Numerical Integration Approach Based on Radial Integration Method for General 3D Polyhedral Finite Elements

Abstract

We construct an efficient quadrature method for the integration of the Galerkin weak form over general 3D polyhedral elements based on the radial integration method (RIM). The basic idea of the proposed method is to convert the polyhedral domain integrals to contour plane integrals of the element by utilizing the RIM which can be used for accurate evaluation of various complicated domain integrals. The quadrature construction scheme for irregular polyhedral elements involves the treatment of the nonpolynomial shape functions as well as the arbitrary geometry shape of the elements. In this approach, the volume integrals for polyhedral elements with triangular or quadrilateral faces are evaluated by transforming them into face integrals using RIM. For those polyhedral elements with irregular polygons, RIM is again used to convert the face integrals into line integrals. As a result, the volume integration of Galerkin weak form over the polyhedral elements can be easily carried out by a number of line integrals along the edges of the polyhedron. Some benchmark numerical examples including the patch tests are utilized to demonstrate the accuracy and convenience of the proposed method.