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Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method. (English) Zbl 1003.65133
Summary: The accurate numerical solution of near singular boundary integrals was an issue of major concern in most of the boundary element analysis next to the singular boundary integrals. The problem was solved in this paper by a kind of nonlinear transformation, namely, the distance transformation for the accurate evaluation of near singular boundary integrals with various kernels for both the two- and three-dimensional problems incorporated with the distance functions defined in the local intrinsic coordinate systems. It is considered that two effects la the role in the transformation. They are the damping out of the near singularity and the rational redistribution of integration points.
The actual numerical computation can be performed by standard Gaussian quadrature formulae and can be easily included in the existing computer code, along with its insensitivity to the kind of the boundary elements. Numerical results of potential problem were presented, showing the effectiveness and the generality of the algorithm, which makes it possible, for the first time, to observe the behavior of various boundary integral values with numerical means, when the source point is moving across the boundary with fine steps.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31A10 Integral representations, integral operators, integral equations methods in two dimensions
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[1] Brebbia, C.A.; Telles, J.C.F.; Wrobel, L.C., Boundary element techniques—theory and applications in engineering, (1984), Springer Heidelberg · Zbl 0556.73086
[2] Tanaka, M.; Matsumoto, T.; Nakamura, M., Boundary element method (in Japanese), (1991), Baifukan Press Tokyo
[3] Cristescu, M.; Loubignac, G., Gaussian quadrature formulas for functions with singularities in 1/R over triangles and quadrangles, (), 375-390
[4] Cruse, T.A.; Aithal, R., Non-singular boundary integral equation implementation, Int J numer meth engng, 36, 237-254, (1993)
[5] Krishnasamy, G.; Rizzo, F.J.; Liu, Y.J., Boundary integral equations for thin bodies, Int J numer meth engng, 37, 107-121, (1994) · Zbl 0795.73076
[6] Liu, Y.J., Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification, Int J numer meth engng, 41, 541-558, (1998) · Zbl 0910.73068
[7] Dirgantara, T.; Aliabadi, M.H., Crack growth analysis of plates loaded by bending and tension using dual boundary element method, Int J fract, 105, 27-74, (2000)
[8] Aliabadi, M.H.; Martin, D., Boundary element hyper-singular formulation for elastoplastic contact problems, Int J numer meth engng, 48, 995-1014, (2000) · Zbl 0974.74072
[9] Zhang, D.; Rizzo, F.J.; Rudolphi, Y.J., Stress intensity sensitivities via hypersingular boundary integral equations, Comput mech, 23, 389-396, (1999) · Zbl 0967.74075
[10] Sladek, N.; Sladek, J.; Tanaka, M., Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity, Int J numer meth engng, 36, 1609-1628, (1993) · Zbl 0772.73091
[11] Doblare, M.; Gracia, L., On non-linear transformations for the integration of weakly-singular and Cauchy principal value integrals, Int J numer meth engng, 40, 3325-3358, (1997) · Zbl 1049.74789
[12] Zhang GH, Lou ZW. Formulations for stress calculation of boundary layer point in BEM. In: Tanaka M, Du QH, editors. Proceedings of the Third Japan-China Symposium on Boundary Element Methods. Tokyo: Pergamon Press, 1990. p. 73-82.
[13] Cerrolaza, M.; Alarcon, E., A bi-cubic transformation of the Cauchy principal value integrals in boundary methods, Int J numer meth engng, 28, 987-999, (1989) · Zbl 0679.73040
[14] Aliabadi, M.H.; Hall, W.S., Taylor expansions for singular kernels in the boundary element method, Int J numer meth engng, 21, 2221-2236, (1985) · Zbl 0599.65011
[15] Ma, H.; Kamiya, N., Domain supplemental approach to avoid boundary layer effect of BEM in elasticity, Engng anal boundary elem, 23, 281-284, (1999) · Zbl 0963.74566
[16] Telles, J.C.F., A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int J numer meth engng, 24, 959-973, (1987) · Zbl 0622.65014
[17] Johnston, P.R., Application of sigmoidal transformations to weakly singular and nearly-singular boundary element integrals, Int J numer meth engng, 45, 1333-1348, (1999) · Zbl 0935.65130
[18] Ma, H.; Kamiya, N., A general algorithm for accurate computation of field variables and its derivatives near boundary in BEM, Engng anal boundary elem, 25, 843-849, (2001)
[19] Liu, Y.J.; Rudolphi, T.J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, Comput mech, 24, 286-292, (1999) · Zbl 0969.74073
[20] Liu, Y.J., On the simple solution and non-singular nature of the BIE/BEM—a review and some new results, Engng anal boundary elem, 24, 789-795, (2000) · Zbl 0974.65110
[21] Krishnasamy, G.; Schmerr, L.W.; Rudolphi, T.J.; Rizzo, F.J., Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering, ASME J appl mech, 57, 404-414, (1990) · Zbl 0729.73251
[22] Guiggiani, M.; Casalini, P., Direct computation of Cauchy principal value integrals in advanced boundary elements, Int J numer meth engng, 24, 1711-1720, (1987) · Zbl 0635.65020
[23] Guiggiani, M.; Gigante, A., A general algorithm for multi-dimensional Cauchy principal value integrals in the boundary element method, ASME J appl mech, 57, 906-915, (1990) · Zbl 0735.73084
[24] Guiggiani, M.; Krishnasamy, G.; Rudolphi, T.J.; Rizzo, F.J., A general algorithm for the numerical solution of hypersingular boundary integral equations, ASME J appl mech, 59, 604-614, (1992) · Zbl 0765.73072
[25] Mukherjee, S., Boundary element methods in creep and fracture, (1982), Applied Science Publishers London · Zbl 0534.73070
[26] Mukherjee, S.; Chati, M.K.; Shi, X., Evaluation of nearly singular integrals in boundary element, contour and node methods for three-dimensional linear elasticity, Int J solids struct, 37, 7633-7654, (2000) · Zbl 0993.74077
[27] Chen, H.B.; Lu, P.; Schnack, E., Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM, Engng anal boundary elem, 25, 851-876, (2001) · Zbl 1051.74050
[28] Granados, J.J.; Gallego, R., Regularization of nearly hypersingular integrals in the boundary element method, Engng anal boundary elem, 25, 165-184, (2001) · Zbl 1015.74073
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