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 suboptimal 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.

Due to the high computational burden and the high nonlinearity 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 suboptimal predictions. This paper introduces the Multi-Fidelity Successive Response Surface (MF-SRS), a Gaussian process-based method, which leverages a nonlinear 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 crossmember and crash box system. 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 14.1 %, revealing its potential for future applications.

Studying complex phenomena in detail by performing real experiments is often an unfeasible task. Virtual experiments using simulations are usually used to support the development process. However, numerical simulations are limited by their computational cost. Metamodeling techniques are commonly used to mimic the behavior of unknown solver functions, especially for expensive black box optimizations. If a good correlation between the surrogate model and the black box function is obtained, expensive numerical simulations can be significantly reduced. The sampling strategy, which selects a subset of samples that can adequately predict the behavior of expensive black box functions, plays an important role in the fidelity of the surrogate model. Achieving the desired metamodel accuracy with as few solver calls as possible is the main goal of global surrogate modeling. In this paper, exploration-oriented adaptive sampling strategies are compared with commonly used one-stage sampling approaches, such as Latin Hypercube Design (LHD). The difference in the quality of approximation is tested on benchmark functions from 2 up to 30 variables. Two novel sampling algorithms to get fine-grained quasi-LHDs will be proposed and an improvement to a well-known, pre-existing sequential input algorithm will be discussed. Finally, these methods are applied to a crash box design to investigate the performance when approximating highly non-linear crashworthiness problems. It is found that adaptive sampling approaches outperform one-stage methods both in terms of mathematical properties and in terms of metamodel accuracy in the majority of the tests. A proper stopping algorithm should also be employed with adaptive methods to avoid oversampling.