# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 5 Multi-Fidelity Successive Response Surface
### Optimization Approach: Differential Evolution, Trust Region, and Verification Step  
Design optimization for expensive-to-evaluate engineering applications involves finding the best set of design variables that meet specific objectives while navigating through a demanding computational environment. Hence, the challenge is to identify the maximum of such costly objective functions with the minimum number of sequential queries, thus reducing time and cost [43]. Numerous single-objective optimization methods have been developed to date, encompassing both local and global optimization techniques. Typically, selecting the right optimization algorithm is crucial for locating the optimum, especially when surface response models are not in use. Yet, as [PERSON] pointed out [15], when Metamodel-Based Optimization approaches are employed, the selection of the optimization algorithm becomes less critical if the surrogate models accurately capture the physical properties. This statement highlights the importance of accurate metamodeling for effective optimization of complex engineering design problems. Our proposed optimization strategy employs a hybrid approach. Initially, we leverage a probability-based global optimization algorithm to rapidly identify an optimal point. Subsequently, using this point as a starting reference, a local method refines the solution. This secondary step ensures that potential local enhancements are duly captured and not overlooked. The Differential Evolution (DE) algorithm is selected as the global optimization method in our approach due to its remarkable convergence properties, capability to address non-linear constraints, adaptability for parallelization, and straightforward implementation. This stochastic global search algorithm excels in managing non-differentiable, non-linear, and multi-modal objective functions. Furthermore, its efficacy is validated in prior studies on structural optimization applications [44; 45; 46]. In our implementation, we chose a population size of 25, a crossover probability of 0.7, and allowed the differential weight to range from 0.5 to 1 (dithering is employed). These parameter values are based on average values as reported in the tests by [PERSON] et al. [47]. It is important to note that while these parameter choices play a role in the optimization process, they are not of primary importance in determining the optimization result in our context. The accuracy of the metamodel is of greater importance in influencing the final results. Based on our experience, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) could be a  
Figure 4: Progressing from left to right, the figures depict the evolution of the RoI and the subsequent addition of samples over three iterations. High-fidelity samples are represented in red, low-fidelity samples in purple, and samples from prior iterations that fall outside the RoI are shown in gray. compelling global method, delivering a performance comparable to the DE algorithm. For detailed information on DE and CMA-ES, please refer to [47; 48; 49], respectively. As a local optimization method, our choice falls on the Trust Region, a popular gradient-based optimization algorithm well suited for non-linear problems. Its use is primarily motivated by the need to manage the quality of the model approximation in non-linear optimization. The basic idea is to iteratively refine an estimate of the optimum based on local information. Like other local optimization methods, Trust Region methods are sensitive to the starting point. If the function has multiple local optima, the method is expected to converge to a different optimum depending on where you start [50]. Therefore, its use must be combined with a global optimization method, such as DE. In our approach, a local optimization is performed starting from each member of the last population of the DE algorithm.