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Evaluating the Origin Intensity Factor in the Singular Boundary Method for Three-Dimensional Dirichlet Problems

  • Linlin Sun (a1) (a2) (a3), Wen Chen (a1) and Alexander H.-D. Cheng (a3)


In this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.


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*Corresponding author. Email: (W. Chen)


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[1] Chen, W. and Wang, F. Z., A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem., 34 (2010), pp. 530532.
[2] Chen, W., Singular boundary method: a novel, simple, meshfree, boundary collocation numerical method, Chinese J. Solid Mech., 30(6) (2009), pp. 592599 (in Chinese).
[3] Nintcheu Fata, S., Explicit expressions for 3D boundary integrals in potential theory, Int. J. Numer. Methods Eng., 78(1) (2009), pp. 3247.
[4] Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C. L., Boundary Element Techniques: Theory and Applications in Engineering, Springer, New York, 1984.
[5] Cheng, A. H. D. and Cheng, D. T., Heritage and early history of the boundary element method, Eng. Anal. Bound. Elem., 29 (2005), pp. 268302.
[6] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 6995.
[7] Golberg, M. A. and Chen, C. S., The method of fundamental solutions for potential, helmholtz and diffusion problems, In Golberg, M.A., editor, Boundary Integral Methods–Numerical and Mathematical Aspects, pages 103176, Computational Mechanics Publications, Southhampton, 1998.
[8] Karageorghis, A., Lesnic, D. and Marin, L., A survey of applications of the MFS to inverse problems, Inverse Probl. Sci. Eng., 19(3) (2011), pp. 309336.
[9] Chen, W. and Gu, Y., An improved formulation of singular boundary method, Adv. Appl.Math. Mech., 4(5) (2012), pp. 543558.
[10] Liu, Y. and Rizzo, F. J., A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems, Comput. Methods Appl. Mech. Eng., 96(2) (1992), pp. 271287.
[11] Sladek, V., Sladek, J. and Tanaka, M., Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity, Int. J. Numer. Methods Eng., 36 (1993), pp. 16091628.
[12] Gu, Y., Chen, W. and He, X. Q., Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media, Int. J.Heat Mass Transfer, 55 (2012), pp. 48374848.
[13] Gu, Y. and Chen, W., Infinite domain potential problems by a new formulation of singular boundary method, Appl. Math. Model., 37 (2013), pp. 16381651.
[14] Fu, Z. J., Chen, W. and Gu, Y., Burton-Miller type singular boundary method for acoustic radiation and scattering, J. Sound Vib., 333(16) (2014), pp. 37763793.
[15] Lin, J., Chen, W. and Chen, C. S., Numerical treatment of acoustic problems with boundary singularities by the singular boundary method, J. Sound Vib., 333(14) (2014), pp. 31773188.
[16] Chen, W., Zhang, J. Y. and Fu, Z. J., Singular boundary method for modified Helmholtz equations, Eng. Anal. Bound. Elem., 44 (2014), pp. 112119.
[17] Qu, W. Z. and Chen, W., Solution of two-dimensional Stokes flow problems using singular boundary method, Adv. Appl. Math.Mech., 7(1) (2015), pp. 1330.
[18] Yang, C. and Li, X. L., Meshless singular boundary methods for biharmonic problems, Eng. Anal. Bound. Elem., 56 (2015), pp. 3948.
[19] Wei, X., Chen, W., Chen, B. and Sun, L. L., Singular boundary method for heat conduction problems with certain spatially varying conductivity, Comput. Math. Appl., 69 (2015), pp. 206222.
[20] Wei, X., Chen, W., Sun, L. and Chen, B., A simple accurate formula evaluating origin intensity factor in singular boundary method for two-dimensional potential problems with Dirichlet boundary, Eng. Anal. Bound. Elem., 58(0) (2015), pp. 151165.
[21] Chen, W., Fu, Z. J. and Wei, X., Potential problems by singular boundary method satisfying moment condition, Comput. Model. Eng. Sci., 54 (2009), pp. 6586.
[22] 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. 290321.
[23] Sun, L. L., Chen, W. and Zhang, C. Z., A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems, Appl. Math. Model., 37(12) (2013), pp. 74527464.
[24] Schaback, R., Adaptive numerical solution of MFS systems, In Chen, C.S., Karageorghis, A., Smyrlis, Y. S., eds., The Method of Fundamental Solutions–A Meshless Method, pages 127, Dynamic Publishers, Inc., Atlanta, 2008.
[25] Shigeta, T., Young, D. L. and Liu, C. S., Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation, J. Comput. Phys., 231 (2012), pp. 71187132.
[26] Li, M., Chen, C. S. and Karageorghis, A., The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions, Comput. Math. Appl., 66 (2013), pp. 24002424.
[27] Chen, C. S., Karageorghis, A. and Li, Yan, On choosing the location of the sources in the MFS, Numer. Algorithms, 72 (2016), pp. 107130.
[28] Banerjee, P. K., The Boundary Element Methods in Engineering, McGRAW-HILL Book Company, Europe, 1994.


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Evaluating the Origin Intensity Factor in the Singular Boundary Method for Three-Dimensional Dirichlet Problems

  • Linlin Sun (a1) (a2) (a3), Wen Chen (a1) and Alexander H.-D. Cheng (a3)


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