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A mapping method for numerical evaluation of two-dimensional integrals with \(1/r\) singularity. (English) Zbl 0776.73073
Summary: Singular integrals occur commonly in applications of the boundary element method (BEM). A simple mapping method is presented here for the numerical evaluation of two-dimensional integrals in which the integrands, at worst, are \(O(1/r)\) \((r\) being the distance from a source to a field point). This mapping transforms such integrals over general curved triangles into regular 2-D integrals. Over flat and curved quadratic triangles, regular line integrals are obtained, and these can be easily evaluated by standard Gaussian quadrature. Numerical tests on some typical singular integrals, encountered in BEM applications, demonstrate the accuracy and efficacy of the method.

74S15 Boundary element methods applied to problems in solid mechanics
65D30 Numerical integration
Full Text: DOI
[1] Aliabadi, M. H.; Hall, W. S.; Phemister, T. G. (1985): Taylor expansions for singular kernels in the boundary element method. Int. J. Numer. Method Engng. 21, 2221-2236 · Zbl 0599.65011
[2] Aliabadi, M. H.; Hall, W. S. (1987a): Weighted Gaussian methods for three-dimensional boundary element kernel integration. Comm. Appl. Numer. Methods 3, 89-96 · Zbl 0605.73081
[3] Aliabadi, M. H.; Hall, W. S. (1987b): Analytical removal of singularities and one-dimensional integration of three-dimensional boundary element method kernels. Engng. Analysis 4, 21-24
[4] Brebbia, C. A.; Telles, J. C. F.; Wrobel, L. C. (1984): Boundary element techniques: Theory and Applications in Engineering, Berlin, Heidelberg, New York: Springer · Zbl 0556.73086
[5] Cristescu, M.; Loubignac, G. (1978): Gaussian quadrature formulas for functions with singularities in 1/R over triangle and quadrangles. In: Brebbia, C. A. (ed): Recent advances in boundary element methods, London, Pentech Press, pp. 375-390 · Zbl 0387.65022
[6] Cruse, T. A. (1969): Numerical solutions in three-dimensional elastostatics. Int. J. Solids Structures 5, 1259-1274 · Zbl 0181.52404
[7] Guiggiani, M.; Gigante, A. (1990): A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. ASME J. Appl. Mech. 57, 906-915 · Zbl 0735.73084
[8] Huang, Q.; Du, Q. (1988): An improved formulation for domain stress evaluation by boundary element methods in elastoplastic problems. Proceedings of the China-U.S. Seminar on Boundary Integral Equation and Boundary Finite Element Methods in Physics and Engineering, Xian, China
[9] Lachat, J. C.; Watson, J. O. (1976): Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics. Int. J. Numer. Method. Engng. 10, 991-1005 · Zbl 0332.73022
[10] Lean, M. H.; Wexler, A. (1985): Accurate numerical integration of singular boundary element kernels over boundaries with curvature. Int. J. Numer. Method. Engng. 21, 211-228 · Zbl 0555.65091
[11] Li, H.-B.; Han, G.-M; Mang, H. A. (1985): A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method. Int. J. Numer. Method. Engng. 21, 2071-2098 · Zbl 0576.65129
[12] Mukherjee, S. (1992) Boundary element methods in creep and fracture. London: Elsevier · Zbl 0534.73070
[13] Nishimura, N.; Kobayashi, S. (1989): A boundary integral equations method for consolidation problems. Int. J. Solids Structures 25, 1-21 · Zbl 0676.73074
[14] Pina, H. L. G.; Fernandes, J. L. M.; Brebbia, C. A. (1981): Some numerical integration formulae over triangles and squares with a 1/R singularity. Appl. Math. Modelling 5, 209-211 · Zbl 0502.65011
[15] Rajiyah, H.; Mukherjee, S. (1987): Boundary element analysis of inelastic axisymmetric problems with large strains and rotations. Int. J. Solids Structures 23, 1679-1698 · Zbl 0627.73045
[16] Rand, R. H. (1984): Computer Algebra in Applied Mathematics: an Introduction to Macsyma. Pitman, Boston · Zbl 0583.68012
[17] Sarihan, V.; Mukherjee, S. (1982): Axisymmetric viscoplastic deformation by the boundary element method. Int. J. Solids and Structures 18, 1113-1128 · Zbl 0505.73055
[18] Zhang, Q.; Mukherjee, S.; Chandra, A. (1992a): Design Sensitivity Coefficients for elastovisco plastic problems by boundary element method. Int. J. Num. Meth. Engng. 34, 947-966 · Zbl 0774.73051
[19] Zhang, Q.; Mukherjee, S.; Chandra, A. (1992b): Shape design sensitivity analysis for geometrically and materially nonlinear problems by the boundary element method. Int. J. Solids Structures 29, 2503-2525 · Zbl 0764.73095
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