A boundary-precise method of numerical integration over implicitly defined regions

Abstract

In this paper, we introduce a boundary-precise method of numerical integration over implicitly defined regions using the idea of isogeometry. This method ensures precise boundary representations during the integration process by establishing proper local parameterizations near these boundaries based on the implicit representations. We establish an adaptive algorithm for the method, and provide two examples for integration over different types of regions. We also calculate the error while integrating by this method and provide detailed error estimates. When solving partial differential equations (PDEs) using weighted extended B-spline (WEB) method, the augmented matrix in the linear system of Galerkin framework is derived through integration, where this boundary-precise method can be applied. Three different types of examples are displayed and the results demonstrate the convergence of error and the stability of order.