An efficient eigensolution method and its implementation for large structural systems
The automated multilevel substructuring (AMLS) method, which was originally designed for efficient frequency response analysis, has emerged as an alternative to the shift-invert block Lanczos method  for very large finite element (FE) model eigenproblems. In AMLS, a FE model of a structure, typically having a ten million degrees of freedom, is automatically and recursively divided into more than ten thousand substructures on dozens of levels. This FE model is projected onto the substructure eigenvector subspace which typically has a dimension of 100,000. Solving the reduced eigenproblem on the substructure eigenvector subspace, however, is unmanageable for modally dense models which typically contain more than 10,000 eigenpairs. In this dissertation, a new eigensolution algorithm for the reduced eigenproblem produced by the AMLS transformation is presented for large structural systems with many eigenpairs. The new eigensolver in combination with AMLS is advantageous for solving the eigenproblems for huge FE models with many eigenpairs because it takes much less computer time and resource than any other existing eigensolvers while maintaining acceptable eigensolution accuracy. Therefore, the new eigensolution algorithm not only makes high frequency analysis possible with acceptable accuracy, but also extends the capability of solving large scale eigenvalue problems requiring many eigenpairs. A reduced eigenvalue problem produced by the AMLS transformation for a large finite element model is defined on the substructure eigenvector subspace. A new distilled subspace is obtained by defining subtrees in the substructure tree, solving subtree eigenproblems, and truncating subtree and branch substructure eigenspaces. Then the reduced eigenvalue problem on the substructure eigenvector subspace is projected onto the smaller distilled subspace, utilizing the sparsity of the stiffness and mass matrices. Using a good guess of a starting subspace on the distilled subspace, which is represented by a sparse matrix, one subspace iteration recovers as much accuracy as needed. Hence, the size of the eigenvalue problem for Rayleigh-Ritz analysis can be greatly minimized. Approximate global eigenvalues are obtained by solving the Rayleigh-Ritz eigenproblem on the refined subspace, computed by one subspace iteration, and the corresponding eigenvectors are recovered by simple matrix-matrix multiplications. For robustness of the implementation of the new eigensolution algorithm, the remedies for a nearly singular stiffness matrix and an indefinite mass matrix are presented. Also, the new eigensolution algorithm is very parallelizable. The parallel implementation of this new eigensolution algorithm for shared memory multiprocessor machines is done by using OpenMP Application Program Interface (API) for performance improvement. Timing and eigensolution accuracy of the implementation of the new eigensolution algorithm are presented, compared with the results from the block Lanczos eigensolver in the commercial software MSC.Nastran. In addition to the new eigensolution algorithm, a new method for an augmented eigenproblem for residual flexibility is developed to mitigate loss of accuracy by paying little computational cost in modal frequency response analysis.