Efficient hypersingular line and surface integrals direct evaluation by complex variable differentiation method

Abstract

We present an efficient numerical scheme to evaluate hypersingular integrals appeared in boundary element methods. The hypersingular integrals are first separated into regular and singular parts, in which the singular integrals are defined as limits around the singularity and their values are determined analytically by taking the finite part values. The remaining regular integrals can be evaluated using rational interpolatory quadrature or complex variable differentiation (CVDM) for the regular function when machine precision like accuracy is required. The proposed method is then generalised for evaluating hypersingular surface integrals, in which the inner integral is treated as the hypersingular line integral via coordinate transformations. The procedure is implemented into 8-node rectangular boundary element and 6-node triangular element for numerical evaluation. Finally, several numerical examples are presented to demonstrate the efficiency of the present method. To the best of our knowledge, the proposed method is more accurate, faster and more generalised than other methods available in the literature to evaluate hypersingular integrals.