Singularities occurring in multimaterials with transparent boundary conditions
Authors:
Philippe Destuynder and Caroline Fabre
Journal:
Quart. Appl. Math. 74 (2016), 443-463
MSC (2010):
Primary 22E46, 53C35, 57S20
DOI:
https://doi.org/10.1090/qam/1429
Published electronically:
June 16, 2016
MathSciNet review:
3518224
Full-text PDF Free Access
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Additional Information
Abstract: Computation of waves at the interface between two materials with different wave velocities is an important engineering problem. Transparent boundary conditions on the input and output boundaries are known for a single velocity wave. Adapting them in case of two velocities, singularities appear in the computation of the waves. We precisely exhibit these singularities both with theoretical and numerical points of view.
References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- H. D. Bui, Mécanique de la rupure fragile, Masson, Paris, 1978.
- Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
- Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1988 (French). Évolution: semi-groupe, variationnel. [Evolution: semigroups, variational methods]; Reprint of the 1985 edition. MR 1016605
- Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 436612, DOI 10.1090/S0025-5718-1977-0436612-4
- P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci. 3 (1981), no. 1, 70–87 (French, with English summary). MR 606849, DOI 10.1002/mma.1670030106
- Ph. Destuynder and C. Fabre, Few remarks on the use of Love waves in non destructive testing, AIMS Journals, Discrete and continuous dynamical systems, to appear.
- Y.C. Fung, Foundations of solid mechanics, Prentice-Hall Inc., Englewood Cliffs, NJ, 1965, p. 182.
- P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
- Laurence Halpern, Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation, Math. Comp. 38 (1982), no. 158, 415–429. MR 645659, DOI 10.1090/S0025-5718-1982-0645659-6
- Lloyd N. Trefethen and Laurence Halpern, Well-posedness of one-way wave equations and absorbing boundary conditions, Math. Comp. 47 (1986), no. 176, 421–435. MR 856695, DOI 10.1090/S0025-5718-1986-0856695-2
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
- A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
- Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
- D. Zagier, The remarkable dilogarithm, J. Math. Phys. Sci. 22 (1988), no. 1, 131–145. MR 940391
References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957 (56 \#9247)
- H. D. Bui, Mécanique de la rupure fragile, Masson, Paris, 1978.
- Haïm Brezis, Analyse fonctionnelle, Theory and applications, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). MR 697382 (85a:46001)
- Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1988 (French). Évolution: semi-groupe, variationnel. [Evolution: semigroups, variational methods]; Reprint of the 1985 edition. MR 1016605 (90m:00006b)
- Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 0436612 (55 \#9555)
- P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci. 3 (1981), no. 1, 70–87 (French, with English summary). MR 606849 (82c:73070), DOI 10.1002/mma.1670030106
- Ph. Destuynder and C. Fabre, Few remarks on the use of Love waves in non destructive testing, AIMS Journals, Discrete and continuous dynamical systems, to appear.
- Y.C. Fung, Foundations of solid mechanics, Prentice-Hall Inc., Englewood Cliffs, NJ, 1965, p. 182.
- P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209 (93h:35004)
- Laurence Halpern, Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation, Math. Comp. 38 (1982), no. 158, 415–429. MR 645659 (83d:65245), DOI 10.2307/2007278
- Lloyd N. Trefethen and Laurence Halpern, Well-posedness of one-way wave equations and absorbing boundary conditions, Math. Comp. 47 (1986), no. 176, 421–435. MR 856695 (88b:65148), DOI 10.2307/2008165
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177 (50 \#2670)
- A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Fourth Ed., Dover Publications, New York, 1944. MR 0010851 (6,79e)
- Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584 (37 \#3168)
- D. Zagier, The remarkable dilogarithm, J. Math. Phys. Sci. 22 (1988), no. 1, 131–145. MR 940391 (89f:33024)
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Additional Information
Philippe Destuynder
Affiliation:
Département d’ingénierie mathématique, laboratoire M2N, Conservatoire National des Arts et Métiers, 292, rue saint Martin, 75003 Paris, France
Email:
philippe.destuynder@cnam.fr
Caroline Fabre
Affiliation:
Laboratoire de mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
Email:
caroline.fabre@u-psud.fr
Keywords:
Wave guides,
transmission,
singularities,
transparent boundary conditions
Received by editor(s):
January 19, 2015
Published electronically:
June 16, 2016
Article copyright:
© Copyright 2016
Brown University
