ann sided polygonal edge based smoothed finite element method nes fem for solid mechanics

8/10/2019 ANN Sided Polygonal Edge Based Smoothed Finite Element Method NES FEM for Solid Mechanics

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. (2010)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1375COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING

An n-sided polygonal edge-based smoothed nite element method(nES-FEM) for solid mechanics

T. Nguyen-Thoi 1,3,, , G. R. Liu 1,2 and H. Nguyen-Xuan 2,3

1 Center for Advanced Computations in Engineering Science ( ACES ), Department of Mechanical Engineering , National University of Singapore , 9 Engineering Drive 1 , Singapore 117576 , Singapore

2 Singapore-MIT Alliance (SMA), E4-04-10 , 4 Engineering Drive 3 , Singapore 117576 , Singapore3 Department of Mechanics , Faculty of Mathematics and Computer Science , University of Science , Vietnam

National UniversityHCM , 227 Nguyen Van Cu , District 5 , Hochiminh City , Vietnam

SUMMARY

An edge-based smoothed nite element method (ES-FEM) using triangular elements was recently proposedto improve the accuracy and convergence rate of the existing standard nite element method (FEM) forthe elastic solid mechanics problems. In this paper the ES-FEM is further extended to a more generalcase, n-sided polygonal edge-based smoothed nite element method ( nES-FEM), in which the problemdomain can be discretized by a set of polygons, each with an arbitrary number of sides. The simpleaveraging point interpolation method is suggested to construct nES-FEM shape functions. In addition,a novel domain-based selective scheme of a combined nES / NS-FEM model is also proposed to avoidvolumetric locking. Several numerical examples are investigated and the results of the nES-FEM are foundto agree well with exact solutions and are much better than those of others existing methods. Copyrightq 2010 John Wiley & Sons, Ltd.

Received 7 July 2009; Revised 23 November 2009; Accepted 23 November 2009

KEY WORDS : numerical methods; nite element method (FEM); smoothed nite element methods

(S-FEM); edge-based smoothed nite element method (ES-FEM); node-based smoothednite element method (NS-FEM); polygonal element

1. INTRODUCTION

The strain smoothing technique has been proposed by Chen et al. [1] to stabilize the solutionsin the context of the meshfree method [2, 3 ] and then applied in the natural element method [4].Liu [5] has generalized the gradient (strain) smoothing technique and applied it in the meshfreecontext to formulate the node-based smoothed point interpolation method (NS-PIM or LC-PIM)[6, 7 ] and the node-based smoothed radial point interpolation method (NS-RPIM or LC-RPIM)[8]. Applying the same idea to the nite element method (FEM), a cell-based smoothed nite

element method (SFEM or CS-FEM) [912 ] and a node-based smoothed nite element method(NS-FEM) [13, 14 ] have also been formulated.

In the CS-FEM, the domain discretization is still based on quadrilateral elements as in FEM;however, the stiffness matrices are calculated based over smoothing cells (SC) located inside thequadrilateral elements as shown in Figure 1. When the number of SC of the elements equals1, the CS-FEM solution has the same properties with those of FEM using reduced integration.

Correspondence to: T. Nguyen-Thoi, Center for Advanced Computations in Engineering Science (ACES), Departmentof Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore.

E-mail: [email protected]

Copyright q 2010 John Wiley & Sons, Ltd.


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T. NGUYEN-THOI, G. R. LIU AND H. NGUYEN-XUAN

2

6

(d)

4

1

3

2

y

(a) x

1

3

4

2

(b)

4

1

6 3

52

: virtual nodes to form the smoothing cells

8

: field nodes

y

(c)

1 x

3

9

7

5

4

Figure 1. Division of quadrilateral element into the smoothing cells (SCs) inCS-FEM by connecting the mid-segment-points of opposite segments of smoothing

cells: (a) 1 SC; (b) 2 SCs; (c) 4 SCs; and (d) 8 SCs.

When SC approaches innity, the CS-FEM solution approaches to the solution of the standarddisplacement compatible FEM model [10 ]. In practical calculation, using four smoothing cellsfor each quadrilateral element in CS-FEM is easy to implement, work well in general and henceadvised for all problems. The numerical solution of CS-FEM (SC = 4) is always stable, accurate,much better than FEM, and often very close to the exact solutions. The CS-FEM has been developedfor general n-sided polygonal elements ( nCS-FEM) [15 ] dynamic analyses [16 ], incompressiblematerials using selective integration [17, 18 ], plate and shell analyses [1923 ], and further extendedfor the extended nite element method (XFEM) to solve fracture mechanics problems in 2Dcontinuum and plates [24 ].

In the NS-FEM, the strain smoothing domains and the integration of the weak form are

performed over the smoothing domains associated with nodes. The NS-FEM works well for trian-gular elements, and can be applied easily to general n-sided polygonal elements [13, 14 ] for 2Dproblems and tetrahedral elements for 3D problems. For n-sided polygonal elements, the smoothingdomain (k ) associated with the node k is created by connecting sequentially the mid-edge-pointto the central points of the surrounding n-sided polygonal elements of the node k as shown inFigure 2. When only linear triangular or tetrahedral elements are used, the NS-FEM produces thesame results as the method proposed by Dohrmann et al. [25 ] or to the LC-PIM [6] using linearinterpolation. The NS-FEM [13, 14 ] has been found immune naturally from volumetric lockingand possesses the upper bound property in strain energy as presented in [26 ]. Hence, by combiningNS-FEM and FEM with a scale factor [0, 1], a new method named as the alpha nite elementmethod ( FEM) [27 ] is proposed to obtain nearly exact solutions in strain energy using triangularand tetrahedral elements. The NS-FEM has been developed for adaptive analysis [28 ], primal-dualshakedown [29 ], 2D and 3D visco-elastoplastic analyses [30 ]. One disadvantage of NS-FEM is itslarger bandwidth of stiffness matrix compared with that of FEM, because the number of nodesrelated to the smoothing domains associated with nodes is larger than that related to the elements.The computational cost of NS-FEM therefore is larger than that of FEM for the same meshes used.

Recently, an edge-based smoothed nite element method (ES-FEM) [31 ] was also been formu-lated for static, free and forced vibration analyses in 2D problems. In the ES-FEM, the problemdomain is also discretized using triangular elements as in FEM; however, the stiffness matricesare calculated based on smoothing domains associated with the edges of the triangles. For trian-gular elements, the smoothing domain (k ) associated with the edge k is created by connectingtwo endpoints of the edge to the centroids of the adjacent elements as shown in Figure 3.The numerical results demonstrated that the ES-FEM [31 ] possesses the following excellent

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Biomed. Engng. (2010)

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(nES-FEM) FOR SOLID MECHANICS

node

: central point of n-sided polygonal elements: field node : mid-edge point

k (k )

(k )

Figure 2. n-sided polygonal elements and the smoothing domain (shaded area)associated with nodes in the NS-FEM.

: centroid of triangles

C(k )

B(k )

: field node

Ainner edge k

D

G

boundaryedge m

(m)

H

E(m)

Figure 3. Triangular elements and the smoothing domains (shaded areas)associated with edges in the ES-FEM.

properties: (1) ES-FEM is often found super-convergent and much more accurate than FEM usingtriangular elements (FEM-T3) and even more accurate than FEM using quadrilateral elements(FEM-Q4) with the same sets of nodes; (2) there are no spurious non-zeros energy modes and henceES-FEM is both spatial and temporal stable and works well for vibration analysis; (3) no additionaldegree of freedom is used; (4) a novel domain-based selective scheme is proposed leading to acombined ES / NS-FEM model that is immune from volumetric locking and hence works verywell for nearly incompressible materials. Note that similar to NS-FEM, the bandwidth of stiffnessmatrix in ES-FEM is larger than that in FEM-T3; hence, the computational cost of ES-FEM islarger than that of FEM-T3. However, when the efciency of computation (computation time forthe same accuracy) in terms of both energy and displacement error norms is considered, ES-FEM ismore efcient [31 ]. ES-FEM has been developed for 2D piezoelectric [32 ], 2D visco-elastoplastic[33 ], plate [34 ] and primaldual shakedown analyses [35 ]. In addition, the idea of the ES-FEM isquite straightforward to extend for the 3D problems using tetrahedral elements to give a so-calledthe face-based smoothed nite element method (FS-FEM) [36, 37 ].

In this paper, we aim to extend the ES-FEM to even more general case, i.e. n-sided polygonaledge-based smoothed nite element method (or nES-FEM) in which the problem domain can bediscretized by a set of polygons, each with an arbitrary number of sides. The simple averagingpoint interpolation method is suggested to construct nES-FEM shape functions. In addition, anovel domain-based selective scheme of a combined nES / NS-FEM model is also proposed toavoid volumetric locking. Several numerical examples are investigated and the results found agreewell with exact solutions and are much better than those of other existing methods. Note that,


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more information about the interpolants as well as the new integration techniques for the n -sidedpolygonal elements can be found in Refs [3843 ].

The paper is outlined as follows. In Section 2, the idea of the nES-FEM is introduced. InSection 3, the construction of nES-FEM shape function is presented and the domain-based selectivescheme nES / NS-FEM for nearly incompressible materials is described briey in Section 4. Section5 presents numerical implementations of the nES-FEM. Some numerical examples are presented

and examined in Section 6 and some concluding remarks are made in the Section 7.

2. BRIEFING OF n ES-FEM

2.1. Brieng on the FEM [4446 ]

The discrete equations of the FEM are generated from the Galerkin weakform and the integrationis performed on the basis of element as follows:

( s u )T D ( su ) d u Tb d t u T t d = 0 (1)where b is the vector of external body forces, D is a symmetric positive-denite (SPD) matrixof material constants, t is the prescribed traction vector on the natural boundary t , u is trialfunctions, u is test functions and s u is the symmetric gradient of the displacement eld.

The FEM uses the following trial and test functions:

u h (x ) = N n

I = 1N I (x )d I , u h (x ) =

N n

I = 1N I (x ) d I (2)

where N n is the total number of nodes of the problem domain, d I is the nodal displacement vectorand N I (x ) is the shape function matrix of I th node.

By substituting the approximations, u h and u h , into the weakform and invoking the arbitrarinessof virtual nodal displacements, Equation (1) yields the standard discretized system of algebraicequation:

K FEM d = f (3)

where K FEM is the stiffness matrix, f is the element force vector that is assembled with entries of

K FEM IJ = B T I DB J d (4)f I = N T I (x )b d + t N T I (x ) t d (5)

with the straindisplacement matrix dened as

B I (

x) = s

N I (

x) (6)

2.2. An nES-FEM

In the nES-FEM, the domain discretization is still based on polygonal elements of arbitrary numberof sides, but the integration using strain smoothing technique [1] is performed based on the edgesof the elements. In such an edge-based integration process, the problem domain is divided intosmoothing domains associated with edges such that = N edk = 1

(k ) and (i ) ( j ) =, i = j , inwhich N ed is the total number of edges of the problem domain. For n -sided polygonal elements,the smoothing domain (k ) associated with the edge k is created by connecting two endpoints of the edge to the two central points of the two adjacent elements as shown in Figure 4.


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I (AB, BI, IA)

: central point of elements (I, O, K): field node

(CK,KD,DO,OC )

C

(CD )

(CKDO )

edge k

(k)

D

O

( ABI )

boundary edge m ( AB)

(m)

B A

K

(m)

(k)

Figure 4. Domain discretization and the smoothing domains (shaded areas) associated with edgesof n-sided polygonal elements in the nES-FEM.

Applying the edge-based smoothing operation, the compatible strains = s u used in Equa-tion (1) is used to create a smoothed strain on the smoothing domain (k ) associated with edge k :

k = (k ) ( x ) k (x ) d = (k ) su (x ) k (x ) d (7)where k (x ) is a given smoothing function that satises at least unity property

(k )k (x ) d = 1 (8)

Using the following constant smoothing function:

k (x ) =1/ A(k ) , x (k )

0, x / (k )(9)

where A(k ) is the area of the smoothing domain (k ) , and applying a divergence theorem, one canobtain the smoothed strain

k =1

A(k ) (k ) n (k ) (x )u (x ) d (10)where (k ) is the boundary of the domain (k ) as shown in Figure 4, and n (k ) (x ) is the outwardnormal vector matrix on the boundary (k ) and has the following form for 2D problems:

n (k ) (x ) =

n (k ) x 0

0 n(k ) y

n (k ) y n(k ) x

(11)

In the nES-FEM, the trial function u h (x ) is the same as in Equation (2) of the FEM and thereforethe force vector f in the n ES-FEM is calculated in the same way as in the FEM.


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Substituting Equation (2) into Equation (10), the smoothed strain on the domain (k ) associatedwith edge k can be written in the following matrix form of nodal displacements:

k = I N (k )n

B I (x k )d I (12)

where N (k )n is the total number of nodes of elements attached with the common edge k and B I (x k )is termed as the smoothed straindisplacement matrix on the domain (k ) ,

B I (x k ) =

b I x (x k ) 0

0 b I y(x k )

b I y(x k ) b I x (x k )

(13)

and it is calculated numerically using

b I h (x k ) =1

A(k ) (k ) n (k )h (x ) N I (x ) d , ( h = x, y) (14)When a linear compatible displacement eld along the boundary (k ) is used, one Gaussian

point is sufcient for line integration along each segment of boundary (k )

i of (k )

, the aboveequation can be further simplied to its algebraic form

b I h (x k ) =1

A(k ) M

i = 1n (k )ih N I (x

GPi )l

(k )i , (h = x, y) (15)

where M is the total number of the boundary segments of (k )i , xGPi is the midpoint (Gaussian

point) of the boundary segment of (k )i , whose length and outward unit normal are denoted as l(k )i

and n(k )ih , respectively.Equation (15) implies that only shape function values at some particular points along segments

of boundary (k )i are needed and no explicit analytical form is required. This gives tremendousfreedom in shape function construction. It is therefore that a simple averaging method is suggested

to construct n

ES-FEM shape functions in Section 3.The stiffness matrix K of the system is then assembled by a similar process as in the FEM

K IJ = N ed

k = 1K (k )

IJ (16)

where K (k ) IJ is the stiffness matrix associated with edge k and is calculated by

K (k ) IJ = (k ) B T I D B J d = B T I D B J A(k ) (17)

3. CONSTRUCTION OF n ES-FEM SHAPE FUNCTION

In the nES-FEM, by using the smoothed strain of the cell (k ) associated with the edge k , thedomain integration becomes line integration along the boundary (k ) of the cell. Only the shapefunctions themselves at some particular points along segments of boundary are used and no explicitanalytical form is required. When a linear compatible displacement eld along the boundary (k )

is used, one Gauss point at midpoint on each edge is sufcient for accurate boundary integration.The values of the shape functions at these Gauss points, e.g. #g1 on the segment 1-A shown inFigure 5, are evaluated averagely using the values of shape functions of two related nodes on thesegment: points #1 and #A.

In order to construct a linear compatible displacement eld along the boundary (k ) of the cell,the following conditions of the shape functions for the discrete points of an n-sided polygonal


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2

3

4

5

7

8 9

: field node : central point of n-sided polygonal element

g1

1

A

6

B

g2

g3

g4

: Gauss point

Figure 5. Gauss points of the smoothing domains associated with edges for n-sidedpolygonal elements in the nES-FEM.

element need to be satised: (i) delta function: N i (x j ) = ij ; (ii) partition of unity:ni = 1 N i (x ) = 1;

(iii) linear consistency: ni = 1 N i (x )x i = x ; (iv) linear shape functions along lines connecting the

central point and eld points, e.g. line A-6 or line B-6; (v) values of the shape functions for thecentral points of n -sided polygonal elements, e.g. points #A or #B, are evaluated as

1n

1n

. . .1n

(size : 1 n ) (18)

with the coordinates of the central points calculated simply using

xc =1n

n

i = 1 xi , yc =

1n

n

i = 1 yi (19)

where the number of nodes n of the n -sided polygonal element may be different from one elementto the other and xi , yi are the coordinates of nodes of the n -sided polygonal element.

The condition (iii) is essential to reproduce the linear polynomial elds such as in the standard

patch test. With conditions (iv) and (v), a linear compatible displacement eld along the boundary(k ) of the cell (k ) associated with the edge k is constructed.Figure 5 and Table I demonstrate explicitly the shape function values at different points of the

smoothing domain associated with the edge 16. The number of support nodes for the smoothingdomain is 9 (from #1 to #9). We have 4 segments

(k )

i on (k )

(1 A, A6, 6 B, B1) . Each segmentneeds only one Gauss point, and therefore, there are a total of four Gauss points ( g1, g2, g3, g4)used for the entire smoothing domain (k ) , and the shape function values at these four Gausspoints can be tabulated in Table I by simple inspection.

The simple averaging shape functions presented above create a continuous displacement eld onthe whole domain of the problem. The strain eld of these non-mapped shape functions includesthe piecewise constant functions that are not continuous on the edges of elements and on theedges of smoothing cells. With such a construction, the simple averaging shape functions of thenES-FEM are similar to those in the nCS-FEM [15 ] and the nNS-FEM [13 ] using the n-sidedpolygonal elements.

It should be mentioned that the purpose of introducing the central points is to facilitate theevaluation of the shape function at the Gauss points and to ensure the linear compatible propertyalong the edges of the cell connecting the central point and eld points. No extra degrees of freedom are associated with these points. In other words, these points carry no additional eldvariables. This means that the nodal unknowns in the nES-FEM are the same as those in the FEMof the same mesh.

It is easy to see that the linear shape function of triangular elements of the standard FEMsatisfy naturally the ve above conditions. Hence, the nES-FEM can be applied easily to traditionaltriangular elements without any modication.


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Table I. Shape function values at different sites on the smoothing domain boundary associatedwith the edge 16 in Figure 5.

Site Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9 Description

1 1.0 0 0 0 0 0 0 0 0 Field node2 0 1.0 0 0 0 0 0 0 0 Field node3 0 0 1.0 0 0 0 0 0 0 Field node4 0 0 0 1.0 0 0 0 0 0 Field node5 0 0 0 0 1.0 0 0 0 0 Field node6 0 0 0 0 0 1.0 0 0 0 Field node7 0 0 0 0 0 0 1.0 0 0 Field node8 0 0 0 0 0 0 0 1.0 0 Field node9 0 0 0 0 0 0 0 0 1.0 Field node A 16

16

16

16

16

16 0 0 0 Centroid of element

B 15 0 0 0 0 1

515

15

15 Centroid of element

g1 7121

121

121

121

121

12 0 0 0 Mid-segment of (k )

i

g2 1121

121

121

121

127

12 0 0 0 Mid-segment of (k )

i

g3 110 0 0 0 0 6

101

101

101

10 Mid-segment of (k )

i

g4 610 0 0 0 0 1

101

101

101

10 Mid-segment of (k )

i

Different from the standard isoparametric nite elements, the nES-FEM obtains shape functionswithout using coordinate transformation or mapping. Hence, the eld domain can be discretizedin a much more exible way.

4. A DOMAIN-BASED SELECTIVE SCHEME: A COMBINED nES / NS-FEM

Volumetric locking appears when the Poissons ratio approaches 0.5. The application of selectiveformulations in the conventional FEM [47 ] has been found effectively to overcome such a lockingand hence the similar idea is employed in this paper. However, different from the FEM using

selective integration [47 ], our selective scheme will use two different types of smoothing domainsselectively for two different material parts ( -part and -part). Since the node-based smoothingdomains used in the NS-FEM were found effective in overcoming volumetric locking [13 ] and the

-part is known as the culprit of the volume locking, we use node-based domains for the -partand edge-based domains for the -part. The details are given below.

The material property matrix D for isotropic materials can be decomposed as

D = D 1 + D 2 (20)

where D 1 relates to the shearing modulus = E / [2(1 + )] and hence is termed as -part of D ,and D 2 relates to Lames parameter = 2 /( 1 2 ) and hence is termed as -part of D . For planestrain cases, we have

D =

+ 2 0

+ 2 0

0 0

=

2 0 0

0 2 0

0 0 1

+

1 1 0

1 1 0

0 0 0

= D 1 + D 2 (21)

and for axis-symmetric problems, we have

D =

2 0 0 0

0 2 0 0

0 0 1 0

0 0 0 2

+

1 1 0 1

1 1 0 1

0 0 0 0

1 1 0 1

= D 1 + D 2 (22)


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In our domain-based selective scheme proposed, we use the nNS-FEM to calculate the stiffnessmatrix related to -part and the nES-FEM to calculate the one related to the -part. The stiffnessmatrix of the combined n ES / NS-FEM model becomes

K = N ed

i = 1( B i1 )

T D 1 Bi1 A

i1

K nES FEM1+

N n

j= 1( B j2 )

TD 2 B j2 A

j2

K nNS FEM2(23)

where B i1 and Ai1 are the smoothed straindisplacement matrix and area of the smoothing domain

(i )ed associated with edge i , correspondingly; B

j2 and A

j2 are the smoothed straindisplacement

matrix and area of the smoothing domain ( j )n associated with node j , correspondingly; N n is thetotal number of nodes of the problem domain.

5. NUMERICAL IMPLEMENTATION

5.1. Domain discretization with polygonal elements of the Voronoi diagram

The discretization of a problem domain into n -sided convex polygonal elements can be performedusing the Voronoi diagram, and the procedure can be described as follows [48 ].

The problem domain and its boundaries are rst discretized by a set of properly scattered pointsP := { p1, p2 , . . . , pn }. Based on the given points, the domain is further decomposed into the samenumber of Voronoi cells C := { C 1, C 2 , . . . , C n } according to the nearest-neighbor rule dened by

C i = { x 2 : d (x , x i )< d (x , x j ), j = i} i (24)

In numerical implementation, as illustrated in Figure 6(a), the obtained Voronoi diagram hasnot cell vertices along the boundaries, then the articial nodes along the boundary outside thedomain need to be added to enclose the cells at the boundaries. With this, the boundary cellsare then bounded, but the vertices of these cells do not fall within the problem domain as shown

in Figure 6(b). Next, these vertices will be shifted inwards onto the boundaries as shown inFigure 6(c). The nal shape of these Voronoi cells is generally irregular but they are all convexpolygons. The initial point pi is regarded as the representative point of the i th element. Once weget the information of these Voronoi diagrams, a set of polygonal elements is then formed for ournumerical analysis.

0 1 2 3 4 5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

boundary of domain

0 1 2 3 4 5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

boundary of domain

0 1 2 3 4 5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

boundary of domain

(a) (b) (c)

Figure 6. (a) Voronoi diagram without adding the articial nodes along the boundary outsidethe domain; (b) Voronoi diagram with the articial nodes added along the boundary outside the

domain; and (c) nal Voronoi diagram.


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The following points need to be noted: (i) The original discrete points P only serve as numericaldevices for domain decomposition and do not function in the following numerical analysis; (ii) If we prefer more regular elements, such as rectangular elements, hexagon elements, we need toarrange a special point pattern P before the generation of Voronoi diagrams; (iii) For demonstrationpurpose, we arrange the initial points in an arbitrary form in the following numerical examplesregardless of the issue of computational cost. As a result, the number of elements sides is generally

changing from element to element; (iv) The nES-FEM does not require the elements to be convex;(v) Triangular elements used in the FEM can be used directly in the nES-FEM without anyalterations.

5.2. Procedure of the nES-FEM

The numerical procedure for the n ES-FEM is briey as follows:

1. Divide the domain into a set of elements and obtain information on nodes coordinates andelement connectivity;

2. Determine the area of cells (k ) associated with edge k and nd neighboring elements of each edge;

3. Loop over all the edges:

(a) Determine the nodal connecting information of cell (k ) associated with edge k ;(b) Determine the outward unit normal of each boundary side for cell (k ) ;(c) Compute the matrix B I (x k ) using Equation (13);(d) Evaluate the stiffness matrix using Equation (17) and force vector of the current cell;(e) Assemble the contribution of the current smoothing domain to form the system stiffness

matrix using Equations (16) and force vector;

4. Implement essential boundary conditions;5. Solve the system of equations to obtain the nodal displacements;6. Evaluate strains and stresses at nodes of interest.

In the n ES-FEM, the value of strains (or stresses) at the node i will be average value of strains(or stresses) of the smoothing domains (k ) associated with edges k around the node i , and iscalculated numerically by

i =1

A(i )

N (i )ed

k = 1k A(k ) (25)

where N (i )ed is the total number of edges connecting directly to node i , k and A(k ) are the strain

and the area of the smoothing domain (k ) associated with edge k around the node i , respectively,

and A(i ) = N (i )edk = 1 A

(k ) .

5.3. Rank test for the stiffness matrixProperty 1 : The nES-FEM possesses only legal zero-energy modes that represents the rigidmotions, and there exists no spurious zero-energy mode. This is ensured by the following reasons:

(i) The numerical integration used to evaluate Equation (17) in the nES-FEM satises thenecessary condition given in Section 6.1.3 in Reference [2]. This is true for all possiblenES-FEM models, as detailed in Table II.

(ii) The edge strain smoothing operation ensures linearly independent columns (or rows) in thestiffness matrix due to the independence of these smoothing domains.

(iii) The shape functions used in the n ES-FEM are of partition of unity.


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Table II. Existence of spurious zero-energy modes in an element.

FEM with reducedType of element nES-FEM integration

Triangle

N R = 3

n Q = 3, N Q = 3 n Q = 9

n t = 3, N u = 2 n t = 6

N Q > N u N R

spurious zero energy modes

not possible

n Q = 1, N Q = 3 n Q = 3

n t = 3, N u = 2 n t = 6

N Q = N u N R

spurious zero energy

modes not possible

Quadrilateral

N R = 3

n Q = 4, N Q = 3 n Q = 12

n t = 4, N u = 2 n t = 8

N Q > N u N R


not possible

n Q = 1, N Q = 3 n Q = 3

n t = 4, N u = 2 n t = 8

N Q < N u N R

spurious zero energy

modes possible

n sided polygonal

(n> 4)

N R = 3

n Q = n, N Q = 3 n Q = 3n

n t = n , N u = 2 n t = 2n

N Q > N u N R


not possible

Not applicable

Note : N R : number of DOFs of rigid motion; nt : number of nodes; n Q : number of quadrature points / cells; N u : number of total DOFs; N Q : number of independent equations.

Therefore, no deformed zero-energy mode will appear in the nES-FEM. In other words, anydeformation (except the rigid motions) will result in strain energy in an nES-FEM model.

5.4. Standard patch test

Satisfaction of the standard patch test requires that the displacements of all the interior nodes followexactly (to machine precision) the same linear function of the imposed displacement. A domaindiscretization of a square patch using 36 n-sided polygonal elements is shown in Figure 7. The


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0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 7. Domain discretization of a square patch using 36 n-sided polygonal elements.

following error norm in displacements is used to examine the computed results

ed =2 N ni = 1 |u i u

hi |

2 N ni = 1 |u i |

100% (26)

where u i is the exact solution and u hi is the numerical solution.The parameters are taken as E = 100, = 0.3 and the linear displacement eld is given by

u = x

v = y(27)

It is found that the nES-FEM can pass the standard patch test within machine precision withthe error norm in displacements: ed = 2.56e 12 (%).

6. NUMERICAL EXAMPLES

In this section, some example will be analyzed to demonstrate the effectiveness, accuracy andconvergence properties of the present method. The results of the nES-FEM will be compared withthose of the n-sided polygonal element-based smoothed nite element method ( nCS-FEM) [15 ]

and the n -sided polygonal node-based smoothed nite element method ( nNS-FEM) [13 ]. In orderto study the convergence rate of the present method, two norms are used here. The displacementerror norm is given in Equation (26) and the energy error norm is calculated by

ee =12

N s

k = 1( (k ) )T D ( (k ) ) A(k )

1/ 2

(28)

where (k ) and A(k ) are the smoothing strain and the area of the smoothing domain (k ) , respec-tively; is the analytical strain calculated at the nodes (for nNS-FEM), or at edge-mid-points (fornES-FEM), or at the central points of smoothing domains (k ) (for nCS-FEM); N s is the total


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number of smoothing domains. Note that the convergence rates of the error norms are calculatedbased on the average length of sides of the elements by

h = A N e (29)where A is the area of the whole problem domain.

6.1. Cantilever loaded at the end

A cantilever with length L and height D is studied as a benchmark here, which is subjected to aparabolic traction at the free end as shown in Figure 8. The cantilever is assumed to have a unitthickness so that plane stress condition is valid. The analytical solution is available and can befound in a textbook by Timoshenko and Goodier [49 ].

u x =Py

6 E I (6 L 3 x) x + (2 + ) y2

D2

4

u y = P

6 E I 3 y2

( L x) + (4 + 5 ) D2 x

4 + (3 L x) x2

(30)

where the moment of inertia I for a beam with rectangular cross section and unit thickness isgiven by I = D3 / 12.

The stresses corresponding to the displacements equation (30) are

xx( x, y) =P ( L x) y

I , yy( x, y) = 0, xy( x, y) =

P2 I

D2

4 y2 (31)

The related parameters are taken as E = 3.0 107 kPa, = 0.3, D = 12m, L = 48m and P =1000 N. In the computations, the nodes on the left boundary are constrained using the exactdisplacements obtained from Equation (30) and the loading on the right boundary uses thedistributed parabolic shear stresses in Equation (31). The domain discretization using n-sidedpolygonal elements is shown in Figure 9.

D

y

L

x P

Figure 8. Cantilever subjected to a parabolic traction at the free end.

0 5 10 15 20 25 30 35 40 45

0

5

5

Figure 9. Domain discretization of the cantilever subjected to a parabolic traction at the free endusing n-sided polygonal elements.


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Figure 10 shows that all the computed stresses using nES-FEM agree well with the analyticalsolutions. The convergence of the strain energy is shown in Figure 11. It is seen that the nCS-FEMmodel behaves overly stiff and hence gives a lower bound, and nNS-FEM behaves overly softand gives an upper bound. The nES-FEM has a very close-to-exact stiffness and hence is mostaccurate. The convergences of displacement and energy norms are demonstrated in Figures 12 and13, respectively. It is observed that nES-FEM not only has smaller errors than those of nNS-FEM

and nCS-FEM, but also has higher convergence rates in both norms.Figure 15 shows the inuence of the point distribution in the nES-FEM by comparing the

evolution of strain energy of two sets of meshes: one set with regular meshes as shown in Figure 14

0 2 4 6

0

1000

2000

y (x=0)

N o r m a

l s t r e s s

x x

0 2 4 6

0

y (x=0)

S h e a r s

t r e s s

x y

Analytical solution

Analytical solution

Figure 10. Normal stress xx and shear stress xy along the section of x = 0 using nES-FEM for thecantilever subjected to a parabolic traction at the free end.

500 1000 1500 2000 25003.6

3.8

4

4.2

4.4

4.6

4.8

5

Degrees of freedom

S

t r a

i n e n e r g y

Analytical solution

Figure 11. Convergence of the strain energy solution of n ES-FEM in comparison with other methods forthe cantilever subjected to a parabolic traction at the free end.


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0 0.1 0.2 0.3 0.4 0.5

0

log 10h

l o g 1 0 e e

Figure 12. Error in energy norm of nES-FEM in comparison with other methods for the cantileversubjected to a parabolic traction at the free end.

and one set with irregular meshes as shown in Figure 9. It is seen that the regular meshes give moreaccurate results than the irregular meshes.

Figure 16 compares the computation time of three methods using the same direct full-matrixsolver. It is found that with the same sets of nodes, the computation time of the nES-FEM is onlyshorter than that of the nNS-FEM. However, when the efciency of computation (CPU time forthe same accuracy) in terms of both displacement and energy norms is considered as shown inFigures 17 and 18, the nES-FEM model stands out clearly as a winner.

Table III compares the evolution of the conditioning number with mesh density between n CS-FEM and nES-FEM. It is seen that the conditioning numbers of nES-FEM are little less than thoseof n CS-FEM.

6.2. Semi-innite plane

The semi-innite plane shown in Figure 19 is studied subjected to a uniform pressure within anite range ( a x a ) . The plane strain condition is considered. The analytical stresses are givenby [49 ]

11 = p2

[2( 1 2 ) sin2 1 + sin2 2 ]

22 = p2

[2( 1 2 ) + sin2 1 sin2 2 ]

12 = p2

[cos2 1 cos2 2 ]

(32)

The directions of 1 and 2 are indicated in Figure 19. The corresponding displacements canbe expressed as

u 1 = p(1 2)

E 1 21

[( x + a ) 1 ( x a ) 2 ]+ 2 y lnr 1r 2

u 2 = p(1 2)

E 1 21

y( 1 2 ) + 2 H arctan1c

+ 2( x a ) ln r 2

2( x + a ) ln r 1 + 4a ln a + 2a ln(1 + c2)

(33)


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0 0.1 0.2 0.3 0.4 0.5

0

0.5

1

log10 h

l o g 1 0 e d

Figure 13. Error in displacement norm of nES-FEM in comparison with other methods for the cantileversubjected to a parabolic traction at the free end.

0 5 10 15 20 25 30 35 40 45

0

5

5

Figure 14. A domain discretization using regular meshes of the cantilever subjected to a parabolic tractionat the free end using n-sided polygonal elements.

0 500 1000 1500 2000 2500 30004.36

4.38

4.4

4.42

4.44

4.46

4.48

4.5

Degrees of freedom

S t r a

i n e n e r g y

Analytical solution

Figure 15. Inuence of the point distribution in the strain energy solution of the nES-FEM for the cantileversubjected to a parabolic traction at the free end.


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2.6 2.8 3 3.2 3.4

0

1

2

log10 Degrees of freedom

l o g 1 0

C P U t i m e ( s e c o n

d s

)

Figure 16. Comparison of the computation time of different methods for solving the cantilever

subjected to a parabolic traction at the free end. For the same distribution of nodes, the n CS-FEM isthe fastest to deliver the results.

0 0.5 1 1.5 2

0

0.5

0.5

1

log 10 CPU times (seconds)

l o g 1 0 e d

Figure 17. Comparison of the efciency (computation time for the solutions of same accuracymeasured in displacement norm) for solving the cantilever subjected to a parabolic traction

at the free end. The nES-FEM stands out as a winner.

where H = ca is the distance from the origin to point O , the vertical displacement is assumed tobe zero and c is a coefcient.

Owing to the symmetry about the y-axis, the problem is modeled with a 5 a 5a square witha = 0.2 m, c = 100 and p = 1MPa. The left side is subjected the symmetric boundary conditionof displacement and the bottom side is constrained using the exact displacements given byEquation (32). The right side is subjected to tractions computed from Equation (33). Figure 20gives the discretization of the domain using n -sided polygonal elements.

From Figures 21 and 22, it is observed that all the computed displacements and stresses usingnES-FEM agree well with the analytical solutions. It is also noticed that the present stresses are


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0 0.5 1 1.5 2

0

log10 CPU times (seconds)

l o g 1 0 e e

Figure 18. Comparison of the efciency of computation time in terms of energy norm of the cantileversubjected to a parabolic traction at the free end. The nES-FEM stands out clearly as a winner.

Table III. Comparison of the evolution of the conditioning number with meshdensity between nCS-FEM and nES-FEM.

Mesh density 16 4 24 6 32 8 40 10 48 12

nCS-FEM 6 .58e + 11 8 .25e + 13 1 .22e + 14 1 .99e + 13 2 .25e + 14nES-FEM 6 .46e + 11 8 .18e + 13 1 .21e + 14 1 .96e + 13 2 .20e + 14

Figure 19. Semi-innite plane subjected to a uniform pressure.

very smooth though no post-processing is performed for them. The convergence of the strain energyis shown in Figure 23. Again, it is seen that the nCS-FEM gives a lower bound, nNS-FEM gives anupper bound and nES-FEM is most accurate. The convergence rates in displacement and stressesare displayed in Figure 24 and Figure 25, respectively. It is observed again that both displacement


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0 0.2 0.4 0.6 0.8 1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 20. Domain discretization of the semi-innite plane problem using n -sided polygonal elements.

and energy norms of the nES-FEM not only have smaller errors than those of nNS-FEM andnCS-FEM, but also have the faster convergence rate.

Figure 26 shows the error in displacement for nearly incompressible material when Poissonsratio is changed from 0.4 to 0.4999999. The results show that the proposed selective scheme inSection 4 can overcome the volumetric locking for nearly incompressible materials and gives betterresults than those of the nNS-FEM.

6.3. A cylindrical pipe subjected to an inner pressure

Figure 27 shows a thick cylindrical pipe, with internal radius a = 0.1m, external radius b = 0.2 m,subjected to an internal pressure p = 600kN / m2. Because of the axi-symmetric characteristic of the problem, we only calculate for one quarter of cylinder as shown in Figure 27, and Figure 28gives the discretization of the domain using n -sided polygonal elements. Plane strain condition isconsidered and Youngs modulus E = 21000kN / m2 , Poissons ratio = 0.3. Symmetric conditionsare imposed on the left and bottom edges, and outer boundary is traction free. The exact solutionfor the stress components is [49 ]

r (r ) =a 2 p

b2 a 21

b2

r 2, (r ) =

a 2 pb2 a 2

1 +b2

r 2, r = 0 (34)

whereas the radial and tangential exact displacements are given by

u r (r ) =(1 + )a 2 p E (b2 a 2)

(1 2 )r +b2

r , u = 0 (35)

where (r , ) are the polar coordinates, and is measured counter-clockwise from the positive x-axis.

From Figure 29, it is observed again that all the computed displacements and stresses agreewell with the analytical solutions. The present stresses are also very smooth. The convergence


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0 0.2 0.4 0.6 0.8 11.5

1

0.5

0

0.5

x (y=0)

D

i s p

l a c e m e n

t u

0 0.2 0.4 0.6 0.8 112

10

8

6

4

x (y=0)

D i s p

l a c e m e n

t v

nESFEMAnalytical solution

nESFEMAnalytical solution

Figure 21. Computed and exact displacements of nodes along the free surface ( y = 0)of the semi-innite plane problem.

0

(a) (b)0.2 0.4 0.6 0.8 1

7

6

5

4

3

2

1

0x 10 4

x (y = 1)

s t r e s s x x

Analytical solutionnESFEM

0 0.2 0.4 0.6 0.8 12.6

2.4

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6x 10 5

x (y = 1)

s t r e s s y y

Analytical solutionnESFEM

Figure 22. Computed and exact stresses of nodes along bottom side ( y = 1) for thesemi-innite plane problem: (a) stress xx and (b) stress yy.

of the strain energy is shown in Figure 30. It is again seen that the nCS-FEM gives a lowerbound, nNS-FEM gives an upper bound and nES-FEM is most accurate. The convergence ratesin displacement and stresses are displayed in Figures 31 and 32, respectively. Once again, it isobserved that both displacement and energy norms of the n ES-FEM not only have smaller errorsthan those of n NS-FEM and n SFEM, but also have the faster convergence rate.

Figure 34 shows the inuence of the point distribution in the nES-FEM by comparing the evolu-tion of strain energy of two sets of meshes: one set with regular meshes as shown in Figure 33and one set with irregular meshes as shown in Figure 28. It is seen that the regular meshes givemore accurate results than the irregular meshes.


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500 1000 1500 2000 2500 3000 3500 40004.5

4.51

4.52

4.53

4.54

4.55

4.56

4.57

4.58

4.59x 10 5

Degrees of freedom

S t r a

i n e n e r g y

nCSFEM

nNSFEM

nESFEM

Analytical solution

Figure 23. Convergence of the strain energy solution of nES-FEM in comparison with

other methods for the semi-innite plane problem.

1.5 1.4 1.3 1.21.2

1.3

1.4

1.5

1.6

1.7

log10 h

l o g 1 0

e e

nCSFEM (r=0.99)nNSFEM (r=1.09)nESFEM (r=1.12)

Figure 24. Error in energy norm of nES-FEM in comparison with other methods forthe semi-innite plane problem.

Figure 35 again shows that the proposed selective scheme in Section 4 can overcome thevolumetric locking for nearly incompressible materials, and gives better results than those of thenNS-FEM.

7. CONCLUSIONS

ES-FEM [31 ] was recently proposed to improve the accuracy and convergence rate of the standardFEM. In this work, the ES-FEM is extended to more general case, i.e. n-sided polygonal edge-basedsmoothed nite element method (or nES-FEM) in which the problem domain can be discretized bya set of polygons, each with an arbitrary number of sides. The simple averaging method is suggested


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1.5 1.4 1.3 1.22

1

0

log10 h

l o g 1 0 e d


Figure 25. Error in displacement norm of nES-FEM in comparison with other methodsfor the semi-innite plane problem.

0.4 0.49 0.499 0.4999 0.49999 0 .499999 0.49999990

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Poissons ratio

D i s p

l a c e m e n

t n o r m

e d

nNSFEM

nESFEM

Selective nES/NSFEM

Figure 26. Error in displacement norm for different Poissons ratios of the selectivenES / NS-FEM in comparison with other methods for the semi-innite plane problem.

r

o x

p

a

b

p

y

x o

Figure 27. A thick cylindrical pipe subjected to an inner pressure and its quarter model.


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0 0.05 0.1 0.15 0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Figure 28. Discretization of the domain using n-sided polygonal elements of the thick cylindrical pipe subjected to an inner pressure.

0.1(a) (b)

0.12 0.14 0.16 0.18 0.23

3.5

4

4.5

5

5.5x 10 5

r

R a

d i a l d i s p l a c e m e n

t u

r

Analytical solu.nESFEM

0.1 0.12 0.14 0.16 0.18 0.26

4

2

0

2

4

6

8

10

r

s t r e s s

Analytical solu. of

nESFEM solu. of Analytical solu. of rnESFEM solu. of r

Figure 29. Computed and exact results of nodes along the x-axis ( y = 0) of the thick cylindrical pipe subjected to an inner pressure: (a) radial displacement ur ; (b) radial

stress r and tangential stress .

to construct nES-FEM shape functions. The novel domain-based selective scheme ES / NS-FEMproposed in the ES-FEM is extended to the nES-FEM to give a combined nES / NS-FEM modelthat is immune from volumetric locking. Several numerical examples are investigated and theresults of the nES-FEM are found to agree well with exact solutions and are much better thanthose of others existing methods in displacement and energy norms, both in the accuracy and theconvergence rate. In addition, in comparison of the stiffness of models, the nES-FEM is softer thanthe n CS-FEM (which gives a lower bound) and stiffer than the nNS-FEM (which gives an upperbound). The nES-FEM is hence between the nCS-FEM and nNS-FEM, and is closer than theexact solution compared with the nCS-FEM and n NS-FEM. Depending on the specic problems,


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200 400 600 800 1000 1200 1400 1600 18000.2555

0.256

0.2565

0.257

0.2575

0.258

0.2585

0.259

0.2595

0.26

Degrees of freedom

S t r a i n e n e r g y

nCSFEMnNSFEMnESFEMAnalytical solution

Figure 30. Convergence of the strain energy solution of nES-FEM in comparison with othermethods for the thick cylindrical pipe subjected to an inner pressure.

2.2 2.1 2 1.9 1.8

2.2

2

1.8

1.6

1.4

log 10h

l o g 1 0

e e


Figure 31. Error in energy norm of nES-FEM in comparison with other methods for thethick cylindrical pipe subjected to an inner pressure.

2.2 2.1 2 1.9 1.8

1.5

1

0.5

0

log10 h

l o g 1 0

e d


Figure 32. Error in displacement norm of nES-FEM in comparison with other methods forthe thick cylindrical pipe subjected to an inner pressure.


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0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Figure 33. A domain discretization using regular meshes of the thick cylindrical pipesubjected to an inner pressure.

200 400 600 800 1000 1200 1400 1600 1800 20000.2566

0.2567

0.2568

0.2569

0.257

0.2571

0.2572

0.2573

Degrees of freedom

S t r a

i n e n e r g y

nESFEM (irregular meshes)

nESFEM (regular meshes)Analytical solution

Figure 34. Inuence of the point distribution in the strain energy solution of the nES-FEMfor the thick cylindrical pipe subjected to an inner pressure.

0 .4 0 .49 0 .499 0.4999 0.49999 0.499999 0 .49999990

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Poissons ratio

D i s p

l a c e m e n

t n o r m

e d

nNSFEMnESFEMSelective nES/NSFEM

Figure 35. Error in displacement norm for different Poissons ratios of the selective n ES / NS-FEMin comparison with other methods for the thick cylindrical pipe subjected to an inner pressure.


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the strain energy of nES-FEM can be overestimated or underestimated compared with the exactsolution.

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