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Variable transformations for nearly singular integrals in the boundary element method. (English) Zbl 1100.65109
Summary: We review variable transformation methods for evaluating nearly singular integrals over curved surfaces, which were proposed by the author and co-workers.
The rest of the paper is organized as follows. Section 2 gives a brief explanation of the boundary element formulation of the three-dimensional potential problem. In section 3, we analyze the nature of integral kernels occuring in such a formulation. In section 4, we present the outline of the projection and angular $ radial transformation method proposed by the author. In section 5, we treat the radial variable transformation, which is particularly important in the method. In section 6, we perform an error analysis of the method using complex function theory, which yields insight regarding the optimal radial variable transformation. In section 7, we mention the use of the double exponential transformation in the radial variable transformation.

65N38 Boundary element methods for boundary value problems involving PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N15 Error bounds for boundary value problems involving PDEs
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[1] Barret, W., Convergence properties of Gaussian quadrature formulae, Comput. J., 3 (1960), 272-277. · Zbl 0098.31703 · doi:10.1093/comjnl/3.4.272
[2] Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C., Boundary Element Techniques: Theory and Applications in Engineering, Springer-Verlag, Berlin, 1984. · Zbl 0556.73086
[3] Choraku, A., On the Optimization and Theoretical Error Analysis of Variable Transformation-Type Numerical Quadrature for the Boundary Element Method, Bach- elor Thesis, Department of Mathematical Engineering and Information Physics, The University of Tokyo, 1996 (in Japanese).
[4] Cruse, T. A. and Aithal, R., Non-singular boundary integral equation implementation, Int. J. Numer. Methods Eng., 36 (1993), 237-254.
[5] Donaldson, J. D. and Elliot, D., A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal., 9 (1972), 573-602. · Zbl 0264.65020 · doi:10.1137/0709051
[6] Davis, P. J. and Rabinowitz, P., Methods of Numerical Integration, Academic Press, 1984. · Zbl 0537.65020
[7] Hackbusch, W. and Sauter, S. A., On numerical cubature of nearly singular surface integrals arising in BEM collocation, Computing, 52 (1994), 139-159. · Zbl 0799.65028 · doi:10.1007/BF02238073
[8] Hayami, K. and Brebbia, C. A., A new coordinate transformation method for singular and nearly singular integrals over general curved boundary elements, in C. A. Breb- bia, W. L. Wendland, G. Kuhn (eds.), Boundary Elements IX, Proc. 9th Int. Conf. on Boundary Elements, Stuttgart, 1987, Computational Mechanics Publications with Springer-Verlag, Berlin, 1 (1987), 375-399. 841
[9] Hayami, K. and Brebbia, C. A., Quadrature methods for singular and nearly singular in- tegrals in 3-D boundary element method (Invited paper), in C. A. Brebbia (ed.), Bound- ary Elements X, Proc. 10th Int. Conf. on Boundary Element Methods, Southampton, 1988, Computational Mechanics Publications with Springer-Verlag, Berlin, 1 (1988), 237-264.
[10] Hayami, K., High precision numerical integration methods for 3-D boundary element analysis, IEEE Trans. Magnetics, 26 (1990), 603-606.
[11] , A robust numerical integration method for three-dimensional boundary element analysis, in M. Tanaka, C. A. Brebbia and T. Honma (eds.), Boundary Elements XII, Proc. 12th Int. Conf. on Boundary Elements in Engineering, Sapporo, 1990, Computa- tional Mechanics Publications with Springer-Verlag, Berlin, 1 (1990), 33-51. · Zbl 0804.65024
[12] , A robust numerical integration method for 3-D boundary element analysis and its error analysis using complex function theory, in T. O. Espelid and A. Genz (eds.), Numerical Integration, Proc. NATO Advanced Research Workshop on Numerical Inte- gration, Bergen, 1991, Kluwer Academic Publishers, (1992), 235-248. · Zbl 0741.65016
[13] , A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals, Thesis Submitted for the Degree of Doctor of Philosophy of the Council for National Academic Awards, Computational Mechanics Institute, Wessex Institute of Technology, Southampton, U.K., 1991. (Also available as C. A. Brebbia and S. A. Orszag (eds.), Lect. Notes Eng., 73, Springer-Verlag, Berlin, 1992.)
[14] Hayami, K., Matsumoto, H. and Moroga, K., Improvement and implementation of PART: Numerical quadrature for nearly singular boundary element integrals, in C. A. Brebbia, J. Dominguez and F. Paris (eds.), Boundary Elements XIV, Proc. 14th Int. Conf. on Boundary Element Methods, Seville, 1992 Computational Mechanics Publica- tions with Elsevier Science Publishers, 1 (1992), 605-617. · Zbl 0829.65028
[15] Hayami, K., Numerical Quadrature for Nearly Singular Integrals in the Three Dimen- sional Boundary Element Method, Ph.D. Thesis, The University of Tokyo, 1992.
[16] Hayami, K. and Matsumoto, H., A numerical quadrature for nearly singular boundary element integrals, Eng. Anal. Bound. Elem., 13 (1994), 143-154.
[17] , Improvement of quadrature for nearly singular integrals in 3D-BEM, in C. A. Brebbia ed., Boundary Elements XVI, Proc. 16th Int. Boundary Element Method Con- ference, (1994), 201-210, Computational Mechanics Publications, Southampton. · Zbl 0812.65014
[18] Higashimachi, T., Okamoto, N., Ezawa, Y., Aizawa, T. and Ito, A., Interactive struc- tural analysis system using the advanced boundary element method, in C. Brebbia, T. Futagami and M. Tanaka eds., Boundary Elements V, Proc. 5th Int. Conf., Hiroshima, (1983), 847-856, Springer-Verlag, Berlin and CML Publications, Southampton.
[19] Jun, L., Beer, G. and Meek, J. L., The application of double exponential formulas in the boundary element method, in C. Brebbia and G. Maier eds., Boundary Elements VII, Proc. 7th Int. Conf., Como, Italy, Springer-Verlag, 2 (1985), 13.3-13.17. · Zbl 0596.65008
[20] Koizumi, M. and Utamura, M., A new approach to singular kernel integration for general curved elements, Boundary Elements VIII, Proc. 8th Int. Conf., Tokyo, Springer-Verlag, (1986), 665-675. · Zbl 0618.73087
[21] , A polar coordinate integration scheme with a hierarchical correction proce- dure to improve numerical accuracy, in T. A. Cruse ed., Proc. IUTAM Symp. on Adv. Boundary Element Methods, San Antonio, Texas, (1987), 215-222.
[22] Kunihiro, N., Hayami, K. and Sugihara, M., Automatic numerical integration of nearly singular boundary element integrals, in T. Ushijima, Z. Shi and T. Kako eds., Adv. Numer. Math., Proc. 2nd Japan-China Seminar on Numerical Mathematics, Tokyo, 1994, Lecture Notes in Numer. Appl. Anal., 14 (1995), 249-252. · Zbl 0835.65045
[23] , Automatic numerical integration of nearly singular integrals in the boundary element method, in S. N. Atluri, G. Yagawa and T. A. Cruse, eds., Computational Me- chanics ’95, Proc. Int. Conf. on Computational Engineering Science, Hawaii, Springer- Verlag, 2 (1995), 2841-2846.
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