The semianalytical analysis of nearly singular integrals in 2D potential problem by isogeometric boundary element method

Abstract

Benefited from the accuracy improvement in modeling physical problem of complex geometry and integrating the discretization and simulation, the isogeometric analysis in boundary element method (IGABEM) has been drawn a great deal of attention. The nearly singular integrals of 2D potential problem in the IGABEM are addressed by a semianalytical scheme in the present work. We use the subtraction technique to separate the integrals to singular and nonsingular parts, where the singular parts can be calculated by the analytical formulae derived by utilizing a series of integration by parts, while the nonsingular parts are calculated numerically with fewer quadrature points. Comparing the present semianalytical results with the ones of exact solutions, we find that the present method can obtain precise potential and flux densities of inner points much closer to the boundary without refining the elements nearby. Sufficient comparisons with other regularization schemes, such as the exponential and sinh transformation methods, are also conducted. The results in the numerical examples show the competitiveness of the present method, especially when calculating the nearly strongly and highly singular integrals during the simulation of the flux density.

Numerical Aspects of Isogeometric Boundary Element Methods: (Nearly) Singular Quadrature, Trimmed NURBS and Surface Crack Modeling

“…This also complicates the singular and nearly-singular behaviors in the integrand of BEM. Efforts have been made to tackle this problem [30,31]. In this work, we investigate the conformal transformation technique [29] to the improve the (hyper-)singular integration in detail, then an L − 1/5 transformation is implemented to handle nearly singular integration.…”

An isogeometric boundary element method for heat transfer problems of multiscale structures in electronic packaging with arbitrary heat sources

“…In [79] the power series expansion method is used to evaluate the strongly and weakly singular integrals in 3D IGABEM. The nearly singular integrals in IGABEM can be computed by a semianalytical analysis method [67], adaptive integration method [80], or transformation method [65]. Recently, Gong et al [68,69] developed a hybrid integration scheme for the nearly singular integrals in the 2D/3D IGABEM.…”

“…In [66], the nearly singular integrals are computed by sinh-transformation method. In [67], a semianalytical method is presented to evaluate the nearly singular integrals in IGABEM. Recently, a hybrid nearly singular integration method is given in [68,69] in the study of coated structures, that is capable of producing accurate and efficient solutions for thin sections.…”