Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields. The singularities are described by eigenpairs of a corresponding operator pencil on a subdomain of the sphere. The solution approach is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitian form. This eigenvalue problem is discretized by the finite element method. The resulting quadratic matrix eigenvalue problem is then solved with the Skew Hamiltonian Implicitly Restarted Arnoldi method (SHIRA) which is specifically adapted to the structure of this problem. Some numerical examples are given that show the performance of this approach.