The Research Of Some Problems In Eigensensitivity Computation And Frequency Response Analysis

Eigensensitivity analysis and frequency response analysis is an active branch inscience and engineering. It can be applied to the fields such as model modification,damage identification and structural optimization, and other fields. With thedevelopment of computer technology and finite element method, related research isbeing considered by many authors. Under such engineering background andrequirement, we study some problems in eigensensitivity analysis and frequencyresponse analysis.For symmetric undamped system, we present the preconditioned conjugate gradientmethod for computing eigensensitivity of both cases of distinct and repeatedeigenvalues. For proportional damping system, we establish two new model orderreduction methods for computing frequency responses, which are based onpreconditioned conjugate gradients and Ritz vectors, respectively. The mainadvantages of new methods in this dissertation can be summarized as follows:computational effort is small; required storage is small; convergence is rapid; thesymmetry and sparse property of the original coefficient matrices is kept; algorithm iseasy to implement and the accuracy of approximate solutions can be adaptively bemonitored. Numerical examples illustrate the efficiency of new methods.This dissertation consists of five parts. In chapter1, we introduce the engineering background of eigensensitivity analysis and frequency response analysis. In chapter2, the PCG method and Nelson-type method for solving the eigensensitivity problem, which include the case of distinct and repeated eienvalues, are first reviewed. The preconditioned conjugate gradient method for computing eigensensitivity of both cases of distinct and repeated eigenvalues are then presented. Comparison of Nelson-type method with the proposed PCG method both in efficiency and in accuracy is also completed in this chapter. In chapters3and4, two new model order reduction methods for solving the frequency response problem are established, which deals with the lower and middle frequency sweep bands, respectively. We also compare brute force approach method with present methods both in efficiency and in accuracy. In chapter5, we summary the new methods proposed in this paper and indicate some problems that should further be studied.1. Eigensensitivity computation for undamped symmetric system byusing preconditioned conjugate gradient methodConsider the following undamped symmetric eigenvalue problem (?) where K and M are the stiffness and mass matrices, respectively; λi and φi are the ith eigenvalue and corresponding eigenvector, respectively; n is number of the total degrees of freedom and δtj is the Kronecker delta symbols. The paper is mainly concerned with the derivatives of eigenvalues and eigenvectors at p=p0. For convenience, p0is omitted for variable evaluated at p=p0.The key problem in calculating the eigenvector derivative is to compute the particular solution. The methods for computing the eigenvector’s derivatives with distinct and repeated eigenvalues are presented as follows.1) The eigensensitivity computation for distinct eigenvaluesSuppose the stiffness matrix K and mass matrix M are differentiable. We differentiate both sides of (1) with respect to the design parameter p at p=p0to obtain whereBased on Fox’s modal superposition method, the particular solution vi can be expressed as The coefficient c, can be written as Since we have some lower eigenpairs only, the particular solution vi can be expressed as vi=Φci+vi(6) For computing the unknown part vi of the particular solution, we use the PCG method to solve the following modified equation where σ≤0and the preconditioner is chosen as K-σM. The effectiveness of the proposed method is verified by numerical examples. 2) The eigensensitivity computation for repeated eigenvaluesIt is assumed that,when the design parameter at p=p0,the eigenvalue problem in Eq.(1)has a m(1<m≤l<<n) repeated eigenvalues, which is denoted as λs+1=λs+2=…=λs+m=λ0.The eigenvalues of the eigenvalue problem are ordered as follows:0≤λ1≤λ2≤…≤λs<λs+1=…=λs+m<λs+m+1≤…≤λl≤λl-1≤…≤λn. Since the coefficient matrix Fi=K-λiM is singular with rank n-m,based on the modal superposition method,the particular solution of the eigenvector derivative can be expressed as where λj-λ0The form of fi is similar with Eq.(3),the unknown part of the particular solution can be written as vi=Φwi,so we haveFor computing vii,in a way similar to the case of the distinct eigenvalue,we use the PCG method to solve the following modified equation where σ≤0and the preconditioner is chosen as K-σM.The effectiveness of the proposed method is verified by numerical examples. 2. Model order reduction methods for computing frequency response problem with proportional damping systemA multi-degree-of-freedom structural system subjected to a time-harmonic external force can be written aswhere F=F1+iF2is the complex vector of applied force amplitudes, F1and F2are real vectors; a dot denotes differentiation with respect to time t; M, C and K are large and sparse n×n mass, damping and stiffness matrices, respectively. In this paper, Rayleigh damping is considered, where α,β are two real scalars that may depend on co.Assume that X(t)=xeiωt is the solution of Eq.(12), its substitution into Eq.(12) yieldsLet x1and x2satisfy the following equationswhere x1and x2are complex vectors. Once solutions x1and x2to equations (15) are achieved, x=x1+ix2is then the solution to Eq.(14).We will solve the first equation in equations (15) for x1as an example, x2can be solved by the same method.For different frequency sweep band, we will propose two different new model order reduction methods:one is based on preconditioned conjugate gradients, the other is based on Ritz vectors. 1) An model order reduction method based on preconditioned conjugate gradientsConsider the frequency sweep band [0,ωR]. Based on the modal superposition method, x1in equations (15) can be expressed as We start from the undamped system equation (17) can be modified as Let ω=ωR, we solve the Eq.(18) by using the PCG method and obtain the conjugate gradient directions Then we use them as the basis of the projection subspace and get In the same method, x2can be obtained. The frequency response x=x1+ix2is finally achieved. The effectiveness of the proposed method is verified by numerical examples.2) An model order reduction method based on Ritz vectors Consider the frequency sweep range Ritz vector can be obtained by solving the equation where More Ritz vectors can be obtained by the iterative procedure The vectors are orthogonalized and normalized at each step by the use of the following procedure: where Since after each vector is found, it is normalized as to get eigenvectors of the reduced system Γ=[γ1,…,γk], where γi(||γi||2=1), i=1,2,…, k. The final set of orthogonalized Ritz vectors is created fromFinally, the approximate solution x1in Eq.(15) can be written as Using the same method, one may obtain x2. The finial frequency response is x=x1+ix2. The effectiveness of the proposed method is verified by numerical examples.