Abstract
Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.
Similar content being viewed by others
References
Brandão, M. P. (1986): Improper integrals in theoretical aerodynamics: the problem revisited. AIAA J. 25(9), 1258–1260
Chien, C. C.; Rajiyah, H.; Atluri, S. N. (1990): An effective method for solving the hyper-singular integral equations in 3D acoustics. J. Acoust. Soc. Am. 88(2), 918–937
Cruse, T. A. (1988): Boundary element analysis in computational fracture mechanics. Boston: Kluer, Academic Publishers
Ervin, V. J.; Kieser, R.; Wendland, W. L. (1990): Numerical approximation for a model 2D hypersingular integral equation. Computational Engineering with Boundary Elements, Vol. 1, S. Gilli, C. A. Brebbia, A. H. D. Cheng (eds.) Southampton: Computational Mechanics Publ.
Guiggiani, M.; Gigante, A. (1991): A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. J. Appl. Mech. 57, 906–915
Gunter, N. M. (1967): Potential theory and its applications to basic problems of mathematical physics, New York: Fredrick Ungar
Hadamard, J. (1923): Lectures on Cauchy's problem in linear partial differential equations. New Haven: Yale University Press
Hartmann, F. (1989): Introduction to boundary element theory and applications. Berlin, Heidelberg, New York: Springer
Kaya, A. C.; Erdogan, F. (1987): On the solution of integral equations with strongly singular kernels. Q. Appl. Math. 45(1), 105–122
Krishnasamy, G.; SchmerrL. W.; RudolphiT. J.; RizzoF. J. (1990): Hypersingular boundary integral equations: Some applications in accoustic and elastic wave scattering. J. Appl. Mech. 57, 404–414
Martin, P. A.; RizzoF. J. (1989): On boundary integral equations for crack problems. Proc. Roy. Soc. A421 341–355
Mathews, I. C. (1986): Numerical techniques for three-dimensional steady-state fluid-structure interaction. J. Acoust. Soc. Am. 79(5), 1317–1325
Mikhlin, S. G. (1965): Multidimensional singular integrals and integral equations. First Edition.New York: Pergamon
Muskhelishvili, N. I. (1953): Singular integral equations. Groningen: Noordhoff
Rudolphi, T. J. (1991): The use of simple solutions in the regularization of hypersingular boundary integral equations. Math. Comput. Modelling 15(3–5), 269–278
Rudolphi, T. J.; Krishnasamy, G.; Schmerr, L. W.; Rizzo, F. J. (1988): On the use of strongly singular integral equations for crack problems. Proc. 10th Int. Conf. Boundary Elements, Southampton, England, September
Author information
Authors and Affiliations
Additional information
Communicated by S. N. Atluri, August 27, 1991
Rights and permissions
About this article
Cite this article
Krishnasamy, G., Rizzo, F.J. & Rudolphi, T.J. Continuity requirements for density functions in the boundary integral equation method. Computational Mechanics 9, 267–284 (1992). https://doi.org/10.1007/BF00370035
Issue Date:
DOI: https://doi.org/10.1007/BF00370035