# Diagonal plus low rank approximation of matrices for solving modal frequency response problems

## Abstract

If a structure is composed mainly of one material but contains a small amount of a second material, and if these two materials have significantly different levels of structural damping, this can increase the cost of solving the modal frequency response problem substantially. Even if the rank of the contribution to the finite element structural damping matrix from the second material is very low, the matrix becomes fully populated when transformed to the modal representation. As a result, the complex-valued modal matrix that represents the structure’s stiffness and structural damping is both full rank, because of the diagonal part contributed by the stiffness, and fully populated, because of off-diagonal imaginary terms contributed by the second material’s structural damping. Solving the modal frequency response problem at many frequencies requires either the factorization of a coefficient matrix at every frequency, or the solution of a complex symmetric eigenvalue problem associated with the modal stiffness/structural damping matrix. The cost of both of these approaches is proportional to the cube of the number of modes included in the analysis. This cost could be reduced greatly if the damping properties of the structure were handled carefully in modeling the structure, but in practical computation of the modal frequency response, the information that could potentially reduce the computational cost is often unavailable. This thesis explores the possibilities of obtaining a representation of the complex modal stiffness/structural damping matrix as a diagonal matrix plus a matrix of minimal rank. An algorithm for computing a “diagonal plus low rank” (DPLR) representation is developed, along with an iterative algorithm for using an inexact DPLR approximation in the solution of the modal frequency response problem. The behavior of these algorithms is investigated on several example problems.