An eigensolution strategy for large systems

Abstract

A solution strategy is presented for the evaluation of frequencies and mode shapes for very large structural systems. The subspace iteration method is modified to calculate the eigenpairs in groups near different shift points. If the bandwidth is large and only a few vectors are required, the cost of factorization of the stiffness matrix will dominate and the standard subspace iteration method is effective. However, for the case where the bandwidth is of comparable size to the number of eigenvectors required, there is significant advantage to shifting and evaluating the eigenpairs in groups. It is demonstrated that a good approximation for the optimum number of iteration vectors is given by the square root of the bandwidth. Therefore, if 80 eigenpairs are required of a system with a bandwidth of 400, 20 iteration vectors would be used and the eigenpairs would be found in approximate groups of 10 near 8 shift points. In addition, for the solution of a special class of dynamic response problems, it has been shown that a direct superposition of Ritz vectors yields a more accurate solution than a superposition of the exact eigenvectors. Since this approach eliminates the need to solve for the exact eigenpairs the numerical algorithm, which automatically generates the series of orthogonal Ritz vectors, is presented as an alternative to the solution for the exert eigenpairs. The method does produce an approximate set of frequencies which are excited by a specified load pattern.