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An Improved Formulation of Singular Boundary Method

Published online by Cambridge University Press:  03 June 2015

Wen Chen  and
Yan Gu
Wen Chen*
Affiliation:
College of Harbour, Coastal and Offshore Engineering, Hohai University, No. 1 Xikang Road, Nanjing, Jiangsu 210098, China
Yan Gu*
Affiliation:
College of Harbour, Coastal and Offshore Engineering, Hohai University, No. 1 Xikang Road, Nanjing, Jiangsu 210098, China
*
Corresponding author. URL:http://em.hhu.edu.cn/chenwen/english.html, Email: [email protected]
Email: [email protected]
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Abstract

This study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.

Keywords

65N1565N38Singular boundary methodfundamental solutionsingularitydesingularization techniquemeshless
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 543 - 558
Copyright
Copyright © Global-Science Press 2012

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References

[1]Karageorghis, A., Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities, Numer. Methods. Partial. Diff. Eq., 8 (1992), pp. 1–19.CrossRefGoogle Scholar
[2]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.Google Scholar
[3]Chen, C. S., Golberg, M. A. and Hon, Y. C., The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, Int. J. Numer. Methods Eng., 43 (1998), pp. 1421–1435.3.0.CO;2-V>CrossRefGoogle Scholar
[4]Chen, C. S. and Liu, G. R., Preface to: mesh reduction techniques-part I, Eng. Anal. Bound. Elem., 28 (2004), pp. 423–424.Google Scholar
[5]Liu, Y. J., Nishimura, N. and Yao, Z. H., A fast multipole accelerated method of fundamental solutions for potential problems, Eng. Anal. Bound. Elem., 29 (2005), pp. 1016–1024.Google Scholar
[6]Chen, C. S., Cho, H. A. and Golberg, M. A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng. Anal. Bound. Elem., 30 (2006), pp. 405–410.CrossRefGoogle Scholar
[7]Chen, W. and Tanaka, M., A meshless, integration-free and boundary-only RBF technique, Comput. Math. Appl., 43 (2002), pp. 379–391.Google Scholar
[8]Chen, W. and 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 (2003), pp. 1859–1875.CrossRefGoogle Scholar
[9]Chen, J. T., Chang, M. H., Chen, K. H. and Chen, I. L., Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function, Comput. Mech., 29 (2002), pp. 392–408.Google Scholar
[10]Fooladi, M., Golbakhshi, H., Mohammadi, M. and Soleimani, A., An improved meshless method for analyzing the time dependent problems in solid mechanics, Eng. Anal. Bound. Elem., 35 (2011), pp. 1297–1302.CrossRefGoogle Scholar
[11]Young, D. L., Chen, K. H. and Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys., 209 (2005), pp. 290–321.Google Scholar
[12]Young, D. L., Chen, K. H. and Lee, C. W., Singular meshless method using double layer potentials for exterior acoustics, J. Acoust. Soc. Am., 119 (2005), pp. 96–107.Google Scholar
[13]Cheng, A. H. D. and Cheng, D. T., Heritage and early history of the boundary element method, Eng. Anal. Bound. Elem., 29 (2005), pp. 268–302.Google Scholar
[14]Hwang, W. S., Hung, L. P. and Ko, C. H., Non-singular boundary integral formulations for plane interior potential problems, Int. J. Numer. Methods Eng., 53 (2002), pp. 1751–1762.CrossRefGoogle Scholar
[15]Chen, K. H., Chen, J. T. and Kao, J. H., Regularized meshless method for antiplane shear problems with multiple inclusions, Int. J. Numer. Methods Eng., 73 (2008), pp. 1251–1273.CrossRefGoogle Scholar
[16]Sarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Eng. Anal. Bound. Elem., 33 (2009), pp. 1374–1382.Google Scholar
[17]Liu, Y. J., A new boundary meshfree method with distributed sources, Eng. Anal. Bound. Elem., 34 (2010), pp. 914–919.Google Scholar
[18]Chen, W., Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method, Chin. J. Solid. Mech., 30 (2009), pp. 592–599.Google Scholar
[19]Chen, W. and Wang, F. Z., A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem., 34 (2010), pp. 530–532.Google Scholar
[20]Chen, W., Fu, Z. J. and Wei, X., Potential problems by singular boundary method satisfying moment condition, Comput. Model. Eng. Sci., 54 (2009), pp. 65–85.Google Scholar
[21]Gu, Y., Chen, W. and Zhang, C. Z., Singular boundary method for solving plane strain elastostatic problems, Int. J. Solids Struct., 48 (2011), pp. 2549–2556.Google Scholar
[22]Golberg, M. A., The method of fundamental solutions for Poisson’s equation, Eng. Anal. Bound. Elem., 16 (1995), pp. 205–213.Google Scholar
[23]Poullikkas, A., Karageorghis, A., Georgiou, G. and Ascough, J., The method of fundamental solutions for stokes flows with a free surface, Numer. Methods Partial Diff. Eq., 14 (1998), pp. 667–678.CrossRefGoogle Scholar
[24]Chen, C. S., Muleshkov, A. S., Golberg, M. A. and Mattheij, R. M. M., A mesh-free approach to solving the axisymmetric Poisson’s equation, Numer. Methods Partial Diff. Eq., 21 (2005), pp. 349–367.Google Scholar
[25]Marin, L., Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity, Int. J. Solids Struct., 47 (2010), pp. 3326–3340.Google Scholar
[26]Zheng, J.-H., Soe, M. M., Zhang, C. and Hsu, T.-W., Numerical wave flume with improved smoothed particle hydrodynamics, Hydrodyn. Ser. B, 22 (2010), pp. 773–781.Google Scholar
[27]Banerjee, P. K., The Boundary Element Methods in Engineering, McGRAW-HILL Book Company Europe, 1994.Google Scholar
[28]Chen, J. T. and Chen, P. Y., Null-field integral equations and their applications, in: Boundary Elements and Other Mesh Reduction Methods XXIX., WIT Press, Southampton, 2007, pp. 88–97.Google Scholar