# A Kriging Model Based Finite Element Model Updating Method for Damage Detection
## 2 Kriging Based Model Updating Method
### Kriging Model in Model Updating  
Model updating technique is to make the theoretical model, usually the FE model, produce a response as closely as possible to the experimental data. The traditional sensitivity-based model updating method uses the sensitivity matrix to correlate the updating parameters with the residuals of the response, which is the first derivative of the response residual with respect to the updating parameters, and then iteratively solves the associated optimization by the (weighted) least-squares method. However, this method usually requires the knowledge of all the degrees of freedom (DoFs) of the finite element model. When some DoFs cannot be measured, the model condensation technique shall be used and if the number of measured response points is much smaller than the number of the model degrees of freedom, this method will fail to give a unique solution. Model updating can be essentially considered an optimization problem which can be formulated in the following form:  
\[\min f(\mathbf{x}),\ \text{s.t.}\quad lb_{j}<x_{j}<ub_{j},\quad j=1,2,\cdots,n \tag{11}\]  
where \(f(\mathbf{x})\) is the objective function and usually the response residual between theoretical prediction and experimental data, design variable \(x_{j}\) is the \(j\)th parameter of the structure to be updated, \(lb_{j}\) and \(ub_{j}\) are the lower and upper bound of \(x_{j}\), respectively. By solving this optimization problem, design variable vector \(\mathbf{x}\) which minimizes the objective function (response residual) gives the right updating parameter values. Besides using the sensitivity method which can also be regarded as a steepest descent method, the artificial intelligence algorithm is also convenient. In this paper, the widely used genetic algorithm (GA) is selected. The advantage of this method is that it is easy to update some special parameters of which the sensitivity matrices are difficult to calculate and to specify the interval of parameters to prevent them from convergence to meaningless results. Of course, due to the strong complementarity among the updating parameters, the optimization problem of model updating is a typical multi-peak problem. The algorithm can easily cause convergence to a local solution if the number of updating parameters is large. In addition, the core problem of this approach is that a huge number of objective functional values at different sets of design variables are required by the GA to find the global minimum. If the computation time taken to evaluate a single value of the objective functional from the theoretical model is long, the total duration of solution by the algorithm may be unacceptable. That is why a Kriging model is required here to calculate the objective functional instead of the original FE model. There are two ways to use the Kriging model to improve the computational efficiency in the model updating procedure. Firstly, the response of the FE model is used as the output response. A precise Kriging model is constructed for every measured response to totally replace the FE analysis in the subsequent model updating procedure. However, this kind of usage is mainly for the frequency as response and for few updating parameters. When it comes to FRF, for Kriging model construction, not only many measured responses are required, but also many sample points are required, since the true response surface of the FRF is much more complicated than that of the frequency. These are the main reasons why, in this paper, the second approach is taken in which the value of the objective function of the optimization is directly used as the output response of the Kriging model. In so doing, only one Kriging model is required to be built and the EGO algorithm is adopted to obtain the model updating results, which is described in the next subsection.