Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The numerical evaluation of a $ 2$-D Cauchy principal value integral arising in boundary integral equation methods

Author: Giovanni Monegato
Journal: Math. Comp. 62 (1994), 765-777
MSC: Primary 65N38; Secondary 65D32
MathSciNet review: 1212268
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the problem of computing 2-D Cauchy principal value integrals of the form

$\displaystyle {\fint_S}F({P_0};P)\,dP,\qquad {P_0} \in S,$

where S is either a rectangle or a triangle, and $ F({P_0};P)$ is integrable over S, except at the point $ {P_0}$ where it has a second-order pole. Using polar coordinates, the integral is first reduced to the form

$\displaystyle \int_{{\theta _1}}^{{\theta _2}} {[unk]_0^{R(\theta )}\frac{{f(r,\theta )}}{r}dr\, d \theta ,} $

where $ [unk]$ denotes the finite part of the (divergent) integral. Then ad hoc products of one-dimensional quadrature rules of Gaussian type are constructed, and corresponding convergence results derived. Some numerical tests are also presented.

References [Enhancements On Off] (What's this?)

  • [1] C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary element techniques, Springer-Verlag, Berlin, 1984.
  • [2] T. A. Cruse, Numerical solutions in three-dimensional elastostatics, Internat. J. Solids and Structures 5 (1969), 1259-1274.
  • [3] -, Application of the boundary-integral equation method to three-dimensional stress analysis, Comput. & Structures 3 (1973), 509-527.
  • [4] T. A. Cruse and R. B. Wilson, Advanced applications of boundary-integral equation methods, Nuclear Engrg. Des. 46 (1978), 223-234.
  • [5] B. G. Gabdulkhaev and L. A. Onezov, Cubature formulas for singular integrals, Izv. Vyssh. Uchebn. Zaved. Mat. 7 (1976), 100-105; English transl. in Soviet Math. (Iz. VUZ). MR 0448815 (56:7120)
  • [6] M. Guiggiani and A. Gigante, A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, ASME J. Appl. Mech. 57 (1990), 907-915. MR 1165521 (93b:73037)
  • [7] J. Hadamard, Lectures on Cauchy's problem in linear partial differential equations, Yale Univ. Press, 1923; Dover Publ., 1952.
  • [8] J. G. Kazantzakis and P. S. Theocaris, The evaluation of certain two-dimensional singular integrals used in three-dimensional elasticity, Internat. J. Solids and Structures 15 (1979), 203-207. MR 526644 (80b:73012)
  • [9] G. Krishnasamy, L. W. Schmerr, T. J. Rudolphi, and F. J. Rizzo, Hypersingular boundary integral equations: some applications in acoustic and elastic scattering, ASME J. Appl. Mech. 57 (1990), 404-414. MR 1058810 (91c:73029)
  • [10] H. R. Kutt, The numerical evaluation of principal value integrals by finite-part integration, Numer. Math. 24 (1975), 205-210. MR 0378366 (51:14534)
  • [11] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 0213785 (35:4642)
  • [12] G. Monegato, Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals, Numer. Math. 43 (1984), 161-173. MR 736078 (85h:65049)
  • [13] -, On the weights of certain quadratures for the numerical evaluation of Cauchy principal value integrals and their derivatives, Numer. Math. 50 (1987), 273-281. MR 871229 (88c:65027)
  • [14] A. G. Ramm and A. van der Sluis, Calculating singular integrals as an ill-posed problem, Numer. Math. 57 (1990), 139-145. MR 1048308 (91m:65065)
  • [15] F. J. Rizzo and D. J. Shippy, An advanced boundary integral equation method for three-dimensional thermoelasticity, Internat. J. Numer. Methods Engrg. 11 (1977), 1753-1768.
  • [16] P. O. Runck, Bemerkungen zu den Approximationssätzen von Jackson und Jackson-Timan, Abstrakte Räume und Approximation (P. L. Butzer, B. Szökefalvi-Nagy, eds.), ISNM, vol. 10, Birkhäuser Verlag, Basel, 1969, pp. 303-308. MR 0265831 (42:740)
  • [17] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975.
  • [18] P. S. Theocaris, N. I. Ioakimidis, and J. G. Kazantzakis, On the numerical evaluation of two-dimensional principal value integrals, Internat. J. Numer. Methods Engrg. 14 (1980), 629-634. MR 571082 (81g:65034)
  • [19] P. S. Theocaris, Modified Gauss-Legendre, Lobatto and Radau cubature formulas for the numerical evaluation of 2-D singular integrals, Internat. J. Math. Math. Sci. 6 (1983), 567-587. MR 712576 (85k:65024)
  • [20] F. Tricomi, Equazioni integrali contenenti il valor principale di un integrale doppio, Math. Z. 27 (1928), 87-133. MR 1544900
  • [21] G. Tsamasphyros and P. S. Theocaris, Cubature formulas for the evaluation of surface singular integrals, BIT 19 (1979), 368-377. MR 548616 (81a:65037)
  • [22] J. Weaver, Three-dimensional crack analysis, Internat. J. Solids and Structures 13 (1977), 321-330.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N38, 65D32

Retrieve articles in all journals with MSC: 65N38, 65D32

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society