Abstract
There is a great need for numerical methods which treat time-dependent elastic wave propagation problems. Such problems appear in many applications, for example in geophysics or non-destructive testing. In particular, seismic wave propagation is one of the areas where intensive scientific computation has been developed and used since the beginning of the 70’s, with the apparition of finite differences time domain methods (FDTD). Although very old, these methods remain very popular and are widely used for the simulation of wave propagation phenomena or more generally for the numerical resolution of linear hyperbolic systems. They consist in obtaining discrete equations whose unknowns are generally field values at the points of a regular mesh with spatial step h and time step Δt. A prototype of these methods is the famous Yee§s scheme introduced in 1966 for Maxwell§s equations. There are several reasons that explain the success of Yee type schemes, among which their easy implementation and their efficiency which are related to the following properties:
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a uniform regular grid is used for the space discretization, so that there is a minimum of information to store and the data to be computed are structured: in other words, one avoids all the complications due to the use of non uniform meshes.
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an explicit time discretization is applied: no linear system has to be solved at each time step.
This text is a modification of a part of a longer text to appear as a chapter of a book (Derveaux et al., 2006).
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Bibliography
S. Abarbanel and D. Gottlieb. A mathematical analysis of the PML method. Journal of Computational Physics, 134:357–263, 1997.
J.D. Achenbach. Wave Propagation in Elastic Solids. Elsevier, 1984.
D. N. Arnold, D. Boffi, and R. S. Falk. Approximation by quadrilateral finite elements. Math. Comp., 71(239):909–922 (electronic), 2002. ISSN 0025-5718.
B.A. Auld. Acoustic Fields and Elastic Waves in Solids. John Wiley, 1973.
G.A. Baker and V.A. Dougalis. The effect of quadrature errors on finite element approximations for the second-order hyperbolic equations. SIAM Journal on Numerical Analysis, 13:577–598, 1976.
A. Bamberger, G. Chavent, and P. Lailly. Etude de schémas numériques de l’élastodynamique linéaire. Technical Report 41, INRIA, Octobre 1980.
A. Bamberger, J.-C. Guillot, and P. Joly. Numerical diffraction by a uniform grid. SIAM Journal on Numerical Analysis, 25:753–783, 1988.
E. Bécache, S. Fauqueux, and P. Joly. Stability of perfectly matched layers, group velocities and anisotropic waves. Journal of Computational Physics, 188:399–433, 2003.
E. Bécache and P. Joly. On the analysis of Berengers perfectly matched layers for Maxwell equations. Mathematical Modelling and Numerical Analysis, 36:87–120, 2002.
E. Bécache, P. Joly, and C. Tsogka. Fictitious domains, mixed FE and PML for 2-D elastodynamics. Journal of Computational Acoustics, 9:1175–1201, 2001.
E. Bécache, P. Joly, and C. Tsogka. A new family of mixed finite elements for the linear elastodynamic problem. SIAM Journal on Numerical Analysis, 39:2109–2132, 2002.
A. Bemberger, B. Engquist, L. Halpern, and P. Joly. Higher order paraxial approximations for the wave equation. SIAM Journal on Applied Mathematics, 48:129–154, 1988.
J.P. Bérenger. A perfectly matched layer for the absorption of electromagnetic waves. Journal Computational Physics, 114:185–200, 1994.
J.P. Bérenger. Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. Journal Computational Physics, 127:363–379, 1996.
J.P. Bérenger. Improved PML for the FDTD solution of wave-structure interaction problems. IEEE Transactions on Antennas and Propagation, 45:466–473, 1997.
B. Chalindar. Conditions aux Limites Artificielles pour les Equations de l’Elastodynamique. PhD thesis, Université de Saint-Etienne, 1987.
M.J.S. Chin-Joe-Kong, W.A. Mulder, and M. Van Veldhuizen. Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. Journal of Engineeting Mathematics, 35:405–426, 1999.
P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, 1982.
G. Cohen. Higher-Order Numerical Methods for Transient Wave Equations. Springer, 2002.
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.
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.
F. Collino. Perfectly matched absorbing layers for the paraxial equations. Journal of Computational Physics, 131:164–180, 1996.
F. Collino and P. Monk. Optimizing the perfectly matched layer. Computational Methods Applied to Mechanical Engineering, 164:157–171, 1998a.
F. Collino and P. Monk. The Perfectly Matched Layer in curvilinear coordinates. SIAM Journal on Scientific Computation, 164:157–171, 1998b.
F. Collino and C. Tsogka. Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heteregeneous media. Geophysics, 66:294–305, 2001.
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.
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.
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.
T. Dupont. L 2-estimates for Galerkin methods for second order hyperbolic equations. SIAM Journal on Numerical Analysis, 10:880–889, 1973.
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.
G. Duvaut and J.L. Lions. Inequalities in Mechanics and Physics. Springer, 1976.
B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation, 31:629–651, 1977.
A. C. Eringen and E. S. Şuhubi. Elastodynamics, Volume I, Finite motions. Academic Press, 1974.
A.C. Eringen and E.S. Şuhubi. Elastodynamics, Volume II, Linear Theory. Academic Press, 1975.
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.
K.O. Friedrichs. On the boundary-value problems of the theory of elasticity and Korn’s inequality. Annals of Mathematics, Second Series, 48:441–471, 1947.
W. Gautschi. Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, 2004.
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.
D. Givoli. High-order local non-reflecting boundary conditions: a review. Wave Motion, 39:319–326, 2004.
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.
M. Grote and J. Keller. Exact nonreflecting boundary conditions for the time dependent wave equation. SIAM Journal on Applied Mathematics, 55:280–297, 1995.
M. Grote and J. Keller. Nonreflecting boundary conditions for time dependent scattering. Journal of Computational Physics, 127:52–81, 1996.
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.
T. Hagstrom. Radiation boundary conditions for the numerical simulation of waves. Acta numerica, 8:47–106, 1999.
T. Hagstrom. New results on absorbing layers and radiation boundary conditions. preprint, 2005.
T. Hagstrom and S.I. Hariharan. A formulation of asymptotic and exact boundary conditions using local operators. Applied Numerical Mathematics, 27:403–416, 1998.
L. Halpern. Étude de Conditions aux Limites Absorbantes pour des Schémas Numériques Relatifs à des Equations Hyperboliques Linéaires. PhD thesis, Université Paris IV, 1980.
F. Hastings, J.B. Schneider, and S.L. Broschat. 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.
R.L. Higdon. Initial-boundary value problems for linear hyperbolic systems. SIAM Review, 28:177–217, 1986.
R.L. Higdon. Radiation boundary conditions for elastic wave propagation. SIAM Journal on Numerical Analysis, 27:831–870, 1990.
R.L. Higdon. Absorbing boundary conditions for elastic waves. Geophysics, 56:231–241, 1991.
R.L. Higdon. Absorbing boundary conditions for acoustic and elastic waves in stratified media. Journal of Computational Physics, 101:386–418, 1992.
L. Hörmander. The analysis of linear partial differential operators. III. In Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], volume 274. Springer, 1994.
M. Israeli and S.A. Orszag. Approximation of radiation boundary conditions. Journal of Computational Physics, 41:115–135, 1981.
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.
V.A. Kondratiev and O.A. Oleinik. On Korn’s inequalities. Comptes Rendus de l’Académie des Sciences de Paris, Série I, 308:483–487, 1989.
H-O. Kreiss and J. Lorenz. Initial-Boundary Value Problems and the Navier-Stokes Equations. Academic Press, 1989.
E. Lindman. Free space boundary conditions for time dependant wave equation. Journal of Computational Physics, 18:66–78, 1975.
J.L. Lions and E. Magenès. Problèmes aux Limites non Homogènes et Applications. Dunod, 1968.
J. Miklowitz. The theory of elastic waves and waveguides. volume 22 of North-Holland Series in Applied Mathematics and Mechanics. North-Holland, 1978.
W.A. Mulder. Higher-order mass-lumped finite elements for the wave equation. Journal of Computational Acoustics, 9:671–680, 2001.
J.A. Nitsche. On Korn’s second inequality. RAIRO Analyse Numérique, 15:237–248, 1981.
C. Peng and M.N. Toksoz. An optimal absorbing boundary condition for elastic wave modeling. Geophysics, 60:296–301, 1995.
P.G. Petropoulos, L. Zhao, and A.C. Cangellaris. 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.
A. Simone and S. Hestholm. Instability in applying absorbing boundary conditions to high-order seismic modeling algorithms. Geophysics, 63:1017–1023, 1998.
J. Sochacki, R. Kubichek, J. George, W.R. Fletcher, and S. Smithson. Absorbing boundary conditions and surface waves. Geophysics, 52:60–71, 1987.
A.H. Stroud. Numerical Quadrature and Solution of Ordinary Differential Equations. Springer, 1974
C.K.W. Tam, L. Auriault, and F. Cambuli. 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.
M.E. Taylor. Partial differential equations. I. volume 115 of Applied Mathematical Sciences. Springer, 1996.
F.L. Teixeira and W.C. Chew. Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves. Microwave and Optical Technology Letters, 17: 231–236, 1998.
F.L. Teixeira and W.C. Chew. Unified analysis of perfectly matched layers using differential forms. Microwave and Optical Technology Letters, 20:124–126, 1999.
N. Tordjman. Eléments Finis d’Ordre Elevé avec Condensation de Masse pour l’Equation des Ondes. PhD thesis, Paris IX, 1995.
L. Trefethen and L. Halpern. Well posedness of one way equations and absorbing boundary conditions. Mathematics of Computation, 47:421–435, 1986.
L.N. Trefethen. Group velocity in finite difference schemes. SIAM Review, 24, 1982.
L.N. Trefethen. Group velocity interpretation of the stability theory of Gustafsson, Kreiss and Sundström. Journal of Computational Physics, 49, 1983.
L.N. Trefethen. Instability of difference models for hyperbolic initial boundary value problems. Communications on Pure and Applied Mathematics, 37:329–367, 1984.
E. Turkel and A. Yefet. Absorbing PML boundary layers for wave-like equations. Absorbing boundary conditions. Applied Numerical Mathematics, 27:553–557, 1998.
L. Zhao and A.C. Cangellaris. 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.
O.C. Zienkiewicz. The Finite Element Method in Engineering Science. McGraw-Hill, 1971.
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Joly, P. (2007). Numerical Methods for Elastic Wave Propagation. In: Destrade, M., Saccomandi, G. (eds) Waves in Nonlinear Pre-Stressed Materials. CISM Courses and Lectures, vol 495. Springer, Vienna. https://doi.org/10.1007/978-3-211-73572-5_6
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