Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties
 Hirshikesh^{1},
 S. Natarajan^{1}Email authorView ORCID ID profile,
 R. K. Annabattula^{1},
 S. Bordas^{2} and
 E. Atroshchenko^{3}
DOI: 10.1186/s4054001700203
© The Author(s) 2017
Received: 10 August 2016
Accepted: 3 May 2017
Published: 25 May 2017
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.
Keywords
Trefftz finite element Polytopes Tcomplete functions Boundary integrationBackground
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. [22–24] 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 [37–39]. Recently, the idea of employing local solutions over arbitrary finite elements has been investigated in [5, 31–33]. 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
Generalization to arbitrary polygons
Overview of hybrid Trefftz finite element method
Numerical examples

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.
Cantilever beam bending
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
Circular beam
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.
Notes
Declarations
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.
Acknowledgements
No funding was received for this project.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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