The Regularization Method With Application To Finite Element Model Updating

The structural dynamics model updating has become increasingly important in the structural dynamics field. Because of the modeling errors, an analytical model obtained by finite element method(FEM) must be corrected by test data from the actual structure into a reliable finite element model. The model updating method can be grouped into two major types: direct matrix methods and the sensitivity-based model updating methods, which often need to solve large-scale ill-posed linear systems. For these systems, the test data contaminated by noise may rarely lead to a physically meaningful updated model.This dissertation is to study the theories and applications of regularization method. The effect of the Tikhonov method and TSVD method is verified in model updating. For large ill-posed problems the computation of SVD decomposition is not feasible.So the following regularization methods are discussed.For lower-scale linear system, the regularized RRQR method is presented and applied to model updating. The choice of regularization parameter based on the Generalized Cross-Validation (GCV) function or L-curve is given and the perturbation errors is also analyzed. Numerical examples show that the updating result of the regularized RRQR method is superior to that of QR method and the minimum-norm method even if the test data are contaminated with additive random noise. It also demonstrates that the minimum-norm least squares solution sometimes produces an erroneous model.For the large-scale problem, some regularized Krylov subspace methods are presented.The regularized LQSR methods based on semi-convergence and Tikhonov method are presented to solve the over- and underdetermined linear systems. The regularization parameters and the convergence of these methods are discussed. The numerical results show that the FEM updated by them is more superior to that of minimum-norm least squares solution.Based on semi-convergence, the regularized FOM method combined with the residual smoothing technique and hybrid method are presented and applied to model updating. The numerical results show that the instability of computation can be overcame and the updated FE model are more superior to that of minimum-norm least squares solution.The regularized QMR and BiCG methods are presented. Using L-curve and GCV function , the regularization parameters are determined. The methods are also applied to model updating. The results show that the FE model corrected by L-curve method is more reliable than that of GCV method and more superior to that of minimum-norm least squares solution.The regularized methods presented in this paper may also be applied to other application fields and have greater significance.