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Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties
Asia Pacific Journal on Computational Engineering volume 4, Article number: 3 (2017)
The Erratum to this article has been published in Asia Pacific Journal on Computational Engineering 2017 4:4
Abstract
In this paper, the accuracy and the convergence properties of Trefftz finite element method over arbitrary polygons are studied. Within this approach, the unknown displacement field within the polygon is represented by the homogeneous solution to the governing differential equations, also called as the Tcomplete set. While on the boundary of the polygon, a conforming displacement field is independently defined to enforce the continuity of the field variables across the element boundary. An optimal number of Tcomplete functions are chosen based on the number of nodes of the polygon and the degrees of freedom per node. The stiffness matrix is computed by the hybrid formulation with auxiliary displacement frame. Results from the numerical studies presented for a few benchmark problems in the context of linear elasticity show that the proposed method yields highly accurate results with optimal convergence rates.
Background
Wachspress [1] introduced the concept of defining basis functions on any wedge form, which also yields interpolants on polytopes of any convex shapes. These shape functions are unconventional when compared to the polynomials used in the conventional finite elements. The use of elements with arbitrary number of sides provides flexibility in automatic mesh manipulation. For example, the domain can be discretized without a need to maintain a particular element topology. This is advantageous in adaptive mesh refinement, where a straightforward subdivision of individual elements usually results in hanging nodes. Traditionally, this is eliminated by introducing additional edges/faces to retain conformity. This can be alleviated if the computations are directly on meshes with hanging nodes. However, until recently elements with arbitrary number of sides did not find their applications in the computational mechanics, partly because of the associated difficulties with mesh generation and numerical integration. With the pioneering work of Alwood and Cornes [2], Sukumar and Tabarraei [3], Dasgupta [4], to name a few, now discretization of the domain with finite elements having arbitrary number of sides has gained increased attention [3, 5,6,7,8,9,10]. This has led to a new area of finite elements called ‘polygonal finite elements.’ There are different techniques to compute the basis functions over arbitrary polygons. Some of them include (a) using length and area measures [1]; (b) natural neighbor interpolants [11]; (c) maximum entropy approximant [12]; and (d) harmonic shape functions [8]. As the approximation functions over arbitrary polygonal elements are usually nonpolynomial (in particular, rational polynomials) which introduces difficulties in the numerical integration, improving numerical integration over polytopes has gained increasing attention [3, 13,14,15]. It is beyond the scope of this paper to review advances in polygonal finite element methods. Interested readers are referred to the literature [16, 17] and references therein for detailed discussion. Once the basis functions are constructed, the conventional Galerkin procedure is normally employed to solve the governing equations over the polygonal/polyhedral meshes.
Sukumar [18] used Voronoï cells and natural neighbor interpolants to develop a finite difference method on unstructured grids. Rashid and Gullet [19] proposed a variable element topology finite element method, in which shape functions for convex and nonconvex elements are computed in the physical space using constrained minimization procedure. Based on the assumed stress hybrid formulation, Ghosh et al. [20] developed the Voronoï cell finite element method. Tiwary et al. [21] studied the behavior of microstructures with irregular geometries. Liu et al. [22,23,24] generalized the concept of strain smoothing technique to arbitrarily shaped polygons. The main idea is to write the strain as the divergence of a spatial average of the compatible strain field. On another front, a fundamental solution less method (Scaled Boundary Method) was introduced by Wolf and Song [25]. It shares the advantages of the FEM and the boundary element method (BEM). Like the FEM, no fundamental solution is required, and like the BEM, the spatial dimension is reduced by one, since only the boundary needs to be discretized, resulting in a decrease in the total degrees of freedom. Ooi et al. [26] employed scaled boundary formulation in polygonal elements to study crack propagation.
Apart from the aforementioned formulations, recent studies, among others, include developing polygonal elements based on the virtual nodes [27] and the virtual element methods [28]. The other possible approach is to employ basis functions that satisfy the differential equation locally [29, 30]. This method has been studied in detail in [31, 32] and extended to higher order polygons in [5, 33]. Zienkiewicz [34] presented a concise discussion on different approximation procedures to differential equations. It was shown that Trefftztype approximation is a particular form of weighted residual approximation. This can be used to generate hybrid finite elements. Earlier studies employed boundarytype approximation associated with Trefftz to develop special type finite elements, for example, elements with holes/voids [35, 36], for plate analysis [37,38,39]. Recently, the idea of employing local solutions over arbitrary finite elements has been investigated in [5, 31,32,33]. However, its convergence properties and accuracy when applied to linear elasticity need to be investigated.
In this paper, hybrid Trefftz arbitrary polygons will be formulated and its convergence properties and accuracy will be numerically studied with a few benchmark problems in the context of linear elasticity. An optimal number of Tcomplete functions are chosen based on the number of nodes of the polygon and degrees of freedom per node. The salient features of the approach are (a) only the boundary of the element is discretized with 1D finite elements, and (b) explicit form of the shape functions and special numerical integration scheme are not required to compute the stiffness matrix.
The paper commences with an overview of the governing equations for elasticity and the corresponding Galerkin form. Section “Overview of hybrid Trefftz finite element method" introduces a hybrid Trefftztype approximation over arbitrary polytopes. The efficiency, the accuracy, and the convergence properties of the HTFEM (Hybrid Trefftz Finite Element Method) are demonstrated with a few benchmark problems in section “Numerical examples”. The numerical results from the HTFEM are compared with the analytical results and with the polygonal FEM with Laplace/Wachspress interpolants, followed by concluding remarks in the last section.
Governing equations and weak form
For a 2D static linear elasticity problem defined in the domain \(\Omega\) bounded by \(\Gamma = \Gamma _u \bigcup \Gamma _t\), \(\Gamma _u \bigcap \Gamma _t = \emptyset\), in the absence of body forces, the governing equation is given by
with the following conditions prescribed on the boundary:
where \(\varvec{\sigma }\) is the stress tensor. The discrete equations for this problem are formulated from the Galerkin weak form:
where \(\mathbf {u}\) and \(\delta \mathbf {u}\) are the trial and the test functions, respectively, and \(\mathbf {D}\) is the material constitutive matrix. The FEM uses the following trial and test functions:
where NP is the total number of nodes in the mesh, \(\mathbf {N}\) is the shape function matrix, and \(\mathbf {d}_I\) is the vector of degrees of freedom associated with node I. Upon substituting Eq. (4) into Eq. (3) and invoking the arbitrariness of \(\delta \mathbf {u}\), we obtain the following discretized algebraic system of equations:
with
where \(\mathbf {K}\) is the stiffness matrix and \(\Omega ^h\) is the discretized domain formed by the union of elements \(\Omega ^e\). The stiffness matrix is computed over each element and assembled to the global matrix. The size of the stiffness matrix depends on the number of nodes in an element.
Generalization to arbitrary polygons
The growing interest in the generalization of FE over arbitrary meshes has opened up a new area of finite elements called ‘polygonal finite elements.’ In polygonal finite elements, the number of sides of an element is not restricted to three or four as in the case of 2D. The Voronoï tessellation is a fundamental geometrical construct to generate a polygonal mesh covering a given domain. Polygonal meshes can be generated from Voronoï diagrams. The Voronoï diagram is a subdivision of the domain into regions \(V(p_I)\), such that any point in \(V(p_I)\) is closer to node \(p_I\) than to any other node. Figure 1 shows a Voronoï diagram of a point P. The firstorder Voronoï V(N) is a subdivision of the Euclidean space \(\mathbb {R}^2\) into convex regions, mathematically:
where \(d(\mathbf {x}_I,\mathbf {x}_J)\), the Euclidean matrix, is the distance between \(\mathbf {x}_I\) and \(\mathbf {x}_J\). The quality of the generated polygonal mesh depends on the randomness in the scattered points. Figure 2 shows a typical Voronoï tessellation of two sets of scattered dataset. The quality of a polygonal mesh determines the accuracy of the solution [3]. To improve the quality of the Voronoï tessellation, the generating point of each Voronoï cell can be used as its center of mass, leading to a special type of Voronoï diagram, called the centroidal Vornoï tessellation (CVT) [40]. Sieger et al. [41] presented an optimizing technique to improve the Voronoï diagrams for use in FE computations.
Overview of hybrid Trefftz finite element method
The basic idea in the Trefftz FEM is to employ the series of the homogeneous solution to the governing differential equation (see Eq. 1) as the approximation function to model the displacement field within the domain and an independent set of functions to represent the boundary and to satisfy interelement compatibility (see Fig. 3). The set of functions that are used to represent the displacement field within the domain are also called as Tcomplete set. The displacement field within an element can be written as
where \(\mathbf {c}\) are the vectors of undetermined coefficients and \(\mathbf {N}_I\) are the approximation functions that are selected from the series solution of the homogeneous part of the governing differential equation (see Eq. 1). For linear elastostatics, based on the Mushelishvili’s complex variable formulation, the \(\mathbf {N}_I\) and the corresponding stress fields are given by [29]
where \(J=1,2,3,4\), \(k = 1,2,\ldots\); \(Z_{1k} = i \kappa z^k + i \kappa z \overline{z}^{k1},~~ Z_{2k} = \kappa z^k  k z \overline{z}^{k1},~~Z_{3k} = i \overline{z}^k\) , and \(Z_{4k} = \overline{z}^k\) ; \(R_{1k}=2ikz^{k1}, R_{2k}=2kz^{k1}, R_{3k} = 0, R_{4k} = 0\) ; and \(S_{1k} = ik(k1)z^{k2}\overline{z}, S_{2k} = k(k1)z^{k2}\overline{z}, S_{3k} = iKz^{k1}, S_{4k} = kz^{k1}\).
However, the intraelement displacement field given by Eq. (8) is nonconforming across the interelement boundary. The unknown coefficients \(\mathbf {c}\) are computed from the external boundary conditions and/or from the continuity conditions on the interelement boundary. Of the various methods available to enforce these conditions, in this study, we use the hybrid technique. In this technique, the elements are linked through an auxiliary conforming displacement frame which has the same form as in the conventional FEM. The displacement field on the element boundary, or otherwise called frame, is given by
where \(\mathbf {q}_I\) are the unknowns of the problem and \(\tilde{\mathbf {N}}_I\) are the standard 1D FE shape functions. To satisfy the interelement continuity, a modified variational form is employed [29], given by:
where \(\Gamma _{eI}\) is the interelement boundary, \(\Gamma _{e1} = \Gamma _u \cap \Gamma _e,\) and \(\Gamma _{e2} = \Gamma _\sigma \cap \Gamma _e\). The minimization of the modified variational principle given by Eq. (11) leads to the following system of algebraic equations:
where
and
where \(\mathbf {Q} = \mathbf {A} \mathbf {T}\), and
where \(n_1\) and \(n_2\) are the outward normals. The integrals in the above equation can be computed by employing standard Gaussian quadrature rules. It is noted that in the computation of the stiffness matrix, we have to compute the inverse of the matrix \(\mathbf {H}\). The necessary condition for the matrix \(\mathbf {H}\) to be of full rank is
where \(N_\mathrm{dof}\) is the total number of degrees of freedom of the element and \(m_\mathrm{min}\) is the minimum number of Tcomplete functions to be used. Additionally, if \(m_\mathrm{min}\) does not guarantee a matrix with full rank, full rank can be achieved by suitably increasing the number of Tcomplete functions.
Numerical examples
In this section, we present the convergence and accuracy of the arbitrary polygons with local Trefftz functions using benchmark problems in the context of linear elasticity. The results from the proposed approach are compared with analytical solution where available and with the conventional polygonal finite element method with Laplace interpolants. To discuss the results, we employ the following convention:

PFEM Polygonal finite element method with Laplace/Wachspress interpolants (conventional approach). The numerical integration within each element is done by subdividing the polygon into triangles and employing a sixthorder Dunavant quadrature rule.

HTPFEM Hybrid Trefftz polygonal finite element method. Within each polygon, Tcomplete functions are employed to compute the stiffness matrix. Onedimensional Gaussian quadrature is employed along the boundary of the polygon and the order of the quadrature depends on the number of Tcomplete functions employed.
The builtin Matlab^{®} function voronoin and Matlab^{®} functions in PolyTop [42] for building the meshconnectivity are used to create the polygonal meshes. Interested readers are referred to the corresponding author to obtain a MATLAB code for all the test cases presented in this manuscript. For the purpose of error estimation, we employ the relative error in the displacement norm and in the energy norm, given by:
Displacement norm
Energy norm
where \(\mathbf {u}, \varvec{\varepsilon }\) are the analytical solution or a reference solution and \(\mathbf {u}^h, \varvec{\varepsilon }^h\) are the numerical solution.
Cantilever beam bending
A twodimensional cantilever beam was subjected to a parabolic shear load at the free end. The domain is discretized with arbitrary polygonal elements. See Fig. 4 for geometry, boundary conditions, and representative polygonal mesh. The geometry of the cantilever: length \(L=\) 10 and height \(D=\) 2. The material properties are as follows: Young’s modulus, \(E=\) 3e\(^7\), Poisson’s ratio \(\nu =\) 0.25, and the parabolic shear force \(P=\) 150. The exact solution for displacements is given by
where \(I = D^3/12\) is the moment of inertia, \(\overline{E} = E\), \(\overline{\nu } = \nu,\) and \(\overline{E} = E/(1\nu ^2)\), \(\overline{\nu } = \nu /(1\nu )\) for plane stress and plane strain, respectively.
The numerical convergence of the relative error in the displacement norm and the relative error in the energy norm are shown in Fig. 5. The results from the HTPFEM and Polygonal FEM are compared with the available analytical solution. Both the Polygonal FEM and the HTPFEM yield optimal convergence in \(L^2\) and \(H^1\) norm. It is seen that with mesh refinement, both the methods converge to the exact solution. An estimation of the convergence rate is also shown. From Fig. 5, it can be observed that the HTPFEM yields more accurate results and better convergence rate.
Infinite plate with a circular hole
In this example, consider an infinite plate with a traction free hole under uniaxial tension \((\sigma = 1)\) along xaxis. See Fig. 6 for geometry description, boundary description, and representative polygonal mesh. The exact solution of the principal stresses in polar coordinates \((r,\theta )\) is given by
where a is the radius of the hole. Owing to symmetry, only one quarter of the plate is modeled. The material properties are as follows: Young’s modulus \(E=\) 10\(^5\) and Poisson’s ratio \(\nu =\) 0.3. In this example, analytical tractions are applied on the boundary. The domain is discretized with polygonal elements, and along each edge of the polygon, the shape function is linear. The convergence rate in terms of the displacement norm is shown in Fig. 7. The relative error in the displacement norm for the PFEM and HTPFEM is shown in Fig. 7. It can be seen that the HTPFEM yields more accurate results when compared with the PFEM. The HTPFEM yields slightly a better convergence rate when compared to the PFEM.
Circular beam
As a last example, consider a circular cantilevered beam subjected to a prescribed displacement \(u_o=\)0.01 at the free end. The material property, boundary conditions considered for this study, and a representative polygonal mesh are shown in Fig. 8. The material is assumed to linear elastic and in a state of plane stress. The exact solution for the elastic energy is given by
The convergence in the relative error in the energy norm is shown in Fig. 9. It can be seen that the HTPFEM yields more accurate results when compared with the PFEM. The HTPFEM yields a convergence rate of 1.82 and the PFEM yields a convergence rate of 1.0. Both the methods converge to the exact energy with mesh refinement.
Next, we consider two problems with complex boundary: (a) a wrench and (b) a twodimensional crane hook, both subjected to a concentrated force, \(P=\) 210 KN. The geometry, loading, and boundary conditions are shown in Figs. 10 and 11 for the wrench and the crane hook, respectively. The material properties are as follows: Young’s modulus \(E=\) 3e\(^7\) and Poisson’s ratio \(\nu =\) 0.3. The domain is discretized with arbitrary polygonal elements. The appropriate number of Tfunctions and integration points are chosen based on the number of sides of the polygonal element. As these two problems do not have a closed form solution, we use the results from a FE simulation having 29,016 and 33,104 nodes for wrench and crane hook domain, respectively, as a reference solution. The convergence of the total strain energy with mesh refinement is shown in Fig. 12 for the wrench and crane hook domain. The results from the present method are compared with the results from the conventional PFEM. It is inferred that the present method converges faster than the conventional PFEM. Moreover, for the same number of degrees of freedom, the present method is more accurate than the PFEM with triangulation.
Concluding remarks
In this paper, we studied the convergence and accuracy of hybrid Trefftz polygonal finite elements. The hybrid Trefftz finite elements were constructed by employing the Tcomplete set of functions and a set of independent auxiliary field on the boundary. From the numerical studies presented, it is seen that hybrid Trefftz finite elements yield more accurate results and better convergence rate when compared to the conventional polygonal finite elements with Laplace interpolants. One of the salient features of the hybrid Trefftz approach is that special finite elements with embedded cracks/voids can be constructed. This can then be combined with the extended finite element method to model strong and weak discontinuities and singularities within the domain. However, the success of the method relies on the knowledge of Tcomplete function. This is a topic for future communication.
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Authors' contributions
H is a Ph.D. student jointly supervised by SN and RKA, Department of Mechanical Engineering, IITMadras. SB and EA are external collaborators who helped in clarifying various aspects of the implementation of the method. All authors read and approved the final manuscript.
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The original version of this article was revised to correct the name of R. K. Annabattula.
An erratum to this article is available at http://dx.doi.org/10.1186/s4054001700221.
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Keywords
 Trefftz finite element
 Polytopes
 Tcomplete functions
 Boundary integration