An iterative method for solving finite element model updating problems

Updating finite element models using measured data is a challenging problem in the area of structural dynamics. The model updating process requires that the updated model can reproduce a given set of measured data by replacing the corresponding ones from the original model, and preserves the symmetry of the original model. The finite element model updating problems can be mathematically formulated as following two problems. Problem 1: Given M(a) is an element of R(nxn), Lambda = diag{lambda 1,..., lambda(p)} is an element of C(pxp), X = [x(1),...,x(p)] is an element of C(nxp), where p < n and both Lambda and X are closed under complex conjugation in the sense that lambda(2j) = <(lambda)over bar>(2j-1) is an element of C, x(2j) = (x) over bar (2j-1) is an element of C(n) for j = 1,...,l, and lambda(k) is an element of R, x(k) is an element of R(n) for k = 2l + 1,...,p, find real-valued symmetric matrices D and K such that M(0)X Lambda(2) + DX Lambda + KX = 0. Problem 2: Given real-valued symmetric matrices D(a), K(a) is an element of R(nxn), find ((D) over cap, (K) over cap) is an element of S(E) such that parallel to(D) over cap - D(a)parallel to(2) + parallel to(K) over cap - K(a)parallel to = min((D,K)is an element of SE) (parallel to D - D(a)parallel to + parallel to K - K(a)parallel to(2)), where S(E) is the solution set of Problem 1 and parallel to . parallel to is the Frobenius norm. This paper presents an iterative method to solve Problems 1 and 2. By the method, a symmetric solution pair can be obtained within finite iteration steps in the absence of round errors, and the minimum Frobenius norm symmetric solution pair can be obtained by choosing a special kind of initial matrix pair. Moreover, the optimal approximation solution ((D) over cap, (K) over cap) of Problem 2 can be obtained by finding the minimum Frobenius norm symmetric solution pair of a changed Problem 1. Numerical examples show that the introduced iterative algorithm is quite efficient. (C) 2010 Elsevier Inc. All rights reserved.