A stable discontinuous Galerkin method for linear elastodynamics in 3D geometrically complex elastic solids using physics based numerical fluxes

Kenneth Duru, Leonhard Rannabauer, Alice-Agnes Gabriel, On Ki Angel Ling, Heiner Igel, Michael Bader

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Time-stable, high order accurate and explicit numerical methods are effective for hyperbolic wave propagation problems. As a result of the complexities of real geometries, internal interfaces and nonlinear boundary and interface conditions, discontinuities and sharp wave fronts may become fundamental features of the solution. Therefore, geometrically flexible and adaptive numerical algorithms are crucial for high fidelity and efficient simulations of wave phenomena in many applications. Adaptive curvilinear meshes hold promise to minimise the effort to represent complicated geometries or heterogeneous material data avoiding the bottleneck of feature-preserving meshing. To enable the design of stable DG methods on three space dimensional (3D) curvilinear elements we construct a structure preserving skew-symmetric coordinate transformation motivated by the underlying physics. Using a physics-based numerical penalty-flux, we develop a 3D provably energy-stable discontinuous Galerkin finite element approximation of the elastic wave equation in geometrically complex and heterogeneous media. By construction, our numerical flux is upwind and yields a discrete energy estimate analogous to the continuous energy estimate. The ability to treat conforming and non-conforming curvilinear elements allows for flexible adaptive mesh refinement strategies. The numerical scheme has been implemented in ExaHyPE, a simulation engine for parallel dynamically adaptive simulations of wave problems on adaptive Cartesian meshes. We present 3D numerical experiments of wave propagation in heterogeneous isotropic and anisotropic elastic solids demonstrating stability and high order accuracy. We demonstrate the potential of our approach for computational seismology in a regional wave propagation scenario in a geologically constrained 3D model including the geometrically complex free-surface topography of Mount Zugspitze, Germany.
Original languageEnglish (US)
Pages (from-to)114386
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Dec 10 2021
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2022-05-25
Acknowledged KAUST grant number(s): ORS-2017-CRG6 3389.02
Acknowledgements: We thank the anonymous reviewer for constructive criticism which significantly improved the clarity and quality of the manuscript and Editor J. Tinsley Oden for guiding the review. The work presented in this paper was enabled by funding from the European Union's Horizon 2020 research and innovation program under grant agreements no. 671698 (ExaHyPE), no. 823844 (ChEESE) and no. 852992 (TEAR). The authors also acknowledge support by the German Research Foundation (DFG) (Grant Nos. GA 2465/2-1, GA 2465/3-1), by KAUST-CRG (Grant No. ORS-2017-CRG6 3389.02) and by KONWIHR (project NewWave). A.L. is supported by the Swiss Federal Institute of Technology grant (project ETH-10 17–2). Computing resources were provided by the Institute of Geophysics of LMU Munich, Germany [59], the Leibniz Supercomputing Centre, Germany (SuperMUC-NG project pr63qo) and the KAUST Shaheen Supercomputing Laboratory (project k1488). The first author KD would like to thank Dimitri Komatitsch (1970–2019) for his help, fruitful discussions and for providing the analytical solution for elastic surface waves in an anisotropic medium.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • General Physics and Astronomy
  • Mechanics of Materials
  • Mechanical Engineering
  • Computational Mechanics
  • Computer Science Applications


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