Abstract
The boundary integral equations for the crack opening displacement in acoustic and elastic scattering problems are discussed in the case of flat cracks by means of the Fourier analysis technique. The pseudo-differential nature of the hypersingular integral operators is shown and their symbols explicited. It is then proved that the variational problems assocaited with these BIE are well-posed in a Sobolev functional framework which is closely linked with the elastic energy. A decomposition of the vector integral equation in the elastic case into scalar integral equations is obtained as a by-product of the variational formulation.
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References
ACHENBACH J.D.: Wave Propagation in Elastic Solids. North Holland, Amsterdam. (1975).
BAMBERGER A.: Approximation de la diffraction d'ondes Elastiques (I). in:Nonliner Partial Differential Equations and their Applications; (eds: H. Brezis and J.L.Lions). Pitman, London 1984, 48–95.
BENDALI A.: Numerical Analysis of the Exterior Boundary Value problem for the Time Harmonic Maxwell Equations by a Boundary Finite Element Method.Math. of Comp. v.43 (1984),p. 29–46 &47–68.
BENDALI A., DEVYS C.: Calcul numérique du rayonnement électromagnétiques.Onde Electrique, v.66 (1986),no1,p. 77–81.
Benjelloun-Touimi Z.: Diffraction par un réseau 1-périodique dans R3. Thèse, Université Paris VI, (1988).
BUDRECK D.E., ACHENBACH J.D.,: Scattering from Three-Dimensional Planar Cracks by the Boundary Integral Equation Method.Journ. Appl. Mech., v.55 (1988),p.405–412
Bui H.D., Loret B., Bonnet M., Régularisation des équations intégrales de l'élastodynamique et de l'élastostatique.CRAS Paris, série II, t. 300 (1985).
CHAZARAIN J., PIRIOU A., Introduction à la théorie des équations aux Dérivées Partielles. Gauthiers Villars, Paris (1981).
COLTON D., KRESS R., Integral Equation Method in Scattering Theory John Wiley, N.Y. (1983).
Cortey-Dumont P.: Thèse Docteur ès Sciences, Université PARIS VI, (1985).
Costabel M., Stephan E.P.: A Boundary Element Method for 3-d crack Problems. (Invited Lecture at the4th International Symposium on Numerical Methods in Engineering, Atlanta, Georgia 1986).
DING Y., FORESTIER A., HA DUONG T., A Galerkin Scheme for the Time Domain Integral Equation of Acoustic Scattering from a Hard Surface.Journ. Acoust. Soc. Am., v.86 (4 (1989),p. 1566–1572.
Giroire J.: Thèse Docteur ès Sciences, Université PARIS VI, (1987).
Ha Duong T.: On the Transient Acoustic Scattering by a Flat Object.Japan J. of Appl. Math (to appear). (1990).
Ha Duong T.: Thèse Docteur ès Sciences, Université Paris VI, (1987).
HAMDI M.: Une Formulation variationnelle par Equations Intégrales pour la Résolution de l'Equation de Helmholtz avec des Conditions aux limites mixtes.CRAS, série II, T292 (1981),p. 17–20.
Hirose S., Niwa Y.: Scattering of Elastic waves by a Three-Dimensional Crack.Proc. of the VIII Conference on BEM, Tokyo, Springer Verlag (1986).
Hsiao G.C., Kopp P., Wendland W.L.: Some Applications of a Galerkin Collocation Method for Boundary Integral Equations of the First Kind.Preprint no 768 (1983)Fach. Mathematik, Technische Hochschule Darmstadt.
JONES D.S.: A New Method for Calculating Scattering with Particular Reference to the Circular Disc.C.P.A.M., v.9 (1956),p.713–746.
KUPRADZE S. et al.: Three Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelacticity. North Holland, Amsterdam (1979).
LIONS J.L., MAGENES E.: Non Homogeneous Boundary Value Problems and Applications. Springer Verlag, Berlin (1972).
MARTIN P.A., WICKHAM G.R.: Diffraction of Elastic Waves by a Penny-shaped Crack; Analytical and Numerical Results.Proc. R. Soc. London A. 390 (1983),p. 91–129.
MARTIN P.A., RIZZO F.J.: On Boundary Integral Equations for Crack Problems.Proc. R. Soc. London A. 421 (1989),p. 341–355
MEISTER E., SECK F.O.: The Explicit Solution of Elastodynamical Diffraction Problems by Symbol Factorization.Zeitschrift. fur Analysis und ihre Anwendungen, Bd8 (1989),p.307–328.
NEDELEC J.C.: Résolution par Potentiel de Double Couche du Problème de Neumann extérieur.CRAS, T286 (1978),serie A, p. 103–106.
NEDELEC J.C.: Curved Finite Element Method for the Solution of Integral Singular Equations on Surfaces inR 3.Comp. Meth. Appl. Mech. Eng. 8 (1976),p.61–80.
NEDELEC J.C., PLANCHARD J.: Une méthode variationnelle d'éléments finis pour la résolution numérique d'un problème extérieur dansR 3.RAIRO, série rouge, v.7 (1973),p.105–129.
NISHIMURA N., KOBAYASHI S. A.: Regularised Boundary Integral Method for Elastodynamic Crack Problem.Computational Mechanics v.4 (1989),p.319–328.
STEPHAN E.P.: A Boundary Integral Equation Method for 3-d Crack Problems in Elasticity.Math. Meth. in the Appl. Sci. v.8 (1986),p.609–623.
STEPHAN E.P., WENDLAND W.L.: An augmented Galerkin Procedure for the Boundary Integral Method applied to Two-dimensional Screen and Crack Problems.Applicable Analysis 18 (1986),p.183–219.