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Section: New Results

Discontinuous Galerkin methods for the elastodynamic equations

DGTD-Im1 $\#8473 _p$ method for the elastodynamic equations

Participants : Nathalie Glinsky-Olivier, Loula Fezoui.

We continue developing high order non-dissipative discontinuous Galerkin methods on simplicial meshes (triangles in the 2D case and tetrahedra in the 3D case) for the numerical solution of the first order hyperbolic linear system of elastodynamic equations. These methods share some ingredients of the DGTD-Im1 $\#8473 _p$ methods developed by the team for the time domain Maxwell equations among which, the use of nodal polynomial (Lagrange type) basis functions, a second order leap-frog time integration scheme and a centered scheme for the evaluation of the numerical flux at the interface between neighboring elements. The resulting DGTD-Im1 $\#8473 _p$ methods have been validated and evaluated in detail in the context of propagation problems in both homogeneous and heterogeneous media including problems for which analytical solutions can be computed. Particular attention was given to the study of the mathematical properties of these schemes such as stability, convergence and dispersion.

In the 2D case, the source modeling has been studied via the Garvin test case i.e the propagation of an explosive source in a half-space with a free surface. A class of high order leap-frog schemes has also been studied. These schemes improve the accuracy of the highest orders spatial schemes (for p$ \ge$3 ) while being efficient since they allow the use of larger time steps as compared to the DGTD-Im1 $\#8473 _p$ method based on the second order leap-frog scheme.

Moreover, a preliminary study of site effects has been realised on a realistic topography of the Rognes area (south of France) where one of the strongest historical earthquake occured in 1909. The objective of this study was to analyze the seismic response at the surface by measuring amplification and the concerned frequency range when considering a 2D profile (sourth-north profile with real topographic data) subject to a vertical P plane wave of central frequency varying from 0.2 to 10.0 Hz. The first results are encouraging and an extension of such a study to the three dimensional case including real source signals is underway and will permit a comparison with real seismic recordings (from the data base of CETE Méditérannée, Nice).

In the 3D case, in the framework of the ANR QSHA project, canonical problems are studied such as semi-spherical or ellipsoidal canyon/basin in order to compare results of several numerical methods. More realistic test cases are examined via our participation to the Euroseistest Numerical Benchmark initiative. The objective of this benchmark, organized in the framework of the Cashima project (CEA Cadarache, the LGIT in Grenoble and Aristotle University of Thessaloniki), is to perform simulations of real events on the Volvi area (a well documented region near Thessaloniki) including complex characteristics of the medium.


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