A hypersingular boundary integral method for two-dimensional screen and crack problems. (English) Zbl 0725.73091

The two-dimensional hypersingular boundary integral equations defined over curved boundaries are discussed in detail for the Helmholtz and Navier equations (the latter is further extended to consider harmonic elastic waves). The existence, uniqueness and especially the regularity of the solutions are demonstrated including its local behaviour close to crack geometries.
The paper is highly recommended to those concerned with the application of boundary elements to fracture mechanics type of problems.


74S15 Boundary element methods applied to problems in solid mechanics
74R99 Fracture and damage
74J20 Wave scattering in solid mechanics
65R10 Numerical methods for integral transforms
74S30 Other numerical methods in solid mechanics (MSC2010)
Full Text: DOI


[1] M. S. Agranovič, Spectral properties of diffraction problems, in The General Method of Eigenvibrations in Diffraction, Theory (N. N. Vojtovich, B. S. Katzenelenbaum &A. N. Sivov, eds.), Nauka, Moscow 1977 (Russian).
[2] J. F. Ahner &A. N. Ssiao, On the two-dimensional exterior boundary-value problems of elasticity, SIAM J. Appl. Math.31 (1976) 677–685. · Zbl 0355.73019
[3] C. Atkinson, Fracture mechanics stress analysis, I and II, in Boundary Elements Techniques in Computer Aided Engineering (C. A. Brebbia, ed.), NATO ASI Ser. E, Appl. Sci.84, M. Nijhoff Publ. Dordrecht, Boston, Lancaster (1984) 355–398.
[4] A. Bamberger, Approximation de la diffraction d’ondes élastiques une nouvelle approche (I), (II), (III) Rapp. Int. 91, 96, 98, Centre Math. Appl. Ecole Polytechnique, 91128 Palaiseau, France (1983). · Zbl 0571.73020
[5] A. Bamberger &Tuong Ha Duong, Formulation pour le calcul de la diffraction d’une onde acoustique par une surface rigide. Rapp. Int. 110, Centre Math. Appl. Ecole Polytechnique, 91128 Palaiseau, France (1985). · Zbl 0636.65119
[6] P. Bonnemay, Equations integrales pour l’élasticité plane, Thèse de 3ème cycle, Université de Paris VI (1979).
[7] C. A. Brebbia, J. C. Telles &L. C. Wrobel, Boundary Element Techniques, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1984). · Zbl 0556.73086
[8] D. Clements, Boundary Value Problems Governed by Second Order Elliptic Systems. Pitman, Boston, London, Melbourne (1981). · Zbl 0499.35004
[9] Ph. Cortey-Dumont, On the numerical analysis of integral equations related to the diffraction of elastic waves by a crack, Thèse d’état, University Paris 6 (1985). · Zbl 0574.65064
[10] M. Costabel, Boundary integral operators on curved polygons, Ann. Mat. Pura Appl.33 (1983) 305–326. · Zbl 0533.45009
[11] M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal.19 (1988) 613–626. · Zbl 0644.35037
[12] M. Costabel &E. P. Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, in (W. Fiszdon &K. Wilmánski, eds.), Math. Models and Meth. in Mech. 1981, Banach Center Publ.15 Warsaw (1985) 175. · Zbl 0655.65129
[13] M. Costabel &E. P. Stephan, Curvature terms in the asymptotic expansion for solutions of boundary integral equations on curved polygons. J. Integral Equations5 (1983) 353–371. · Zbl 0538.35022
[14] M. Costabel &E. P. Stephan, The normal derivative of the double layer potential on polygons and Galerkin approximations, Appl. Anal.16 (1983) 205–228. · Zbl 0515.35036
[15] M. Costabel &E. P. Stephan, The method of Mellin transformation for boundary integral equations on curves with corners, in (A. Gerasulis &R. Vichnevetsky, eds.) Numerical Solution of Singular Integral Equations, IMACS, New Brunswick, N.Y. (1984) 95–102.
[16] M. Costabel, E. P. Stephan &W. L. Wendland, On the boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, Ann. Scuola Norm. Sup. Pisa, Ser. IV10 (1983) 197–242. · Zbl 0563.45007
[17] M. Costabel, E. P. Stephan &W. L. Wendland, Zur Randintegralmethode für das erste Fundamentalproblem der ebenen Elastizitätstheorie auf Polygongebieten, in (H. Kurke et al., eds.) Recent Trends in Mathematics, Reinhardsbrunn, B. G. Teubner, Leipzig (1982) 56–68.
[18] M. Durand, Layer potentials and boundary value problems for the Helmholtz equation in the complement of a thin obstacle, Math. Meth. Appl. Sci.5 (1983) 389–421. · Zbl 0527.76076
[19] G. I. Eskin, Boundary Problems for Elliptic Pseudodifferential Operators, Transl. of Math. Mon., American Math. Soc.52, Providence, Rhode Island (1981). · Zbl 0458.35002
[20] P. J. T. Filippi, Layer potentials and acoustic diffraction, J. Sound Vibration54 (1977) 1–29. · Zbl 0368.76073
[21] J. Giroire &J. C. Nedelec, Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comp.3 (1978) 973–990. · Zbl 0405.65060
[22] T. Ha Duong, A finite element method for the double-layer potential solutions for the Neumann exterior problem, Math. Meth. in the Appl. Sci.2 (1980) 191–208. · Zbl 0437.65083
[23] M. Hamdi, Une formulation variationnelle par équations intégrales pour la résolution de l’équation de Helmholtz avec des conditions aux limites mixtes. C.R. Acad. Sci. Paris, Série II292 (1981) 17–21. · Zbl 0479.76088
[24] S. Hildebrandt &E. Wienholtz, Constructive proofs of representation theorems in separable Hilbert space, Comm. Pure Appl. Math.17 (1964) 369–373. · Zbl 0131.13401
[25] G. C. Hsiao, P. Kopp &W. L. Wendland, Some applications of a Galerkin-collocation method for integral equations of the first kind, Math. Meth. in the Appl. Sci.6 (1984) 280–235. · Zbl 0546.65091
[26] G. C. Hsiao, E. P. Stephan &W. L. Wendland, An integral equation formulation for a boundary value problem of elasticity in the domain exterior to an are: In: Singularities and their Constructive Treatment (P. Grisvard, J. Whiteman &W. Wendland, eds.) Lecture Notes in Math.112, Springer-Verlag, Heidelberg (1985) 153–165.
[27] G. C. Hsiao, E. P. Stephan & W. L. Wendland, On the Dirichlet problem in elasticity for a domain exterior to arc, to appear in J. Computational and Applied Math. · Zbl 0742.73026
[28] G. C. Hsiao &W. L. Wendland, On a boundary integral method for some exterior problems in elasticity, (Preprint 769, FB Math. TH Damstadt 1983). Proc. Tbilisi Univ. UDK 539.3, Mat. Mech. Astron.257 (18) (1985) 31–60.
[29] G. C. Hsiao & W. I. Wendland, Boundary integral equations methods for exterior problems in elastostatics and elastodynamics, in preparation.
[30] V. D. Kupradze, Potential Methods in the Theory of Elasticity, Jerusalem, Israel Program Scientific Transl. (1965). · Zbl 0188.56901
[31] U. Lamp, K.-T. Schleicher, E. P. Stephan &W. L. Wendland, Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem, Computing33 (1984) 269–296. · Zbl 0546.65080
[32] J. L. Lions &E. Magenes, Non-homogeneous Boundary Value Problems and Applications I, Springer-Verlag, Berlin, Heidelberg, New York (1972). · Zbl 0223.35039
[33] R. C. MacCamy, On singular integral equations with logarithmic and Cauchy kernels, J. Math. Mech.7 (1958) 355–376. · Zbl 0084.32201
[34] J. Mason &R. N. L. Smith, Boundary integral equation methods for a variety of curved crack problems, in (C. T. H. Baker &G. F. Miller, (eds.) Treatment of Integral Equations by Numerical Methods, Academic Press, London, New York (1982) 239–252.
[35] J. C. Nedelec, Approximation par potentiel de double couche du problème de Neumann extérieur, C. R. Acad. Sci. Paris, Série A286 (1977) 616–619.
[36] J. C. Nedelec, Le potential de double couche pour les ondes élastiques, Rapp. Int. 99, Centre Math. Appl. Ecole Polytechnique, 91128 Palaiseau, France (1983).
[37] E. P. Stephan, Solution procedures for interface problems in acoustics and electromagnetics, in (P. Filippi, ed.) Theoretical Acoustics and Numerical Techniques, CISM No.277, Springer-Verlag, Wien (1983) 291–348. · Zbl 0578.76078
[38] E. P. Stephan, Boundary integral equations for mixed boundary value problems in \(\mathbb{R}\)3, Math. Nachr.134 (1987) 21–53. · Zbl 0649.35022
[39] E. P. Stephan, Boundary integral equations for mixed boundary value problems, screen and transmission problems in \(\mathbb{R}\)3, Habilitationsschrift, TH Darmstadt, Germany (1984) (Preprint 848, FB Math. TH Darmstadt).
[40] E. P. Stephan &W. L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Analysis18 (1984) 183–219. · Zbl 0564.73087
[41] M. Stern, Formulation of cracks in plate bending, in (C. A. Brebbia, ed.) Boundary Element Techniques in Computer Aided Engineering, NATO ASI Ser. E, Appl. Sci.84, M. Nijhoff Publ. Dordrecht, Boston, Lancaster (1984) 345–354.
[42] M. Taylor, Pseudodifferential Operators, Princeton University Press (1981).
[43] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York, London, Vol. 1 (1980).
[44] W. L. Wendland, Strongly elliptic boundary integral equations. In: The State of the Art in Numerical Analysis (A. Jserless, M Powell, eds.) IMA, Oxford Univ. Press (1987). · Zbl 0615.65119
[45] W. L. Wendland, E. P. Stephan &G. C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. in Appl. Sci.1 (1979) 265–321. · Zbl 0461.65082
[46] C. H. Wilcox, Scattering Theory for the d’Alembert Equation in Exterior Domains, Lecture Notes Math.442 Springer-Verlag, Berlin, Heidelberg, New York (1975). · Zbl 0299.35002
[47] P. Wolfe, An integral operator connected with the Helmholtz equation, J. Functional Anal.36 (1980) 105–113. · Zbl 0434.45019
[48] A. Ziani &C. Devys, Méthode integrale pour le calcul des modes quidés d’une fibre optique, Rapp. Int. 129, Centre Math. Appl. École Polytechnique, 91128 Palaiseau, France (1985).
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