High Order Projection Plane Method for Evaluation of Supersingular Curved Boundary Integrals in BEM

Abstract

Boundary element method (BEM) is a very promising approach for solving various engineering problems, in which accurate evaluation of boundary integrals is required. In the present work, the direct method for evaluating singular curved boundary integrals is developed by considering the third-order derivatives in the projection plane method when expanding the geometry quantities at the field point as Taylor series. New analytical formulas are derived for geometry quantities defined on the curved line/plane, and unified expressions are obtained for both two-dimensional and three-dimensional problems. For the two-dimensional boundary integrals, analytical expressions for the third-order derivatives are derived and are employed to verify the complex-variable-differentiation method (CVDM) which is used to evaluate the high order derivatives for three-dimensional problems. A few numerical examples are given to show the effectiveness and the accuracy of the present method.

Numerical Investigation on Convergence Rate of Singular Boundary Method

“…The boundary element method (BEM) [1][2][3][4] and the method of fundamental solutions (MFS) [5][6][7][8] are two important numerical methods for science and engineering applications. However, it is a sophisticated mathematical and timeconsuming issue for numerical integration over the singularities in the BEM.…”