite element meshes for creating models. In particular, for structures containing discontinuities such as cracks, delaminations and voids, meshes cannot be constructed easily, even if the automatic mesh generation technique is used. Recently, Belytschko et al. (1999; Moes et

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246

al, 1999) proposed the extended finite element method (X-FEM) based on the concept of partition of unity (Melenk et al, 1996; Babuska et al, 1997). X-FEM was applied to the evaluation of stress intensity factors and was used to perform crack extension simulation. X-FEM can be used to simplify the modeling of continua containing several cracks, and hence can be used to perform effective stress analyses related to fracture mechanics. In addition, X-FEM can exploit finite element techniques, so implementation and application are relatively simple.

Belytschko et al (1999; Moes, 1999) evaluated the stress intensity factors of cracks in homogeneous isotropic material using X-FEM. One of the present authors applied X-FEM to evaluate bi-material interface cracks (Nagashima et al, 2001). In addition, X-FEM has been applied to crack analysis in three-dimensional problems (Sukumar et al., 2000) The present authors have applied X-FEM to effectively evaluate the fracture mechanics of composite laminates containing cracks, interlaminar delaminations and open holes (Nagashima, 2002). In the present study, X-FEM is applied to stress analysis of composite laminates, which are composed of several orthotropic plates. In the present paper, the interpolation functions of plate and solid elements used in conventional FEM analyses are extended using the concept of the partition of unity. Eigenvalue analyses for buckling load are performed for composite laminates with a hole and for a delamination, and the results obtained by X-FEM are compared to those obtained by conventional FEM.

The present paper is organized as follows. An outline of modeling by X-FEM is presented in section 2, and the finite elements utilized in the analysis of composite laminates are outlined in section 3. Numerical results of the linear buckling analysis of a composite laminate containing an open hole and an interlaminar delamination by X-FEM are presented in section 4, and the results of the present study are summarized in section 5.

X-FEM

In the extended finite element method (X-FEM), the finite element approximation is enriched by additional functions through the notation of partition of unity (Melenk et al., 1996; Babuska et al, 1997). In X-FEM, the approximate displacement function uh of the distributed displacement u is generally expressed as:

u * ( x ) = 2 > / ( x ) u / + 2 > ( x ) a , / ( x ) / J " J E N «jetid (1)

where N is the set of all nodes in the finite element mesh, Uj is the degree of freedom at node I, (#/is the shape function associated with node I, A^ is the subset of nodes enriched for the discontinuity, aj is the corresponding additional degree of freedom, and f(x) is the enrichment function.

The first term on the right-hand side of Eq. (1) is the classical term of the interpolation function and the second term is the enriched term. In this section, the interpolation functions used in X-FEM to model cracks and free surfaces are described.

Crack modeling

In X-FEM, the approximate displacement function wh of the distributed displacement u

247

near the crack is expressed as:

u*(x) = j > , ( x ) u , + 2 > 7 ( x ) £ n ( x ) a * + X ^ ( x ) ^ ( x ) b ; (2) /=1 / k=\ I

where fa is the interpolation function as used in the formulation of the conventional FEM, m is the number of nodes in the finite element, C and J denote the node set considering the singularity of the stress and the discontinuity of displacement near the crack, respectively, and m, aik, and bi denote the vector of freedoms assigned to each node. Here, C n J=§ is satisfied. In addition, y(i=l,4) are the basis for reconstructing the asymptotic solution of the displacement, and H(x) is the Heaviside function used to express the discontinuity of displacement.

Although cracks in orthotropic materials are treated in the present paper, the basis y* is determined from the asymptotic solution of a crack in homogeneous isotropic material and is defined as follows:

y, = 4r cos(—), y2 = 4r sin(—), yz=4r sin(—) sin 0,y^=4r cos(—) sin 9 (3)

where r and #are polar coordinates in a plane defined near the crack front.

Free surface modeling

In order to express the discontinuity of displacement near the surface of voids in the analyzed domain and free surfaces of geometry boundaries, the step function V(x) can be utilized as an enrichment function (Sukumar et al, 2001). The step function V(x) takes the value one inside the domain and the value zero outside the domain. The approximate displacement function uh of the distributed displacement u near free surfaces is expressed as:

uh(x)=fi^l{x)V(x)nI ( 4 )

where fa is the interpolation function as used in the formulation of conventional FEM, m is the number of nodes in the finite element, F denotes the node set considering the discontinuity of displacement near free surfaces, and ui denotes the vector of freedoms assigned to each node. Here, V(x) is the step function used to express the discontinuity of displacement. This interpolation function, the value of which vanishes outside the analyzed domain, can express the displacement field near free surfaces.

Enriched Finite Element

In this section, the finite elements, which are extended by X-FEM based on the partition of unity concept are outlined. In the present paper, the enriched plate and solid elements are discussed only briefly.

Plate element

The plate element examined herein is a four-node quadrilateral element that has five degrees of freedom for each node. This plate element (MITC element) (Dvorkin et al, 1984) is based on Mindlin plate theory, and in the formulation of the element, the out-of-plane shear strain is directly interpolated in order to avoid shear locking even if the full

248

integration is performed. The classical laminate theory, which assumes that the distribution of strain in each laminate is continuous throughout the thickness of the composite laminate, is used. This type of plate element can also be degraded to a three-node triangular plate.

Solid element

The solid element examined herein is an eight-node isoparametric hexahedral element that has three degrees of freedom for each node. In solid modeling of the composite laminates, the aspect ratio of the elements tends to be large in order to avoid the excessively increasing the number of elements. Therefore, the incompatible mode, which can express bending deformations, is considered to obtain appropriate stiffness (Macneal, 1994). The freedoms for the incompatible deformation modes can be removed by static condensation in linear elastic analysis before the components of their local stiffness are added to the system equations.

Numerical Examples

Numerical examples for the composite laminates solved by X-FEM will be presented in this section. In the present calculation, a four-node plate element and an eight-node solid element are used, and the 6th-order Gauss integration are adopted to evaluate the local stiffness, mass and initial stress matrices containing any enriched elements. Fiber-reinforced material is assumed in this example. The elastic properties of the unidirectional ply are £i=142 GPa, ET =10.8 GPa, GLT =5.49 GPa, GTT=3.71 GPa, vlT

=0.3, VTT=0.45 and p=1.5><10"6 kg/mm3, and its stacking sequence is [45/-45/0/90]s. for the present analysis.

Eigenvalue analysis for buckling load of a composite laminate with a hole

In this example, a laminated square plate with an open hole was analyzed as shown in Figure 1, Eigenvalue analysis for buckling load was performed. The square plate with an open hole, which was clumped at the bottom under compressive load, as shown in Figure 1, was analyzed by X-FEM using plate elements (Number of nodes: 441; Number of elements: 400). The conventional finite element analysis using a three-node triangular plate elements (Number of nodes: 551; Number of elements: 989) was also performed for comparison. The finite element meshes for both X-FEM and FEM analyses are shown in Figure 2. The structured mesh was used for X-FEM analysis, and the open hole was defined independent of the finite elements. In addition, the same problem was solved by X-FEM using solid elements (Number of nodes: 3,969; Number of elements: 3,200). The mesh of the solid model was generated by copying the node distribution of the plate model. The linear buckling eigenvalue of the laminated plate with an open hole, which was clumped at the bottom under compressive load, as shown in Figure 1, was calculated by X-FEM using both plate and solid elements. The conventional finite element analysis using three-node triangular plate elements was also performed for comparison. The five smallest buckling loads were calculated. The results were summarized in Table 1. The plate and solid analyses by X-FEM were found to provide solutions as appropriate as those obtained by conventional FEM.

249

Buckling analysis of a laminated square plate with a delamination

The eigenvalue analysis for linear buckling load was performed for the laminated square plate with an interlaminar delamination under compressive load, as shown in Figure 3. A square 50 mm x 50 mm delamination located 0.25 mm from the surface, between a 45°-layer and a 0°-layer, was assumed. Solid elements can model the interlaminar delamination directly. Moreover, X-FEM can model the delamination without "double nodes". The examples of the distribution of nodal properties for modeling the delamination are shown in Figure 4. The nodes indicated by "J" were enriched by the basis function H(x) (Heaviside function), which can express the discontinuity near a crack. In Model-B, shown in Figure 4(b), the node indicated by "C" was enriched by the basis function, which can reconstruct the asymptotic displacement solution near a crack tip. X-FEM analyses using Model-A and Model-B, as shown in Figure 4, were performed. Conventional FEM analysis using double nodes was also performed for comparison. The results for buckling load are summarized in Table 2. No differences were observed between the X-FEM (Model-A) and conventional FEM analyses. The more accurate solution appears to be obtained by Model-B, as shown in Figure 4(b), using the enrichment function, which can reconstruct the asymptotic displacement solution near a crack tip in homogeneous isotropic materials. The modeling of delaminations by X-FEM is much easier than that by conventional FEM.

Concluding and Remarks

In the present paper, X-FEM was applied to the structural analysis of composite laminates with discontinuity such as an open hole and an interlaminar delamination. The MITC-type plate element based on Mindlin plate theory and the solid element including incompatible deformation modes were extended to express the discontinuity within the interpolation function independent of the finite element mesh, and the eigenvalue analysis for buckling load of laminated composite plate with an open hole and an interlaminar delamination were performed. Fairly good agreement was observed between the results obtained by X-FEM and those obtained by conventional FEM. X-FEM can model discontinuities such as cracks and free surfaces much easier than conventional FEM. Using such advantages allow adequate consideration for damage evaluation in composite materials.

References

Babuska, I. and Melenk, J.M., The partition of unity methods, Int. j . numer. methods eng., 40 (1997), 727-758. Belytschko, T. and Black, T., Elastic crack growth in finite elements with minimal remeshing, Int. j . numer.

methods eng., 45(1999), 601-620. Dvorkin, E. N. and Bathe, K. J., A continuum mechanics based four node shell element for general nonlinear

analysis, Eng., Comput. 1 (1984), 77-88. Macneal, R., H., Finite elements: their design and performance, Marcel Dekker (1994). Melenk, J. M. and Babuska, I., The partition of unity finite element method: Basic theory and applications, Comput.

Methods Appl. Mech. Engrg., 139 (1996), 289-314. Moes, N. , Dolbow, J. and Belytschko, T., A finite element method for crack growth without remeshing, Int. j .

numer. methods eng., 46(1999), 131-150. Nagashima, T., Omoto, Y. and Tani, S., Stress analysis of structures containing interface cracks by X-FEM,

abstracts of 6TH USNCC, (2001), 29. Nagashima, T., Stress Analysis of Composite Materials using X-FEM, Proceedings of the 43rd

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2002).

250

Suemasu, H., Kumagai, T. and Gozu, K., Compressive behavior of multiply delaminated composite laminates part 1: experimental and analytical development, AIAA Journal, Vol. 36, No.7, 1998, 1279-1285.

Suemasu, H. and Kumagai, T., Compressive behavior of multiply delaminated composite laminates part 2: finite

element analysis, AIAA Journal, Vol. 36, No.7, 1998, 1286-1290.

Sukumar, N., Moes, N., Moran, B. and Belytschko, T., Extended finite element method for three-dimensional crack

modeling, Int. j . numer. methods eng., 48(2000), 1549-1570.

Sukumar, N., Chopp, D. L., Moes, N. and Belytschko, T., Modeling holes and inclusions by level sets in the extended finite element method, Comput. Methods Appl. Mech. Engrg., 190(2001), 6183-6200.

i

1.0mm

Figure 1 Laminated plate with a circular hole

(a)X-FEM (b) FEM

Figure 2 Finite Element Mesh

Compressive Load : P

1.0mm

50m

m

*-

Wm9 VmmB, (a) Model -A

ielamination

ill :".::; i W>

\ m £^pH

mm

delamination

\ teifcW ??M:J

ids f.-.yrf. :.•;:.;;•

dm li i • r

•::«$;#

(b) Model -B

Figure 3 Laminated plate with a delamination Figure 4 Nodal properties for modeling a delamination

Table 1 Buckling load for a laminated plate with an open hole

Table2 Buckling load for a laminated plate with/without a delamination

order

1 2 3 4 5

X-FEM Plate

53.242 516.37 821.56 1199.3 1532.2

Solid 52.944 518.66 780.11 1156.4 1538.9

FEM Plate

53.520 518.60 827.94 1219.1 1539.1

Unit: N

order

1 2 3 4 5

Without delamination

FEM Solid

73.042 656.49 1183.8 1783.9 2131.0

Plate

71.665 642.72 1222.5 1782.1 2105.1

With delamination 50mm x 50mm at 0.25mm

FEM Double Node 72.125 626.33 915.81 1152.7 1563.5

X-FEM Model-A

72.125 626.33 915.81 1152.7 1563.5

Model-B

71.641 620.49 924.19 1144.8 1563.7

Unit: N

251


BOUNDARY CONDITION ENFORCEMENT IN VOXEL-TYPE FEM

T. Nagashima Faculty of Science and Technology, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo

102-8554, JAPAN The Institute of Physical and Chemical Research, 2-1 Hirosawa, Wako, Saitama

351-0198 JAPAN [email protected]. ac.jp

Abstract

The present paper introduces a stress analysis method which discretizes the weak-formed governing equations based on the modified variational method using both structured cellular data and an interpolation function enriched using the step function. In the proposed method, nodes located on the vertexes of cells and sub-cells, which are used as a unit for numerical integration of the weak form, are generated for each cell. Moreover, the definition of boundaries having complex geometries can be performed independently of the cell using the enriched interpolation function. The modified variational method eliminates the subjective constraint conditions due to the essential boundaries. The proposed method is applied to analyses of two-dimensional elastic problems having small deformation, and appropriate results are obtained.

Keywords:, X-FEM, Modified variational method, Finite element, MLSM.

Introduction

The extended finite element method (X-FEM ), which uses interpolation functions that satisfy the partition of unity condition (Melenk et al., 1996; Babuska et al, 1997), has been proposed by Belytschko et a/.(1999; Moes et al., 1999), and X-FEM has been applied to the stress analyses of continua containing discontinuity of displacement, such as cracks. X-FEM, which uses the interpolation function enriched by the step function, can also model the discontinuity of the free surface independent of the finite element mesh (Sukumar et al, 2001). This method is essentially identical to the stress analysis method using several cubic solid finite elements, which are often referred to as "Voxels", to model structures having complex geometry. The authors have been investigating a stress analysis method using voxel-type finite elements generated by the volume CAD (V-CAD) developed in RIKEN (Nagashima, 2002), which can handle the volume data directly.

Structured data, such as voxel data, for modeling the analyzed domain simplifies the definition of the analysis model, because the location of the cell including the specified point can be searched easily by simple calculation related to the coordinates, since the information of element-connectivity, which should be described as input data in the conventional FEM, is not required. Moreover, the interpolation function enriched by the step function used in X-FEM analysis can simplify the definition of the boundaries

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252

having complex geometries. In addition, the modified principle of virtual work(Washizu, 1975),enables the simple treatment of the boundary conditions, particularly the essential boundary conditions (Lu et al., 1994). Therefore, the present paper proposes a stress analysis method using the aforementioned procedures. Namely, the proposed method has several characteristics, which are described as follows:

• The modified principle of virtual work is used as the guiding principle for discretizing the governing equation.

• The proposed method does not require mesh data, i.e. the element-connectivity information, as input data. The proposed method generates the structured cellular data automatically and performs the numerical integration for the weak-formed governing equations using this data.

• The proposed method uses the interpolation function enriched by the step function, which can model boundaries having complex geometry.

In the proposed method, the conventional interpolation function used in FEM can be derived by assuming a cell to be a quadrilateral finite element in two dimensions (or a hexahedral finite element in three dimensions). However, the proposed method does not determine the type of interpolation function because a cell is a utilized only for the integration of the weak-formed governing equations. Therefore, in the proposed method, the interpolation function produced by MLSM can be used. Moreover, the proposed method can solve the three-dimensional problem.

Governing equations

The governing equation for the two-dimensional problem with small displacement on the domain Q bounded T is described as follows:

V»o + b = 0 in Q (1) where cr is the stress tensor and b is a body force vector.

The boundary conditions are given as follows:

a • n = t on T, (2.1)

u = u on Tu (2.2) where t is the traction vector, u is the displacement vector, n is the unit vector normal to the domain Q and the superposed bar denotes prescribed boundary values. Tt and Tu

represent the natural boundary and the essential boundary, respectively.

A modified variational principle, which is equivalent to Eqs. (1), (2.1) and (2.2), is introduced as follows:

f Vs6\:<sdXl- jdv'bdn- J3v•IdT- jd\»(u-a)aT- jdvtaT = 0 (3) n r, r„ r„

where Sv is a test function andVs6\ is the symmetric part of V<5v.

Equation (3) is accompanied by no subjective conditions.

Moreover, for the linear elasticity, the following equations are used:

E = VIu, O = D : E (4)

253

where e is the strain tensor and D is the elastic tensor.

Equations (3) and (4) provide the principle by which to discretize the governing equations in the proposed method.

Numerical method

In this section, the stress analysis method based on the modified variational method shown in the previous section is described for two-dimensional elastic problems with small displacement. Numerical examples using the method described in this section are shown in next section.

Definition of boundaries and cells

In the proposed method, the boundary line segments are used to define the two-dimensional shape. The cells, which cover the entire region enclosed by the boundary line segments, are generated. These cellular structures make it possible to search the location of a cell quickly. Moreover, nodes are generated at the vertex location of each cell. These cells have almost the same function as background cells, as used in EFGM (Belytschko et al, 1994) for the purpose of integrating the weak form. The primary difference is that nodes exist at the four vertexes when constructing a cell in the proposed method. Based on the location of the four nodes of a cell, the cell is classified as one of three types: boundary cell, inner cell or outer cell. A cell having all four nodes located inside the region is classified as an inner cell, and the cell. A cell having all four nodes located outside the region is classified as an outer cell. Other cells are classified as boundary cells, as shown in Figure 1.

Numerical integration of the weak form

In order to integrate the weak-formed governing equations shown in the previous section, the aforementioned cells and boundary line segments are used. The ordinary Gauss-Legendre numerical integration method is adopted for domain integration of inner cells. Moreover each boundary cell is divided into several sub-cells. The sub-cell division can be performed arbitrary; however, in the rectangular cell, the edge is equal to the length. In boundary cells, domain integral is performed for each sub-cell. Thus, the weak form is evaluated at the center of the sub-cell. In order to determine whether the center point of a sub-cell lies inside or outside the region, boundary line segments are utilized. In order to discretize the boundary integration term of the weak form based on the modified variational method, several integration points on the boundary line segments are used.

Approximation function

In the proposed method, any approximation functions, which can be evaluated using the values of neighbor nodes, can be used. If each cell is treated as a conventional quadrilateral finite element, the standard interpolation function as used in the conventional FEM is available. Alternately the interpolation function based on MLSM (Lancaster et al, 1981; Belytschko et al, 1994) using the nodes surrounding the evaluation point can be used. Because both interpolation functions satisfy the partition of unity condition, these functions can be enriched using the step function, the value of

254

which is one inside the domain and zero outside the domain, in order to describe an arbitrary boundary shape. In the present paper, the interpolation function based on the finite element approximation and MLSM approximation is utilized. In general, the approximation function «h of any point x is expressed as follows:

u*(x) = 5 > / « r « « , (5)

where V (x), which is referred to as the step function, takes the value of one inside the domain and zero outside the domain.

Nodal properties

In the proposed method, all of the vertexes of each structured cell correspond to nodes. The interpolation functions as described above can descretize the weak-formed govering equations and generate the stiffness matrix associated with nodal freedoms in the same manner as the conventional finite element method. However, for some nodes, the degree of freedom is not associated with the global stiffness. The degrees of freedom of such nodes are perfectly removed from the system equations, and the node is called a "dead node". Therefore, the nodal properties associated with the cell type, as shown in Figure 1, are defined as follows:

• A node belonging to at least one boundary cell is defined as an enriched node. • A node which is never referred to from any integration point is defined as a dead

node. • Other nodes are defined as normal nodes.

Numerical examples

In this section, numerical examples of two-dimensional problems are solved using the proposed method. The finite element approximation and moving least squares approximations are used in the calculation. In the analyses presented in this section, two-dimensional problems having small displacement are solved. An isotropic elastic material having a Young's modulus of 21,000 kgf/mm2 and a Poisson's ratio of 0.3 is assumed. Moreover, the order of Gaussian integration points is set to two for the normal cell and each boundary cell is divided into 36 equally divided squares and then integrated. For the boundary integration, the integration point is set in the interval length, which is determined by the size of each cell.

Analysis of a plate with a hole under tension load

The square plate with a hole was analyzed under the loading condition shown in Figure 2. The plane stress problem was solved using structured cells. Conventional finite element analysis was also performed for comparison. The cell and finite element mesh used in the analysis are shown in Figure 3. Both the finite element approximation and the moving least squares interpolation were used as interpolation functions. The maximum and minimum stress components obtained by these calculations are compared in Figure 4. The solutions obtained using the proposed method were found to be appropriate.

255

Analysis of a cylinder under internal pressure

A cylinder under internal pressure, as shown in Figure 5, was analyzed. The plane strain problem was solved using 31x31 structured cells. Conventional finite element analysis was also performed for comparison. The cell and finite element mesh used in the analysis are shown in Figure 6. Both the finite element approximation and the moving least squares interpolation were used as interpolation functions. The maximum and minimum stress components obtained by these calculations are compared in Figure 7. The solutions obtained using the proposed method were found to be appropriate.

Concluding remarks

The present paper proposed a stress analysis method based on the modified variational method. The proposed method descretized the weak formed governing equation using structured cells, which cover the entire domain of the model. Because the essential boundary conditions are introduced into the system equations, the obtained equations can be solved directly without the subjective conditions. As numerical examples, two-dimensional plane stress and plane strain problems are solved using the proposed method and the results were compared to those obtained by conventional FEM. The solutions obtained using the proposed method were found to be appropriate.

References

Babuska, I. and Melenk, J. M., The partition of unity methods, Int. j . numer. methods eng., 40 (1997), 727-758.

Belytschko, T., Lu, Y.Y. and Gu, L., Element-free Galerkin methods, Int.j.numer.methods eng., 37 (1994), 229-256.

Belytschko, T. and Black, T., Elastic crack growth in finite elements with minimal remeshing, Int. j . numer. methods eng., 45(1999), 601-620.

Lancaster, P. and Salkauskas, K., Surface generated by moving least squares methods, Math. Comp. 37 (1981) 141-158.

Lu, Y.Y., Belytschko, T. and Gu, L., A new implementation of the element free Galerkin method, Comput. Methods Appl. Mech. Engrg., 113 (1994), 397-414.

Melenk, J.M. and Babuska, I., The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg., 139 (1996), 289-314.

MoSs, N., Dolbow, J. and Belytschko, T., A finite element method for crack growth without remeshing, Int. j . numer. methods eng., 46(1999), 131-150.

Nagashima, T., Ishihara, Y., Niiyama, K. and Makinouchi, A., Development of Stress Analysis System by X-FEM with Voxel-Type Mesh, Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), July 7-12, 2002, Vienna, Austria, Editors: Mang, H.A., Rammerstorfer, F.G. and Eberhardsteiner, J., Publisher: Vienna University of Technology, Austria, ISBN 3-9501554-0-6, http://wccm.tuwien.ac.at.

Sukumar, N., Mo6s, N., Moran, B. and Belytschko, T., Extended finite element method for three-dimensional crack modeling, Int. j . numer. methods eng., 48 (2000), 1549-1570.

Washizu, K., Variational Methods in Elasticity and Plasticity, 2nd edition (Pergamon, New York, 1975).

http://wccm.tuwien.ac.at

256

• Enriched Node • Dead Node • Normal Node

Boundary

1 kgf/mm2

T f f t thickness: lmm

Plane stress

I Inner Cell H I Boundary Cell • Outer Cell

Figure 1 Nodal properties Figure 2 Plate with a hole under tension load

Number of nodes:l,680 Number of elements: 1,600

(a) FEM model

Number ofiiodes:l,024 Number of cells: 961

I f ff*T::3ffi iffl |l.]L^i.:l.|p..iiijj jf»i;3;;:;f:p;ii| Mff:l!l^p4S i ipfcii l f||:if||m:lP

(b) Cell model

I hEMfquad) j frb-typc I MTSr-typc

3ig-X MAX sig-X MIN jig-Y MAX sig-Y MINau-XY M A » u - X Y MJNUises MAXMises MIN

Figure 3 Models for the analysis of a plate Figure 4 Calculated components of stress with a hole

Plane strain

«tf emal pressure p

in the plate Number of nodes:336 Number of elements:300

Mil (a) FEM model

Number of nodes: 1,746 Number of cells: 1,681

IffiM

SITUT

fell

wr

I f frln*

#

1 n

;|l

a, H * l i t

m

11M : asi

t ' 4"fi

3 *!H

p

ti\

^fr

(b) Cell model

Figure 5 Cylinder under internal pressure Figure 6 Models for the analysis of a cylinder

Figure 7 Calculated components of stress in the cylinder

Author Index

257

A

Ando,T. 194

Arimoto,S. 109

Karageorghis, A. 17

Karim,Md. R. 115

Kawashima, T. 77

Chen, C. S. 15 Chen,X.L. 84 Cheng, J.Q. 49 Chin, G. L. 217,223

D Dai,K.Y. 29,43

Fairweather, G. 17 Fan,S.C. 229 Frijters,E. 200

Gu,Y.T. 23,96,167,200

H

Hagihara, S. 63 Hon, B. Y. C. 16

I Ikeda,T. 63 Imasato, J. 69 Iraha,S. 57,123

Lam, K. Y. 35,49,140, 211, 217,223

Li,H. 49

Li, J. 15

Lim,K.M. 29,43

Lim,S.P. 84

Liu, G. R. 29,35,43,84, 96,135,140,

167,200,211,217,223

Liu,L. 90,96

Liu,M.B. 211

Liu,X. 35,140

M

Martin, P. A. 17 Matsubara, H. 57 Miyazaki, N. 63 Murakami, A. 109

N Nagashima,T. 245,251 Nakama,Y. 194 Nakasumi,S. 239 Ng,T.Y. 49 Nogami,T. 115,161 Noguchi,H. 77

J J i n , C R . 63

O Ohtsubo,H. 7,239

K Kanok-Nukulchai, W. 179 Kanto,Y. 194

Pepper, D. W. 15

258

Q Qiang,H.F. 229

S Sakai,Y. 69 Sato,Y. 77 Shimada, A. 194 Suemasu, H. 245 Suzuki, K. 7,239

T Tai,K. 35,140 Tan, V. B.C. 90,96 Tomiyama, J. 57,123 Tsunori, M. 63

W Wang, J. G. 115,161 Watanabe,T. 63 Wu,Y. L. 129,135

X Xie,H. 161

Y

Yagawa,G. 3,57,123,194 Yamada,Y. 123 Yao,Z.H. 151 Yew,Y.K. 49 Yin,X.P. 179

Z Zhang, J. M. 151

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