Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals for Plane Isotropic Elastic Problems

Transcription

1 International Journal of Applied Mechanics Vol. 9, No. 3 (2017) (24 pages) c World Scientific Publishing Europe Ltd. DOI: /S Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals for Plane Isotropic Elastic Problems Hui Wang College of Civil Engineering and Architecture Henan University of Technology, Zhengzhou , P. R. China Research School of Engineering, Australian National University Canberra, ACT 2600, Australia huiwang china@163.com Qing-Hua Qin Research School of Engineering, Australian National University Canberra, ACT 2600, Australia qinghua.qin@anu.edu.au Received 11 November 2016 Revised 20 March 2017 Accepted 21 March 2017 Published 17 May 2017 Polygonal finite elements with high level of geometric isotropy provide greater flexibility in 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 by centroidal Voronoi tessellation for two-dimensional isotropic elastic problems and then is formulated with element boundary integrals only. For the present n-sided polygonal finite element, two independent fields are introduced: (i) displacement and stress fields inside the element; (ii) frame displacement field along the element boundary. The interior fields are approximated by fundamental solutions so that they exactly satisfy the governing equations to convert element domain integral in the two-field functional into element boundary integrals to reduce integration dimension. While the frame displacement field is approximated by the conventional shape functions to satisfy the conformity requirement between adjacent elements. The two independent fields are coupled by the weak functional to form the stiffness equation. This hybrid formulation enables the construction of n-sided polygons and extends the potential applications of finite elements to convex polygons of arbitrary order. Finally, five examples including patch tests in square domain, thick cylinder under internal pressure, beam bending and composite with clustered holes are provided to illustrate convergence, accuracy and capability of the present Voronoi polygonal finite elements. Keywords: Isotropic elasticity; polygonal finite element; fundamental solutions; boundary integration. Corresponding author

2 H. Wang & Q.-H. Qin 1. Introduction As an alternative to conventional finite elements like triangles and quadrilaterals for two-dimensional problems or tetrahedrals and hexahedrals for threedimensional problems, there has been growing interest in developing nontraditional finite element over arbitrary polygonal and polyhedral meshes over the past decade [Francis et al., 2016; Manzini et al., 2014; Sukumar and Malsch, 2006]. In convex polygonal or polyhedral finite elements, the number of element side is not restricted to three (triangles) or four (quadrilaterals) as in the two-dimensional cases, and four (tetrahedrals) or eight (hexahedrals) as in the three-dimensional cases, so that they are capable of possessing higher degrees of geometric isotropy and thus the meshing effort can be significantly simplified for describing complex geometries without introducing numerical instability and quality of meshes can be improved [Tabarraei and Sukumar, 2006; Weyer et al., 2002]. Moreover, the potential use of convex polygonal finite elements with a large number of sides can provide greater flexibility for the modeling of crystalline materials [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 usually employed as shape functions for approximated displacement fields in the finite element [Floater et al., 2006; Hiyoshi and Sugihara, 1999; Tabarraei and Sukumar, 2006; Wachspress, 1975; Warren et al., 2007]. However, the construction of Laplace/Wachspress shape functions requires complicated mathematical transformations, especially for polygonal elements with curved edges. Moreover, the numerical domain integration associated with arbitrary polygonal finite element 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 developed to model heterogeneous microstructures of composites. [Ghosh, 2011; Ghosh and Moorthy, 2004], in which the domain integral over whole polygonal element is still needed. Apart from the polygonal finite element technique with Laplace/Wachspress interpolants or Airy s stress function basis, the hybrid Trefftz finite element method (HT-FEM) using T-complete solutions of problem as approximation kernels can be utilized for developing polygonal finite elements, because of distinctive characteristic of element boundary integral in the HT-FEM [de Freitas, 1998; Jirousek and Zieliński, 1997; Qin, 2000; Qin and Wang, 2009]. However, the expressions of T-complete solutions of some problems are either complex or difficult to be derived. Moreover, one needs more truncated terms for hybrid polygonal finite

3 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals elements with large numbers of sides to prevent spurious energy modes and keep the solving matrix be of full rank [de Freitas, 1998; Jirousek and Zieliński, 1997; Qin, 1995, 2000, 2003; Qin and Wang, 2009]. In this study, a novel hybrid finite element formulation for Voronoi convex polygons with any number of sides (n-sided convex polygons) is formed with the help of fundamental solutions of problem which have unified expressions in practice, and its performances on convergence and accuracy are numerically studied via a few benchmark problems and composite problems with clustered holes in the context of two-dimensional linear isotropic elasticity. To generate convex polygons in various shaped geometric domains, the PolyMesher written in Matlab codes based on Voronoi diagrams is employed and modified by introducing new distance functions [Talischi et al., 2012a]. For each Voronoi n-sided polygonal finite element, two types of independent fields are introduced into the double-variable hybrid variational functional. One is the intra-element displacement and stress fields, which are approximated by the linear combination of fundamental solutions associated with several fictitious source points so that they can naturally satisfy the elastic equilibrium equations. Another is the auxiliary conforming element displacement frame field, which is defined along the element boundary and interpolated by the conventional shape 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 conformity of displacement field on the common interface of adjacent elements. The independence of the intra-element fields and the frame field makes us conveniently construct arbitrary n-sided polygonal elements. Moreover, the mathematical definition of the intra-element fields allows the domain integral in the hybrid functional be converted into integrals on element boundary wireframe, which are very suitable for n-sided polygonal finite elements and can be easily evaluated by summing Gaussian numerical quadrature values on segments of the element wireframe. This means that multiple types of polygonal elements with different number of sides can be used together to model a specific domain with same kernel functions, i.e., fundamental solutions in unified form. This is the main advantage of the present hybrid polygonal finite element over the conventional polygonal finite element with Laplace/Wachspress interpolants or Trefftz polygonal finite element with T-complete functions. The outline of this paper is as follows. The governing equations for plane linear elasticity are reviewed in Sec. 2, and the conventional finite element formulation is simply revisited in Sec. 3 for the purpose of comparison. In Sec. 4, Voronoi polygonal finite element formulation with fundamental solution kernels are developed and numerical examples are presented in Sec. 5. Finally, some concluding remarks are drawninsec Governing Equations for Plane Elasticity For simplicity, our attention in this study is restricted to the classic linear isotropic elasticity in two dimensions, which have been solved by various numerical methods,

4 H. Wang & Q.-H. Qin Fig. 1. Schematic diagram for plane elastic problems. i.e., FEM [Zienkiewicz and Taylor, 2005], BEM [Brebbia et al., 1984] and meshless methods [Peng et al., 2009; Ren and Cheng, 2011]. As indicated in Fig. 1, a two-dimensional (2D) static linear isotropic elasticity domain Ω is bounded by the boundary Γ = Γ u Γ t, Γ u Γ t = 0. Γ u and Γ t are displacement and traction boundaries, respectively. Referred to the Cartesian coordinate system (x 1,x 2 ), the static equilibrium equation for the dashed linear elastic element around an arbitrary point x (x 1,x 2 ) (see Fig. 1) in the absence of body force is given in matrix form by [Timoshenko and Goodier, 1987] L T σ =0 (1) where σ = {σ 11,σ 22,σ 12 } T is the stress vector, and L is the strain displacement operator matrix 0 L T = x 1 x 2 (2) 0 x 2 x 1 The strain vector ε = {ε 11,ε 22,γ 12 } T is defined by the kinematic relation as ε = Lu (3) where u = {u 1,u 2 } T 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)

5 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals where D is the constitutive matrix and has the form E νe 1 ν 2 1 ν 2 0 νe E D = 1 ν 2 1 ν 2 0 (5) E 0 0 2(1 + ν) for plane stress cases. Besides, the following displacement and traction boundary conditions prescribed on the displacement boundary Γ u and the traction boundary Γ t u = ū on Γ u (6) t = t on Γ t should be augmented to form a complete solving system. In Eq. (6), ū and t are respectively the specified displacement and traction constraints. According to the equilibrium of the dashed triangle shown in Fig. 1, the traction vector t = {t 1,t 2 } T is expressed by t = Aσ (7) where [ ] n1 0 n 2 A = (8) 0 n 2 n 1 and n i (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 reviewed for the purpose of comparison. In the conventional polygonal finite element theory [Zienkiewicz and Taylor, 2005], the displacement field at point with coordinate x Ω e is approximated for a typical polygonal finite element occupying the domain Ω e by n u = U i d i = U e d e (9) i=1 where n is the total number of element nodes, d i = {u 1i,u 2i } T is the column vector of nodal degrees of freedom related to the ith node, d e = {d T 1, d T 2,...,d T n } T is the final nodal displacement vector of the element e, andu e =[U 1, U 2,...,U n ]isthe resulted finite element shape function matrix in which [ ] φi 0 U i = (10) 0 φ i

6 H. Wang & Q.-H. Qin is the element shape submatrix associated with the ith node and usually consists of two-dimensional element shape functions φ i expressed in the following general form [Sukumar and Malsch, 2006] φ i (x) = w i (x) n j=1 w j(x) (11) In Eq. (11), w i (x) are non-negative weight functions, which have differently defined for difference shape functions, i.e., Wachspress shape functions and Laplace shape functions [Sukumar and Malsch, 2006]. Subsequently, the strain and stress fields defined by Eqs. (3) and (4) can be expressed in terms of nodal displacement vector d e,thatis where ε = Lu = B e d e, σ = Dε = DB e d e (12) B e = LU e =[LU 1, LU 2,...,LU n ] (13) The final discrete equations can be formulated from the Galerkin weak or variational form δε T σdω δu T tdγ = 0 (14) Ω e Γ T e where δ denotes the variational operator and Γ T e =Γ e Γ t is the element traction boundary. Substituting the variational forms of the strain and displacement fields δu = U e δd e, δε = B e δd e (15) and Eq. (12) into Eq. (14) yields ( ) ( ) δd T e B T e DB e dω d e δd T e U T tdγ e = 0 (16) Ω e Γ T e Invoking the arbitrariness of nodal variation δd e,wehave K e d e = f e (17) where K e = B T e DB e dω, f e = U T tdγ e (18) Ω e Γ T e It is obviously seen that the shape function and its derivatives are vital for the implementation of conventional finite element. The shape functions are defined for the entire element domain to locate and relate element nodes, so different shape functions bring different element matrices and different degrees of precision. For polygonal finite elements with large number of sides and nodes, it is very

7 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals complicated to construct suitable weight functions so that the shape function satisfies all required properties [Sukumar and Malsch, 2006], especially for polygonal elements with curved edges. This is the main reason that the topology of conventional finite element is usually restricted to triangle and quadrilateral for two-dimensional problems or tetrahedral and hexahedral for three-dimensional problems. Another key issue to be addressed is the evaluation of such domain integral in Eq. (18). So far, the numerical quadrature rule over arbitrary polygons has not yet reached a mature stage and the most popular approach is to partition the n-sided polygonal finite element (n >4) into n triangles by the centroid of the element and then use well-known quadrature rules on each triangle [Sukumar and Tabarraei, 2004]. In this study, a different Voronoi polygonal hybrid finite element formulation based on the fundamental solutions of the two-dimensional linear elastic problem is presented below, which is fully independent of the construction of shape functions and the polygonal element domain integration. 4. Voronoi Polygonal Hybrid Finite Element Formulation The implementation of polygonal hybrid finite elements involves two important issues: (i) the geometrical description and mesh discretization of the enclosed computing domain with finite number of convex polygons and (ii) element-level approximations of physical fields to accurately compute the design response. For the first issue, the advanced Voronoi polygon meshing technique developed by Talischi et al. [2012a] can be utilized to represent flexible mesh generation in arbitrary geometries. Mathematically, every common edge of a Voronoi polygonal cell is defined as being normal to the line connecting two neighboring seed points and has equivalent distance to them, so that Voronoi cells can easily possess more connected neighbors. Figure 2(a) shows a Voronoi diagram and its Delaunay triangulation generated by the Voronoi tessellation technique and a particular polygonal Voronoi cell associated with seed point p is hatched as an example in the figure. As one of Voronoi cells, the centroidal Voronoi tessellation possessing the added attribution that the seed points are coincident with the cell centroids is employed in the study to produce high-quality convex polygonal discretization in the computing domain [Du et al., 1999]. Moreover, to approximate the practical boundary of the domain during Voronoi meshing, the signed distance function is defined in Paulino s meshing scheme [Talischi et al., 2012a] to provide all essential information of the domain geometry so that one can flexibly construct the desired domain by algebraic expressions. Second, after convex polygonal meshing is obtained, the fundamental solution based hybrid finite element technique is formulated here to convert the element domain integral into element boundary integrals and obtain the final solving system of equations. For a typical Voronoi polygonal hybrid finite element e occupying the domain Ω e, as shown in Fig. 2(b), the linear combinations of displacement and stress fundamental solutions of the problem are respectively used as the approximation

8 H. Wang & Q.-H. Qin (a) Voronoi diagram of polygonal cells Fig. 2. (b) Approximation related to Voronoi 6-sided polygonal element Illustration of Voronoi polygonal hybrid finite elements. functions to model the intra-element displacement and stress fields within the element domain Ω e m m u(x) = N k (x)c k = N e (x)c e, σ(x) = T k (x)c k = T e (x)c e, x Ω e k=1 k=1 (19)

9 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals with N k = u 11 (x, x s k ) u 21 (x, x s k ) σ111 (x, x s k ) σ 211 (x, x s k ) u 12 (x, x s k ) u 22 (x, x s k ), T k = σ122 (x, x s k ) σ 222 (x, x s k ), σ112 (x, x s k ) σ 212 (x, x s k ) c k = {c k 1,ck 2 }T (20) where m is the number of source points x s k (k =1,,m) and in practice it can be chosen to be same as the number of nodes, as done in literature for general and special quadrilateral case [Qin and Wang, 2015; Wang and Qin, 2011, 2012]. c e = {c T 1, ct 2,...,cT m }T is the unknown coefficient vector. N e =[N 1, N 2,...,N m ]and T e =[T 1, T 2,...,T m ] denote the matrices consisting of displacement fundamental solution u li (x, xs k )(l, i =1, 2) and stress fundamental solution σ lij (l, i, j =1, 2) at the field point x due to the unit force along the lth direction at the source points x s k, respectively [Wang and Qin, 2011]. It is evident that the intra-element displacement and stress fields (19) can naturally satisfy the linear elastic governing equations (1) because of the physical definition of fundamental solutions, if a series of source points are placed outside the element as they are well done in the standard meshless method of fundamental solutions (MFS) [Fairweather and Karageorghis, 1998; Wang et al., 2006]. However, the intra-element displacement field given by Eq. (19) is nonconforming across the inter-element boundary, as indicated by the shaded region in Fig. 2(b). To deal with such problem, the hybrid technique popularly used in the hybrid finite element method pioneered by Pian [Pian and Wu, 2005] is employed to introduce an auxiliary conforming displacement frame field which has similar form as that in the conventional FEM [Zienkiewicz and Taylor, 2005]. Here, the independent displacement frame field defined along the element boundary Γ e is written as ũ(x) =Ñ e (x)d e, x Γ e (21) where d e is the nodal displacement vector same as that in Eq. (9), and Ñ e is the standard FE shape function matrix with one-dimensional shape functions for the two-dimensional problem considered in the paper. For example, if there are two nodes on a particular edge for the linear case, the shape function matrix over this edge can be written by ] [Ñ1 0 Ñ 2 0 Ñ e = (22) 0 Ñ 1 0 Ñ 2 where Ñ1 =(1 ξ)/2, Ñ2 =(1+ξ)/2 are respectively the classic one-dimensional linear 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]

10 H. Wang & Q.-H. Qin To link these two independent fields, the double-variable weak variational form originally developed in literature [Qin and Wang, 2015; Wang and Qin, 2011, 2012] for traditional eight-node quadrilateral elements is employed Π me = 1 σ T εdω tũdγ + t (ũ u) dγ (23) 2 Ω e Γ T e Γ e where Γ T e =Γ e Γ t and t is the traction field on the element boundary Γ e and may be approximated by considering Eqs. (7) and (19) as t = AT e c e = Q e c e (24) Due to the natural feature of the intra-element fields, Eq. (23) can be further simplified by applying the Gaussian theorem to the domain integral in it Π me = 1 tudγ tũdγ + tũdγ (25) 2 Γ e Γ T e Γ e Substituting the intra-element fields (19), (24) and the frame field (21) into the functional (25) yields Π me = 1 2 ct e H ec e d T e g e + c T e G ed e (26) where H e = Q T e N edγ, G e = Q T e ÑedΓ, g e = Ñ T tdγ e (27) Γ e Γ e Γ T e To enforce inter-element continuity on the common element boundary, the unknown vector c e should be expressed in terms of nodal degree of freedom d e. The minimization of the functional Π me in Eq. (26) with respect to c e and d e, respectively, yields Π me Π me c T = H e c e + G e d e = 0, e d T = G T e c e g e = 0 (28) e from which we can obtain the element stiffness equation K e d e = g e (29) and the optional relationship of c e and d e c e = H 1 e G ed e (30) where the element stiffness matrix K e = G T e H 1 e G e only consists of numerical integrals of the symmetric matrix H e and the matrix G e over the element boundary Γ e. In practice, they can be evaluated by the well-known one-dimensional Gaussian quadrature rule along the element sides of the polygon one by one, without any difficulty, as indicated in Reference [Qin and Wang, 2009], thus the present hybrid strategy is very suitable for constructing n-sided polygonal finite elements. Besides, we observe that the introduction of conforming frame displacement field permits the direct imposition of essential boundary conditions and the direct evaluation of effect of traction boundary conditions, as done in the classic FEM [Zienkiewicz and Taylor, 2005]

11 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals 5. Numerical Examples In this section, the behaviors like convergence and accuracy of the present arbitrary polygonal element with local fundamental solution kernels are assessed through the following five examples including displacement and equilibrium patch tests, cylinder under internal pressure, cantilever beam bending, and composites with clustered holes in the context of two-dimensional isotropic linear elasticity. The elastic properties E = 1000 and ν =0.3are chosen in the analysis. The Voronoi polygonal mesh is generated in the study by directly using or modifying the Matlab function 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 L 2 u e L 2 Er(σ) = σe σ L 2 σ e L 2 = = M k=1 [(ue k1 u k1) 2 +(u e k2 u k2) 2 ] M k=1 (ue k1 2 + u e k2 2 ) (31) M k=1 [(σe k11 σ k11) 2 +(σ e k22 σ k22) 2 +(σ e k12 σ k12) 2 ] M k=1 (σe k σ e k σ e k12 2 ) In Eqs. (31) and (32), the vectors u e ki (i = 1, 2) and σe kij (i = 1, 2) are the displacement and stress analytical solutions at node k(k =1,...,M), and u ki and σ kij 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 polygonal element Displacement patch test First, the ability of the present polygonal hybrid finite elements to represent linear displacement fields is addressed. The computing domain is a unit square. In this example, a linear displacement fields satisfying the governing equations is considered [Tabarraei and Sukumar, 2006] (32) u 1 = u 2 = x 1 + x 2 (33) which is applied on the boundary of the unit square to produce the essential boundary conditions. For the plane stress application, the corresponding constant stress solutions are σ 11 = σ 22 = E 1 ν, σ 12 = E (34) 1+ν During the computation, total four mesh configurations ranging from coarse mesh 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 sides and it is obviously seen that the five-sided elements and six-sided elements are

12 H. Wang & Q.-H. Qin (a) (c) 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 fivesided 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-sided elements, 29 six-sided elements, and 4 seven-sided elements are employed to model the unit square domain. Besides, Fig. 3 shows that the five-sided elements generated in the Voronoi polygonal meshing strategy are usually close to the domain boundary. Using the present Voronoi polygonal meshes, the convergent results of displacement and 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 )to O(10 4 ) in displacement and from O(10 2 )too(10 3 ) in stress, as the number of elements increases from 6 to 64. (b) (d)

13 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals Fig. 4. Fig. 5. Convergent results of displacement and stress for the displacement patch test. Schematic diagram of square plate under uniform tension along the x 1 direction Equilibrium patch test Next, the ability of the present polygonal hybrid finite elements to represent a uniaxial plane stress field is verified by the equilibrium patch test. The computing domain is still the unit square plate used in the first example. A uniaxial stress σ 0 in the x 1 -direction is applied on the right edge of the square, as shown in Fig. 5. Correspondingly, the exact displacement and stress solutions are given by [Qin and Wang, 2009] u 1 = σ 0x 1 E, u 2 = σ 0νx 2 (35) E σ 11 = σ 0, σ 22 =0, σ 12 = 0 (36) In the computational procedure, the same mesh configurations as those in the displacement patch test are employed to model the square plate. Correspondingly, the convergent demonstrations of displacement and stress are respectively given in

14 H. Wang & Q.-H. Qin 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 element can produce convergent results, as expect, when the mesh changes from coarse case to dense case Thick cylinder under internal pressure To demonstrate the ability of the present Voronoi polygonal element for dealing with 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 many researchers to demonstrate the efficiency of the developed numerical methods such as radial basis collocation methods [Hu et al., 2007; Wang and Zhong, 2013], in which the strong RBF interpolation can produce exponential convergence rate. Due to axisymmetric feature of the cylinder model, only one quarter of it, the shaded region Fig. 7. Schematic diagram of thick cylinder under internal pressure

15 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals in Fig. 7, is chosen for computation and the corresponding boundary conditions are also displayed in the figure. For this particular problem, the theoretical solutions of displacements and stresses in the polar coordinate system (r, θ) are expressed as [Timoshenko and Goodier, 1987] where u r = 1+ν E [ Ar +2B(1 2ν)r ], u θ = 0 (37) σ r = A r 2 +2B, σ θ = A r 2 +2B, σ rθ = 0 (38) A = R2 a R2 b Rb 2 p, B = R2 a 2(Rb 2 (39) R2 a )p In the practical computation, the inner and outer radii are R a =5andR b = 10, respectively. The applied internal uniform pressure is chosen as p = 10. In Fig. 8, total three polygonal mesh configurations are used to model the computing domain: (a) 150 Voronoi polygonal elements including 3 four-sided elements, 46 fivesided elements, 96 six-sided elements and 5 seven-sided elements; (b) 400 Voronoi polygonal elements including 5 four-sided elements, 97 five-sided elements, 273 sixsided elements and 25 seven-sided elements; (c) 560 Voronoi polygonal elements including 9 four-sided elements, 119 five-sided elements, 398 six-sided elements and 34 seven-sided elements. For comparison, in Fig. 8, the mesh divisions using general four-node quadrilateral finite elements (CPE4R) in ABAQUS are also provided. It is noted that the general finite element mesh is produced by setting same number of 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 shown in Fig. 9. It is seen from Fig. 9 that both the present Voronoi polygonal elements and the general four-node quadrilateral elements yield optimal convergence with mesh refinement. From Fig. 9, it can be observed that the present hybrid polygonal elements yields more accurate results than general four-node quadrilateral elements. Next, the variations of radial displacement and radial and hoop stresses along the bottom edge of the computing domain using 150 polygonal elements are displayed in Fig. 10, from which it s found that the numerical results from the present polygonal elements agree well with the available exact results, except for the radial stress σ r at r = 5. The main reason is that the linear approximation of equivalent nodal loads brings large error along the curved edge. Same problem can be found when we solve this example using the commercial finite element software ABAQUS with linear general element. To improve the numerical accuracy at r =5,wecanusemore elements in the computing domain. For clarifying this, comparison of exact solutions and 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 that with mesh refinement, both the two methods converge to the exact solution. Again, one observes that the present hybrid Voronoi polygonal element can produce better accuracy than the general four-node quadrilateral finite element. R 2 a

16 H. Wang & Q.-H. Qin 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) Beam bending In the fourth example, we consider a beam bending problem, in which the beam is subjected to a parabolic shear load at the free end, and the left edge is constrained by the given displacement distributions, as shown in Fig. 11. The top and bottom edges of the beam are traction free. Correspondingly, the exact displacement and

17 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals 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] u 1 = Px 2 6EI z u 2 = P 6EI z [ (6L 3x 1 )x 1 +(2+ν) (x 22 D2 4 )] [ 3νx 2 2 (L x 1)+(4+5ν) D2 x 1 +(3L x 1 )x ] (40) σ 11 = P I z (L x 1 )x 2, σ 22 =0, σ 12 = P 8I z (D 2 4x 2 2 ) (41)

18 H. Wang & Q.-H. Qin Table 1. Comparison of exact solutions and different numerical results. r =5(PointA) r =10(PointB) EXACT σ r σ θ Voronoi polygonal element σ r (150 elements) (150 elements) (560 elements) (560 elements) σ θ (150 elements) (150 elements) (560 elements) (560 elements) ABAQUS σ r (144 elements) (144 elements) (560 elements) (560 elements) σ θ (144 elements) (144 elements) (560 elements) (560 elements) Fig. 12. Fig. 11. Cantilever beam bending under the parabolic shear load at the free end. Mesh configuration for the beam bending problem with 600 Voronoi polygonal elements. where I z = D 3 /12 is the moment of inertia for the beam with rectangular crosssection and unit thickness, and P is the resultant shear force on the free end of the beam. In the practical numerical computation, it is assumed that the cantilever has length L =4,heightD = 1. The resultant shear force on the right edge is P = 10. Figure 12 displays the mesh configuration with the present Voronoi polygonal elements including 2 four-sided elements, 126 five-sided elements, 455 six-sided elements and 17 seven-sided elements. With the present mesh, the deflection u 2 of the beam on the bottom edge and the normal stress σ 11 on the left edge are respectively plotted in Fig. 13, in which the exact solutions are also given for comparison. It is clearly seen that the present Voronoi polygonal element can accurately capture the variation of vertical displacement and normal stress along the horizontal direction

19 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals Fig. 13. Variations of deflection u 2 along the bottom edge and stress σ 11 ontheleftedgeforthe beam bending problem Composite with clustered circular holes In the last example, a unit cell of composite square weaken with four circular holes is considered, as shown in Fig. 14. For this composite problem, it is interesting to determine its effective transverse elastic modulus. To do so, the unit cell under tension is accounted for. For such case, a specific positive displacement δ is applied on the right-hand side of the square to represent tension, whilst the left-hand side of the square is constrained. The remaining top and bottom sides keep free. With the specific boundary conditions, the unit cell can be solved by the present Voronoi polygonal 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 mesh configuration

20 H. Wang & Q.-H. Qin average tensile stress along the right-hand side can be given by L σ 11 = 1 σ 11 (L, x 2 )dx 2 (42) L 0 which can be numerically evaluated. Correspondingly, the average strain along the x 1 -direction can be given by the applied displacement ε 11 = δ L (43) According to the elastic theory of isotropic medium, the effective elastic modulus of the composite in the transverse direction is thus determined by [Kaw, 2005] L E1 c = σ 11 = 1 σ 11 (L, x 2 )dx 2 (44) ε 11 δ 0 From the above procedure, it is found that the accurate stress distribution on the right-hand side is important to evaluate the effective elastic modulus. Here, the present Voronoi polygonal elements are employed to solve the unit cell domain. In the computation, the length of the square side is L =1.0. The four circular holes with same radius 0.15 locate (0.76,0.50), (0.24,0.50), (0.50,0.76) and (0.50,0.24), respectively, to form a hole cluster. The applied tensile displacement δ = 0.1. The unit cell with hole cluster is modeled with 1200 Voronoi polygonal elements including 11 four-sided elements, 339 five-sided elements, 780 six-sided elements, 69 seven-sided Fig. 15. Variation of tensile stress σ 11 along the right side of the unit cell

21 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals elements and 1 eight-sided element, as displayed in Fig. 14. The total number of nodes is With the present Voronoi polygonal mesh division, the variation of tensile stress on the right-hand side of the cell is plotted in Fig. 15, in which the results from standard linear quadrilateral finite elements with similar number of nodes (2676 nodes) implemented by ABAQUS is provided for comparison. Results in Fig. 15 dedicate that the present Voronoi polygonal elements can accurately capture the variation of tensile stress. Further, the average value of it can be evaluated and then the effective transverse elastic modulus of the composite given Eq. (44) is E1 c = for Voronoi polygonal elements and for standard linear quadrilateral finite elements. 6. Conclusion Voronoi cells can easily possess more connected neighbors and thus are suitable for generating unstructured polygonal mesh with high level of geometric isotropy. In the paper, as an alternative to the conventional triangle and quadrilateral elements, a new unconstructed polygonal finite element model originating from centroidal Voronoi tessellation technique is developed for modeling elastic response of twodimensional isotropic elastic media. Different to the conventional conforming finite element which is based on shape function interpolation in the whole element level and only possesses one-node connection, the present Voronoi polygonal element possessing higher degrees of geometric isotropy is formulated by introducing two independent fields: the interior displacement and stress fields consisting of fundamental solution kernels inside the element and the frame displacement fields approximated through linear shape functions along the element boundary. The attractive property of element boundary integrals in the numerical formulation permits versatile construction of convex polygons of arbitrary order to model the computing domain, and the conforming frame displacement approximation enables us to directly impose the essential displacement boundary conditions on the element boundary and evaluate the equivalent nodal forces caused by the natural boundary conditions, as done in the classical FEM. It is demonstrated from numerical experiments that the present Voronoi polygonal element has good convergence and accuracy for handling two-dimensional linear elastic analysis and hence significantly extends the potential applications of finite elements to convex n-sided polygons. In addition, it s straightforward to integrate the present technique with conventional finite elements when necessary, and also it is not difficult to apply it to other problems, like anisotropic and three-dimensional problems. Acknowledgments The work described in this paper was partially supported by the National Natural Science Foundation of China (Grant Nos and )

22 H. Wang & Q.-H. Qin References Barbier, C., Michaud, P., Baillis, D., Randrianalisoa, J. and Combescure, A. [2014] New laws for the tension/compression properties of Voronoi closed-cell polymer foams in relation to their microstructure, European Journal of Mechanics-A/Solids 45, Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C. [1984] Boundary Element Techniques: Theory and Applications in Engineering (Springer, Berlin). Dasgupta, G. [2003] Integration within polygonal finite elements, Journal of Aerospace Engineering 16(1), de Freitas, J. A. T. [1998] Formulation of elastostatic hybrid-trefftz stress elements, Computer Methods in Applied Mechanics and Engineering 153(1), Diaz, A. R. and Benard, A. [2003] Designing materials with prescribed elastic properties using polygonal cells, International Journal for Numerical Methods in Engineering 57(3), Du, Q., Faber, V. and Gunzburger, M. [1999] Centroidal Voronoi tessellations: Applications and algorithms, SIAM Review 41(4), Fairweather, G. and Karageorghis, A. [1998] The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics 9(1 2), Floater, M. S., Hormann, K. and Kós, G. [2006] A general construction of barycentric coordinates over convex polygons, Advances in Computational Mathematics 24(1 4), Francis, A., Ortiz-Bernardin, A., Bordas, S. and Natarajan, S. [2016] Linear smoothed polygonal and polyhedral finite elements, International Journal for Numerical Methods in Engineering, doi: /nme Fritzen, F., Böhlke, T. and Schnack, E. [2009] Periodic three-dimensional mesh generation for crystalline aggregates based on Voronoi tessellations, Computational Mechanics 43(5), Ghosh, S. [2011] Micromechanical Analysis and Multi-Scale Modeling Using the Voronoi Cell Finite Element Method (CRC Press, New York). Ghosh, S. and Moorthy, S. [2004] Three dimensional Voronoi cell finite element model for microstructures with ellipsoidal heterogeneties, Computational Mechanics 34(6), Ghosh, S., Nowak, Z. and Lee, K. [1997] Quantitative characterization and modeling of composite microstructures by Voronoi cells, Acta Materialia 45(6), Hiyoshi, H. and Sugihara, K. [1999] Two generalizations of an interpolant based on Voronoi diagrams, International Journal of Shape Modeling 5(02), Hu, H. Y., Chen, J. S. and Hu, W. [2007] Weighted radial basis collocation method for boundary value problems, International Journal for Numerical Methods in Engineering 69(13), Jafari, R. and Kazeminezhad, M. [2011] Microstructure generation of severely deformed materials using Voronoi diagram in Laguerre geometry: Full algorithm, Computational Materials Science 50, Jirousek, J. and Zieliński, A. [1997] Survey of Trefftz-type element formulations, Computers & Structures 63(2), Kaw, A. K. [2005] Mechanics of Composite Materials (CRC Press, New York). Manzini, G., Russo, A. and Sukumar, N. [2014] New perspectives on polygonal and polyhedral finite element methods, Mathematical Models and Methods in Applied Sciences 24(8),

23 Voronoi Polygonal Hybrid Finite Elements with Boundary Integrals Nguyen-Hoang, S. and Nguyen-Xuan, H. [2016] A polytree-based adaptive polygonal finite element method for topology optimization, International Journal for Numerical Methods in Engineering, doi: /nme Nguyen-Xuan, H., Nguyen-Hoang, S., Rabczuk, T. and Hackl, K. [2017] A polytree-based adaptive approach to limit analysis of cracked structures, Computer Methods in Applied Mechanics and Engineering 313, Peng, M., Liu, P. and Cheng, Y. [2009] The complex variable element-free galerkin (CVEFG) method for two-dimensional elasticity problems, International Journal of Applied Mechanics 01(02), Pian, T. H. H. and Wu, C. C. [2005] Hybrid and Incompatible Finite Element Methods (CRC Press, New York). Qin, Q. H. [1995] Hybrid-Trefftz finite element method for Reissner plates on an elastic foundation, Computer Methods in Applied Mechanics and Engineering 122(3 4), Qin, Q. H. [2000] The Trefftz Finite and Boundary Element Method (WIT Press, Southampton). Qin, Q. H. [2003] Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach, Computational Mechanics 31(6), Qin, Q. H. and Mai, Y.-W. [2002] BEM for crack-hole problems in thermopiezoelectric materials, Engineering Fracture Mechanics 69(5), Qin, Q. H. and Wang, H. [2009] Matlab and C Programming for Trefftz Finite Element Methods (CRC Press, New York). Qin, Q. H. and Wang, H. [2015] Special elements for composites containing hexagonal and circular fibers, International Journal of Computational Methods 12(04), Quey, R., Dawson, P. R. and Barbe, F. [2011] Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing, Computer Methods in Applied Mechanics and Engineering 200(17), Rashid, M. and Gullett, P. [2000] On a finite element method with variable element topology, Computer Methods in Applied Mechanics and Engineering 190(11), Ren, H. and Cheng, Y. [2011] The interpolating element-free galerkin (IEFG) method for two-dimensional elasticity problems, International Journal of Applied Mechanics 03(04), Sukumar, N. and Malsch, E. A. [2006] Recent advances in the construction of polygonal finite element interpolants, Archives of Computational Methods in Engineering 13(1), Sukumar, N. and Tabarraei, A. [2004] Conforming polygonal finite elements, International Journal for Numerical Methods in Engineering 61(12), Tabarraei, A. and Sukumar, N. [2006] Application of polygonal finite elements in linear elasticity, International Journal of Computational Methods 3(4), Talischi, C., Paulino, G. H., Pereira, A. and Menezes, I. F. [2010] Polygonal finite elements for topology optimization: A unifying paradigm, International Journal for Numerical Methods in Engineering 82(6), Talischi, C., Paulino, G. H., Pereira, A. and Menezes, I. F. [2012a] PolyMesher: A generalpurpose mesh generator for polygonal elements written in Matlab, Structural and Multidisciplinary Optimization 45(3), Talischi, C., Paulino, G. H., Pereira, A. and Menezes, I. F. [2012b] PolyTop: A Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes, Structural and Multidisciplinary Optimization 45(3),

24 H. Wang & Q.-H. Qin Teferra, K. and Graham-Brady, L. [2015] Tessellation growth models for polycrystalline microstructures, Computational Materials Science 102, Timoshenko, S. P. and Goodier, J. N. [1987] Theory of Elasticity (McGraw-Hill, New York). Wachspress, E. L. [1975] A Rational Finite Element Basis (Academic Press, New York). Wang, H. and Qin, Q. H. [2011] Fundamental-solution-based hybrid FEM for plane elasticity with special elements, Computational Mechanics 48(5), Wang, H. and Qin, Q. H. [2012] Boundary integral based graded element for elastic analysis of 2D functionally graded plates, European Journal of Mechanics-A/Solids 33(1), Wang, H., Qin, Q. H. and Kang, Y. [2006] A meshless model for transient heat conduction in functionally graded materials, Computational Mechanics 38(1), Wang, L. and Zhong, Z. [2013] Radial basis collocation method for nearly incompressible elasticity, Journal of Engineering Mechanics 139(4), Warren, J., Schaefer, S., Hirani, A. N. and Desbrun, M. [2007] Barycentric coordinates for convex sets, Advances in Computational Mathematics 27(3), Weyer, S., Fröhlich, A., Riesch-Oppermann, H., Cizelj, L. and Kovac, M. [2002] Automatic finite element meshing of planar Voronoi tessellations, Engineering Fracture Mechanics 69(8), Zienkiewicz, O. C. and Taylor, R. L. [2005] The Finite Element Method for Solid and Structural Mechanics (Butterworth-Heinemann, Amsterdam)