Gauss quadrature rules for numerical integration over a standard tetrahedral element by decomposing into hexahedral elements

摘要

In recent years hexahedral elements have gained more importance than compared to tetrahedral elements (e.g. importance in the study of aero-acoustic equations using hexahedral elements to check the computational efficiency between tetrahedral and hexahedral elements). Also among the various integration schemes, Gauss Legendre quadrature which can evaluate exactly the (2n−1)th order polynomial with n-Gaussian points is most commonly used in view of the accuracy and efficiency of calculations. In this paper, we present a Gauss quadrature method for numerical integration over a standard tetrahedral element T[0,1]3 by decomposing into hexahedral elements H[−1,1]3. The method can be used for computing integrals of smooth functions, as well as functions with end-point singularities. The performance of the method is demonstrated with several numerical examples. By the proposed method, with less number of divisions we are obtaining the exact solutions with minimum errors and number of computations is reduced drastically. We have evaluated the aspect ratio value of each hexahedral element which is in the range 1–5, as per the element quality check these elements can be used for mesh generation in FEM.