A Research Of Nonsymmetric Generalized Eigenvalue And An Application In Flow In A Collapsible Channel

Large scale eigenvalue problems are of increasing importance in scientific and engineering computing. Considerable progress has been made over past several decades towards the numerical solution of large scale nonsymmetric problems,for example, Subspace iteration method,ABLE method,QZ method,Arnoldi method and so on. In most cases, we may not care all eigenvalues, but some eigenvalues which you need in your problems, such as the eigenvalues that have largest (smallest) real part, the eigenvalues that have largest(smallest) magnitude, or the eigenvalues that have largest(smallest) imaginary part.Flow in collapsible channel has a number of application in physiological flows and medical divices. The model of flow in collapsible channel is so small that the elements are also small, and the stiffness matrix and the mass matrix are both large sparse matrices, and it is hard to solve this eigenproblem. Arnoldi method is a very useful method which has been used to solving some eigenvalues of a large sparse matrix, and restarting technique that has been used in it make Arnoldi method better and better, and if we solve eigenproblems by Arnoldi method, it has an advantage in the storage and computing time. So it is necessary to used Arnoldi method in the research of flow in collapsible channel.In Chapter I, it introduces the background of large scale eigenproblems and basic numerical algorithms for solving them. You may see the theory of QZ method in Chapter II and the theory of Arnoldi method in Chapter III, respectively. In Chapter IV, it has several methods about Matrix-vector Product. In Chapter V, it would be the application of IRA in the research of flow in collapsible channel.Major work in the paper: After knowing IRA and all formulas well, and using IRA code well, we announced a new matrix-vector product which can start from element matrices, and completed a code. It can also be used in solving generalized eigenproblem, whose stiffness matrix is nonsymmetric and mass matrix is symmetric and self-definite. But the mass matrix got from the eigenvalue equation can not be suitable the condition. Towards these cases we have some technique, finally we got the suitable code which can be used in research of flow in collapsible channel. This code can be also used to solve generalized eigenproblem, whose stiffness matrix and mass matrix are both nonsymmetric. The eigenproblem which has been got in before is solved by QZ method and IRA, respectively. The several eigenvalues that have largest real part is the keypoint that we care, and it has a comparison of computing time and precision between the results got by QZ method and IRA. At last by Clamda and Re that we got from keypoints ,a Clamda-Re curve has been made. After an analysis of the curve, we should know the curve has divide the plate into two part: one is stabile part and the other is instabile part.