Efficient methods for eigenvalue and dynamic response analysis of structures

Abstract

This dissertation proposes an efficient eigenvalue solution method for structures by improving Lanczos method. An improved Lanczos vector superposition method for efficient dynamic response analysis of structures is also proposed in this disseration. The Lanczos method is an efficient eigenvalue solution method. In the field of quantum physics, the modified Lanczos algorithm using the technique of matrix power was presented to more efficiently obtain the eigenstate of quantum systems. Similar power technique was also applied to the subspace iteration method. However, the matrix-powered algorithm has not been applied to the Lanczos method for structural eigenproblems. This dissertation proposes an improved Lanczos method for eigenvalue analysis of structures by applying the power of the dynamic matrix. The convergence of proposed matrix-powered Lanczos method is better than that of the conventional Lanczos method because the matrix-powered Lanczos algorithm can reduce the number of required Lanczos vectors. The number of operations of proposed method is also smaller than that of the conventional method. However, in some cases, high power value of the dynamic matrix causes numerical instability. Therefore, special care must be taken in the selection of power value. By analyzing four numerical examples such as a simple spring-mass system, a plane frame structure, a three-dimensional frame structure and a three-dimensional building structure, the efficiency of the proposed matrix-powered Lanczos method is verified and the suitable power value of the dynamic matrix is presented. The proposed matrix-powered Lanczos method is also compared with the matrix-powered subspace iteration method through numerical examples. The Lanczos vector superposition method is efficient in the dynamic response analysis of structures. However, it has some drawback. For example, if multi-input loads such as moving loads on bridges, winds acting on high-rise buildings and wave forces appl...