A Novel Shape-Free Plane Quadratic Polygonal Hybrid Stress-Function Element

A novel plane quadratic shape-free hybrid stress-function (HS-F) polygonal element is developed by employing the principle of minimum complementary energy and the fundamental analytical solutions of the Airy stress function. Without construction of displacement interpolation function, the formulations of the new model are much simpler than those of the displacement-based polygonal elements and can be degenerated into triangular or quadrilateral elements directly. In particular, it is quite insensitive to various mesh distortions and even can keep precision when element shape is concave. Furthermore, the element does not show any spurious zero energy modes. Numerical examples show the excellent performance of the new element, denoted by HSF-AP-19β, in both displacement and stress solutions.


Introduction
The conventional triangular and quadrilateral elements are widely used in 2D finite element analyses [1,2]. However, when simulating microstructure problems, inconveniences may still exist. For instance, the shape of one crystal of plane polycrystalline materials is usually regarded as a polygon. If these microstructures are meshed by those conventional triangular and quadrilateral elements, the density of the elements will be extremely high. But if each crystal is modeled only by one polygonal element, the related computation costs can be dramatically reduced [3]. Actually, the polygonal elements are more flexible for meshing structures with complex shapes [4][5][6] or connecting different kinds of meshes [7,8], so that they possess obvious advantages for simulating heterogeneous materials [9][10][11][12][13].
Compared with the conventional finite elements, it is more difficult to formulate the interpolation functions for polygonal elements because of the arbitrariness of the element shape. Many researchers have made great efforts on developing effective techniques to solve this problem. In 1975, Wachspress [14] firstly suggested to take the rational functions as the element shape functions. These functions satisfy Kronecker-delta property and continuity requirement along the polygon and can keep linear on the adjacent edges and zero on the opposite edge. However, their construction procedures were relatively complicated. Later, a series of improvements on the Wachspress interpolation were successfully proposed [2,[15][16][17][18][19][20][21][22][23]. Beside the Wachspress method, the Laplace functions were also utilized to build the interpolation functions for polygonal elements, such as the works proposed by Wang and Li [24] and Sukumar and Tabarraei [25]. Another mentionable approach is the barycentric coordinate method [26][27][28] which can be applied to develop both conventional and polygonal elements. During recent years, new polygonal elements are still emerging in many literatures. Dohrmann et al. [7] developed a transition element for connecting dissimilar finite element meshes; Peters and Heymsfield [29] developed constant strain elements consisting of an arbitrary number of nodes; Liu et al. [30,31] constructed several linear 2 Mathematical Problems in Engineering polygonal elements by the smoothed finite element method; Chen et al. [32] proposed a quadratic spline element based on area coordinates and B-net method; Sukumar [33] proposed an arbitrary quadratic polygonal element by applying quadratic maximum-entropy serendipity shape functions. All the elements mentioned above belong to displacementbased elements, in which the displacement interpolation functions satisfying compatibility requirements must be considered.
In order to avoid the difficulties in constructions of displacement-based polygonal elements, some researchers also utilized the hybrid finite element method to develop polygonal elements. Based on the principle of minimum complementary energy, Ghosh and Mallett [34] proposed the Voronoi cell finite element method (VCFEM), in which no displacement interpolation function is needed. By applying the Hellinger-Reissner variational principle, Zhang and Katsube [35,36] developed the hybrid polygonal element (HPE) to simulate materials with inclusions and holes. Peng et al. [37,38] proposed a novel base force element method by using the principle of minimum complementary energy and developed quadratic polygonal elements to solve problems with concave polygonal meshes.
Recently, Cen et al. [39][40][41][42][43][44] proposed a hybrid stressfunction (HS-F) element method based on the principle of minimum complementary energy and the fundamental analytical solutions of Airy stress function. They constructed several elements with excellent performance, such as 8node and 12-node plane quadrilateral elements and 4node plane quadrilateral element with drilling degrees of freedom. These quadrilateral elements are not sensitive to severely distorted meshes and can work well even when the element shapes degenerate into concave quadrilaterals or triangles. So, they are called shape-free finite elements since their performances are independent to the element shapes. Furthermore, this HS-F method can be directly and easily extended to develop arbitrary polygonal element due to its flexible theoretical frame. In the paper, a quadratic polygonal element will be developed by the above HS-F approach.
The arrangements of this paper are as follows. In Section 2, the 2D HS-F element method is briefly reviewed. Then, a new quadratic polygonal HS-F element, denoted by HSF-AP-19 , is formulated in Section 3. Several standard numerical examples are performed in Section 4 to validate the high performance of the new element. Finally, some conclusion remarks are given in Section 5.

Brief Reviews on the Plane HS-F Element Method
For a 2D finite element model, the complementary energy functional can be written as [39][40][41][42][43][44] in which is the thickness of the element; is the element area; Γ is the element boundary; is the element stress vector; T is the traction force vector along the element boundaries; and are the direction cosines of the outer normal ⃗ of the element boundaries; C is the elasticity matrix of compliances, and, for isotropic cases, it can be expressed by ] , plane stress: = , = , plane strain: , Young's modulus; , Poisson's ratio. (3) u is the assumed displacement vector along element boundaries and can be interpolated by the element nodal displacement vector q : where N| Γ is the interpolation function matrix for element boundary displacements. The element stress fields are assumed as follows: where ( = 1 ∼ ) are unknown stress parameters; S is the stress solution matrix: ]3× .
The components (stress interpolation functions) in S are all derived from fundamental solutions of stress function . The first nineteen solutions are given in Table 1.
Therefore, these components are also fundamental stress solutions which satisfy all governing equations. This is the key point for developing the hybrid stress-function (HS-F) element models. According to the principle of minimum complementary energy [39][40][41][42][43][44], we have By applying the principle of the minimum complementary energy again, the final finite element equation of the hybrid stress-function (HS-F) element method can be written as where K is the element stiffness matrix of 2D hybrid stressfunction elements: with p is the element equivalent load vector. Its part caused by concentrated and distributed line forces can be determined by the standard procedure for the conventional finite elements. And the part caused by body forces is given by Once the element nodal displacement vector q is solved, the stresses at any point within the element can be given by

Formulations of a New Plane Quadratic Polygonal HS-F Element
As shown in Figure 1, consider an arbitrary -sided quadratic polygonal HS-F element; the element nodal displacement vector q is defined as in which and V ( = 1 − 2 ) are the nodal displacements in -and -directions, respectively.

Displacement Interpolation along the Element Boundary.
The element boundary displacement vector can be described by the element nodal displacement vector (see (4)). For element edge with three nodes , = + 1, and = + 1 (if > 2 , let = 1), let where is the local coordinate along each element boundary and −1 ≤ ≤ 1.
Then, for th edge Γ , the matrix N| Γ in (4) can be expressed by 3.2. Evaluation Procedure for the Matrix H in (11). For an arbitrary -sided polygonal element, the evaluation of matrix H in (11) should be performed along the whole element boundaries. Thus, (11) can be rewritten as And the direction cosines of the outer normal of each element edge, and in (2), are given by Substitution of (15) and (16) into (18) yields Five Gauss integration points are used to evaluate (18).

Evaluation
Procedure for the Matrix M in (8). As shown in Figure 2, any polygonal element can be divided into some subtriangles. The matrix M in (8) will be firstly evaluated in each subtriangle by standard numerical integration technique. And the sum of these values is the final matrix M.
The whole procedure can be performed automatically by computer code. The resulting new polygonal element is denoted by HSF-AP-19 .

Numerical Examples
In this section, several problems are solved to test the performance of the new quadratic polygonal HSF-AP-19 element. The results are compared with those obtained by the following two elements: PS2: the quadratic polygonal spline element based on area coordinates and B-net method, Chen et al. [32]; BFEM: the base force element method for quadratic polygonal mesh problems, Peng et al. [37,38].

Patch Test.
A small patch is divided into some arbitrary polygonal elements in Figure 3. The displacement fields and the exact stress solution corresponding to the constant strain are    The displacements of boundary nodes are taken as the displacement boundary conditions. No matter whether the inner element edges are straight or curved and no matter whether the shapes of the elements are convex or concave, the exact results of the displacements at each node and stresses at any point can be obtained by the present HSF-AP-15 element. This demonstrates that the new element passes the patch test and thus can ensure solution convergence. Figure 4, several polygonal meshes are applied to analyze a cantilever rectangular plate subjected to uniformly distributing shearing forces. Consider the plane stress condition; the parameters of the model are taken as = 8, = 5, = 1, = 0.3, and 0 = 1.0. It can be seen that no matter whether the element shapes in the meshes are convex or concave polygonal, the following exact solutions [2],

Pure Shear Test of a Rectangular Plate. As shown in
can be obtained.

Pure Bending
Problem of a Cantilever Beam. As shown in Figure 5, a cantilever beam under plane stress condition is The dimensions of the beam are = 10 and = 1. Other parameters are given in Figure 5. Eighteen mesh divisions containing polygonal elements with concave and convex shapes are employed for the calculations. The displacement and stress results of select points are listed in Table 2.
It can be seen that, so long as all element edges keep straight, exact displacements and stress solutions can always be obtained by the element HSF-AP-19 . However, even if a large number of meshes are employed to solve this problem, the exact solutions cannot be obtained by the BFEM [37]. Figure 6, a cantilever under plane stress condition beam is subjected to a linear bending moment caused by a shear force at the free end. The theoretical solutions for this problem are given by [

Linear Bending for a Cantilever Beam. As shown in
The mesh divisions employed are given in Figure 6. All numerical results are listed in Table 3. For this linear bending problem, the exact solutions cannot be directly obtained by the present HSF-AP-19 element. However, once the upper and lower boundaries of the beam are divided into more segments, the results will rapidly converge to the exact solutions. [45]. As shown in Figure 7, six new mesh divisions are designed to solve thin beam problem proposed by MacNeal and Harder [45]. Two loading cases are considered: pure bending and transverse linear bending. Young's modulus = 10 7 , Poisson's ratio = 0.3, and the thickness of the beam = 0.1. The results of the tip deflection are given in Table 4. It can be seen that the present HSF-AP-19 element exhibits good performance: it can not only provide the exact solutions for the pure bending case but also produce the high precision results for the linear bending case. [49]. As shown in Figure 8, a skew cantilever under plane stress condition is subjected to a shear distributed load at the free edge. The typical meshes are given in Figure 8. The results of vertical deflection at point (48,52), the maximum principal stress at point (24,22), and the minimum principal stress at point (24,52) are all listed in Table 5, in which the stresses are the average values at the corresponding nodes. It can be seen that the new HSF-AP-19 element has a good convergence. Figure 9, a curved cantilever beam under plane stress condition is subjected to a transverse force at the upper end in the radial direction. The dimensions of the model are as follows: the inner radius = 0.7, the outer radius = 1.0, and other constants are given in Figure 9. Two Poisson's ratio cases are considered: = 0.3 and = 0.4999. Furthermore, two mesh division types are adopted: elements with curved edges and straight edges. The theoretical radial displacement of point at the upper end is given by [48]

Curved Beam. As shown in
The results of the tip displacement V are given in Table 6. It can be observed that the new HSF-AP-19 element exhibits good convergence. Moreover, the errors produced by the HSF-AP-19 element are quit small, no matter whether the meshes have curved or straight edges.

Infinite
Plate with a Circular Hole [3]. As shown in Figure 10, a plate with a central circular hole is under plane stress state and is subjected to a uniform tensile load of 1.0 in -direction. In order to simulate the properties of the infinite plate, the plate-to-hole aspect ratio of 50 is applied, and the radius of hole is taken as 1.0, while the half-width of plate is 50. The related parameters are as follows: = 1.0 × 10 3 , = 0.3, and thickness = 1.0. The meshes employed are given in Figure 10.
The results at points and are listed in Table 7. Besides the HSF-AP-19 element, the results obtained by the PS2 [32] element are also given for comparison. It can be seen that the accuracy of the results obtained by the HSF-AP-19 element is higher than those obtained by the PS2 element, especially for stress solutions. Figure 11, a cantilever beam with large aspect ratio is subjected to a concentrated load = 1 at free end. The geometric parameters of the cantilever beam are as follows:

A Cantilever Beam under a Concentrated Load. As shown in
= 5, = 1, and ℎ = 0.1. Young's modulus is = 10 9 , and Poisson's ratio is zero. Six mesh divisions are shown in Figure 11, in which Meshes a, b, and c are employed for the HSF-AP-19 element and Meshes d, e, and f for the BFEM [37].
The vertical displacements of point are presented in Table 8. It can be seen that the HSF-AP-19 element can provide better results by using much less elements.

Cook's Skew Beam Problem with Heterogeneous Material.
As shown in Figure 12, a Cook's skew beam with heterogeneous material (under plane stress condition) is subjected to a shear distributed load at the free edge. The beam is 8  Table 2: Results at selected locations for the pure bending problem ( Figure 5).
With the mesh refinement, the results obtained by Abaqus 8-node elements CPS8 and CPS8R [47] will tend to the exact solutions. As to a specified precision, the total element number needed by Abaqus is much more than that required by the HSF-AP-19 element. This example is a very simple problem of heterogeneous materials. However, when solids with more complicated materials are modeled, such as the crystal structures, the computational     advantages in the analysis of complex heterogeneous material problems.

Concluding Remarks
In the paper, a novel quadratic hybrid stress-function polygonal element model, HSF-AP-19 , is developed based on the principle of minimum complementary energy and the fundamental analytical solutions of the Airy stress function. The present element has the following characters.
(1) It is simple to build this new element without construction of displacement interpolation functions, which are very complex for polygonal elements compared with quadrilateral elements. This polygonal HSF-AP-19 element can be degenerated into triangular or quadrilateral elements directly.
(2) Spurious zero energy modes can be avoided by selecting suitable number of fundamental analytical solutions for stresses (from constant terms to higherorder terms). For quadratic HS-F model, the first nineteen stress solutions must be employed.
(3) The present HSF-AP-19 element exhibits better accuracy and convergence in both displacement and stress solutions. Even though meshes containing convex or concave elements are employed, the precisions of the results will not be affected. So, it is a shape-free element model. For constant stress/strain problem, the exact solutions can be obtained for all cases. So long as element edges keep straight, all elements can produce the exact solutions for pure bending problem. Moreover, the new element can produce better results using relatively coarse meshes for some problems, such as high-order problems, problems