Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties
© The Author(s) 2017
Received: 10 August 2016
Accepted: 3 May 2017
Published: 25 May 2017
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 T-complete 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 T-complete 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.
KeywordsTrefftz finite element Polytopes T-complete functions Boundary integration
Sukumar  used Voronoï cells and natural neighbor interpolants to develop a finite difference method on unstructured grids. Rashid and Gullet  proposed a variable element topology finite element method, in which shape functions for convex and non-convex elements are computed in the physical space using constrained minimization procedure. Based on the assumed stress hybrid formulation, Ghosh et al.  developed the Voronoï cell finite element method. Tiwary et al.  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 . 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.  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  and the virtual element methods . 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  presented a concise discussion on different approximation procedures to differential equations. It was shown that Trefftz-type approximation is a particular form of weighted residual approximation. This can be used to generate hybrid finite elements. Earlier studies employed boundary-type 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 T-complete 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 Trefftz-type 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
PFEM Polygonal finite element method with Laplace/Wachspress interpolants (conventional approach). The numerical integration within each element is done by sub-dividing the polygon into triangles and employing a sixth-order Dunavant quadrature rule.
HT-PFEM Hybrid Trefftz polygonal finite element method. Within each polygon, T-complete functions are employed to compute the stiffness matrix. One-dimensional Gaussian quadrature is employed along the boundary of the polygon and the order of the quadrature depends on the number of T-complete 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 HT-PFEM and Polygonal FEM are compared with the available analytical solution. Both the Polygonal FEM and the HT-PFEM 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 HT-PFEM yields more accurate results and better convergence rate.
Infinite plate with a circular hole
Next, we consider two problems with complex boundary: (a) a wrench and (b) a two-dimensional 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 T-functions 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.
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 T-complete 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 T-complete function. This is a topic for future communication.
H is a Ph.D. student jointly supervised by SN and RKA, Department of Mechanical Engineering, IIT-Madras. 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.
No funding was received for this project.
The authors declare that they have no competing interests.
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- Wachspress E (1971) A rational basis for function approximation. Springer, New YorkView ArticleMATHGoogle Scholar
- Alwood R, Cornes G (1969) A polygonal finite element for plate bending problems using the assumed stress approach. Int J Numer Methods Eng 1:135View ArticleGoogle Scholar
- Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61:2045MathSciNetView ArticleMATHGoogle Scholar
- Dasgupta G (2003) Interpolants within convex polygons: Wachspress’ shape functions. J Aerosp Eng (ASCE) 16(1):1View ArticleGoogle Scholar
- Rjasanow S, Weißer S (2012) Higher order BEM-based FEM on polygonal meshes. SIAM J Numer Anal 50:2357MathSciNetView ArticleMATHGoogle Scholar
- Barros FB, de Barcellos CS, Duarte CA (2007) p-Adaptive C k generalized finite element method for arbitrary polygonal clouds. Comput Mech 41:175View ArticleMATHGoogle Scholar
- da Veiga LB, Brezzi F, Cangiani A, Manzini G, Marini L, Russo A (2013) Basic principles of virtual element methods. Maths Models Methods Appl Sci 23:199MathSciNetView ArticleMATHGoogle Scholar
- Bishop J (2013) A displacement-based finite element formulation for general polyhedra using harmonic shape functions. Int J Numer Methods Eng 97:1. doi:10.1002/nme.4562 MathSciNetView ArticleMATHGoogle Scholar
- Biabanaki S, Khoei A (2012) A polygonal finite element method for modeling arbitrary interfaces in large deformation problems. Comput Mech 50:19. doi:10.1007/s00466-011-0668-4 MathSciNetView ArticleMATHGoogle Scholar
- Biabanaki S, Khoei A, Wriggers P (2013) Polygonal finite element methods for contact-impact problems on non-conformal meshes. Comput Methods Appl Mech Eng 269:198. doi:10.1016/j.cma.2013.10.025 MathSciNetView ArticleMATHGoogle Scholar
- Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43(5):839MathSciNetView ArticleMATHGoogle Scholar
- Sukumar N (2013) Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons. Comput Methods Appl Mech Eng 263:27MathSciNetView ArticleMATHGoogle Scholar
- Natarajan S, Bordas S, Mahapatra DR (2009) Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping. Int J Numer Methods Eng 80:103MathSciNetView ArticleMATHGoogle Scholar
- Mousavi S, Xiao H, Sukumar N (2010) Generalized Gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng 82(1):99MathSciNetMATHGoogle Scholar
- Talischi C, Paulino GH (2013) http://arxiv.org/pdf/1307.4423v1.pdf. (In review)
- Fries T, Matthies H (2003) Classification and overview of meshfree methods. Tech. Rep. D38106, Institute of Scientific Computing, Technical University, Braunschweig, Hans-Sommer-StrasseGoogle Scholar
- Sukumar N, Malsch E (2006) Recent advances in the construction of polygonal finite element interpolants. Arch Comput Methods Eng 13(1):129MathSciNetView ArticleMATHGoogle Scholar
- Sukumar N (2003) Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids. Int J Numer Methods Eng 57:1MathSciNetView ArticleMATHGoogle Scholar
- Rashid M, Gullet P (2000) On a finite element method with variable element topology. Comput Methods Appl Mech Eng 190(11–12):1509MathSciNetView ArticleMATHGoogle Scholar
- Moorthy S, Ghosh S (2000) Adaptivity and convergence in the Voronoi cell finite element model for analyzing heterogeneous materials. Comput Methods Appl Mech Eng 185:37View ArticleMATHGoogle Scholar
- Tiwary A, Hu C, Ghosh S (2007) Numerical conformal mapping method based Voronoi cell finite element model for analyzing microstructures with irregular heterogeneities. Finite Elem Anal Design 43:504MathSciNetView ArticleGoogle Scholar
- Liu G, Nguyen T, Dai K, Lam K (2007) Theoretical aspects of the smoothed finite element method (SFEM). Int J Numer Methods Eng 71(8):902MathSciNetView ArticleMATHGoogle Scholar
- Dai K, Liu G, Nguyen T (2007) An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics. Finite Elem Anal Design 43:847View ArticleGoogle Scholar
- Nguyen-Thoi T, Liu G, Nguyen-Xuan H (2011) An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics. Int J Numer Methods Eng 27:1446MATHGoogle Scholar
- Wolf J, Song C (2001) The scaled boundary finite-element method—a fundamental solution-less boundary—element method. Comput Methods Appl Mech Eng 190:5551View ArticleMATHGoogle Scholar
- Ooi ET, Song C, Tin-Loi F, Yang Z (2012) Polygon scaled boundary finite elements for crack propagation modelling. Int J Numer Methods Eng 91:319MathSciNetView ArticleMATHGoogle Scholar
- Tang X, Wu S, Zheng C, Zhang J (2009) A novel virtual node method for polygonal elements. Appl Math Mech 30(10):1233MathSciNetView ArticleMATHGoogle Scholar
- da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini L, Russo A (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23:199MathSciNetView ArticleMATHGoogle Scholar
- Qin Q (2005) Trefftz finite element method and its applications. Appl Mech Rev 58:316View ArticleGoogle Scholar
- Copeland D, Langer U, Pusch D (2009) In: Bercovier M, Gander M, Komhuber R, Widlund O (eds). Lecture notes in Computational Science and Engineering, vol. 70Google Scholar
- Hofreither C, Langer U, Pechstein C (2010) Analysis of a non-standard finite element method based on boundary integral operators. Elect Trans Num Anal 37:413MathSciNetMATHGoogle Scholar
- Weißer S (2011) Residual error estimate for BEM-based FEM on polygonal meshes. Numerische Math 118:765MathSciNetView ArticleMATHGoogle Scholar
- Weißer S (2012) Finite element methods with local trefftz trial functions. Ph.D. thesis, Universität des Saarlandes, SaarbrückenGoogle Scholar
- Zienkiewicz OC (1997) Trefftz type approximation and the generalized finite element method—history and development. Comput Assis Mech Eng Sci 4:305MATHGoogle Scholar
- Piltner R (1985) Special finite elements with holes and internal cracks. Int J Numer Methods Eng 21:1471MathSciNetView ArticleMATHGoogle Scholar
- Qin QH, He XQ (2009) Special elliptic hole elements of Trefftz FEM in stress concentration analysis. J Mech MEMS 1:335Google Scholar
- Jirousek J, Wróblewski A, Qin Q, He X (1995) A family of quadrilateral hybrid-Trefftz p-elements for thick plate analysis. Comput Methods Appl Mech Eng 127:315View ArticleMATHGoogle Scholar
- Qin Q (1995) Hybrid-Trefftz finite element method for Reissner plates on an elastic foundation. Comput Methods Appl Mech Eng 122:379MathSciNetView ArticleMATHGoogle Scholar
- Choo YS, Choi N, Lee BC (2010) A new hybrid-Trefftz triangular and quadrilateral plate elements. Appl Math Model 34:14MathSciNetView ArticleMATHGoogle Scholar
- Du Q, Wang D (2005) Anisotropic centroidal Voronoi tessellations and their applications. SIAM J Sci Comput 26(3):737MathSciNetView ArticleMATHGoogle Scholar
- Sieger D, Alliez P, Botsch M (2010) In: Proceedings of the 19th International Meshing Roundtable, pp. 335–350Google Scholar
- Talischi C, Paulino GH, Pereira A, Menezes IF (2012) PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45:329MathSciNetView ArticleMATHGoogle Scholar