## 4 Vibration Reduction Optimization Approach Accelerated by Global POD ROM  
### Efficient Global Optimization Algorithm Based on Surrogate Model  
By combining Optimization with Kriging surrogate model, Efficient Global Optimization (EGO) can significantly improve the efficiency of optimization design. It is convenient to solve the problem in the global scope [33]. Firstly, the mathematical expression of Kriging surrogate model [34] is:  
\[y(\mathbf{x})=\mathbf{\beta}^{\mathrm{T}}f(\mathbf{x})+Z(\mathbf{x}), \tag{19}\]  
where \(\mathbf{\beta}^{\mathrm{T}}f(\mathbf{x})\) is the global regression term, \(f(\mathbf{x})\) is usually a polynomial function, and \(\mathbf{\beta}^{\mathrm{T}}\) is the regression parameter; \(Z(\mathbf{x})\) is the error of random distribution, which provides an approximation to the local simulation deviation, with mean 0, variance \(\sigma^{2}\) and covariance:  
\[\mathrm{cov}[Z(\mathbf{x}_{i}),Z(\mathbf{x}_{j})]=\sigma^{2}R(\mathbf{x}_{i},\mathbf{x}_{j}), \tag{20}\]  
where \(R\) is the correlation function, which represents the spatial correlation between any two points. Based on the established Kriging model, the expected improvement (\(EI\)) function is used as the criterion to determine the position of the optimal design sample in the design space. The minimum value of the objective function is the current optimal result, which is expressed as follows:  
\[y_{\min}=\min[y(\mathbf{x}_{1}),y(\mathbf{x}_{2}),\cdots y(\mathbf{x}_{N})]. \tag{21}\]  
As an improvement function of the optimal result in the new design state, \(I(\mathbf{x})\) is defined as follows:  
\[I(\mathbf{x})=\max((y_{\min}-y(\mathbf{x})),0). \tag{22}\]  
\(EI(\mathbf{x})\) is the expectation of \(I(\mathbf{x})\), which can be calculated according to the equation:  
\[EI(\mathbf{x})=(y_{\min}-\hat{y}(\mathbf{x}))\Psi(\frac{y_{\min}-\hat{y}(\mathbf{x})}{s(\bm {x})})+s(\mathbf{x})\Phi(\frac{y_{\min}-\hat{y}  
probability density function and cumulative distribution function of the standard normal distribution, respectively. The calculation cost of the \(EI\) function is much lower than the actual response calculation cost. Intelligent algorithms such as Genetic Algorithm and Simulated Annealing Algorithm can be used to obtain the maximum expected value of \(EI\), and the point where \(EI\) function reaches the maximum value is taken as the update point [35]. The optimization framework of EGO is shown in Figure 2, and the steps are as follows. Step 1: The initial sample configurations set is generated, the corresponding responses are calculated, and the sample database is initialized. Step 2: The Kriging surrogate model is constructed between design variables and objective function. Step 3: Taking the value of \(EI\) function as the objective function, the point that maximizes it is selected as the update point. Step 4: The real finite element model is used to calculate the response of the updated points, and the updated points and responses are put into the sample library to update the surrogate model. Step 5: The above steps 2-4 are repeated and iterated until convergence conditions are satisfied, and then the optimization results are outputted. An Optimization Framework for Vibration Reduction of Complex Thin-Walled Shells Accelerated by the Global POD ROM  
The process of vibration reduction optimization based on the global POD ROM is shown in Figure 3, which is divided into offline stage and online stage. The offline stage is the preparation stage, which is used to build and update the global POD ROM, and the online stage is the application stage, which uses the built global POD ROM for vibration reduction optimization design. The steps are as follows:  
Step 1: In offline stage, the initial configurations are obtained by sampling in the design space using LHS. Step 2: The mode shapes corresponding to the initial configurations are calculated and the snapshot matrix is assembled.