Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T17:22:34.062Z Has data issue: false hasContentIssue false
  • English
  • Français

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

Published online by Cambridge University Press:  03 June 2015

N. Anders Petersson  and
Björn Sjögreen
N. Anders Petersson*
Affiliation:
Center for Applied Scientific Computing, L-422, LLNL, P.O. Box 808, Livermore, CA 94551, USA
Björn Sjögreen*
Affiliation:
Center for Applied Scientific Computing, L-422, LLNL, P.O. Box 808, Livermore, CA 94551, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

Keywords

65M0674S2086A15Far field boundary conditionshigher order methodselastic wave equationsummation by parts
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 913 - 955
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Appelö, D. and Colonius, T.A high order super-grid-scale absorbing layer and its application to linear hyperbolic systems. J. Comput. Phys., 228:4200–4217, 2009.Google Scholar
[2]Bécache, E., Fauqueux, S., and Joly, P.Stability of perfectly matched layers, group velocities and anisotropic waves. J. Compt. Phys., 188:399–433, 2003.Google Scholar
[3]Berenger, J. P.A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114:185–200, 1994.Google Scholar
[4]Clayton, R. and Engquist, B.Absorbing boundary conditions for acoustic and elastic wave equations. Bull. Seismo. Soc. Amer., 67, 1977.Google Scholar
[5]Eringen, A. Cemal and Şuhubi, Erdoğan S.Elastodynamics, Volume II. Elsevier, 1975.Google Scholar
[6]Higdon, R. L.Radiation boundary conditions for elastic wave propagation. SIAM J. Numer. Anal., 27:831–870, 1990.Google Scholar
[7]Kreiss, H.-O. and Lorenz, J.Initial-Boundary Value Problems and the Navier-Stokes Equations. Academic Press, 1989.Google Scholar
[8]Lamb, Horace. On the propagation of tremors over the surface of an elastic solid. Phil. Trans. Roy. Soc. London, Ser. A, 203, 1904.Google Scholar
[9]Mattsson, K., Ham, F., and Iaccarino, G.Stable and accurate wave-propagation in discontinuous media. J. Comput. Phys., 227:8753–8767, 2008.CrossRefGoogle Scholar
[10]Mattsson, K. and Nordström, J.Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys., 199:503–540, 2004.CrossRefGoogle Scholar
[11]Mooney, Harold M.Some numerical solutions for Lamb’s problem. Bull. Seismo. Soc. Amer., 64, 1974.Google Scholar
[12] Netlib. Repository of scientific computing software. http://www.netlib.org.Google Scholar
[13]Petersson, N. A. and Sjögreen, B.An energy absorbing far-field boundary condition for the elastic wave equation. Comm. Comput. Phys., 6:483–508, 2009.Google Scholar
[14]Petersson, N. A. and Sjögreen, B.Stable grid refinement and singular source discretization for seismic wave simulations. Comm. Comput. Phys., 8(5):1074–1110, November 2010.Google Scholar
[15]Petersson, N. A. and Sjögreen, B.User’s guide to SW4, version 1.0. Technical Report LLNL-SM-642292, Lawrence Livermore National Laboratory, 2013. (Source code available from computation.llnl.gov/casc/serpentine).Google Scholar
[16]Sjögreen, B. and Petersson, N. A.Perfectly matched layers for Maxwell’s equations in second order formulation. J. Comput. Phys., 209:19–46, 2005.Google Scholar
[17]Sjögreen, B. and Petersson, N. A.A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation. J. Sci. Comput., 52:17–48, 2012. DOI 10.1007/s10915-011-9531-1.Google Scholar
[18]Skelton, E. A., Adams, S. D. M., and Craster, R. V.Guided elastic waves and perfectly matched layers. Wave Motion, 44:573–592, 2007.Google Scholar