Non‐singular boundary integral formulations for plane interior potential problems

Abstract

SUMMARYIn this article, a non-singular formulation of the boundary integral equation is developed to solve smooth and non-smooth interior potential problems in two dimensions. The subtracting and adding-back technique is used to regularize the singularity of Green's function and to simplify the calculation of the normal derivative of Green's function. After that, a global numerical integration is directly applied at the boundary, and those integration points are also taken as collocation points to simplify the algorithm of computation. The result indicates that this simple method gives the convergence speed of order N −3 in the smooth boundary cases for both Dirichlet and mix-type problems. For the non-smooth cases, the convergence speed drops at O(N −1=2 ) for the Dirichlet problems.

Novel meshless method for solving the potential problems with arbitrary domain

“…In which the terms of P N m¼1 A ðiÞ ðs m ; x i Þ and P N m¼1 B ðiÞ ðs m ; x i Þ are the adding-back terms and the terms of A (i) (s i , x i ) and B (i) (s i , x i ) are the subtracting terms in the two brackets for the special treatment technique. After using the desingularization of subtracting and adding-back technique [13,17], we are able to remove the singularity and hypersingularity of the kernel functions. Therefore, the diagonal coefficients for the interior problems can be extracted out as: Table 1 The properties of the influence matrices for the Laplace equation …”

“…Eqs. (4) and (5) for the interior problems need to be regularized by using special treatment of the desingularization of subtracting and adding-back technique [13,17] as follows:…”

“…Even though these methods can locate the source points on the physical boundary and use the non-singular kernels, there still accompanies some difficulty at the ill-posed problems. Therefore, the purpose of this paper is to develop a novel meshless method for solving the potential problems based on the potential theory as well as the desingularization of subtracting and adding-back technique [13,17] to regularize the singularity and hypersingularity of the kernel functions. The proposed method is to distribute the observation and source points on the coincident locations of the real boundary even using the singular kernels (double layer potentials) instead of non-singular kernels and still maintains the spirit of the MFS.…”

Regularized meshless method for multiply-connected-domain Laplace problems

“…Eqs. (4) and (5) for the multiply-connected problems need to be regularized by using the regularization of subtracting and addingback technique [18][19][20] as follows:…”

“…The RMM eliminates the perplexing artificial boundary in the MFS, which can be arbitrary. The subtracting and adding-back technique [18][19][20] can regularize the singularity and hypersingularity of the kernel functions. This method can simultaneously distribute the observation and source points on the physical boundary even using the singular kernels instead of non-singular kernels [21,22].…”