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On numerical cubatures of singular surface integrals in boundary element methods. (English) Zbl 0761.65012

Cubature formulae for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in \(\mathbb{R}^ 3\) are derived. Asymptotic estimates of the integration error are also presented.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
65N38 Boundary element methods for boundary value problems involving PDEs
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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[1] Aliabadi, M.H., Hall, W.S. (1987): Weighted Gaussian methods for three-dimensional boundary element kernel integration. Comm. Appl. Num. Math.30, 89-96 · Zbl 0605.73081 · doi:10.1002/cnm.1630030203
[2] Aliabadi, M.H., Hall, W.S., Phemister, T.G. (1985): Taylor expansions for singular kernels in the boundary element method. Int. J. Numer. Methods Eng.21, 2221-2236 · Zbl 0599.65011 · doi:10.1002/nme.1620211208
[3] Bendali, A., Devys, C. (1984). Calcule num?rique du rayonnement de cornets ?lectromagn?tiques dont l’ouverture est partiellement remplie par un di?lectrique. In: P. Bossavit, ed., GAMNI Modelec. Pluralis, Paris, pp. 13-24
[4] Brebbia, C.A., Telles, J.C.I., Wrobel, L.C. (1984): Boundary Element Techniques. Springer, Berlin Heidelberg New York · Zbl 0556.73086
[5] Chazarain, J., Piriou, A. (1982): Introduction to the Theory of Linear Partial Differential Operators. North Holland, Amsterdam · Zbl 0487.35002
[6] Costabel, M., Wendland, W.L. (1986): Strong ellipticity of boundary integral operators. J. Reine Angew. Math.372, 34-63 · Zbl 0628.35027 · doi:10.1515/crll.1986.372.34
[7] Cristescu, M., Laugignac, G. (1978): Gaussian quadrature formulas for functions with singularities in 1/R over triangles and quadrangles. In: C.A. Brebbia, ed., Recent Advances in Boundary Element Methods. Pentech Press, London, pp. 375-390
[8] Davis, P.J., Rabinowitz, P. (1980): Methods of Numerical Integration. Academic Press, London · Zbl 1139.65016
[9] De Doncker, E., Piessens, R. (1976): Automatic computation of integrals with singular integrand over a finite or an infinite range. Computing17, 265-279 · Zbl 0342.65014 · doi:10.1007/BF02259651
[10] Djaoua, M. (1975): M?thode d’?l?ments finis pour la r?solution d’un probl?me ext?rieur dans ?3. Rapport Int. 3, Ecole Polytechnique, Centre Math. Appl., Palaiseau, France
[11] Duffy, M.G. (1982): Quadrature over a pyramid or cube of integrals with a singularity at a vertex. SIAM J. Numer. Anal.19, 1260-1262 · Zbl 0493.65011 · doi:10.1137/0719090
[12] Ervin, V.J., Wendland, W.L. (1992): Singular behaviour of the hypersingular kernel for a 3-D problem. In preparation
[13] Gray, L.J., Martha, L.F., Ingraffea, A.R. (1990): Hypersingular integrals in boundary element fracture analysis. Int. J. Numer. Methods Eng.29, 1135-1158 · Zbl 0717.73081 · doi:10.1002/nme.1620290603
[14] Guiggiani, M., Gigante, A. (1990): A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. ASME J. Appl. Mech.57, 907-915 · Zbl 0735.73084 · doi:10.1115/1.2897660
[15] Guiggiani, M., Krishnasamy, G., Rudolphi, T.J., Rizzo, F.J. (1992): A general algorithm for numerical solution of hypersingular boundary integral equations. ASME J. Appl. Mech. (to appear) · Zbl 0765.73072
[16] Hartmann, F. (1989): Introduction to Boundary Elements. Springer, Berlin Heidelberg New York · Zbl 0693.73054
[17] Hayami, K., Brebbia, C.A. (1988): Quadrature methods for singular and nearly singular integrals in 3-D boundary element methods. In: C.A. Brebbia, ed., Boundary Elements X, Vol. 1. Springer, Berlin Heidelberg New York, pp. 237-264
[18] Johnson, C.G.L., Scott, L.R. (1989): An analysis of quadrature errors in second-kind boundary integral equations. SIAM J. Numer. Anal.26, 1356-1382 · Zbl 0695.65067 · doi:10.1137/0726079
[19] Kalik, K. (1992): Berechnung schwach singul?rer und fast schwach singul?rer Integrale auf r?umlichen Dreiecken. In preparation
[20] Kalik, K., Wendland, W.L. (1992): The approximation of closed manifolds by triangulated manifolds and the triangulation of closed manifolds. Computing47, 255-275 · Zbl 0757.41032 · doi:10.1007/BF02320196
[21] Kieser, R. (1991): ?ber einseitige Sprungrelationen und hypersingul?re Operatoren in der Methode der Randelemente. Doctoral Thesis, University Stuttgart · Zbl 0774.35005
[22] Kieser, R., Schwab, C., Wendland, W.L., (1992): Numerical evaluation of hypersingular integrals on curved surfaces using symbolic manipulation. In preparation · Zbl 0770.65008
[23] Klees, R. (1991): L?sung des fixen geod?tischen Randwertproblems mit Hilfe der Randelementmethode. Doctoral Thesis, University Karlsruhe
[24] Korhut, L., Mikeli?, A. (1986): The potential integral over a polynomial distribution over a curved triangular domain. Int. J. Numer. Methods. Eng.23, 2277-2285 · Zbl 0603.65014 · doi:10.1002/nme.1620231209
[25] Krishnasamy, G., Rizzo, F.J., Rudolphi, T.J. (1991): Hypersingular boundary integral equations: Their occurrence, interpretation, regularization and computation. In: P.K. Banerjee, S. Kobayashi, eds., Developments in Boundary Element Methods, Vol. 7, Chap. 7. Appl. Sci. Publ., Barking, Essex, England
[26] Kupradze, V.D., Burchuladze, T.V., Gegelia, T.G., Bacheleishvilii, M.O. (1979): Three Dimensional Problems of Elasticity and Thermoelasticity. North-Holland, Amsterdam (Nauka, Moscow, 1976)
[27] Kutt, H.R. (1975): On the numerical evaluation of finite-part integrals involving an algebraic singularity. WISK-report No. 179, Pretoria · Zbl 0327.65026
[28] Kutt, H.R. (1975): Quadrature formulae for finite-part integrals. WISK-report No. 178, Pretoria · Zbl 0327.65027
[29] Kutt, H.R. (1975): The numerical evaluation of principal value integrals by finite part integration. Numer. Math.24, 205-210 · Zbl 0306.65010 · doi:10.1007/BF01436592
[30] Lachat, J.C., Watson, J.O. (1976): Effective numerical treatment of boundary integral equations: A formulation for 3-dimensional elastostatics. Int. J. Numer. Methods Eng.10, 991-1005 · Zbl 0332.73022 · doi:10.1002/nme.1620100503
[31] Lean, M.H., Wexler, A. (1985): Accurate numerical integration of singular boundary element kernels over boundaries with curvature. Int. J. Numer. Methods Eng.21, 211-228 · Zbl 0555.65091 · doi:10.1002/nme.1620210203
[32] 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 BEM. Int. J. Numer. Methods Eng.21, 2071-2098 · Zbl 0576.65129 · doi:10.1002/nme.1620211109
[33] Lyness, J.N. (1976): An error functional expansion for N-dimensional quadrature with an integrand function singular at a point. Math. Comput.30, 1-23 · Zbl 0343.65007
[34] Lyness, J.N. (1980): Quadrature error functional expansions for the simplex when the integrand function has singularities at vertices. Math. Comput.34, 213-225 · Zbl 0442.41023 · doi:10.1090/S0025-5718-1980-0551299-8
[35] Lyness, J.N., Gatteschi, L. (1982): On quasi degree quadrature rules. Numer. Math.39, 259-267 · Zbl 0499.41023 · doi:10.1007/BF01408699
[36] Lyness, J.N., Gatteschi, L. (1982): A note on cubature over a triangle of a function having specified singularities. In: G. H?mmerlin, ed., Numerical Integration. Proc. 1981 Oberwolfach Conf., Intern. Ser. Numer. Math.57, Birkh?user, Basel, pp. 164-169 · Zbl 0488.41029
[37] Mikhlin, S.G., Pr?ssdorf, S. (1986): Singular Integral Operators. Springer, Berlin Heidelberg New York
[38] Monegato, G. (1992): The numerical evaluation of a 2-D Cauchy principal value integral arising from a BIE-method. To appear
[39] Ne?as, J. (1967): Les m?thodes directes en th?orie des ?quations elliptiques. Masson, Paris
[40] Nedelec, J.C. (1976): Curved finite element methods for the solution of singular integral equations on surfaces in ?3. Comput. Methods Appl. Mech. Eng.8, 61-80 · Zbl 0333.45015 · doi:10.1016/0045-7825(76)90053-0
[41] Nedelec, J.C. (1977): Approximation des Equations Int?grales en Mecanique et en Physique. Lecture Notes. Centre de Math?matiques Appliqu?es, Ecole Polytechnique, Palaiseau, France
[42] Nedelec, J.C. (1982): Integral equations with non integrable kernels. Integral Equations and Operator Theory5, 562-572 · Zbl 0479.65060 · doi:10.1007/BF01694054
[43] Nowak, P. (1986): A new type of higher-order boundary integral approximation for three dimensions. In: E.H. Hirschel, ed., Finite Approximation in Fluid Mechanics. Vieweg, Braunschweig, pp. 218-231
[44] Paget, D.F. (1981): A quadrature rule for finite-part integrals. BIT21, 212-220 · Zbl 0457.41027 · doi:10.1007/BF01933166
[45] Rabinowitz, P., Richter, N. (1970): New error coefficients for estimating quadrature errors for analytic functions. Math. Comp.24, 561-570 · Zbl 0217.52402 · doi:10.1090/S0025-5718-1970-0275675-X
[46] Robinson, I., De Doncker, E. (1981): Automatic computation, of improper integrals over a bounded or unbounded planar region. Computing16, 252-284 · Zbl 0457.65008
[47] Romate, J.E. (1989): The Numerical Simulation of Nonlinear Gravity Waves in Three Dimensions Using a Higher Order Panel Method. Doctoral Thesis, University Twente, Netherlands
[48] Schwab, C., Wendland, W.L. (1985): 3D BEM numerical integration. In: C.A. Brebbia, G. Maier, eds., Boundary Elements VII II. Springer, Berlin Heidelberg New York, pp. 13-85 to 13-101 · Zbl 0608.65016
[49] Schwab, C., Wendland, W.L. (1992): Kernel properties and representation of boundary integral operators. Math. Nachr. (to appear) · Zbl 0805.35168
[50] Stroud, A.H. (1971): Approximate Calculation of Multiple Integrals. Prentice Hall, Englewood Cliffs, N.J. · Zbl 0379.65013
[51] Wendland, W.L. (1987): Strongly elliptic boundary integral equations. In: A. Iserles, M. Powell, eds., The State of the Art in Numerical Analysis. Clarendon Press, Oxford, pp. 511-561
[52] Yserentant, H. (1983): A. remark on the numerical computation of improper integrals. Computing30, 179-183 · Zbl 0498.65009 · doi:10.1007/BF02280788
[53] Atkinson, K. (1985): Solving integral equations on surfaces in space. In: G. H?mmerlin, K.-H. Hoffmann, eds., Constructive Methods for the Practical Treatment of Integral Equations. Proc. 1984 Oberwolfach Conf., Intern. Ser. Numer. Math.73. Birkh?user, Basel, pp. 20-43. · Zbl 0557.65086
[54] Fairweather, G., Rizzo, F.Y., Shippy, D.Y. (1979): Computation of double integrals in the boundary integral method. In: R. Vichnevetsky, R. Stepleman, eds., Advances in Computer Methods for Partial Differential Equations?III. IMACS Symp. Rutgers Univ. Computer Sc., New Brunswick, NY, pp. 331-334
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