13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- In Step 2, the Bayesian model updating method is further used to identify the detailed damage parameters. The damage parameters are identified only from the damage suspicious region of Step 1. There are several advantages for this new damage identification strategy. First, the number of the parameters to be identified has been reduced, which make the cost of the computation greatly drop. Second, the search targets of the damage identification only focus to the damage suspicious area, rather other the entire structure. The “output-equivalent” issue in the damage mechanism modelling can be effectively relieved, which refer to the problem that different damage assumes may produce identical out parameters [1]. 2.2. Bayesian Model Updating Framework Assume the damage could be expressed by the reduction of the element stiffness, but independent from the element mass. Therefore, introducing the parameter vector θ = [θ1, ..., θi, ..., θN], which represents the contribution of the element stiffness to the system stiffness matrix, the system stiffness matrix K can be written as: 0 1 ( ) N i i i Kθ = = +∑ K θ K (1) where N is the degrees of freedom (DOFs) of the linear discrete system, θi (0 < θi < 1) is non-dimensional, and the smaller the size of θi , the more serious the damage of element, other words, deeper the crack. The combination of adjacent damage element constitutes the shape and direction of the crack. Obviously, the accuracy of crack identification depends on the sizes of finite elements which can be controlled artificially but usually at the cost of the computation effort [5]. Based on Bayesian theorem, when given the measured data D and the probabilistic models M, the post PDF of θ can be expressed as: ( | , ) ( | , ) ( | ) p M cp M p M = θ D D θ θ (2) where c is a normalizing constant, p(θ| M) is the prior PDF of θ, and p(D| θ, M) is the likehood function. Because that the measured modal frequency usually has the more precision than the other modal parameters. Moreover, the most algorithms including the mode shapes have to deal with the problems of the finite element (FE) model reduction or mode shape expansion to bridge the gap between the real structure and the simulated model. So the modal frequencies are used to construct the likelihood function, assume the Nm (≤ N) modes of natural frequency are considered here: µ( ) ( ) µ( ) ( ) /2 1/2 1 1 1 1 ( | ) exp (2 ) 1 2 s m s N N N T n p M n n π ψ ψ ψ ψ − = = ∑ ⎧ ⎫ ⎡ ⎤ ⎡ ⎤ − − ∑ − ⎨ ⎬ ⎣ ⎦ ⎣ ⎦ ⎩ ⎭ ∑ D θ θ (3) where ∑ is the variance matrix of the measured modal frequency, Ns is the number of the measured modal frequency sets, µψis the measured modal frequency of the monitoring structure under unknown health status, and the ψ(θ) is computed modal frequency of the FE model. The computation of the high-dimensional integral in the Bayesian method is difficult and has attracted the attention of many researchers over the decades. Several improved stochastic simulation methods have been developed to solve the high-dimensional and complex posterior PDF, such as the adaptive Metropolis-Hastings (AMH) [6], the transitional Markov chain Monte Carlo (TMCMC) [7], the Hybrid Monte Carlo (HMC) [8], and so on. Inherited from the AMH that introduced a series of intermediate PDFs, the TMCMC can not only automatically select the intermediate PDFs but
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