article

A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media

Authors:

Lucas C. Wilcox,

Carsten Burstedde,

Omar GhattasAuthors Info & Claims

Journal of Computational Physics, Volume 229, Issue 24

Pages 9373 - 9396

https://doi.org/10.1016/j.jcp.2010.09.008

Published: 01 December 2010 Publication History

Abstract

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic-acoustic media. A velocity-strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic-acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic-acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.

References

[1]

}}Ainsworth, M., Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. Journal of Computational Physics. v198 i1. 106-130.

[2]

}}Ainsworth, M., Monk, P. and Muniz, W., Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. Journal of Scientific Computing. v27 i1-3. 5-40.

[3]

}}Bao, H., Bielak, J., Ghattas, O., Kallivokas, L.F., O'Hallaron, D.R., Shewchuk, J.R. and Xu, J., Large-scale simulation of elastic wave propagation in heterogeneous media on parallel computers. Computer Methods in Applied Mechanics and Engineering. v152 i1-2. 85-102.

[4]

}}Appelö, D. and Petersson, N.A., A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Communications in Computational Physics. v5 i1. 84-107.

[5]

}}Lombard, B., Piraux, J., Gélis, C. and Virieux, J., Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves. Geophysical Journal International. v172. 252-261.

[6]

}}Lombard, B. and Piraux, J., Numerical treatment of two-dimensional interfaces for acoustic and elastic waves. Journal of Computational Physics. v195 i1. 90-116.

[7]

}}R.W. Hoberecht, A finite volume approach to modeling injury mechanisms of blast-induced traumatic brain injury, Master's thesis, University of Washington, 2009.

[8]

}}Voinovich, P., Merlen, A., Timofeev, E. and Takayama, K., A Godunov-type finite-volume scheme for unified solid-liquid elastodynamics on arbitrary two-dimensional grids. Shock Waves. v13 i3. 221-230.

[9]

}}Komatitsch, D., Barnes, C. and Tromp, J., Wave propagation near a fluid-solid interface: a spectral element approach. Geophysics. v65 i2. 623-631.

[10]

}}Chaljub, E., Capdeville, Y. and Vilotte, J., Solving elastodynamics in a fluid-solid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids. Journal of Computational Physics. v187 i2. 457-491.

[11]

}}Chaljub, E. and Valette, B., Spectral element modelling of three-dimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core. Geophysical Journal International. v158 i1. 131-141.

[12]

}}Komatitsch, D., Tsuboi, S. and Tromp, J., The spectral-element method in seismology. In: Levander, A., Nolet, G. (Eds.), Geophysical Monograph, vol. 157. American Geophysical Union, Washington DC, USA. pp. 205-228.

[13]

}}Riviere, B. and Wheeler, M.F., A discontinuous Galerkin method applied to nonlinear parabolic equations. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (Eds.), Lecture Notes in Computational Science and Engineering, vol. 11. Springer-Verlag. pp. 231-244.

[14]

}}Grote, M.J., Schneebeli, A. and Schötzau, D., Discontinuous Galerkin finite element method for the wave equation. SIAM Journal on Numerical Analysis. v44 i6. 2408-2431.

[15]

}}Chung, E.T. and Engquist, B., Optimal discontinuous Galerkin methods for wave propagation. SIAM Journal on Numerical Analysis. v44 i5. 2131-2158.

[16]

}}De Basabe, J.D., Sen, M.K. and Wheeler, M.F., The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion. Geophysical Journal International. v175 i1. 83-93.

[17]

}}S. Delcourte, L. Fezoui, N. Glinsky-Olivier, A high-order discontinuous Galerkin method for the seismic wave propagation, in: ESAIM: Proceedings, vol. 27, 2009, pp. 70-89.

[18]

}}Dumbser, M. and Käser, M., An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-II. The three-dimensional isotropic case. Geophysical Journal International. v167 i1. 319

[19]

}}Käser, M. and Dumbser, M., A highly accurate discontinuous Galerkin method for complex interfaces between solids and moving fluids. Geophysics. v73 i3. T23-T35.

[20]

}}Hesthaven, J.S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. 2008. Texts in Applied Mathematics, 2008.Springer.

Digital Library

[21]

}}LeVeque, R., Finite Volume Methods for Hyperbolic Problems. 2002. Cambridge University Press.

[22]

}}Kopriva, D.A., Metric identities and the discontinuous spectral element method on curvilinear meshes. Journal of Scientific Computing. v26 i3. 301-327.

[23]

}}Kopriva, D.A., Implementing Spectral Methods for Partial Differential Equations. 2009. Springer.

[24]

}}Dumbser, M. and Käser, M., An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-II. The three-dimensional isotropic case. Geophysical Journal International. v167 i1. 319

[25]

}}Gonzalez, O. and Stuart, A.M., . 2008. Cambridge Texts in Applied Mathematics, 2008.Cambridge University Press.

[26]

}}Feng, K.-A., Teng, C.-H. and Chen, M.-H., A pseudospectral penalty scheme for 2D isotropic elastic wave computations. Journal of Scientific Computing. v33 i3. 313-348.

[27]

}}Deville, M., Fischer, P. and Mund, E., High-Order Methods for Incompressible Fluid Flow. 2002. Cambridge Monographs on Applied and Computational Mathematics, 2002.Cambridge University Press.

[28]

}}S.S. Collis, M. Heinkenschloss, Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems, Technical Report TR02-01, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, 2002. <http://www.caam.rice.edu/heinken>.

[29]

}}Kopriva, D.A., Woodruff, S.L. and Hussaini, M.Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. International Journal for Numerical Methods in Engineering. v53 i1. 105-122.

[30]

}}Kopriva, D.A. and Gassner, G., On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. Journal of Scientific Computing. v44. 136-155.

Digital Library

[31]

}}Kennedy, C.A., Carpenter, M.H. and Lewis, R.M., Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Applied Numerical Mathematics. v35 i3. 177-219.

[32]

}}Kaufman, A.A. and Levshin, A.L., Acoustic and Elastic Wave Fields in Geophysics, III. 2005. Methods in Geochemistry and Geophysics, 2005.Elsevier B.V.

[33]

}}Achenbach, J.D., . 1973. North-Holland Series in Applied Mathematics and Mechanics, 1973.North-Holland Publishing Company.

[34]

}}Vinh, P.C. and Ogden, R.W., On formulas for the Rayleigh wave speed. Wave Motion. v39 i3. 191-197.

[35]

}}C. Burstedde, L.C. Wilcox, O. Ghattas, p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees, SIAM Journal on Scientific Computing, submitted for publication.

[36]

}}Burstedde, C., Ghattas, O., Gurnis, M., Tan, E., Tu, T., Stadler, G., Wilcox, L.C. and Zhong, S., Scalable adaptive mantle convection simulation on petascale supercomputers. In: SC'08: Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis, ACM/IEEE.

[37]

}}Burstedde, C., Ghattas, O., Gurnis, M., Isaac, T., Stadler, G., Warburton, T. and Wilcox, L.C., Extreme-scale AMR. In: SC'10: Proceedings of the International Conference on High Performance Computing, Networking, Storage, and Analysis, ACM/IEEE.

[38]

}}Dziewonski, A.M. and Anderson, D.L., Preliminary reference earth model. Physics of the Earth and Planetary Interiors. v25 i4. 297-356.

[39]

}}Carrington, L., Komatitsch, D., Laurenzano, M., Tikir, M.M., Michéa, D., Goff, N.L., Snavely, A. and Tromp, J., High-frequency simulations of global seismic wave propagation using SPECFEM3D GLOBE on 62K processors. In: SC'08: Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis, ACM/IEEE.

[40]

}}Aki, K. and Richards, P.G., Quantitative Seismology. 2002. second ed. University Science Books.

[41]

}}Performance applications programming interface (PAPI). <http://icl.cs.utk.edu/papi/>.

Cited By

Kamalinia HBarbarulo ATie B(2024)A coupled acoustic/elastic discontinuous Galerkin finite element methodComputers and Structures10.1016/j.compstruc.2023.107208291:COnline publication date: 15-Jan-2024
https://dl.acm.org/doi/10.1016/j.compstruc.2023.107208
Ravelo FPeixoto PSchreiber M(2024)An Explicit Exponential Integrator Based on Faber Polynomials and its Application to Seismic Wave ModelingJournal of Scientific Computing10.1007/s10915-023-02413-098:2Online publication date: 3-Jan-2024
https://dl.acm.org/doi/10.1007/s10915-023-02413-0
Hanindhito BGourounas DFathi ATrenev DGerstlauer AJohn LRauchwerger LCameron KNikolopoulos DPnevmatikatos D(2022)GAPSProceedings of the 36th ACM International Conference on Supercomputing10.1145/3524059.3532373(1-13)Online publication date: 28-Jun-2022
https://dl.acm.org/doi/10.1145/3524059.3532373
Show More Cited By

A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

Recommendations

A Discontinuous Galerkin Finite Element Method for Hamilton--Jacobi Equations

In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton--Jacobi equations. This method is based on the Runge--Kutta discontinuous Galerkin finite element method for solving conservation laws. The method ...
A high-order discontinuous Galerkin method with time-accurate local time stepping for the Maxwell equations

We present an explicit numerical method to solve the time-dependent Maxwell equations with arbitrary high order of accuracy in space and time on three-dimensional unstructured tetrahedral meshes. The method is based on the discontinuous Galerkin finite ...
Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method

This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 229, Issue 24

December, 2010

410 pages

ISSN:0021-9991

Issue’s Table of Contents

Copyright © Elsevier Inc. © 2010.

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 01 December 2010

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

34
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 19 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Kamalinia HBarbarulo ATie B(2024)A coupled acoustic/elastic discontinuous Galerkin finite element methodComputers and Structures10.1016/j.compstruc.2023.107208291:COnline publication date: 15-Jan-2024
https://dl.acm.org/doi/10.1016/j.compstruc.2023.107208
Ravelo FPeixoto PSchreiber M(2024)An Explicit Exponential Integrator Based on Faber Polynomials and its Application to Seismic Wave ModelingJournal of Scientific Computing10.1007/s10915-023-02413-098:2Online publication date: 3-Jan-2024
https://dl.acm.org/doi/10.1007/s10915-023-02413-0
Hanindhito BGourounas DFathi ATrenev DGerstlauer AJohn LRauchwerger LCameron KNikolopoulos DPnevmatikatos D(2022)GAPSProceedings of the 36th ACM International Conference on Supercomputing10.1145/3524059.3532373(1-13)Online publication date: 28-Jun-2022
https://dl.acm.org/doi/10.1145/3524059.3532373
Wolf SGalis MUphoff CGabriel AMoczo PGregor DBader M(2022)An efficient ADER-DG local time stepping scheme for 3D HPC simulation of seismic waves in poroelastic mediaJournal of Computational Physics10.1016/j.jcp.2021.110886455:COnline publication date: 15-Apr-2022
https://dl.acm.org/doi/10.1016/j.jcp.2021.110886
Dassi FFumagalli AMazzieri IScotti AVacca G(2022)A Virtual Element Method for the Wave Equation on Curved Edges in Two DimensionsJournal of Scientific Computing10.1007/s10915-021-01683-w90:1Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1007/s10915-021-01683-w
O’Reilly ONordström J(2022)Provably non-stiff implementation of weak coupling conditions for hyperbolic problemsNumerische Mathematik10.1007/s00211-021-01263-y150:2(551-589)Online publication date: 1-Feb-2022
https://dl.acm.org/doi/10.1007/s00211-021-01263-y
Hanindhito BLi RGourounas DFathi AGovil KTrenev DGerstlauer AJohn L(2021)Wave-PIM: Accelerating Wave Simulation Using Processing-in-MemoryProceedings of the 50th International Conference on Parallel Processing10.1145/3472456.3472512(1-11)Online publication date: 9-Aug-2021
https://dl.acm.org/doi/10.1145/3472456.3472512
Krenz LUphoff CUlrich TGabriel AAbrahams LDunham EBader Mde Supinski BHall MGamblin T(2021)3D acoustic-elastic coupling with gravityProceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis10.1145/3458817.3476173(1-14)Online publication date: 14-Nov-2021
https://dl.acm.org/doi/10.1145/3458817.3476173
Kopriva DGassner G(2021)A Split-form, Stable CG/DG-SEM for Wave Propagation Modeled by Linear Hyperbolic SystemsJournal of Scientific Computing10.1007/s10915-021-01618-589:1Online publication date: 1-Oct-2021
https://dl.acm.org/doi/10.1007/s10915-021-01618-5
Duru KRannabauer LGabriel AIgel H(2021)A New Discontinuous Galerkin Method for Elastic Waves with Physically Motivated Numerical FluxesJournal of Scientific Computing10.1007/s10915-021-01565-188:3Online publication date: 1-Sep-2021
https://dl.acm.org/doi/10.1007/s10915-021-01565-1
Show More Cited By

View Options

View options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents