Elasticity Mtensors and the strong ellipticity condition Abstract   In this paper, we establish two sufficient conditions for the strong ellipticity of any fourthorder elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity    M   tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourthorder elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity    M   tensor is defined with respect to the Meigenvalues of elasticity tensors. We prove that any nonsingular elasticity    M   tensor satisfies the strong ellipticity condition by employing a PerronFrobeniustype theorem for Mspectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity    M   tensors are also established.

Efficient methods of computing interior transmission eigenvalues for the elastic waves Abstract   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 xcfx81. Let us denote    (    C    0    ,    xcfx81    0    )    and    (    C    1    ,    xcfx81    1    )    the background and the perturbed medium parameters, respectively. We consider two cases of perturbations,      C    0        C    1    ,    xcfx81    1    xe2x89xa0    xcfx81    0      (case 1) and      C    0    xe2x89xa0    C    1    ,    xcfx81    1        xcfx81    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 JacobiDavidson method combining with partial locking or partial deflation techniques to compute 500 positive eigenvalues.