Abstract Based on strong and weak forms of elastic wave equations, a Chebyshev spectral element method (SEM) using the Galerkin variational principle is developed by discretizing the wave equation in the spatial and time domains and introducing the preconditioned conjugate gradient (PCG)-element by element (EBE) method in the spatial domain and the staggered predictor/corrector method in the time domain. The accuracy of our proposed method is verified by comparing it with a finite-difference method (FDM) for a homogeneous solid medium and a double layered solid medium with an inclined interface. The modeling results using the two methods are in good agreement with each other. Meanwhile, to show the algorithm capability, the suggested method is used to simulate the wave propagation in a layered medium with a topographic traction free surface. By introducing the EBE algorithm with an optimized tensor product technique, the proposed SEM is especially suitable for numerical simulation of wave propagations in complex models with irregularly free surfaces at a fast convergence rate, while keeping the advantage of the finite element method.
This work is supported by the National Natural Science Foundation of China (Grant No. 40774099, 10874202) and the National High Technology Research and Development Program of China (Grant No. 2008AA06Z205).
Cite this article:
CHE Cheng-Xuan,WANG Xiu-Ming,LIN Wei-Jun. The Chebyshev spectral element method using staggered predictor and corrector for elastic wave simulations[J]. APPLIED GEOPHYSICS, 2010, 7(2): 174-184.
[1]
Alford, R. M., Kelly, K. R., and Boore, D. M., 1974, Accuracy of finite difference modeling of the acoustic wave equation: Geophysics, 39(6), 834 - 842.
[2]
Alterman, Z. S., and Loewenthal, D., 1970, Seismic waves in a quarter and three-quarter plane: Geophys. J. Roy. Astr. Soc., 20(2), 101 - 126.
[3]
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., 1988, Spectral methods in fluid dynamics: Springer-Verlag, 65 - 75.
[4]
Cerjan, C., Kosloff, D., Kosloff, R., and Reshef, M., 1985, A non-reflecting boundary condition for discrete acoustic and elastic wave equations: Geophysics, 50(4), 705 - 708.
[5]
Deng, S. Z., Zhou, S. Q., and Zeng, L., 1995, Application of EBE strategy in structural analysis (I)-EBE computational method for matrix-vector multiply and its parallel/vector implementations: Journal of Astronautics, 16(2), 13-19.
[6]
Gazdag, J., 1981, Modeling of the acoustic wave equation with transform method: Geophysics, 46(6), 854 - 859.
[7]
Hilber, H. M., Hughes, T. J. R., and Taylor, R. L., 1977, Improved numerical dissipation for time integration algorithms in structural dynamics: Earthquake Engineering and Structural Dynamics, 5, 283 - 292.
[8]
Hughes, T. J. R., 1987, The finite element method－Linear static and dynamic finite element analysis: Prentice-Hall, Inc, Englewood Cliffs, New Jersey.
[9]
Hughes, T. J. R., Levit, I., and Winget, J., 1983a, Element-by-element implicit algorithms for heat conduction: Journal of Engineering Mechanics, 109(2), 576 - 585.
[10]
Hughes, T. J. R., Levit, I., and Winget, J., 1983b, An element-by-element solution algorithm for problems of structural and solid mechanics: Computer Methods in Applied Mechanics and Engineering, 36(2), 241 - 254.
[11]
Komatitsch, D., and Barnes, C., 2002, Spectral-element simulations of global seismic wave propagation—II. Three-dimensional models, oceans, rotation and self-gravitation: Geophys. J. Int., 150, 303 - 318.
[12]
Komatitsch, D., Barnes, C., and Tromp. J., 2000a, Wave propagation near a fluid-solid interface: A spectral-element approach: Geophysics, 65(2), 623 - 631.
[13]
Komatitsch, D., Barnes, C., and Tromp. J., 2000b, Simulation of isotropic wave propagation based upon a spectral element method: Geophysics, 65(4), 1251 - 1260.
[14]
Komatitsch, D., Barnes, C., and Tromp. J., 2002, Spectral-element simulations of global seismic wave propagation-I. Validation: Geophys. J. Int., 149, 390 - 412.
[15]
Komatitsch, D., and Vilotte, J. P., 1998, The spectral element method: an efficient tool to simulate the seismic response of 2-D and 3-D geological structures: Bull. Seis. Soc. Am., 88(2), 368 - 392.
[16]
Komatitsch, D., Vilotte, J. P., Vai, R., Castillo-Covarrubias,J. M., and Sanchez-Sesma,F. J., 1999, The spectral element method for elastic wave equations-application to 2-D and 3-D seismic problems: Int. J. Meth. Eng., 45, 1139 - 1164.
[17]
Kosloff, D., and Baysal, E., 1982, Forward modeling by a Fourier method: Geophysics, 47(10), 1402 - 1412.
[18]
Kosloff, D., Kessler, D., Filho, A. Q., Behle, A., and Strabilevitz, R., 1990, Solution of the equations of dynamic elasticity by a Chebyshev spectral method: Geophysics, 55(6), 734 - 748.
[19]
Lin, W. J., Wang, X. M., and Zhang, H. L., 2006, An element by element spectral element method for elastic wave modeling: Progress in Natural Science, 16(1), 21 - 29.
[20]
Patera, A. T., 1984, A spectral element method for fluid dynamics: laminar flow in a channel expansion: Journal of Computational Physics, 54(3), 468 - 488.
[21]
Seriani, G., 1997, A parallel spectral element method for acoustic wave modeling: J. Computation Acoustics, 5(1), 53 - 69.
[22]
Seriani, G., 1998, 3-D large-scale wave propagation modeling by spectral element method on Cray T3E: Comp. Meth. Appl. Mech. and Eng., 164, 235 - 247.
[23]
Seriani, G., and Priolo, E., 1994, Spectral element method for acoustic wave simulation in heterogeneous media: Finite element in analysis and design, 16, 337 - 348.
[24]
Seriani, G., Priolo, E., Carcione, J. M., and Padovani, E., 1992, High-order spectral element method for elastic wave modeling: 62th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1285 - 1288.
[25]
Wang, X. M., 2001, Seismic wave simulation in anisotropic media with heterogeneity using high-order finite-difference method: In Proceedings of the 5th SEGJ International Symposium: Imaging Technology, Tokyo, Japan, 113 - 120.
[26]
Wang, X. M., Dodds, K., and Zhao, H. B., 2006, An improved high-order rotated staggered finite-difference algorithm for simulating elastic waves in heterogeneous viscoelastic/anisotropic media: Exploration Geophysics, 37(2), 160 - 174.
[27]
Wang, X. M., Seriani, G., and Lin, W. J., 2007, Some theoretical aspects of elastic wave modeling with a recently developed spectral element method: Science in China (Series G), 50 (2), 185 - 207.
[28]
Wang, X. M., and Zhang, H. L., 2004, Modeling of elastic wave propagation on a curved free surface using an improved finite-difference algorithm: Science in China (Series G), 47(5), 633 - 648.
[29]
Winget, J., and Hughes, T. J. R., 1985, Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies: Computer Methods in Applied Mechanics and Engineering, 52, 711 - 815.
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