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Scattering of flexural wave in thin plate with multiple holes by using the null-field integral equation method

Wei-Ming Lee1, Jeng-Tzong Chen2

Ching-Lun Chien1, Yung-Cheng Wang1

1 Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan

2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan

2008 年 05 月 14日 台北科技大學

National Taiwan Ocean UniversityMSVLAB ( 海大河工系 )

Department of Harbor and River Engineering

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Introduction

Circular holes can reduce the weight of the whole structure or to increase the range of inspection.

Geometric discontinuities result in the stress concentration, which reduce the load carrying capacity.

The deformation and corresponding stresses produced by the dynamic force are propagated through the structure in the form of waves.

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Scattering

At the irregular interface of different media, stress wave reflects in all directions scattering

The scattering of the stress wave results in the dynamic stress concentration

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Overview of numerical methods

Finite Difference M ethod Finite Element M ethod Boundary Element M ethod

M esh M ethods M eshless M ethods

Numerical M ethods

6

PDE- variational IEDE

Domain

BoundaryMFS,Trefftz method MLS, EFG

開刀 把脈

針灸

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Literature review

From literature reviews, few papers have been published to date reporting the scattering of flexural wave in plate with more than one hole.

Kobayashi and Nishimura pointed out that the integral equation method (BIEM) seems to be most effective for two-dimensional steady-state flexural wave.

Improper integrals on the boundary should be handled particularly when the BEM or BIEM is used.

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MotivationMotivation

Numerical methods for engineering problemsNumerical methods for engineering problems

FDM / FEM / BEM / BIEM / Meshless methodFDM / FEM / BEM / BIEM / Meshless method

BEM / BIEMBEM / BIEM

Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity

Boundary-layer Boundary-layer effecteffect

Ill-posed modelIll-posed modelConvergence Convergence raterate

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Objective

For the plate problem, it is more difficult to calculate the principal values

Our objective is to develop a semi-analytical approach to solve the scattering problem of flexural waves and dynamic moment concentration factors in an infinite thin plate with multiple circular holes by using the null-field integral formulation in conjunction with degenerate kernels and Fourier series.

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Flexural wave of plate

4 4( ) ( ),u x k u x xÑ = Î WGoverning Equation:

u is the out-of-plane displacement k is the wave number

4 is the biharmonic operator

is the domain of the thin plates

u(x)

24

3

12(1 )

hk

D

E hD

w r

n

=

=-

ω is the angular frequencyρ is the surface density

D is the flexural rigidityh is the plates thickness

E is the Young’s modulusν is the Poisson’s ratio

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Problem Statement

Problem statement for an infinite plate with multiple circular holes subject to an incident flexural wave

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The integral representation for the plate problem

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Kernel function

The kernel function is the fundamental solution which satisfies

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The slope, moment and effective shear operators

slope

moment

effective shear

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Kernel functions

In the polar coordinate of

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Direct boundary integral equations

Among four equations, any two equations can be adopted to solve the problem.

displacement

slope

with respect to the field point x

with respect to the field point x

with respect to the field point x

normal moment

effective shear force

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x

s

eU

O

iUr

qf

xr

Rf

Expansion

Degenerate kernel (separate form)

Fourier series expansions of boundary data

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Boundary contour integration in the adaptive observer system

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Vector decomposition

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Transformation of tensor components

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Linear system

where H denotes the number of circular boundaries

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Techniques for solving scattering problems

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Case 1: An infinite plate with one hole

Geometric data:a =1mthickness=0.002mBoundary condition:Inner edge : free

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Distribution of DMCF on the circular boundary by using different

methods, the present method, analytical solution and FEM

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Case 2: An infinite plate with two holes

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Distribution of DMCF on the circular boundary by using different methods, the present method and FEM

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Outlines

4. Concluding remarks

3. Illustrated examples

2. Methods of solution

1. Introduction

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Concluding remarks

A semi-analytical approach to solve the scattering problem of flexural waves and to determine DMCF in an infinite thin plate with multiple circular holes was proposed

The present method used the null BIEs in conjugation with the degenerat

e kernels, and the Fourier series in the adaptive observer system.

The improper integrals in the direct BIEs were avoided by employing the

degenerate kernels and were easily calculated through the series sum.

The DMCFs have been solved by using the present method in comparison with the available exact solutions and FEM results using ABAQUS.

1.

2.

3.

4.

5.

Numerical results show that the closer the central distance is, the larger

the DMCF is.

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Thanks for your kind attention

The End