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International Journal of Applied MechanicsVol. 9, No. 3 (2017) 1750031 (24 pages)c© World Scientific Publishing Europe Ltd.DOI: 10.1142/S1758825117500314
Voronoi Polygonal Hybrid Finite Elementswith Boundary Integrals for Plane
Isotropic Elastic Problems
Hui Wang
College of Civil Engineering and ArchitectureHenan University of Technology, Zhengzhou 450001, P. R. China
Research School of Engineering, Australian National UniversityCanberra, ACT 2600, Australia
huiwang [email protected]
Qing-Hua Qin∗
Research School of Engineering, Australian National UniversityCanberra, ACT 2600, Australia
Received 11 November 2016Revised 20 March 2017Accepted 21 March 2017Published 17 May 2017
Polygonal finite elements with high level of geometric isotropy provide greater flexibilityin mesh generation and material science involving topology change in material phase.In this study, a hybrid finite element model based on polygonal mesh is constructed bycentroidal Voronoi tessellation for two-dimensional isotropic elastic problems and thenis formulated with element boundary integrals only. For the present n-sided polygonalfinite element, two independent fields are introduced: (i) displacement and stress fieldsinside the element; (ii) frame displacement field along the element boundary. The inte-rior fields are approximated by fundamental solutions so that they exactly satisfy thegoverning equations to convert element domain integral in the two-field functional intoelement boundary integrals to reduce integration dimension. While the frame displace-ment field is approximated by the conventional shape functions to satisfy the conformityrequirement between adjacent elements. The two independent fields are coupled by theweak functional to form the stiffness equation. This hybrid formulation enables the con-struction of n-sided polygons and extends the potential applications of finite elements toconvex polygons of arbitrary order. Finally, five examples including patch tests in squaredomain, thick cylinder under internal pressure, beam bending and composite with clus-tered holes are provided to illustrate convergence, accuracy and capability of the presentVoronoi polygonal finite elements.
Keywords: Isotropic elasticity; polygonal finite element; fundamental solutions; boundaryintegration.
∗Corresponding author.
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1. Introduction
As an alternative to conventional finite elements like triangles and quadrilat-erals for two-dimensional problems or tetrahedrals and hexahedrals for three-dimensional problems, there has been growing interest in developing nontraditionalfinite element over arbitrary polygonal and polyhedral meshes over the pastdecade [Francis et al., 2016; Manzini et al., 2014; Sukumar and Malsch, 2006]. Inconvex polygonal or polyhedral finite elements, the number of element side is notrestricted to three (triangles) or four (quadrilaterals) as in the two-dimensionalcases, and four (tetrahedrals) or eight (hexahedrals) as in the three-dimensionalcases, so that they are capable of possessing higher degrees of geometric isotropyand thus the meshing effort can be significantly simplified for describing com-plex geometries without introducing numerical instability and quality of meshescan be improved [Tabarraei and Sukumar, 2006; Weyer et al., 2002]. Moreover,the potential use of convex polygonal finite elements with a large numberof sides can provide greater flexibility for the modeling of crystalline mate-rials [Fritzen et al., 2009; Quey et al., 2011; Teferra and Graham-Brady, 2015],material design [Barbier et al., 2014; Diaz and Benard, 2003; Ghosh et al., 1997;Jafari and Kazeminezhad, 2011], cracked structures [Nguyen-Xuan et al., 2017]and topology optimization [Nguyen-Hoang and Nguyen-Xuan, 2016; Talischi et al.,2010, 2012b].
In order to achieve polygonal finite elements with arbitrary number of sides,the Laplace/Wachspress interpolants based on barycentric coordinates are usu-ally employed as shape functions for approximated displacement fields in the finiteelement [Floater et al., 2006; Hiyoshi and Sugihara, 1999; Tabarraei and Sukumar,2006; Wachspress, 1975; Warren et al., 2007]. However, the construction ofLaplace/Wachspress shape functions requires complicated mathematical trans-formations, especially for polygonal elements with curved edges. Moreover, thenumerical domain integration associated with arbitrary polygonal finite ele-ment is a non-trivial task and usually needs special integration rule [Dasgupta,2003; Rashid and Gullett, 2000]. Besides, the Voronoi cell finite element method(VCFEM) using complete polynomials for Airy’s stress functions was devel-oped to model heterogeneous microstructures of composites. [Ghosh, 2011;Ghosh and Moorthy, 2004], in which the domain integral over whole polygonal ele-ment is still needed.
Apart from the polygonal finite element technique with Laplace/Wachspressinterpolants or Airy’s stress function basis, the hybrid Trefftz finite elementmethod (HT-FEM) using T-complete solutions of problem as approximation ker-nels can be utilized for developing polygonal finite elements, because of distinc-tive characteristic of element boundary integral in the HT-FEM [de Freitas, 1998;Jirousek and Zielinski, 1997; Qin, 2000; Qin and Wang, 2009]. However, the expres-sions of T-complete solutions of some problems are either complex or difficult tobe derived. Moreover, one needs more truncated terms for hybrid polygonal finite
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Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals
elements with large numbers of sides to prevent spurious energy modes and keepthe solving matrix be of full rank [de Freitas, 1998; Jirousek and Zielinski, 1997;Qin, 1995, 2000, 2003; Qin and Wang, 2009].
In this study, a novel hybrid finite element formulation for Voronoi convex poly-gons with any number of sides (n-sided convex polygons) is formed with the helpof fundamental solutions of problem which have unified expressions in practice,and its performances on convergence and accuracy are numerically studied via afew benchmark problems and composite problems with clustered holes in the con-text of two-dimensional linear isotropic elasticity. To generate convex polygons invarious shaped geometric domains, the PolyMesher written in Matlab codes basedon Voronoi diagrams is employed and modified by introducing new distance func-tions [Talischi et al., 2012a]. For each Voronoi n-sided polygonal finite element, twotypes of independent fields are introduced into the double-variable hybrid variationalfunctional. One is the intra-element displacement and stress fields, which are approx-imated by the linear combination of fundamental solutions associated with severalfictitious source points so that they can naturally satisfy the elastic equilibriumequations. Another is the auxiliary conforming element displacement frame field,which is defined along the element boundary and interpolated by the conventionalshape functions same as that in the conventional FEM [Zienkiewicz and Taylor,2005] and BEM [Brebbia et al., 1984; Qin and Mai, 2002] to enforce the conformityof displacement field on the common interface of adjacent elements. The indepen-dence of the intra-element fields and the frame field makes us conveniently constructarbitrary n-sided polygonal elements. Moreover, the mathematical definition of theintra-element fields allows the domain integral in the hybrid functional be convertedinto integrals on element boundary wireframe, which are very suitable for n-sidedpolygonal finite elements and can be easily evaluated by summing Gaussian numeri-cal quadrature values on segments of the element wireframe. This means that multi-ple types of polygonal elements with different number of sides can be used togetherto model a specific domain with same kernel functions, i.e., fundamental solutionsin unified form. This is the main advantage of the present hybrid polygonal finiteelement over the conventional polygonal finite element with Laplace/Wachspressinterpolants or Trefftz polygonal finite element with T-complete functions.
The outline of this paper is as follows. The governing equations for plane linearelasticity are reviewed in Sec. 2, and the conventional finite element formulation issimply revisited in Sec. 3 for the purpose of comparison. In Sec. 4, Voronoi polygonalfinite element formulation with fundamental solution kernels are developed andnumerical examples are presented in Sec. 5. Finally, some concluding remarks aredrawn in Sec. 6.
2. Governing Equations for Plane Elasticity
For simplicity, our attention in this study is restricted to the classic linear isotropicelasticity in two dimensions, which have been solved by various numerical methods,
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Fig. 1. Schematic diagram for plane elastic problems.
i.e., FEM [Zienkiewicz and Taylor, 2005], BEM [Brebbia et al., 1984] and mesh-less methods [Peng et al., 2009; Ren and Cheng, 2011]. As indicated in Fig. 1, atwo-dimensional (2D) static linear isotropic elasticity domain Ω is bounded by theboundary Γ = Γu ∪ Γt, Γu ∩ Γt = 0. Γu and Γt are displacement and tractionboundaries, respectively. Referred to the Cartesian coordinate system (x1, x2), thestatic equilibrium equation for the dashed linear elastic element around an arbitrarypoint x ≡ (x1, x2) (see Fig. 1) in the absence of body force is given in matrix formby [Timoshenko and Goodier, 1987]
LTσ = 0 (1)
where σ = σ11, σ22, σ12T is the stress vector, and L is the strain–displacementoperator matrix
LT =
∂
∂x10
∂
∂x2
0∂
∂x2
∂
∂x1
(2)
The strain vector ε = ε11, ε22, γ12T is defined by the kinematic relation as
ε = Lu (3)
where u = u1, u2T is the displacement vector.For the case of linear elastic solid body, the stress vector is related to the strain
vector by the Hooke’s law in matrix form
σ = Dε (4)
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where D is the constitutive matrix and has the form
D =
E
1 − ν2
νE
1 − ν20
νE
1 − ν2
E
1 − ν20
0 0E
2(1 + ν)
(5)
for plane stress cases.Besides, the following displacement and traction boundary conditions prescribed
on the displacement boundary Γu and the traction boundary Γt
u = u on Γu
t = t on Γt
(6)
should be augmented to form a complete solving system. In Eq. (6), u and t arerespectively the specified displacement and traction constraints. According to theequilibrium of the dashed triangle shown in Fig. 1, the traction vector t = t1, t2T
is expressed by
t = Aσ (7)
where
A =
[n1 0 n2
0 n2 n1
](8)
and ni(i = 1, 2) are the unit outward normal components.
3. Conventional Finite Element Formulation
In this section, the finite element formulation with polygonal elements is reviewedfor the purpose of comparison. In the conventional polygonal finite element the-ory [Zienkiewicz and Taylor, 2005], the displacement field at point with coordinatex ∈ Ωe is approximated for a typical polygonal finite element occupying the domainΩe by
u =n∑
i=1
Uidi = Uede (9)
where n is the total number of element nodes, di = u1i, u2iT is the column vectorof nodal degrees of freedom related to the ith node, de = dT
1 ,dT2 , . . . ,dT
nT is thefinal nodal displacement vector of the element e, and Ue = [U1,U2, . . . ,Un] is theresulted finite element shape function matrix in which
Ui =
[φi 0
0 φi
](10)
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is the element shape submatrix associated with the ith node and usually consistsof two-dimensional element shape functions φi expressed in the following generalform [Sukumar and Malsch, 2006]
φi(x) =wi(x)∑n
j=1 wj(x)(11)
In Eq. (11), wi(x) are non-negative weight functions, which have differently definedfor difference shape functions, i.e., Wachspress shape functions and Laplace shapefunctions [Sukumar and Malsch, 2006].
Subsequently, the strain and stress fields defined by Eqs. (3) and (4) can beexpressed in terms of nodal displacement vector de, that is
ε = Lu = Bede, σ = Dε = DBede (12)
where
Be = LUe = [LU1,LU2, . . . ,LUn] (13)
The final discrete equations can be formulated from the Galerkin weak or vari-ational form ∫
Ωe
δεTσdΩ −∫
ΓTe
δuTtdΓ = 0 (14)
where δ denotes the variational operator and ΓTe = Γe ∩ Γt is the element traction
boundary.Substituting the variational forms of the strain and displacement fields
δu = Ueδde, δε = Beδde (15)
and Eq. (12) into Eq. (14) yields
δdTe
(∫Ωe
BTe DBedΩ
)de − δdT
e
(∫ΓT
e
UTe tdΓ
)= 0 (16)
Invoking the arbitrariness of nodal variation δde, we have
Kede = fe (17)
where
Ke =∫
Ωe
BTe DBedΩ, fe =
∫ΓT
e
UTe tdΓ (18)
It is obviously seen that the shape function and its derivatives are vital for theimplementation of conventional finite element. The shape functions are definedfor the entire element domain to locate and relate element nodes, so differentshape functions bring different element matrices and different degrees of preci-sion. For polygonal finite elements with large number of sides and nodes, it is very
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Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals
complicated to construct suitable weight functions so that the shape function satis-fies all required properties [Sukumar and Malsch, 2006], especially for polygonal ele-ments with curved edges. This is the main reason that the topology of conventionalfinite element is usually restricted to triangle and quadrilateral for two-dimensionalproblems or tetrahedral and hexahedral for three-dimensional problems. Anotherkey issue to be addressed is the evaluation of such domain integral in Eq. (18). Sofar, the numerical quadrature rule over arbitrary polygons has not yet reached amature stage and the most popular approach is to partition the n-sided polygonalfinite element (n > 4) into n triangles by the centroid of the element and then usewell-known quadrature rules on each triangle [Sukumar and Tabarraei, 2004].
In this study, a different Voronoi polygonal hybrid finite element formulationbased on the fundamental solutions of the two-dimensional linear elastic problem ispresented below, which is fully independent of the construction of shape functionsand the polygonal element domain integration.
4. Voronoi Polygonal Hybrid Finite Element Formulation
The implementation of polygonal hybrid finite elements involves two importantissues: (i) the geometrical description and mesh discretization of the enclosed com-puting domain with finite number of convex polygons and (ii) element-level approx-imations of physical fields to accurately compute the design response.
For the first issue, the advanced Voronoi polygon meshing technique developedby Talischi et al. [2012a] can be utilized to represent flexible mesh generation inarbitrary geometries. Mathematically, every common edge of a Voronoi polygonalcell is defined as being normal to the line connecting two neighboring seed pointsand has equivalent distance to them, so that Voronoi cells can easily possess moreconnected neighbors. Figure 2(a) shows a Voronoi diagram and its Delaunay trian-gulation generated by the Voronoi tessellation technique and a particular polygonalVoronoi cell associated with seed point p is hatched as an example in the figure. Asone of Voronoi cells, the centroidal Voronoi tessellation possessing the added attri-bution that the seed points are coincident with the cell centroids is employed inthe study to produce high-quality convex polygonal discretization in the computingdomain [Du et al., 1999]. Moreover, to approximate the practical boundary of thedomain during Voronoi meshing, the signed distance function is defined in Paulino’smeshing scheme [Talischi et al., 2012a] to provide all essential information of thedomain geometry so that one can flexibly construct the desired domain by algebraicexpressions.
Second, after convex polygonal meshing is obtained, the fundamental solutionbased hybrid finite element technique is formulated here to convert the elementdomain integral into element boundary integrals and obtain the final solving systemof equations. For a typical Voronoi polygonal hybrid finite element e occupying thedomain Ωe, as shown in Fig. 2(b), the linear combinations of displacement and stressfundamental solutions of the problem are respectively used as the approximation
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(a) Voronoi diagram of polygonal cells
(b) Approximation related to Voronoi 6-sided
polygonal element
Fig. 2. Illustration of Voronoi polygonal hybrid finite elements.
functions to model the intra-element displacement and stress fields within the ele-ment domain Ωe
u(x) =m∑
k=1
Nk(x)ck = Ne(x)ce, σ(x) =m∑
k=1
Tk(x)ck = Te(x)ce, x ∈ Ωe
(19)
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Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals
with
Nk =
u∗
11 (x,xsk) u∗
21 (x,xsk)
u∗12 (x,xs
k) u∗22 (x,xs
k)
, Tk =
σ∗111 (x,xs
k) σ∗211 (x,xs
k)
σ∗122 (x,xs
k) σ∗222 (x,xs
k)
σ∗112 (x,xs
k) σ∗212 (x,xs
k)
,
ck = ck1 , ck
2T (20)
where m is the number of source points xsk(k = 1, · · · , m) and in practice it can
be chosen to be same as the number of nodes, as done in literature for general andspecial quadrilateral case [Qin and Wang, 2015; Wang and Qin, 2011, 2012]. ce =cT
1 , cT2 , . . . , cT
mT is the unknown coefficient vector. Ne = [N1,N2, . . . ,Nm] andTe = [T1,T2, . . . ,Tm] denote the matrices consisting of displacement fundamentalsolution u∗
li(x,xsk)(l, i = 1, 2) and stress fundamental solution σ∗
lij(l, i, j = 1, 2) atthe field point x due to the unit force along the lth direction at the source pointsxs
k, respectively [Wang and Qin, 2011].It is evident that the intra-element displacement and stress fields (19) can nat-
urally satisfy the linear elastic governing equations (1) because of the physical def-inition of fundamental solutions, if a series of source points are placed outside theelement as they are well done in the standard meshless method of fundamentalsolutions (MFS) [Fairweather and Karageorghis, 1998; Wang et al., 2006].
However, the intra-element displacement field given by Eq. (19) is non-conforming across the inter-element boundary, as indicated by the shaded region inFig. 2(b). To deal with such problem, the hybrid technique popularly used in thehybrid finite element method pioneered by Pian [Pian and Wu, 2005] is employed tointroduce an auxiliary conforming displacement frame field which has similar formas that in the conventional FEM [Zienkiewicz and Taylor, 2005]. Here, the inde-pendent displacement frame field defined along the element boundary Γe is writtenas
u(x) = Ne(x)de, x ∈ Γe (21)
where de is the nodal displacement vector same as that in Eq. (9), and Ne is thestandard FE shape function matrix with one-dimensional shape functions for thetwo-dimensional problem considered in the paper. For example, if there are twonodes on a particular edge for the linear case, the shape function matrix over thisedge can be written by
Ne =
[N1 0 N2 0
0 N1 0 N2
](22)
where N1 = (1 − ξ)/2, N2 = (1 + ξ)/2 are respectively the classic one-dimensionallinear shape functions in terms of the natural coordinate ξ varying from −1 to 1,whose definition can be found in most of books on FEM [Zienkiewicz and Taylor,2005].
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To link these two independent fields, the double-variable weak variational formoriginally developed in literature [Qin and Wang, 2015; Wang and Qin, 2011, 2012]for traditional eight-node quadrilateral elements is employed
Πme =12
∫Ωe
σTεdΩ −∫
ΓTe
tudΓ +∫
Γe
t (u− u) dΓ (23)
where ΓTe = Γe ∩Γt and t is the traction field on the element boundary Γe and may
be approximated by considering Eqs. (7) and (19) as
t = ATece = Qece (24)
Due to the natural feature of the intra-element fields, Eq. (23) can be furthersimplified by applying the Gaussian theorem to the domain integral in it
Πme = −12
∫Γe
tudΓ −∫
ΓTe
tudΓ +∫
Γe
tudΓ (25)
Substituting the intra-element fields (19), (24) and the frame field (21) into thefunctional (25) yields
Πme = −12cT
e Hece − dTe ge + cT
e Gede (26)
where
He =∫
Γe
QTe NedΓ, Ge =
∫Γe
QTe NedΓ, ge =
∫ΓT
e
NTe tdΓ (27)
To enforce inter-element continuity on the common element boundary, the unknownvector ce should be expressed in terms of nodal degree of freedom de. The mini-mization of the functional Πme in Eq. (26) with respect to ce and de, respectively,yields
∂Πme
∂cTe
= −Hece + Gede = 0,∂Πme
∂dTe
= GTe ce − ge = 0 (28)
from which we can obtain the element stiffness equation
Kede= ge (29)
and the optional relationship of ce and de
ce= H−1e Gede (30)
where the element stiffness matrix Ke = GTe H−1
e Ge only consists of numericalintegrals of the symmetric matrix He and the matrix Ge over the element bound-ary Γe. In practice, they can be evaluated by the well-known one-dimensionalGaussian quadrature rule along the element sides of the polygon one by one,without any difficulty, as indicated in Reference [Qin and Wang, 2009], thus thepresent hybrid strategy is very suitable for constructing n-sided polygonal finiteelements. Besides, we observe that the introduction of conforming frame displace-ment field permits the direct imposition of essential boundary conditions and thedirect evaluation of effect of traction boundary conditions, as done in the classicFEM [Zienkiewicz and Taylor, 2005].
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5. Numerical Examples
In this section, the behaviors like convergence and accuracy of the present arbitrarypolygonal element with local fundamental solution kernels are assessed throughthe following five examples including displacement and equilibrium patch tests,cylinder under internal pressure, cantilever beam bending, and composites withclustered holes in the context of two-dimensional isotropic linear elasticity. Theelastic properties E = 1000 and ν = 0.3 are chosen in the analysis. The Voronoipolygonal mesh is generated in the study by directly using or modifying the Matlabfunction Polymesher [Talischi et al., 2012a]. For the purpose of error estimation,the following errors are introduced for displacement and stress analysis
Er(u) =‖ue − u‖L2
‖ue‖L2=
√∑Mk=1 [(ue
k1 − uk1)2 + (uek2 − uk2)2]√∑M
k=1 (uek1
2 + uek2
2)(31)
Er(σ) =‖σe − σ‖L2
‖σe‖L2=
√∑Mk=1 [(σe
k11 − σk11)2 + (σek22 − σk22)2 + (σe
k12 − σk12)2]√∑Mk=1 (σe
k112 + σe
k222 + σe
k122)
(32)
In Eqs. (31) and (32), the vectors ueki(i = 1, 2) and σe
kij(i = 1, 2) are the dis-placement and stress analytical solutions at node k(k = 1, . . . , M), and uki andσkij are the displacement and stress numerical solution vectors, respectively. M
denotes the total number of sample points in the computing domain. Particularly,the sample points can be chosen as all nodes and the centroids of each polygonalelement.
5.1. Displacement patch test
First, the ability of the present polygonal hybrid finite elements to represent lineardisplacement fields is addressed. The computing domain is a unit square. In thisexample, a linear displacement fields satisfying the governing equations is consid-ered [Tabarraei and Sukumar, 2006]
u1 = u2 = x1 + x2 (33)
which is applied on the boundary of the unit square to produce the essential bound-ary conditions. For the plane stress application, the corresponding constant stresssolutions are
σ11 = σ22 =E
1 − ν, σ12 =
E
1 + ν(34)
During the computation, total four mesh configurations ranging from coarsemesh to refined mesh are taken into account, as illustrated in Fig. 3. In the figure,different colors are used to distinguish elements having different number of sidesand it is obviously seen that the five-sided elements and six-sided elements are
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(a) (b)
(c) (d)
Fig. 3. Square domain discretization with Voronoi polygonal elements: (a) 6 elements (14 nodes);(b) 16 elements (34 nodes); (c) 36 elements (74 nodes); (d) 64 elements (130 nodes).
dominant for the Voronoi polygonal meshing strategy. For example, in Fig. 3(a),total six Voronoi polygonal elements including 1 four-sided elements and five five-sided elements are employed to model the unit square domain, and in Fig. 3(d),total 64 Voronoi polygonal elements including 2 four-sided elements, 29 five-sidedelements, 29 six-sided elements, and 4 seven-sided elements are employed to modelthe unit square domain. Besides, Fig. 3 shows that the five-sided elements generatedin the Voronoi polygonal meshing strategy are usually close to the domain boundary.Using the present Voronoi polygonal meshes, the convergent results of displacementand stress defined by Eqs. (31) and (32) are displayed in Fig. 4. As expected,the numerical accuracy of both displacement and stress increases from O(10−3) toO(10−4) in displacement and from O(10−2) to O(10−3) in stress, as the number ofelements increases from 6 to 64.
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Fig. 4. Convergent results of displacement and stress for the displacement patch test.
Fig. 5. Schematic diagram of square plate under uniform tension along the x1 direction.
5.2. Equilibrium patch test
Next, the ability of the present polygonal hybrid finite elements to represent auniaxial plane stress field is verified by the equilibrium patch test. The computingdomain is still the unit square plate used in the first example. A uniaxial stress σ0 inthe x1-direction is applied on the right edge of the square, as shown in Fig. 5. Corre-spondingly, the exact displacement and stress solutions are given by [Qin and Wang,2009]
u1 =σ0x1
E, u2 = −σ0νx2
E(35)
σ11 = σ0, σ22 = 0, σ12 = 0 (36)
In the computational procedure, the same mesh configurations as those in thedisplacement patch test are employed to model the square plate. Correspondingly,the convergent demonstrations of displacement and stress are respectively given in
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Fig. 6. Convergent results of displacement and stress for the equilibrium patch test.
Fig. 6, from them it is obviously found that the present Voronoi polygonal elementcan produce convergent results, as expect, when the mesh changes from coarse caseto dense case.
5.3. Thick cylinder under internal pressure
To demonstrate the ability of the present Voronoi polygonal element for dealingwith curved boundaries, a long thick circular cylinder under internal pressure p
is accounted for, as indicated in Fig. 7. This problem has been studied by manyresearchers to demonstrate the efficiency of the developed numerical methods such asradial basis collocation methods [Hu et al., 2007; Wang and Zhong, 2013], in whichthe strong RBF interpolation can produce exponential convergence rate. Due toaxisymmetric feature of the cylinder model, only one quarter of it, the shaded region
Fig. 7. Schematic diagram of thick cylinder under internal pressure.
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in Fig. 7, is chosen for computation and the corresponding boundary conditions arealso displayed in the figure. For this particular problem, the theoretical solutionsof displacements and stresses in the polar coordinate system (r, θ) are expressedas [Timoshenko and Goodier, 1987]
ur =1 + ν
E
[−A
r+ 2B(1 − 2ν)r
], uθ = 0 (37)
σr =A
r2+ 2B, σθ = − A
r2+ 2B, σrθ = 0 (38)
where
A = − R2aR2
b
R2b − R2
a
p, B =R2
a
2(R2b − R2
a)p (39)
In the practical computation, the inner and outer radii are Ra = 5 and Rb =10, respectively. The applied internal uniform pressure is chosen as p = 10. InFig. 8, total three polygonal mesh configurations are used to model the computingdomain: (a) 150 Voronoi polygonal elements including 3 four-sided elements, 46 five-sided elements, 96 six-sided elements and 5 seven-sided elements; (b) 400 Voronoipolygonal elements including 5 four-sided elements, 97 five-sided elements, 273 six-sided elements and 25 seven-sided elements; (c) 560 Voronoi polygonal elementsincluding 9 four-sided elements, 119 five-sided elements, 398 six-sided elements and34 seven-sided elements. For comparison, in Fig. 8, the mesh divisions using generalfour-node quadrilateral finite elements (CPE4R) in ABAQUS are also provided.It is noted that the general finite element mesh is produced by setting same numberof segments as that in Voronoi mesh along the boundary of the computing domain.First, the numerical convergence of the relative error in the stress norm is shownin Fig. 9. It is seen from Fig. 9 that both the present Voronoi polygonal elementsand the general four-node quadrilateral elements yield optimal convergence withmesh refinement. From Fig. 9, it can be observed that the present hybrid polygonalelements yields more accurate results than general four-node quadrilateral elements.Next, the variations of radial displacement and radial and hoop stresses along thebottom edge of the computing domain using 150 polygonal elements are displayed inFig. 10, from which it’s found that the numerical results from the present polygonalelements agree well with the available exact results, except for the radial stress σr
at r = 5. The main reason is that the linear approximation of equivalent nodalloads brings large error along the curved edge. Same problem can be found whenwe solve this example using the commercial finite element software ABAQUS withlinear general element. To improve the numerical accuracy at r = 5, we can use moreelements in the computing domain. For clarifying this, comparison of exact solutionsand numerical results from the present method and ABAQUS at two key positions(Points A and B in Fig. 7) is performed in Table 1, from which it is illustrated thatwith mesh refinement, both the two methods converge to the exact solution. Again,one observes that the present hybrid Voronoi polygonal element can produce betteraccuracy than the general four-node quadrilateral finite element.
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150 polygonal elements 144 quadrilateral elements
400 polygonal elements 392 quadrilateral elements
560 polygonal elements 560 quadrilateral elements
Fig. 8. Various mesh configurations of the thick cylinder with hybrid polygonal elements (left)and general 4-node quadrilateral element CPE4R in ABAQUS (right).
5.4. Beam bending
In the fourth example, we consider a beam bending problem, in which the beam issubjected to a parabolic shear load at the free end, and the left edge is constrainedby the given displacement distributions, as shown in Fig. 11. The top and bottomedges of the beam are traction free. Correspondingly, the exact displacement and
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Fig. 9. Convergent results of stress for the thick cylinder under internal pressure.
Fig. 10. Variations of radial displacement and stresses along the bottom edge.
stress solutions are given by [Timoshenko and Goodier, 1987]
u1 = − Px2
6EIz
[(6L − 3x1)x1 + (2 + ν)
(x2
2 −D2
4
)]
u2 =P
6EIz
[3νx2
2(L − x1) + (4 + 5ν)D2x1
4+ (3L − x1)x2
1
] (40)
σ11 = −P
Iz(L − x1)x2, σ22 = 0, σ12 =
P
8Iz(D2 − 4x2
2) (41)
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Table 1. Comparison of exact solutions and different numerical results.
r = 5(Point A) r = 10 (Point B)
EXACT σr −10.000 0.0000σθ 16.667 6.6667
Voronoi polygonal element σr −8.3379 (150 elements) −0.1284 (150 elements)−9.6818 (560 elements) −0.0107 (560 elements)
σθ 16.8756 (150 elements) 6.6746 (150 elements)16.6193 (560 elements) 6.6591(560 elements)
ABAQUS σr −8.4733 (144 elements) −0.2087 (144 elements)−9.2030 (560 elements) −0.1041 (560 elements)
σθ 15.1400 (144 elements) 6.8753 (144 elements)15.8697 (560 elements) 6.7707 (560 elements)
Fig. 11. Cantilever beam bending under the parabolic shear load at the free end.
Fig. 12. Mesh configuration for the beam bending problem with 600 Voronoi polygonal elements.
where Iz = D3/12 is the moment of inertia for the beam with rectangular cross-section and unit thickness, and P is the resultant shear force on the free end of thebeam.
In the practical numerical computation, it is assumed that the cantilever haslength L = 4, height D = 1. The resultant shear force on the right edge is P = 10.Figure 12 displays the mesh configuration with the present Voronoi polygonal ele-ments including 2 four-sided elements, 126 five-sided elements, 455 six-sided ele-ments and 17 seven-sided elements. With the present mesh, the deflection u2 of thebeam on the bottom edge and the normal stress σ11 on the left edge are respectivelyplotted in Fig. 13, in which the exact solutions are also given for comparison. It isclearly seen that the present Voronoi polygonal element can accurately capture thevariation of vertical displacement and normal stress along the horizontal direction.
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Fig. 13. Variations of deflection u2 along the bottom edge and stress σ11 on the left edge for thebeam bending problem.
5.5. Composite with clustered circular holes
In the last example, a unit cell of composite square weaken with four circular holesis considered, as shown in Fig. 14. For this composite problem, it is interestingto determine its effective transverse elastic modulus. To do so, the unit cell undertension is accounted for. For such case, a specific positive displacement δ is appliedon the right-hand side of the square to represent tension, whilst the left-hand sideof the square is constrained. The remaining top and bottom sides keep free. Withthe specific boundary conditions, the unit cell can be solved by the present Voronoipolygonal elements to obtain displacement and stress fields in it. After this, the
Fig. 14. Schematic diagram of square unit cell weaken with hole cluster and related meshconfiguration.
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average tensile stress along the right-hand side can be given by
σ11 =1L
∫ L
0
σ11(L, x2)dx2 (42)
which can be numerically evaluated. Correspondingly, the average strain along thex1-direction can be given by the applied displacement
ε11 =δ
L(43)
According to the elastic theory of isotropic medium, the effective elastic modulusof the composite in the transverse direction is thus determined by [Kaw, 2005]
Ec1 =
σ11
ε11=
1δ
∫ L
0
σ11(L, x2)dx2 (44)
From the above procedure, it is found that the accurate stress distribution onthe right-hand side is important to evaluate the effective elastic modulus. Here, thepresent Voronoi polygonal elements are employed to solve the unit cell domain. Inthe computation, the length of the square side is L =1.0. The four circular holes withsame radius 0.15 locate (0.76,0.50), (0.24,0.50), (0.50,0.76) and (0.50,0.24), respec-tively, to form a hole cluster. The applied tensile displacement δ = 0.1. The unitcell with hole cluster is modeled with 1200 Voronoi polygonal elements including 11four-sided elements, 339 five-sided elements, 780 six-sided elements, 69 seven-sided
Fig. 15. Variation of tensile stress σ11 along the right side of the unit cell.
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elements and 1 eight-sided element, as displayed in Fig. 14. The total number ofnodes is 2408. With the present Voronoi polygonal mesh division, the variation oftensile stress on the right-hand side of the cell is plotted in Fig. 15, in which theresults from standard linear quadrilateral finite elements with similar number ofnodes (2676 nodes) implemented by ABAQUS is provided for comparison. Resultsin Fig. 15 dedicate that the present Voronoi polygonal elements can accurately cap-ture the variation of tensile stress. Further, the average value of it can be evaluatedand then the effective transverse elastic modulus of the composite given Eq. (44) isEc
1 = 375.59 for Voronoi polygonal elements and 375.04 for standard linear quadri-lateral finite elements.
6. Conclusion
Voronoi cells can easily possess more connected neighbors and thus are suitable forgenerating unstructured polygonal mesh with high level of geometric isotropy. Inthe paper, as an alternative to the conventional triangle and quadrilateral elements,a new unconstructed polygonal finite element model originating from centroidalVoronoi tessellation technique is developed for modeling elastic response of two-dimensional isotropic elastic media. Different to the conventional conforming finiteelement which is based on shape function interpolation in the whole element leveland only possesses one-node connection, the present Voronoi polygonal element pos-sessing higher degrees of geometric isotropy is formulated by introducing two inde-pendent fields: the interior displacement and stress fields consisting of fundamentalsolution kernels inside the element and the frame displacement fields approximatedthrough linear shape functions along the element boundary. The attractive prop-erty of element boundary integrals in the numerical formulation permits versatileconstruction of convex polygons of arbitrary order to model the computing domain,and the conforming frame displacement approximation enables us to directly imposethe essential displacement boundary conditions on the element boundary and eval-uate the equivalent nodal forces caused by the natural boundary conditions, asdone in the classical FEM. It is demonstrated from numerical experiments that thepresent Voronoi polygonal element has good convergence and accuracy for handlingtwo-dimensional linear elastic analysis and hence significantly extends the potentialapplications of finite elements to convex n-sided polygons. In addition, it’s straight-forward to integrate the present technique with conventional finite elements whennecessary, and also it is not difficult to apply it to other problems, like anisotropicand three-dimensional problems.
Acknowledgments
The work described in this paper was partially supported by the National NaturalScience Foundation of China (Grant Nos. 11472099 and 11372100).
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