# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 3 Successive Response Surface  
Due to the high computational burden and the challenge of globally capturing the non-linearities within the variable domain, the problem, as formulated above, is not yet ready for resolution. The SRSM can navigate us toward a more feasible formulation for such a problem by employing two main adjustments. Initially, the \"expensive-to-evaluate\" response functions associated with FEA simulations are substituted with the response surface models. Specifically, \(f\) and \(g_{j}\) are replaced with \(\hat{f}\) and \(\hat{g}_{j}\) respectively, such that \(\hat{f}:\mathbb{R}^{d}\to\mathbb{R}\) and \(\hat{g}_{i}:\mathbb{R}^{d}\to\mathbb{R}\). These surrogate models are nothing but regression models constructed on an initial dataset of \(n_{p}\) observations, commonly referred to as the Design of Experiment (DoE): \(\mathcal{D}=\{(x_{p},y_{p})|p=1,\ldots,n_{p}\}\). Many authors have used Polynomial Response Surface models as a simple but effective way of generating a surrogate model [22; 25; 26]. The function is given for a general quadratic polynomial surface approximation:  
\[y_{p}=\beta_{0}+\sum_{i=1}^{d}\beta_{i}x_{pi}+\sum_{i=1}^{d}\sum_{k=1}^{d} \beta_{ik}x_{pi}x_{pk}+\varepsilon_{p}\hskip 28.452756 ptp=1,\ldots,n_{p}, \tag{4}\]  
where \(x_{p}\) are the design points used to train the model, \(y_{p}\) are the associated evaluated response values, \(\beta_{i}\) are the constants to be determined, and \(\varepsilon_{p}\) includes both the bias errors and random errors. The minimum number of numerical simulations, denoted as \(n_{p,min}\), required to construct a given approximation depends on the number of design variables. For instance, in the case of a quadratic approximation, \(n_{p,min}=(d+1)(d+2)/2\). The \(D\)-optimal sampling is frequently employed to explore the variable domain. For a deeper understanding of the PSR and \(D\)-optimal criterion, we recommend readers refer to the work of [PERSON] and [PERSON] [8]. Moreover, we substitute the original optimization problem with a succession of less complex and smaller problems. Ideally, these subproblems should be located in a subregion of the variable domain and should still provide the same optimal solution to the original problem. We can formulate the \(k\)th subproblem by readjusting Equations (1)-(3) as follows:  
\[\min_{x} \hat{f}^{(k)}(x)\] (5) subject to \[\hat{g}^{(k)}_{j}(x)\leq 0,\ j=1,\ldots,n_{c} \tag{6}\] \[x^{(k)}_{il}\leq x_{i}\leq x^{(k)}_{ul},\ i=1,\ldots,d\] (7) where \[x^{(k)}_{il}\geq x_{il},\quad x^{(k)}_{iu}\leq x_{iu}, \tag{8}\]  
where \(x^{(k)}_{il}\) and \(x^{(k)}_{il}\) define the \(k\)th subregion, called the Region of Interest (RoI) by some authors [27; 28]. The domain of the variables can be translated and narrowed by adjusting the bounds of each variable. In this way, the focus of optimization can be shifted to only the new simplified domain of interest. Several heuristic schemes using different measures to evaluate the accuracy of the metamodel have been proposed and used to automate the so-called \"panning and zooming\" strategy. In this article, we briefly introduce the bounds adjustment scheme, which has been successfully used to solve a variety of crashworthiness optimization problems [20; 21; 29]. As shown in Figure 1, the \(\Omega_{k+1}\) subproblem is centered on the optimal design of the \(\Omega_{k}\) subproblem, i.e., \(\mathbf{x}^{(k)}_{opt}\). Moreover, the size of the \(\Omega_{k+1}\) subregion is a fraction of the \(\Omega_{k}\) subregion. This reduction in the feature domain along a generic \(i\)th variable is determined by the fraction parameter \(\lambda_{i}\), which is computed as follows:  
\[\lambda^{(k+1)}_{i}=\eta+(\gamma-\eta)\frac{|x^{(k)}_{i\,opt}-\frac{x^{(k)}_{ il}+x^{(k)}_{il}}{2}|}{\frac{x^{(k)}_{iu}-x^{(k)}_{il}}{2}}. \tag{9}\]  
From Equation (9), it is quite clear that \(\lambda^{(k+1)}_{i}\) is equal to \(\gamma\) when the optimum is located at the lower and upper bounds.