We study the interior transmission eigenvalue problem for the elastic wave scattering in this paper. We aim to show the distribution of positive eigenvalues by efficient numerical algorithms. Here the elastic waves are scattered by the perturbations of medium parameters, which include the elasticity tensor C and the density ρ. Let us denote (C0,ρ0) and (C1,ρ1) the background and the perturbed medium parameters, respectively. We consider two cases of perturbations, C0=C1,ρ1≠ρ0 (case 1) and C0≠C1,ρ1=ρ0 (case 2). After discretizing the associated PDEs by FEM, we are facing the computation of generalized eigenvalues problems (GEP) with matrices of large size. These GEPs contain huge number of nonphysical zeros (for case 1) or nonphysical infinities (for case 2). In order to locate several hundred positive eigenvalues effectively, we then convert GEPs to suitable quadratic eigenvalues problems (QEP). We then implement a quadratic Jacobi-Davidson method combining with partial locking or partial deflation techniques to compute 500 positive eigenvalues.
- Elastic waves
- Generalized eigenvalue problems
- Interior transmission eigenvalues
- Quadratic eigenvalue problems
- Quadratic Jacobi-Davidson method