## 2 Hybrid Model Updating Method
### Kriging Meta-Model  
For a Kriging meta-model, supposing the unknown function is combined by a polynomial part and a stochastic process which provide a global model and localized departures respectively. The formula is as follows  
\[y(\mathbf{X})=f^{\mathrm{T}}(\mathbf{X})\boldsymbol{\beta}+z(\mathbf{X}) \tag{5}\]  
where \(\mathbf{X}\) is variable vector of \(d\) parameters \([x_{1}\cdot\cdot\cdot x_{d}]\), for example, the \(i\)th vector of \(d\) parameters \(\mathbf{X}_{i}=[x_{i1}\cdot\cdot\cdot x_{id}];y(\mathbf{X})\) is the response of FE model with parameter condition of \(\mathbf{X}\); \(f^{\mathrm{T}}(\mathbf{X})\) is a vector of \(m\) polynomial functions, namely \(f^{\mathrm{T}}(\mathbf{X})=[f_{1}(\mathbf{X})\cdot\cdot\cdot f_{m}(\mathbf{X})]\); \(\boldsymbol{\beta}\) is a vector of \(m\) regression coefficients, \(\boldsymbol{\beta}=[\beta_{1}\cdot\cdot\cdot\beta_{m}]^{\mathrm{T}}\); \(z(\mathbf{X})\) is a Gaussian stationary stochastic process with mean 0 and variance \(\sigma^{2}\). The covariance of arbitrary two elements in \(z(\mathbf{X})\) is as follows  
\[\mathbf{cov}(z(\mathbf{X}_{i})z(\mathbf{X}_{j}))=\sigma^{2}R(\mathbf{X}_{i}, \mathbf{X}_{j})=\sigma^{2}\exp(-\sum\limits_{k=1}^{d}\theta_{k}|x_{i,k}-x_{j,k }|^{2}) \tag{6}\]  
where \(R(\mathbf{X}_{i},\mathbf{X}_{j})\) is the spatial correlation of two samples which is usually implemented Gaussian function \(\exp(-\sum\limits_{k=1}^{d}\theta_{k}|x_{i,k}-x_{j,k}|^{2})\) as a core function because its calculation effect is the most welcome; \(\theta_{k}\) is coefficient with value over zero in computation of spatial correlation. Supposing there are \(N\) samples, the matrices of responses and polynomial values are \(\mathbf{Y}=[y(\mathbf{X}_{1})\cdot\cdot\cdot y(\mathbf{X}_{N})]\) and \(\mathbf{F}=\left[f^{\mathrm{T}}(\mathbf{X}_{1})\cdot\cdot\cdot f^{\mathrm{T} }(\mathbf{X}_{N})\right]^{\mathrm{T}}\), respectively. The estimated values of \(\boldsymbol{\beta}\) and \(\sigma^{2}\) are as follows  
\[\boldsymbol{\beta}=(\mathbf{F}^{\mathrm{T}}\mathbf{R}^{-1}\mathbf{F})^{-1} \mathbf{F}^{\mathrm{T}}\mathbf{R}^{-1}\mathbf{Y} \tag{7}\]  
\[\boldsymbol{\phi}^{2}=\frac{1}{N}(\mathbf{Y}-\mathbf{F}\boldsymbol{\hat{ \beta}})^{\mathrm{T}}\mathbf{R}^{-1}(\mathbf{Y}-\mathbf{F}\boldsymbol{\hat{ \beta}}) \tag{8}\]  
where spatial correlation matrix  
\[\mathbf{R}=\left[ \begin{array}{cccc}R(\mathbf{X}_{1},\mathbf{X}_{1})&R( \mathbf{X}_{1},\mathbf{X}_{2})&\dots&R(\mathbf{X}_{1},\mathbf{X}_{N})\\ R(\mathbf{X}_{2},\mathbf{X}_{1})&R(\mathbf{X}_{2},\mathbf{X}_{2})&\dots&R( \mathbf{X}_{2},\mathbf{X}_{N})\\ \vdots&\vdots&\ddots&\vdots\\ R(\mathbf{X}_{N},\mathbf{X}_{1})&R(\mathbf{X}_{N},\mathbf{X}_{2})&\dots&R( \mathbf{X}_{N},\mathbf{X}_{N})\end{array} \right] \tag{9}\]  
In above equations, before obtaining values of \(\boldsymbol{\hat{\beta}}\), \(\boldsymbol{\phi}^{2}\) and \(\mathbf{R}\) through Equations (7)-(9), \(\theta_{1}\), \(\dots\) and \(\theta_{d}\) should be known first. By implementing maximum likelihood estimation method, \(\theta_{1}\), \(\dots\) and \(\theta_{d}\) should be selected to get minimum value of expression below  
\[\min_{\theta_{1}>0,\dots,\theta_{d}>0}\left\{\frac{1}{2}[N\ln(\boldsymbol{\theta }^{2})+\ln|\mathbf{R}|]\right\} \tag{10}\]  
where \(|\mathbf{R}|\) is determinant of matrix \(\mathbf{R}\), the optimal values of \(\theta_{1}\), \(\dots\) and \(\theta_{d}\) can be obtained by optimization method such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), etc. with an objective function of Equation (10).