Novel meshless method for solving the potential problems with arbitrary domain

• Published in: Journal of computational physics 2005-10, Vol.209 (1), p.290-321
• Main Authors: Young, D.L., Chen, K.H., Lee, C.W.
• Format: Article
• Language: English
• Subjects: Boundaries; Circulants; Computational techniques; Conventional MFS; Desingularization technique; Dirichlet problem; Double layer potential; Exact sciences and technology; Exact solutions; Finite element method; Hypersingularity; Kernel function; Mathematical analysis; Mathematical methods in physics; Mathematical models; Matrices; Meshless method; Meshless methods; Mixed-type BC; Non-singular; Physics; Singular fundamental solution; Singularity
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2005.03.007

In this article, a non-singular and boundary-type meshless method in two dimensions is developed to solve the potential problems. The solution is represented by a distribution of the kernel functions of double layer potentials. By using the desingularization technique to regularize the singularity and hypersingularity of the kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. The main difficulty of the coincidence of the source and collocation points then disappears. By employing the two-point function, the off-diagonal coefficients of influence matrices are easily obtained. The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the results of exact solution, conventional MFS and BEM for the Dirichlet, Neumann and mix-type boundary conditions (BCs) of interior and exterior problems with simple and complicated boundaries. Good agreements with exact solutions are observed.