Section:
New Results
Discontinuous Galerkin methods for the
elastodynamic equations
DGTD- method for the elastodynamic equations
Participants :
Nathalie Glinsky, Fabien Peyrusse.
We continue developing high order non-dissipative discontinuous
Galerkin methods on simplicial meshes for the numerical solution of
the first order hyperbolic linear system of elastodynamic equations.
These methods share some ingredients of the DGTD-
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. Recent results concern two
particular points.
The first novelty is the extension of the DGTD- method
initially introduced in [5] to the numerical
treatment of viscoelastic attenuation. For this, the velocity-stress
first order system is completed by additional equations for the
anelastic functions describing the strain history of the
material. These additional equations result from the rheological model
of the generalized Maxwell body and permit the incorporation of
realistic attenuation properties of viscoelastic material accounting
for the behaviour of elastic solids and viscous fluids. In practice,
one needs to add 3L additional equations in 2D and 6L in 3D, where L
is the number of relaxation mechanisms of the generalized Maxwell
body. This method has been implemented in 2D and validated thanks to
comparisons with a FDTD method.
The second contribution is concerned with the numerical assessment of
site effects especially topographic effects. The study of
measurements and experimental records proved that seismic waves can be
amplified at some particular locations of a topography. Numerical
simulations are exploited here to understand further and explain this
phenomenon. The DGTD- method has been applied to a
realistic topography of Rognes area (where the Provence earthquake
occured in 1909) to model the observed amplification and the
associated frequency. Moreover, the results obtained on several
homogeneous and heterogeneous configurations prove the influence of
the medium in-depth geometry on the amplifications measures at the
surface [26] , [25] .