×

zbMATH — the first resource for mathematics

Novel meshless method for solving the potential problems with arbitrary domain. (English) Zbl 1073.65139
Summary: 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 the exact solution, the conventional method of fundamental solutions and the boundary element method for the Dirichlet, Neumann and mix-type boundary conditions of interior and exterior problems with simple and complicated boundaries. Good agreements with exact solutions are observed.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions with formulation, graphs and mathematical tables, (1972), Dover New York · Zbl 0543.33001
[2] Belytschko, T.; Gu, L.; Lu, Y., Fracture and crack growth by element-free Galerkin methods, Model. simul. mater. sci. eng., 2, 519-534, (1994)
[3] Chen, J.T.; Kuo, S.R.; Chen, K.H.; Cheng, Y.C., Comments on vibration analysis of arbitrary shaped membranes using non-dimensional dynamic influence function, J. sound vibr., 235, 1, 156-171, (2000)
[4] Chen, J.T.; Chang, M.H.; Chen, K.H.; Lin, S.R., The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, J. sound vibr., 257, 4, 667-711, (2002)
[5] Chen, J.T.; Chang, M.H.; Chen, K.H., Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function, Comput. mech., 29, 392-408, (2002) · Zbl 1146.76622
[6] J.T. Chen, I.L. Chen, C.S. Wu, On the equivalence of MFS and Trefftz method for Laplace problems, in: Proceedings of the Global Chinese Workshop on Boundary Element and Meshless Method, Hebei, China, 2003
[7] Chen, J.T.; Chen, I.L.; Chen, K.H.; Yeh, Y.T.; Lee, Y.T., A meshless method for free vibration of arbitrarily shaped plates with clamped boundaries using radial basis function, Eng. anal. bound elem., 28, 535-545, (2004) · Zbl 1130.74488
[8] Chen, W.; Tanaka, M., A meshfree, integration-free and boundary-only RBF technique, Comput. math. appl., 43, 379-391, (2002) · Zbl 0999.65142
[9] 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. methods appl. mech. eng., 192, 1859-1875, (2003) · Zbl 1050.76040
[10] Cheng, A.H.D.; Young, D.L.; Tsai, C.C., The solution of poisson’s equation by iterative DRBEM using compactly supported, positive definite radial basis function, Eng. anal. bound. elem., 24, 7, 549-557, (2000) · Zbl 0966.65089
[11] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. comput. math., 9, 69-95, (1998) · Zbl 0922.65074
[12] Gingold, R.A.; Maraghan, J.J., Smoothed particle hydrodynamics: theory and applications to non-spherical stars, Man. not. astron. soc., 181, 375-389, (1977) · Zbl 0421.76032
[13] Hwang, W.S.; Hung, L.P.; Ko, C.H., Non-singular boundary integral formulations for plane interior potential problems, Int. J. numer. meth. eng., 53, 1751-1762, (2002) · Zbl 0994.65130
[14] Kang, S.W.; Lee, J.M.; Kang, Y.J., Vibration analysis of arbitrary shaped membranes using non-dimensional dynamic influence function, J. sound vibr., 221, 1, 117-132, (1999)
[15] Kang, S.W.; Lee, J.M., Application of free vibration analysis of membranes using the non-dimensional dynamic influence function, J. sound vibr., 234, 3, 455-470, (2000)
[16] Liu, W.K.; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle methods for structural dynamics, Int. J. numer. meth. eng., 38, 1655-1679, (1995) · Zbl 0840.73078
[17] Tournour, M.A.; Atalla, N., Efficient evaluation of the acoustic radiation using multipole expansion, Int. J. numer. meth. eng., 46, 825-837, (1999) · Zbl 1041.76548
[18] C.C. Tsai, Meshless numerical methods and their engineering applications, Ph.D. Dissertation, National Taiwan University, Taipei, Taiwan, 2002
[19] Tsai, C.C.; Young, D.L.; Cheng, A.H.D., Meshless BEM for three-dimensional Stokes flows, Computer modeling in engineering and science (CMES), 3, 117-128, (2002) · Zbl 1147.76595
[20] Smyrlis, Y.S.; Karageorghis, A., Some aspects of the method of fundamental solutions for certain harmonic problems, J. scientific comput., 16, 3, 341-371, (2001) · Zbl 0995.65116
[21] Young, D.L.; Tsai, C.C.; Eldho, T.I.; Cheng, A.H.D., Solution of Stokes flow using an iterative DRBEM based on compactly-supported, positive definite radial basis function, Comput. math. appl., 43, 607-619, (2002) · Zbl 1073.76595
[22] Mukherjee, Y.X.; Mukherjee, S., The boundary node method for potential problems, Int. J. numer. meth. eng., 40, 797-815, (1997) · Zbl 0885.65124
[23] Zhang, J.M.; Tanaka, M.; Matsumoto, T., Meshless analysis of potential problems in three dimensions with the hybrid boundary node method, Int. J. numer. meth. eng., 59, 1147-1160, (2004) · Zbl 1048.65121
[24] Zhang, J.M.; Yao, Z.H.; Li, H., A hybrid boundary node method, Int. J. numer. meth. eng., 53, 751-763, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.