Parallel Computation Of Eigenvalue Problems

Dynamic structural analysis is the basic problem in the vibration theory and its applications. In this dissertation the parallel computing methods for eignevalue problems in dynamic structural analysis are studied, and the methods in constructing the parallel environment by PVM and MPI are given.First, the method of parallel algorithm for computing the stiffness matrix and the mass matrix is studied, each processor computes several element stiffness matries and the element mass matries, then the total stiffness matrix and the total mass matrix are computed.Second, the parallel subspace iterative methods for computing the generalized eigenvalue problem are studied. One method is transforming the generalized eigenvalue problem to the standard eigenvalue problem, then solve the eigenvalue problem by parallel subspace iterative method of standard eigenvalue problem, and the another method is projected the generalized eigenvalue problem into the subspace directly. The methods are used for computing the dynamic characteristic of the plane wing and the hang of guided missile, and the results of the numerical experiment in the cluster and PAR2000 show that the algorithm are very effective.Third, the parallel Davidson method, the parallel refined Davidson method and the parallel Jacobi-Davidson method for computing generalized eigenvalue problem are presented, the correction equation of the methods are preconditioned by using the Neumann series, the methods are also used for computing the dynamic characteristic of the plane wing and the hang of guided missile at the environment of IBM-P650 and the Cluster, the parallel efficient and the Speedup is very high.Forth, the parallel algorithms for computing quadratic eigenvalue problem are presented, including parallel Jacobi-Davidson method and parallel refined Jacobi-Davidson method for computing the quadratic eigenvalue problem, and the algorithms are used for computing the quadratic eigenvalue problem with proportional damping of the plane wing. The experiment results show that both of the methods are very effective, and the refined Jacobi-Davidson method is not only convergence in less iterative step, but also has higher parallel efficient and use less time. At last, the parallel subspace iterative method and the parallel refined Jacobi-Davidson method for solving the eigenvalue problem of gyroscopic system are presented. For the parallel subspace iterative method, the eigenvalue problem of gyroscopic systems is first transformed to the generalized eigenvalue problem of symmetric matrices, and then solving the eigenvalue problem by using the parallel subspace iterative method, and the order of the matries is reduced to the scale of oringinal problem in the process of computing.