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Vehicle restraint system design is a difficult optimization problem to solve because (1) the nature of the problem is highly nonlinear, non-convex, noisy, and discontinuous; (2) there are large numbers of discrete and continuous design variables; (3) a design has to meet safety performance requirements for multiple crash modes simultaneously, hence there are a large number of design constraints. Based on the above knowledge of the problem, it is understandable why design of experiment (DOE) does not produce a high-percentage of feasible solutions, and it is difficult for response surface methods (RSM) to capture the true landscape of the problem. Furthermore, in order to keep the restraint system more robust, the complexity of restraint system content needs to be minimized in addition to minimizing the relative risk score to achieve New Car Assessment Program (NCAP) 5-star rating.

Available methodologies for model bias identification are mainly regression-based approaches, such as Gaussian process, Bayesian inference-based models and so on. Accuracy and efficiency of these methodologies may degrade for characterizing the model bias when more system inputs are considered in the prediction model due to the curse of dimensionality for regression-based approaches. This paper proposes a copula-based approach for model bias identification without suffering the curse of dimensionality. The main idea is to build general statistical relationships between the model bias and the model prediction including all system inputs using copulas so that possible model bias distributions can be effectively identified at any new design configurations of the system. Two engineering case studies whose dimensionalities range from medium to high will be employed to demonstrate the effectiveness of the copula-based approach.

Design optimization techniques are widely used to drive designs toward a global or a near global optimal solution. However, the achieved optimal solution often appears to be the only choice that an engineer/designer can select as the final design. This is caused by either problem topology or by the nature of optimization algorithms to converge quickly in local/global optimal or both. Problem topology can be unimodal or multimodal with many local and/or global optimal solutions. For multimodal problems, most global algorithms tend to exploit the global optimal solution quickly but at the same time leaving the engineer with only one choice of design. The paper explores the application of genetic algorithms (GA), simulated annealing (SA), and mixed integer problem sequential quadratic programming (MIPSQP) to find multiple local and global solutions using single objective optimization formulation.