Absorbing Boundary Conditions for 2D Tilted Transverse Isotropic elastic media

Abstract

Abstract. This work deals with the construction of a low-order absorbing boundary condition (ABC) for 2D elliptic TTI media, preserving the system stability. The construction is based on comparing and then connecting the slowness curves for isotropic and elliptic TTI waves. Numerical experiments illustrate the performance of the new ABC. They are performed by integrating the ABC in a DG formulation of Elastodynamics. When applied in a TTI medium, the new ABC performs well with the same level of accuracy than the standard isotropic ABC set in an isotropic medium. The condition demonstrates also a good robustness when applied for large times of simulation.Résumé. Ce travail porte sur la construction d'une condition aux limites absorbante (CLA) pour les milieux TTI elliptiques, préservant la stabilité du système. La mise en oeuvre repose sur la comparaison puis la mise en relation des courbes de lenteurs pour les ondes isotropes et TTI elliptiques. Des expériences numériques illustrent la performance de ces nouvelles CLA. Elles sont réalisées en intégrant la CLA dans une formulation de type Galerkin discontinue pour l'Élastodynamique. Pour un milieu TTI, la nouvelle CLA est efficace avec le même niveau de précision que la CLA isotrope standard pour un milieu isotrope. On observe aussi que la condition est robuste en temps long.

Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problems

“…,Ω + (Cε(u), ε(u)) and by Korn's inequality we conclude that b 0 is coercive on H 1 (Ω) 3 . Consequently, problem (48) is well-posed in H 1 (Ω) 3 .…”

“…While our analysis applies to a wider range of problems, it is currently limited in the sense that the constants C 1 and C 2 we obtain in (1) depends on p. As a result, we recover the same result for a fixed p: in particular (2) and (3). However, our analysis does not cover varying p, and thus, a result like (4) is out of the scope of the present article.…”

“…In the following we assume that the family of meshes is regular and that for each h > 0, conditions (31)- (32) are satisfied by all elements K ∈ T h . 3 Under such assumptions the next interpolation error estimates are valid.…”

“…It is probably possible to remove such hypothesis by employing quasi-interpolation operators. 3 as the singularities are local, the refinement conditions can be defined locally near each corner with different grading conditions, we avoid this specification to lighten the presentation Let us point out that in the case of smooth C γ+1,1 subdomains Ωq, it is not required to perform any kind of refinement.…”

“…The elasticity properties of the material are represented by a constant second-order tensor C = (c nmpq ) 3 n,m,p,q=1 satisfying the following symmetry and ellipticity properties c nmpq = c mnpq = c pqnm , 1 ≤ n, m, p, q ≤ 3, and…”