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In numerical analysis, the singular boundary method (SBM) belongs to a family of meshless boundary collocation techniques which include the method of fundamental solutions (MFS),〔(method of fundamental solutions (MFS) )〕〔Golberg MA, Chen CS, Ganesh M, "Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions", ''Eng Anal Bound Elem'' 2000;24(7–8): 539–47.〕〔Fairweather G, Karageorghis A, "The method of fundamental solutions for elliptic boundary value problems", ''Adv Comput Math'' 1998;9(1): 69–95.〕 boundary knot method (BKM),〔Chen W, Tanaka M, "(A meshless, integration-free, and boundary-only RBF technique )", ''Comput Math Appl '' 2002;43(3–5): 379–91.〕 regularized meshless method (RMM),〔D.L. Young, K.H. Chen, C.W. Lee, "Novel meshless method for solving the potential problems with arbitrary domain", ''J Comput Phys'' 2005;209(1): 290–321.〕 boundary particle method (BPM),〔(boundary particle method (BPM) )〕 modified MFS,〔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'' 2009;33(12): 1374–82.〕 and so on. This family of strong-form collocation methods is designed to avoid singular numerical integration and mesh generation in the traditional boundary element method (BEM) in the numerical solution of boundary value problems with boundary nodes, in which a fundamental solution of the governing equation is explicitly known. The salient feature of the SBM is to overcome the fictitious boundary in the method of fundamental solution, while keeping all merits of the latter. The method offers several advantages over the classical domain or boundary discretization methods, among which are: * meshless. The method requires neither domain nor boundary meshing but boundary-only discretization points; * integration-free. The numerical integration of singular or nearly singular kernels could be otherwise troublesome, expensive, and complicated, as in the case, for example, the boundary element method; * boundary-only discretization for homogeneous problems. The SBM shares all the advantages of the BEM over domain discretization methods such as the finite element or finite difference methods; * to overcome the perplexing fictitious boundary in the method of fundamental solutions (see Figs. 1 and 2), thanks to the introduction of the concept of the origin intensity factor, which isolates the singularity of the fundamental solutions. The SBM provides a significant and promising alternative to popular boundary-type methods such as the BEM and MFS, in particular, for infinite domain, wave, thin-walled structures, and inverse problems. == History of the singular boundary method == The methodology of the SBM was firstly proposed by Chen and his collaborators in 2009.〔Chen W, "(Singular boundary method: A novel, simple, meshfree, boundary collocation numerical method )", ''Chin J Solid Mech'' 2009;30(6): 592–9.〕〔Chen W, Wang FZ, "(A method of fundamental solutions without fictitious boundary )", ''Eng Anal Bound Elem'' 2010;34(5): 530–32.〕 The basic idea is to introduce a concept of the origin intensity factor to isolate the singularity of the fundamental solutions so that the source points can be placed directly on the real boundary. In comparison, the method of fundamental solutions requires a fictitious boundary for placing the source points to avoid the singularity of fundamental solution. The SBM has since been successfully applied to a variety of physical problems, such as potential problems,〔Wei X, Chen W, Fu ZJ, "Solving inhomogeneous problems by singular boundary method", ''J Mar SCI Tech'' 2012; 20(5).〕〔Chen W, Fu ZJ, Wei X, "(Potential Problems by Singular Boundary Method Satisfying Moment Condition )", ''Comput Model Eng Sci'' 2009;54(1): 65–85.〕 infinite domain problem,〔Chen W, Fu Z, "(A novel numerical method for infinite domain potential problems )", ''Chin Sci Bull'' 2010;55(16): 1598–603.〕 Helmholtz problem,〔Fu ZJ, Chen W, "A novel boundary meshless method for radiation and scattering problems", ''Advances in Boundary Element Techniques XI, Proceedings of the 11th international Conference'', 12–14 July 2010, 83–90, Published by EC Ltd, United Kingdom (ISBN 978-0-9547783-7-8)〕 and plane elasticity problem.〔Gu Y, Chen W, Zhang CZ., "(Singular boundary method for solving plane strain elastostatic problems )", ''Int J Solids Struct'' 2011;48(18): 2549–56.〕 There are the two techniques to evaluate the origin intensity factor. The first approach is to place a cluster of sample nodes inside the problem domain and to calculate the algebraic equations. The strategy leads to extra computational costs and makes the method is not as efficient as expected compared to the MFS. The second approach〔Chen W, Gu Y, "(Recent advances on singular boundary method )", ''Joint International Workshop on Trefftz Method VI and Method of Fundamental Solution II'', Taiwan 2011.〕〔Gu Y, Chen, W, "(Improved singular boundary method for three dimensional potential problems )", ''Chinese Journal of Theoretical and Applied Mechanics'', 2012, 44(2): 351-360 (in Chinese)〕 is to employ a regularization technique to cancel the singularities of the fundamental solution and its derivatives. Consequently, the origin intensity factors can be determined directly without using any sample nodes. This scheme makes the method more stable, accurate, efficient, and extends its applicability. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「singular boundary method」の詳細全文を読む スポンサード リンク
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