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A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems. (English) Zbl 1426.65201
Summary: This paper proposes a new formulation of regularized meshless method (RMM), which differs from the traditional RMM in that the traditional formulation generates the diagonal elements of influence matrix via null-field integral equations, while our new one directly employs the boundary integral equations at the domain point to evaluate the diagonal elements. We test the present RMM formulation to two-dimensional anisotropic potential problems in finite and infinite domains in comparison with the traditional RMM. Numerical results show that the present RMM sharply outperforms the traditional RMM in the solution of interior problems, while the latter is clearly superior for exterior problems. A rigorous theoretical analysis of circular domain case also corroborates such numerical experiment observations and is provided in the appendix of this paper.

65N99 Numerical methods for partial differential equations, boundary value problems
65N38 Boundary element methods for boundary value problems involving PDEs
74E10 Anisotropy in solid mechanics
74S15 Boundary element methods applied to problems in solid mechanics
Full Text: DOI
[1] Young, D. L.; Chen, K. H.; Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domains, J. Comput. Phys., 209, 290-321, (2005) · Zbl 1073.65139
[2] Young, D. L.; Chen, K. H.; Lee, C. W., Singular meshless method using double layer potentials for exterior acoustics, J. Acoust. Soc. Am., 119, 96-107, (2006)
[3] Chen, K. H.; Lu, M. C.; Hsu, H. M., Regularized meshless method analysis of the problem of obliquely incident water wave, Eng. Anal. Bound. Elem., 35, 355-362, (2011) · Zbl 1259.76040
[4] Golberg, M. A., The method of fundamental solutions for poisson’s equation, Eng. Anal. Bound. Elem., 16, 205-213, (1995)
[5] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9, 69-95, (1998) · Zbl 0922.65074
[6] Tyler, W. D.; Ashley, L. M.; Leevan, L., Applicability of the method of fundamental solutions, Eng. Anal. Bound. Elem., 33, 637-643, (2009) · Zbl 1244.65220
[7] Chen, W.; Hon, Y. C., Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems, Comput. Meth. Appl. Mech. Eng., 192, 1859-1875, (2003) · Zbl 1050.76040
[8] Wang, F. Z.; Ling, L.; Chen, W., Effective condition number for boundary knot method, CMC-Comput. Mater. Con., 12, 57-70, (2009)
[9] Chen, W.; Lin, J.; Wang, F. Z., Regularized meshless method for nonhomogeneous problems, Eng. Anal. Bound. Elem., 35, 253-257, (2011) · Zbl 1259.65200
[10] Song, R. C.; Chen, W., An investigation on the regularized meshless method for irregular domain problems, CMES-Comput. Model. Eng. Sci., 42, 59-70, (2009) · Zbl 1357.65305
[11] Chen, W.; Song, R. C., Analytical diagonal elements of regularized meshless method for regular domains of 2D Dirichlet Laplace problems, Eng. Anal. Bound. Elem., 34, 2-8, (2010) · Zbl 1244.65226
[12] Chen, W.; Fu, Z. J.; Wei, X., Potential problems by singular boundary method satisfying moment condition, CMES-Comput. Model. Eng. Sci., 54, 65-85, (2009) · Zbl 1231.65245
[13] Chen, W.; Wang, F. Z., A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem., 34, 530-532, (2010) · Zbl 1244.65219
[14] Mantič, V.; París, F.; Berger, J., Singularities in 2D anisotropic potential problems in multi-material corners real variable approach, Int. J. Solids. Struct., 40, 5197-5218, (2003) · Zbl 1060.74548
[15] Zhou, H. L.; Niu, Z. R.; Cheng, C. Z.; Guan, Z. W., Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems, Comput. Struct., 86, 1656-1671, (2008)
[16] Gao, X. W., Source point isolation boundary element method for solving general anisotropic potential and elastic problems with varying material properties, Eng. Anal. Bound. Elem., 34, 1049-1057, (2010) · Zbl 1244.74156
[17] Logan, J. D., Applied partial differential equations, (2004), Springer New York · Zbl 1055.35001
[18] Davis, P. J.; Rabinowitz, P., Methods of numerical integration, (1984), Academic Press New York · Zbl 0154.17802
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