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 $({\mathbf{C}}_0,{\rho }_0)$ and $({\mathbf{C}}_1,{\rho }_1)$ the background and the perturbed medium parameters, respectively. We consider two cases of perturbations, ${\mathbf{C}}_0={\mathbf{C}}_1,{\rho }_1\ne {\rho }_0$ (case 1) and ${\mathbf{C}}_0\ne {\mathbf{C}}_1,{\rho }_1={\rho }_0$ (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.

本文研究了弹性波散射的内部传递特征值问题。我们旨在通过有效的数值算法来显示正特征值的分布。在这里，弹性波被介质参数的扰动散射，介质参数包括弹性张量C和密度ρ。让我们来表示$（{\mathbf{C}}_0，{\rho }_0）$ 和 $（{\mathbf{C}}_{\mathrm{1个}}，{\rho }_{\mathrm{1个}}）$背景和扰动的介质参数。我们考虑两种情况的摄动，${\mathbf{C}}_0={\mathbf{C}}_{\mathrm{1个}}，{\rho }_{\mathrm{1个}}\ne {\rho }_0$ （案例1）和 ${\mathbf{C}}_0\ne {\mathbf{C}}_{\mathrm{1个}}，{\rho }_{\mathrm{1个}}={\rho }_0$（情况2）。在通过FEM离散化关联的PDE之后，我们面临着具有大尺寸矩阵的广义特征值问题（GEP）的计算。这些GEP包含大量非物理零（对于情况1）或非物理无穷大（对于情况2）。为了有效地定位数百个正特征值，我们然后将GEP转换为合适的二次特征值问题（QEP）。然后，我们结合部分锁定或部分放气技术实施二次Jacobi-Davidson方法，以计算500个正特征值。