Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpola
- PDF / 1,995,522 Bytes
- 23 Pages / 595.276 x 790.866 pts Page_size
- 108 Downloads / 201 Views
O R I G I NA L
Márton Petö · Fabian Duvigneau · Daniel Juhre · Sascha Eisenträger
Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants Received: 15 May 2020 / Accepted: 28 August 2020 © The Author(s) 2020
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. Keywords Fictitious domain methods · Polygonal finite element method · Polygonal finite cell method · Discontinuous integrands · Compressed quadtree-decomposition
1 Introduction 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]. Nonetheless, 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. In this context, we propose an efficient solution for computing piece-wise rational integrals arising in the expressions for the element matrices.
1.1 Fictitious domain methods In the conventional FEM, the computational mesh has to conform to the boundary of the domain of interest. As this is one particular bottleneck in the simulation pipeline, extensive research has been devoted to extending M. Petö (B) · F. Duvigneau · D. Juhre Otto von Guericke University Magdeburg, Magdeburg, Germany E-mail: [email protected] S. Eisenträger University of New South Wales, Sydney, Australia
M. Petö et al.
the FEM to unfitted discretizations that do not necessarily conform to geometric features of the given problem. One possibility to exploit this idea is utilized in the extended finite element (XFEM) [3,4] and the generalized finite element methods (GFEM) [4,5], wh
Data Loading...
