# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
###### Abstract  
Due to the high computational burden and the high non-linearity of the responses, crashworthiness optimizations are notoriously hard-to-solve challenges. Among various approaches, methods like the Successive Response Surface Method (SRSM) have stood out for their efficiency in enhancing baseline designs within a few iterations. However, these methods have limitations that restrict their application. Their minimum iterative resampling required is often computationally prohibitive. Furthermore, surrogate models are conventionally constructed using Polynomial Response Surface (PRS), a method that is poorly versatile, prone to overfitting, and incapable of quantifying uncertainty. Furthermore, the lack of continuity between successive response surfaces results in sub-optimal predictions. This paper introduces the Multi-Fidelity Successive Response Surface (MF-SRS), a Gaussian process-based method, which leverages a non-linear multi-fidelity approach for more accurate and efficient predictions compared to SRSM. After initial testing on synthetic problems, this method is applied to a real-world crashworthiness task: optimizing a bumper cross member and crash box system. The results, benchmarked against SRSM and the Gaussian Process Successive Response Surface (GP-SRS)--a single-fidelity Gaussian process-driven extension of SRSM--show that MF-SRS offers distinct advantages. Specifically, it improves upon the specific energy absorbed optimum value achieved by SRSM by 14%, revealing its potential for future applications. crashworthiness optimization; MF-SRS; SRSM; multi-fidelity; design optimization; Gaussian process; NARGP; concept design +
Footnote †: journal: Applied  
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Footnote †: journal: Applied  
To cope with this issue, surrogate models have emerged as an innovative solution to reduce the vast number of function evaluations required, hence optimizing resource utilization. These models operate as an abstraction of an underlying phenomenon, generating a \"model of the model\" as referred to by [PERSON] [5]. Surrogate modeling operates by utilizing a set of sampling evaluations. Typically, this process begins with the design space being sampled through Design of Experiment (DoE) techniques. Subsequent numerical simulations at these sampled points are executed and results are used to fit surrogate models over the variable domain [6]. A broad spectrum of metamodeling techniques have been advanced over recent years to aid complex engineering designs that are reliant on simulation-based methods. These include Gaussian Process (GP) [7], Polynomial Response Surface (PRS) [8; 9], Support Vector Regression (SVR) [10], Radial Basis Function (RBF) [11], and Neural Network (NN) [12; 13]. In the specific context of optimizing crashworthiness problems, PRS, RBF, and GP stand out as the most commonly used and most successful metamodels [14; 15]. Alongside regression methods, various global search strategies can be useful in attempting to identify an optimal global solution in crashworthiness optimization. Given the highly non-linearity of these problems, which commonly results in the presence of multiple local minima, methodologies such as Genetic Algorithms (GAs) [16], Evolutionary Algorithms (EAs) [17; 18], and Simulated Annealing (SA) [19] are particularly suitable, but they often come with a high computational cost that might be unaffordable. In contrast, gradient-based methods are generally not recommended, particularly for the optimization of frontal collisions, as they exhibit increased sensitivity to bifurcations compared to lateral impacts. However, these methods may prove useful as local search strategies for refining the solution identified by a global optimization algorithm. Regardless of the regression model considered, the use of global response surfaces is generally discouraged due to the high non-linearity of crash load cases.