Double Integration Over a Complex Domain Using Triangular Mesh and Gaussian Quadrature

Abstract

This paper introduces a novel composite numerical integration method specifically designed for domains with complex nonlinear boundaries, where nonlinear functions define the boundaries. The proposed method aims to evaluate double integrals over such complex domains efficiently. The domain is initially divided into a mesh of uniform or non-uniform triangles, each of which is transformed into a standard triangular finite element using basis functions in a local coordinate system. The standard triangle is further subdivided into right isosceles triangles, facilitating composite numerical integration. Each right isosceles triangle is mapped onto a unit square finite element, where the Gauss-Legendre quadrature rule is employed to evaluate the double integrals. The integrals across the entire domain are computed by summing the contributions from all sub-triangles. Numerical examples are provided to demonstrate the effectiveness and accuracy of the proposed method.

Jagannath University Journal of Science, Volume 11, Number 2, Dec. 2024, pp. 58−66