Nonlinear coordinate transformation methods for numerical evaluation of singular integrals

Abstract

This thesis presents a study of the performance of the nonlinear coordinate transformations in the numerical evaluation of singular integrals. Accurate numerical scheme for singular integrals is of importance to reliable implementation of the boundary element method. In Chapter 2, we review the traditional nonlinear coordinate transformations, the polynomial and the parametric transformation. We also propose a new nonlinear coordinate transformation, a parametric sigmoidal transformation, containing a parameter b which has most properties of the sigmoidal transformation. In Chapter 3, we consider the weakly singular integrals. It is shown that the new transformation together with the Gauss-Legendre quadrature can better the asymptotic truncation error of the approximation effectively by controlling the value of b. In Chapter 4, we deal with the numerical evaluation of the Cauchy principal value and the Hadamard finite-part integrals by using the Euler-Maclaurin formula. Through the asymptotic error analysis of the Euler-Maclaurin formula using the parametric sigmoidal transformation, it can be observed that it provide a distinct improvement on its predecessors using traditional sigmoidal transformations.