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# Hypersingular meshless method for solving 3D potential problems with arbitrary domain

D. L. Young1,3, K. H. Chen2, T. Y. Liu3, L. H. Shen3, C. S. Wu3

Corresponding author. E-mail: dlyoung@ntu.edu.tw, Tel and Fax: 886-2-23626114.
Department of Civil Engineering, National I-Lan University, I-Lan, Taiwan.
Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan.

Computer Modeling in Engineering & Sciences 2009, 40(3), 225-270. https://doi.org/10.3970/cmes.2009.040.225

## Abstract

In this article, a hypersingular meshless method (HMM) is extended to solve 3D potential problems for arbitrary domains after a 2D model was successfully developed (Young et al. 2005a). The solutions are represented by a distribution of the double layer potentials instead of the single layer potentials as generally used in the conventional method of fundamental solutions (MFS). By using the desingularization technique to regularize the singularity and hypersingularity of the double layer potentials, the source points can be located exactly on the real boundary to avoid the sensitivity of locating fictitious boundary for putting the singularity outside the computational domain as usually faced by the conventional MFS. As a result the diagonal terms of influence matrices are easily determined, and the main singular difficulty of the coincidence of the source and collocation points is then overcome. The numerical evidences of the proposed HMM demonstrate the accuracy of solutions after comparing results with analytical solution, conventional MFS, finite element and local differential quadrature methods for the Dirichlet, Neumann and mixed-type boundary conditions of interior problems with simple and complicated domains. Good agreement with analytical solutions and other numerical results is observed.

## Keywords

Young, D. L., Chen, K. H., Liu, T. Y., Shen, L. H., Wu, C. S. (2009). Hypersingular meshless method for solving 3D potential problems with arbitrary domain. CMES-Computer Modeling in Engineering & Sciences, 40(3), 225–270.

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