# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 5 Multi-Fidelity Successive Response Surface
### Adaptive Sampling: OLHD and MIPT  
Since it is assumed that no a priori information is available about the nature of the response functions, the first stage of the workflow is necessarily a step of pure exploration of the variable domain. In this sampling step, we distribute samples according to a \"hybrid\" sampling strategy. The highest fidelity points are distributed according to a scheme of pre-optimized Latin Hypercube Designs [37]. As demonstrated by the work of [PERSON] et al. [38], these datasets have been optimized for several hours in order to maximize their projective- and space-filling properties. Please note that space-filling properties refer to the ability of a design to uniformly cover the domain of interest, ensuring that the entire input space is adequately represented without leaving large gaps. A dataset with good projective properties, instead, ensures that if we consider only a subset of its dimensions (i.e., we project the design onto a lower dimensional space), the resulting dataset still has desirable space-filling properties in that lower dimensional space. Therefore, unlike the statistical factorial experimental designs commonly used in the SRSM [20], pre-optimized Latin hypercubes allow extracting as much information as possible from each individual sample while avoiding space-near points and repetitive values. We use the database developed by [PERSON] et al. [37], available at [39]. Although this database is extensive, it has its limitations; it only provides datasets for specific combinations of a number of samples and design variables. If the required DoE is not available in this database, we rely on the translational propagation algorithm of [PERSON] et al. to quickly generate Optimal or Near-Optimal Latin Hypercube Designs [40].  
Once the highest fidelity DoE is generated, we employ a sequential sampling strategy to build all the remaining lower-fidelity designs.  
For this purpose, we use a Monte Carlo method originally developed by [PERSON] et al. [38] and extended by [PERSON] et al. [6]: mc-inter-proj-th (MIPT). Starting from  
Figure 2: Overview of the MF-SRS framework.  
a pre-existing design, this method is able to add samples with unit granularity, ensuring an optimal trade-off between space-filling and projective properties. As shown in Figure 3, by using the OLHD points, the MIPT algorithm is able to construct an optimal second layer of samples of lower fidelity. This process is repeated iteratively if more than two levels of fidelity are required. Please see Appendix A for more details on the formulation of the MIPT algorithm. The versatility of this algorithm allows us to apply the same adaptive sampling logic across iterative steps. Unlike one-shot sampling methods, such as LHDs, the MIPT method enables the inheritance and reutilization of samples used in previous iterations. This feature is critical in crashworthiness optimization, both to extract the full potential of each individual FEM simulation and to avoid running unnecessarily expensive crashworthiness simulations.  
To conclude on the topic of sampling, it is important to emphasize that the MIPT samples are introduced in a purely exploratory fashion, independent of the response function values. While this approach may seem less than optimal at first, it remains one of the most robust and effective strategies, especially when dealing with multiple response functions. Balancing different functions within the same domain can pose challenging problems.