Abstract
Polygonal finite elements offer an increased freedom in terms of mesh generation at the price of more complex, often rational, shape functions. Thus, the numerical integration of rational interpolants over polygonal domains is one of the challenges that needs to be solved. If, additionally, strong discontinuities are present in the integrand, e.g., when employing fictitious domain methods, special integration procedures must be developed. Therefore, we propose to extend the conventional quadtree-decomposition-based integration approach by image compression techniques. In this context, our focus is on unfitted polygonal elements using Wachspress shape functions. In order to assess the performance of the novel integration scheme, we investigate the integration error and the compression rate being related to the reduction in integration points. To this end, the area and the stiffness matrix of a single element are computed using different formulations of the shape functions, i.e., global and local, and partitioning schemes. Finally, the performance of the proposed integration scheme is evaluated by investigating two problems of linear elasticity.
Highlights
The traditional finite element method (FEM) offers a robust and well-studied approach for simulating a large variety of physical phenomena governed by partial differential equations [1,2]
Following the poly-finite cell method (FCM) approach proposed by Duczek and Gabbert [36] and the fundamental concept of polygonal elements based on generalized barycentric coordinates [27,28], the enhanced numerical integration scheme based on compressed quadtree-decompositions introduced by Petö et al [38] was investigated in the context of unfitted polygonal meshes
Since the compression procedure leads to a decreased integration point density for integrating discontinuous nonpolynomial functions, the effect of the compression on the integration quality was investigated, with a particular focus on the rational shape functions proposed by Wachspress
Summary
The traditional finite element method (FEM) offers a robust and well-studied approach for simulating a large variety of physical phenomena governed by partial differential equations [1,2]. Throughout the years, several extensions have been developed in order to increase the accuracy, widen the field of application and decrease the computation time. In this contribution, we focus on the combination of two such extensions, namely the fictitious domain approach and polygonal elements employing shape functions based on generalized barycentric coordinates. We focus on the combination of two such extensions, namely the fictitious domain approach and polygonal elements employing shape functions based on generalized barycentric coordinates In this context, we propose an efficient solution for computing piece-wise rational integrals arising in the expressions for the element matrices
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