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Numerical methods for elastic wave propagation. (English) Zbl 1173.74020

Destrade, Michel (ed.) et al., Waves in nonlinear pre-stressed materials. Papers based on the presentation at CISM course, Udine, Italy, September 2006. Wien: Springer (ISBN 978-3-211-73571-8/hbk; 978-3-211-73572-5/ebook). CISM Courses and Lectures 495, 181-281 (2007).
From the introduction: We focus ourselves to two attractive and competing techniques that have in common the feature of being local in space and time, contrary to some exact methods which involve integral operators.
Local absorbing boundary conditions. This type of method has been developed since the late 70s. The idea is to write on the boundary of the computational domain a boundary condition which represents the effect of the presence of the exterior medium. Such a boundary is called local if this boundary condition can be expressed in terms of differential operators (which makes them compatible with traditional discretization techniques). Then, this boundary condition is necessarily not exact and implies some degree of approximation that needs to be quantified in some way. Moreover, the question of the stability of the coupling between the physical propagation interior model and that “unphysical” boundary conditions raises delicate and challenging mathematical questions.
Perfectly matches layers. The idea of an absorbing layer is quite old: the principle is to surround the computational domain with a layer in which the waves are artificially damped. The concept of the perfectly matched layer is much more recent (it has appeared in the middle of the 90’s in electromagnetics). Roughly speaking, the new idea is to build a particular (non-physical) absorbing medium and the absorbing layer. This technique has generated a considerable literature and is now considered as a very attractive alternative to local absorbing boundary conditions.
We try, in Section 4, to give the state-of-the-art about these two techniques with a particular emphasis on the application to elastic wave propagation, which is a cause of specific (sometimes still unsolved) difficulties. Sections 3 to 4 will be preceded, in Section 2, by an introductory presentation of the mathematical model for elastic wave propagation (linear elastodynamics equations) and of its main properties.
For the entire collection see [Zbl 1161.74002].

MSC:

74J10 Bulk waves in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
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[1] Abarbanel, S.; Gottlieb, D., A mathematical analysis of the PML method, Journal of Computational Physics, 134, 357-263 (1997) · Zbl 0887.65122 · doi:10.1006/jcph.1997.5717
[2] J.D. Achenbach. Wave Propagation in Elastic Solids. Elsevier, 1984. · Zbl 0657.73019
[3] Arnold, D. N.; Boffi, D.; Falk, R. S., Approximation by quadrilateral finite elements, Math. Comp., 71, 239, 909-922 (2002) · Zbl 0993.65125 · doi:10.1090/S0025-5718-02-01439-4
[4] B.A. Auld. Acoustic Fields and Elastic Waves in Solids. John Wiley, 1973.
[5] Baker, G. A.; Dougalis, V. A., The effect of quadrature errors on finite element approximations for the second-order hyperbolic equations, SIAM Journal on Numerical Analysis, 13, 577-598 (1976) · Zbl 0337.65060 · doi:10.1137/0713049
[6] A. Bamberger, G. Chavent, and P. Lailly. Etude de schémas numériques de l’élastodynamique linéaire. Technical Report 41, INRIA, Octobre 1980.
[7] Bamberger, A.; Guillot, J.-C.; Joly, P., Numerical diffraction by a uniform grid, SIAM Journal on Numerical Analysis, 25, 753-783 (1988) · Zbl 0659.65079 · doi:10.1137/0725045
[8] Bécache, E.; Fauqueux, S.; Joly, P., Stability of perfectly matched layers, group velocities and anisotropic waves, Journal of Computational Physics, 188, 399-433 (2003) · Zbl 1127.74335 · doi:10.1016/S0021-9991(03)00184-0
[9] Bécache, E.; Joly, P., On the analysis of Berengers perfectly matched layers for Maxwell equations, Mathematical Modelling and Numerical Analysis, 36, 87-120 (2002) · Zbl 0992.78032 · doi:10.1051/m2an:2002004
[10] Bécache, E.; Joly, P.; Tsogka, C., Fictitious domains, mixed FE and PML for 2-D elastodynamics, Journal of Computational Acoustics, 9, 1175-1201 (2001) · Zbl 1360.74156
[11] Bécache, E.; Joly, P.; Tsogka, C., A new family of mixed finite elements for the linear elastodynamic problem, SIAM Journal on Numerical Analysis, 39, 2109-2132 (2002) · Zbl 1032.74049 · doi:10.1137/S0036142999359189
[12] Bemberger, A.; Engquist, B.; Halpern, L.; Joly, P., Higher order paraxial approximations for the wave equation, SIAM Journal on Applied Mathematics, 48, 129-154 (1988) · Zbl 0654.35056 · doi:10.1137/0148006
[13] Bérenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, Journal Computational Physics, 114, 185-200 (1994) · Zbl 0814.65129 · doi:10.1006/jcph.1994.1159
[14] Bérenger, J. P., Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, Journal Computational Physics, 127, 363-379 (1996) · Zbl 0862.65080 · doi:10.1006/jcph.1996.0181
[15] Bérenger, J. P., Improved PML for the FDTD solution of wave-structure interaction problems, IEEE Transactions on Antennas and Propagation, 45, 466-473 (1997) · doi:10.1109/8.558661
[16] B. Chalindar. Conditions aux Limites Artificielles pour les Equations de l’Elastodynamique. PhD thesis, Université de Saint-Etienne, 1987.
[17] Chin-Joe-Kong, M. J.S.; Mulder, W. A.; Van Veldhuizen, M., Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation, Journal of Engineeting Mathematics, 35, 405-426 (1999) · Zbl 0948.74057 · doi:10.1023/A:1004420829610
[18] P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, 1982.
[19] G. Cohen. Higher-Order Numerical Methods for Transient Wave Equations. Springer, 2002. · Zbl 0985.65096
[20] G. Cohen, P. Joly, J. E. Roberts, and N. Tordjman. Higher order triangular finite elements with mass lumping for the wave equation. SIAM Journal on Numerical Analysis, 38, 2001. · Zbl 1019.65077
[21] F. Collino. High order absorbing boundary conditions for wave propagation models: straight line boundary and corner cases. In PA SIAM, Philadelphia, editor, Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993), pages 161-171, 1993. · Zbl 0814.35065
[22] Collino, F., Perfectly matched absorbing layers for the paraxial equations, Journal of Computational Physics, 131, 164-180 (1996) · Zbl 0866.73013 · doi:10.1006/jcph.1996.5594
[23] Collino, F.; Monk, P., Optimizing the perfectly matched layer, Computational Methods Applied to Mechanical Engineering, 164, 157-171 (1998) · Zbl 1040.78524 · doi:10.1016/S0045-7825(98)00052-8
[24] Collino, F.; Monk, P., The Perfectly Matched Layer in curvilinear coordinates, SIAM Journal on Scientific Computation, 164, 157-171 (1998) · Zbl 1040.78524
[25] Collino, F.; Tsogka, C., Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media, Geophysics, 66, 294-305 (2001) · doi:10.1190/1.1444908
[26] G. Derveaux, P. Joly, and C. Tsogka. Numerical methods for elastic wave propagation. In V.A. Dougalis, J.A. Ekaterinaris, and N.A. Kampanis, editors, Numerical Insight. CRC Press, 2006.
[27] J. Diaz. Approches Analytiques et Numériques de Problèmes de Transmission en Propagation d’Ondes en Régime Transitoire. Application au Couplage Fluide-Structure et aux Méthodes de Couches Parfaitement Adaptées. PhD thesis, Universitée Versailles Saint-Quentin, 2005.
[28] T. Ha Duong and P. Joly. On the stability analysis of boundary conditions for the wave equation by energy methods. Part 1: The homogeneous case. Technical Report 1306, INRIA, 1990. · Zbl 0829.35063
[29] Dupont, T., L^2-estimates for Galerkin methods for second order hyperbolic equations, SIAM Journal on Numerical Analysis, 10, 880-889 (1973) · Zbl 0239.65087 · doi:10.1137/0710073
[30] M. Durufle. Intégration Numérique et Eléments Fins d’Ordre Elevé Appliqués aux Equations de Maxwell en Régime Harmonique. PhD thesis, Université Paris IX Dauphine, 2006. · Zbl 1185.65004
[31] G. Duvaut and J.L. Lions. Inequalities in Mechanics and Physics. Springer, 1976. · Zbl 0331.35002
[32] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation, 31, 629-651 (1977) · Zbl 0367.65051 · doi:10.2307/2005997
[33] A. C. Eringen and E. S. Şuhubi. Elastodynamics, Volume I, Finite motions. Academic Press, 1974. · Zbl 0291.73018
[34] A.C. Eringen and E.S. Şuhubi. Elastodynamics, Volume II, Linear Theory. Academic Press, 1975. · Zbl 0344.73036
[35] S. Fauqueux. Eléments Finis Mixtes Spectraux et Couches Absorbantes Parfaitement Adaptées pour la Propagation d’Ondes Elastiques en Régime Transitoire. PhD thesis, Université Paris IX Dauphine, 2003.
[36] Friedrichs, K. O., On the boundary-value problems of the theory of elasticity and Korn’s inequality, Annals of Mathematics, Second Series, 48, 441-471 (1947) · Zbl 0029.17002
[37] W. Gautschi. Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, 2004. · Zbl 1130.42300
[38] G. Geymonat and G. Gilardi. Contre-exemples à l’inégalité de Korn et au lemme de Lions dans des domaines irréguliers. In Équations aux Dérivées Partielles et Applications, pages 541-548. Gauthier-Villars, 1998. · Zbl 0915.35018
[39] Givoli, D., High-order local non-reflecting boundary conditions: a review, Wave Motion, 39, 319-326 (2004) · Zbl 1163.74356 · doi:10.1016/j.wavemoti.2003.12.004
[40] P. Grisvard. Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. In Numerical solution of partial differential equations (Proceedings of the Third Symposium SYNSPADE), pages 207-274. Academic Press, 1976. · Zbl 0361.35022
[41] Grote, M.; Keller, J., Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM Journal on Applied Mathematics, 55, 280-297 (1995) · Zbl 0817.35049 · doi:10.1137/S0036139993269266
[42] Grote, M.; Keller, J., Nonreflecting boundary conditions for time dependent scattering, Journal of Computational Physics, 127, 52-81 (1996) · Zbl 0860.65080 · doi:10.1006/jcph.1996.0157
[43] T. Hagstrom. On high-order radiation boundary condition. In B. Engquist and G.A. Kriegsmann, editors, Computational Wave Propagation, volume 86, pages 1-21. Springer, 1997. · Zbl 0864.65065
[44] Hagstrom, T., Radiation boundary conditions for the numerical simulation of waves, Acta numerica, 8, 47-106 (1999) · Zbl 0940.65108 · doi:10.1017/S0962492900002890
[45] T. Hagstrom. New results on absorbing layers and radiation boundary conditions. preprint, 2005. · Zbl 1059.78040
[46] Hagstrom, T.; Hariharan, S. I., A formulation of asymptotic and exact boundary conditions using local operators, Applied Numerical Mathematics, 27, 403-416 (1998) · Zbl 0924.35167 · doi:10.1016/S0168-9274(98)00022-1
[47] L. Halpern. Étude de Conditions aux Limites Absorbantes pour des Schémas Numériques Relatifs à des Equations Hyperboliques Linéaires. PhD thesis, Université Paris IV, 1980.
[48] Hastings, F.; Schneider, J. B.; Broschat, S. L., Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation, Journal of the Acoustical Society of America, 100, 3061-3069 (1996) · doi:10.1121/1.417118
[49] Higdon, R. L., Initial-boundary value problems for linear hyperbolic systems, SIAM Review, 28, 177-217 (1986) · Zbl 0603.35061 · doi:10.1137/1028050
[50] Higdon, R. L., Radiation boundary conditions for elastic wave propagation, SIAM Journal on Numerical Analysis, 27, 831-870 (1990) · Zbl 0718.35058 · doi:10.1137/0727049
[51] Higdon, R. L., Absorbing boundary conditions for elastic waves, Geophysics, 56, 231-241 (1991) · doi:10.1190/1.1443035
[52] Higdon, R. L., Absorbing boundary conditions for acoustic and elastic waves in stratified media, Journal of Computational Physics, 101, 386-418 (1992) · Zbl 0800.76402 · doi:10.1016/0021-9991(92)90016-R
[53] L. Hörmander. The analysis of linear partial differential operators. III. In Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], volume 274. Springer, 1994. · Zbl 1115.35005
[54] Israeli, M.; Orszag, S. A., Approximation of radiation boundary conditions, Journal of Computational Physics, 41, 115-135 (1981) · Zbl 0469.65082 · doi:10.1016/0021-9991(81)90082-6
[55] P. Joly. Variational methods for time-dependent wave propagation problems. In Topics in Computational Wave Propagation, Direct and Inverse Problems, pages 201-264. LNCSE, 2003. · Zbl 1049.78028
[56] Kondratiev, V. A.; Oleinik, O. A., On Korn’s inequalities, Comptes Rendus de l’Académie des Sciences de Paris, Série I, 308, 483-487 (1989) · Zbl 0698.35067
[57] H-O. Kreiss and J. Lorenz. Initial-Boundary Value Problems and the Navier-Stokes Equations. Academic Press, 1989. · Zbl 0689.35001
[58] Lindman, E., Free space boundary conditions for time dependant wave equation, Journal of Computational Physics, 18, 66-78 (1975) · Zbl 0417.73042 · doi:10.1016/0021-9991(75)90102-3
[59] J.L. Lions and E. Magenès. Problèmes aux Limites non Homogènes et Applications. Dunod, 1968. · Zbl 0165.10801
[60] J. Miklowitz. The theory of elastic waves and waveguides. volume 22 of North-Holland Series in Applied Mathematics and Mechanics. North-Holland, 1978.
[61] Mulder, W. A., Higher-order mass-lumped finite elements for the wave equation, Journal of Computational Acoustics, 9, 671-680 (2001) · Zbl 1360.65239 · doi:10.1142/S0218396X0100067X
[62] Nitsche, J. A., On Korn’s second inequality, RAIRO Analyse Numérique, 15, 237-248 (1981) · Zbl 0467.35019
[63] Peng, C.; Toksoz, M. N., An optimal absorbing boundary condition for elastic wave modeling, Geophysics, 60, 296-301 (1995) · doi:10.1190/1.1443758
[64] Petropoulos, P. G.; Zhao, L.; Cangellaris, A. C., A reflectionless sponge layer absorbing boundary condition for the solution of Maxwell’s equations with high-order staggered finite difference schemes, Journal of Computational Physics, 139, 184-208 (1998) · Zbl 0915.65123 · doi:10.1006/jcph.1997.5855
[65] Simone, A.; Hestholm, S., Instability in applying absorbing boundary conditions to high-order seismic modeling algorithms, Geophysics, 63, 1017-1023 (1998) · doi:10.1190/1.1444379
[66] Sochacki, J.; Kubichek, R.; George, J.; Fletcher, W. R.; Smithson, S., Absorbing boundary conditions and surface waves, Geophysics, 52, 60-71 (1987) · doi:10.1190/1.1442241
[67] A.H. Stroud. Numerical Quadrature and Solution of Ordinary Differential Equations. Springer, 1974 · Zbl 0298.65018
[68] Tam, C. K.W.; Auriault, L.; Cambuli, F., Perfectly matched layer as an absorbing boundary condition for the linearized Euler equations in open and ducted domains, Journal of Computational Physics, 144, 213-234 (1998) · Zbl 1392.76054 · doi:10.1006/jcph.1998.5997
[69] M.E. Taylor. Partial differential equations. I. volume 115 of Applied Mathematical Sciences. Springer, 1996. · Zbl 0869.35002
[70] Teixeira, F. L.; Chew, W. C., Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves, Microwave and Optical Technology Letters, 17, 231-236 (1998) · doi:10.1002/(SICI)1098-2760(199803)17:4<231::AID-MOP3>3.0.CO;2-J
[71] Teixeira, F. L.; Chew, W. C., Unified analysis of perfectly matched layers using differential forms, Microwave and Optical Technology Letters, 20, 124-126 (1999) · doi:10.1002/(SICI)1098-2760(19990120)20:2<124::AID-MOP12>3.0.CO;2-N
[72] N. Tordjman. Eléments Finis d’Ordre Elevé avec Condensation de Masse pour l’Equation des Ondes. PhD thesis, Paris IX, 1995.
[73] Trefethen, L.; Halpern, L., Well posedness of one way equations and absorbing boundary conditions, Mathematics of Computation, 47, 421-435 (1986) · Zbl 0618.65077 · doi:10.2307/2008165
[74] L.N. Trefethen. Group velocity in finite difference schemes. SIAM Review, 24, 1982. · Zbl 0487.65055
[75] L.N. Trefethen. Group velocity interpretation of the stability theory of Gustafsson, Kreiss and Sundström. Journal of Computational Physics, 49, 1983. · Zbl 0501.65046
[76] Trefethen, L. N., Instability of difference models for hyperbolic initial boundary value problems, Communications on Pure and Applied Mathematics, 37, 329-367 (1984) · Zbl 0575.65095 · doi:10.1002/cpa.3160370305
[77] Turkel, E.; Yefet, A., Absorbing PML boundary layers for wave-like equations. Absorbing boundary conditions, Applied Numerical Mathematics, 27, 553-557 (1998) · Zbl 0933.35188
[78] Zhao, L.; Cangellaris, A. C., A general approach to for developping unsplit-field time-domain implementations of perfectly matched layers for FDTD grid truncation, IEEE Transactions on Microwave Theory and Technology, 44, 2555-2563 (1996) · doi:10.1109/22.554601
[79] O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, 1971. · Zbl 0237.73071
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