Abstract
Designing efficient and stable algorithm for finding the eigenvalues and eigenvectors is very important from the static as well as the dynamic aspect in coupled problems. Modal analysis requires first few significant eigenvectors and eigenvalues while direct integration requires the highest value to ascertain the length of the time step that satisfies the stability condition. The paper first presents the modification of the well known WYD method for a solution of single field problems: an efficient and numerically stable algorithm for computing eigenvalues and the corresponding eigenvectors. The modification is based on the special choice of the starting vector. The starting vector is the static solution of displacements for the applied load, defined as the product of the mass matrix and the unit displacement vector. The starting vector is very close to the theoretical solution, which is important in cases of small subspaces. Additionally, the paper briefly presents the adopted formulation for solving the fluid-structure coupled systems problems which is based on a separate solution for each field. Individual fields (fluid and structure) are solved independently, taking in consideration the interaction information transfer between them at every stage of the iterative solution process. The assessment of eigenvalues and eigenvectors for multiple fields is also presented. This eigen problem is more complicated than the one for the ordinary structural analysis, as the formulation produces non-symmetrical matrices. Finally, a numerical example for the eigen solution coupled fluidstructure problem is presented to show the efficiency and the accuracy of the developed algorithm.
Highlights
In dynamic analysis of structures such as: dams, underwater structures, shore and off shore structures and similar, it is necessary to simulate the fluid-structure interaction to ascertain the real behaviour of such a complex system
As presented the modified WYD method can be applied for solving large eigen tasks in coupled fluid/structure problems, for a small number of eigenvalues and vectors
In cases of system symmetry or if identical properties occur in two orthogonal directions, corresponding paired solutions are produced as a single solution
Summary
In dynamic analysis of structures such as: dams, underwater structures, shore and off shore structures and similar, it is necessary to simulate the fluid-structure interaction to ascertain the real behaviour of such a complex system. The greatest need for efficient and effective algorithms exists in the area of solving large eigen problems, those of the order of several thousand degrees of freedom and greater, such as coupled problems. To date many procedures have been developed for computing partial eigen solutions of such large eigen problems [12], for example: simultaneous iteration method, subspace iteration method, determinant search method, Lanczos method, Ritz vector method etc. Each of these methods have their advantages and disadvantages. The following is a presentation of the modified WYD method which is essentially a variant of the Ritz vector method [13]
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