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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.

In vehicle design, response surface model (RSM) is commonly used as a surrogate of the high fidelity Finite Element (FE) model to reduce the computational time and improve the efficiency of design process. However, RSM introduces additional sources of uncertainty, such as model bias, which largely affect the reliability and robustness of the prediction results. The bias of RSM need to be addressed before the model is ready for extrapolation and design optimization. This paper further investigates the Bayesian inference based model extrapolation method which is previously proposed by the authors, and provides a systematic and integrated stochastic bias corrected model extrapolation and robustness design process under uncertainty. A real world vehicle design example is used to demonstrate the validity of the proposed method.

Model validation is a process to assess the validity and predictive capabilities of a computer model by comparing simulation results with test data for its intended use of the model. One of the key difficulties for model validation is to evaluate the quality of a computer model at different test configurations in design space, and interpolate or extrapolate the evaluation results to untested new design configurations. In this paper, an integrated model interpolation and extrapolation framework based on Bayesian inference and Response Surface Models (RSM) is proposed to validate the designs both within and outside of the original design space. Bayesian inference is first applied to quantify the distributions' hyper-parameters of the bias between test and CAE data in the validation domain. Then, the hyper-parameters are extrapolated from the design configurations to untested new design. They are then followed by the prediction interval of responses at the new design points.

Sampling-based methods are general but time consuming for solving a Reliability-Based Design Optimization (RBDO) problem. In order to alleviate the computation burden, score function together with the Monte Carlo method was used to compute the stochastic sensitivities of reliability functions. In literature, re-weighting schemes were shown to converge faster than the regular Monte Carlo method. In this paper, a reweighting scheme together with score function is employed to perform sampling-based stochastic sensitivity analysis to improve the computational efficiency and accuracy. An analytical example is used to show the advantages of the proposed method. Comparisons to the conventional methods are made and discussed. Two RBDO problems are solved to demonstrate the use of the proposed method.

Multidisciplinary design optimization (MDO) methods are commonly used for weight reduction in automotive industry. The design variables for MDO are often selected based on critical parts, which usually are close to optimal after many design iterations. As a result, the real weight reduction benefit may not be fully realized due to poor selection of design parameters. In addition, most applications require running design of experiments (DOE) to explore the full design space and to build response surfaces for optimization. This approach is often too costly if too many design variables are simultaneously considered. In this research, an alternative approach to address these issues is presented. It includes two optimization phases. The first phase uses critical parts for design iterations and the second phase use non-critical for weight reduction. A vehicle body structure is used to demonstrate the proposed strategy to show its effectiveness.

This paper presents an enhanced Bayesian based model validation method together with probabilistic principal component analysis (PPCA). The PPCA is employed to address multivariate correlation and to reduce the dimensionality of the multivariate functional responses. The Bayesian hypothesis testing is used to quantitatively assess the quality of a multivariate dynamic system. Unlike the previous approach, the differences between test and CAE results are used for dimension reduction though PPCA and then to assess the model validity. In addition, physics-based thresholds are defined and transformed to the PPCA space for Bayesian hypothesis testing. This new approach resolves some critical drawbacks of the previous method and provides desirable properties of a validation method, e.g., symmetry. A dynamic system with multiple functional responses is used to demonstrate this new approach.

Response Surface Model (RSM)-based optimization is widely used in engineering design. The major strength of RSM-based optimization is its short computational time. The expensive real simulation models are replaced with fast surrogate models. However, this method may have some difficulties to reach the full potential due to the errors between RSM and the real simulations. RSM's accuracy is limited by the insufficient number of Design of Experiments (DOE) points and the inherent randomness of DOE. With recent developments in advanced optimization algorithms and High Performance Computing (HPC) capability, Direct Multidisciplinary Design Optimization (DMDO) receives more attention as a promising future optimization strategy. Advanced optimization algorithm reduces the number of function evaluations, and HPC cut down the computational turnaround time of function evaluations through fully utilizing parallel computation.