An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics

We present an automated multilevel substructuring (AMLS) method for eigenvalue computations in linear elastodynamics in a variational and algebraic setting. AMLS first recursively partitions the domain of the PDE into a hierarchy of subdomains. Then AMLS recursively generates a subspace for approximating the eigenvectors associated with the smallest eigenvalues by computing partial eigensolutions associated with the subdomains and the interfaces between them. We remark that although we present AMLS for linear elastodynamics, our formulation is abstract and applies to generic H1-elliptic bilinear forms. In the variational formulation, we define an interface mass operator that is consistent with the treatment of elastic properties by the familiar Steklov--Poincare operator. With this interface mass operator, all of the subdomain and interface eigenvalue problems in AMLS become orthogonal projections of the global eigenvalue problem onto a hierarchy of subspaces. Convergence of AMLS is determined in the continuous setting by the truncation of these eigenspaces, independent of other discretization schemes. The goal of AMLS, in the algebraic setting, is to achieve a high level of dimensional reduction, locally and inexpensively, while balancing the errors associated with truncation and the finite element discretization. This is accomplished by matching the mesh-independent AMLS truncation error with the finite element discretization error. Our report ends with numerical experiments that demonstrate the effectiveness of AMLS on a model problem and an industrial problem.