Optimal Numerical Integration Schemes for a Family of Polygonal Finite Elements with Schwarz–Christoffel Conformal Mapping

Abstract

This paper presents optimal numerical integration schemes for a family of Polygonal Finite Elements with Schwarz–Christoffel (SC) conformal mapping with improved convergence properties. With Wachspress interpolation, the arbitrary polygons of a finite element mesh are first mapped to regular polygons using isoparametric mapping and then to the unit-disk using SC mapping. Numerical integration is done over this unit-disk by determining integration points which are optimal with respect to Frobenius norm of stiffness matrix and infinity norm of displacement field. Lowest order Gauss quadrature points on isoparametrically mapped domain are considered as initial conditions for optimization. Hence this method achieves better accuracy and faster convergence compared to other contemporary methods with fewer computational resources. The method automatically reduces to standard Gauss quadrature-based integration when polygons are quadrilaterals as a special case. In general, only the element boundary poses C0 continuity whereas the element interior has higher order spectral properties. Numerical results presented for a few benchmark problems show good accuracy of the proposed schemes. The method is validated considering structured as well as unstructured meshes consisting of pentagons and hexagons which is a significant step forward to meshing and solving elasticity problem in complicated geometry and interfaces, material microstructure, and so forth.