Consistent Regularization for Damage Detection with Noise and Model Errors

Abstract

NUMBER of methods have been developed for damage detection in the last two decades. Many of the methods are based on the observation of dynamic behavior of a structure (Doebling et al. [1], Brincker et al. [2], Maeck and DeRoeck [3]), among which the sensitivity approach via a model-updating technique is commonly accepted and applied extensively in the engineering industry because of its clear mathematical background andquantitativeindications.However,thistypeofmethodisweakat accommodating the influence of measurement errors, leading to illconditioned problems, as demonstrated by Friswell et al. [4] and Humar et al. [5]. This shortcoming means that the existence and uniqueness of the solution is not ensured and numerical instability is likely to take place in the course of the solution process [6,7]. Investigations have since been conducted to deal with the illconditioned problems in model updating. Hansen [8,9] and Vogel [10] proposed regularization methods for obtaining a solution of the inverse problem. It is recognized in regularization theory that the conventional output error, which is usually the vector of differences between thecomputedandmeasuredresponses,canbemadeunrealistically small if the process of damage identification is allowed to behave badly, such that the variable has arbitrarily large deviations from the true set of parameter change, or there may be infinite sets of ill-posed solutions. A stable solution scheme can be achieved by imposing certain constraints in the form of added penalty terms with adjustable weighting parameters based on posterior knowledge. Recently,TiturusandFriswell[11]presentedthesensitivity-based model-updating method with an additional regularization criterion and computed the solutions based on the generalized singular value decomposition.Specificfeaturesoftheparameterandresponsepaths were explored when the regularization parameter varies. The four different types of spaces that arise in the solution were discussed, together with the characteristics of the families of the regularized solution. Weber et al. [12] applied the Tikhonov regularization and truncated singular value decomposition consistently to a nonlinear updating problem. Line-search and stopping criteria known from numericaloptimizationwereadaptedtotheregularizedproblem.The optimal regularization parameter was determined by generalized cross-validation. From experiences gained in model updating with laboratory test structures, the authors found that Tikhonov regularization can give the optimal solution when there is no noise or very small noise in the measurement. But in the updating procedure for a nonlinear inverse problem with the inclusion of noise and model error, the signal-tonoise ratio is getting smaller [13,14] with each iteration, and the solution obtained from a poor regularization parameter is usually unacceptable. The algorithm does not accurately converge, and the results depend strongly on the convergence criteria and tolerances. In this paper, two techniques are proposed in a new regularization method for the identification of local damages in a structure. One technique proposed a new side condition that classified the elements aspossiblydamagedandundamagedelements,whichwillbetreated differently later. The other technique restricts the range of variation of the regularization parameter, such that the regularization solution will be in a realistic range, and the correct optimal point from the curvature of the L-curve is consistently chosen to ensure a continuousconvergingprocess.Bothtechniquesmakefulluseofthe information from results obtained in previous iteration steps. A plane frame structure is studied with two damaged elements and different levels of noise and model errors to illustrate the application of the proposed method. Numerical results show that the proposed consistent regularization method is very effective at improving the results in the inverse problem with ill-conditioning, compared with the conventional Tikhonov regularization.