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| Titel |
A high order discontinuous Galerkin method for elastic wave propagation in arbitrary heterogeneous media |
| VerfasserIn |
Nathalie Glinsky, Diego Mercerat |
| Konferenz |
EGU General Assembly 2013
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 15 (2013) |
| Datensatznummer |
250081437
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| Zusammenfassung |
| High-order numerical methods allow accurate simulations of ground motion using
unstructured and relatively coarse meshes. In realistic media (sedimentary basins for
example), we have to include strong variations of the material properties. For such
configurations, the hypothesis that material properties are set constant in each element of the
mesh can be a severe limitation since we need to use very fine meshes resulting in very small
time steps for explicit time integration schemes. Moreover, smooth models are approximated
by piecewise constant materials. For these reasons, we present an improvement of a nodal
discontinuous Galerkin method (DG) allowing non constant material properties in
the elements of the mesh for a better approximation of arbitrary heterogeneous
media.
We consider an isotropic, linearly elastic two-dimensional medium (characterized by Ï, λ
and μ) and solve the first-order velocity-stress system. As the stress tensor is symmetrical, let
W-ă = (-ăV,-ăÏă)t contain the velocity vector -ăV = (vx,vy)t and the stress components
-ăÏă = (Ïăxx,Ïăyy,Ïăxy)t, then, the system writes
-t -ăW + Ax (Ï,λ,μ) -xW-ă + Ay(Ï,λ,μ) -yW -ă = 0,
where Ax and Ay are 5x5 matrices depending of the material properties.
We apply a discontinuous Galerkin method based on centered fluxes and a leap-frog time
scheme to this system. We consider a bounded polyhedral domain discretized by triangles.
The approximation of W-ă is defined locally on each element by considering the Lagrange
nodal interpolants.
The system is multiplied by a test function φt and integrated on each element Ti. To
avoid computing extra terms, related to the variable properties within Ti, we introduce a
change of variables on the stress components
( )t
-ăÏă = (Ïăxx,Ïăyy,Ïăxy)t - -ăËÏă = 1(Ïăxx + Ïăyy), 1 (Ïăxx - Ïăyy),Ïăxy
2 2
which allows writing the system in a pseudo-conservative form in the variable -ăWË = (-ăV,-ăËÏă)t
Î (Ï,λ,μ) -tW -ăË + ËAx -x -ăËW + AËy -y -ăËW = 0 ,
where the constant matrices Ãx and Ãy do not depend anymore on the material properties and
Î is a diagonal matrix Î(Ï,λ,μ) = diag( -1- -1 1)
Ï,Ï,λ+μ,μ ,μ. Then, the introduction of non
constant material properties inside a triangle Ti is simply realised by the calculation, via
quadrature formulae, of a modified local mass matrix depending on the material properties
and approximating the integral on Ti
-«
φtiÎ (Ï,λ,μ)-tW-ă dV .
Ti
The method is applied to several numerical examples including smooth velocity
variations and a strong jump of the material properties and results in a clear improvement in
accuracy and CPU time when compared to the initial DG method. |
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