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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.

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.

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.

Reliability-based design optimization (RBDO) has been widely used to obtain a reliable design via an existing CAE model considering the variations of input variables. However, most RBDO approaches do not consider the CAE model bias and uncertainty, which may largely affect the reliability assessment of the final design and result in risky design decisions. In this paper, the Gaussian Process Modeling (GPM) approach is applied to statistically correct the model discrepancy which is represented as a bias function, and to quantify model uncertainty based on collected data from either real tests or high-fidelity CAE simulations. After the corrected model is validated by extra sets of test data, it is integrated into the RBDO formulation to obtain a reliable solution that meets the overall reliability targets while considering both model and parameter uncertainties.

Finite Element (FE) models are widely used in automotive for vehicle design. Even with increasing speed of computers, the simulation of high fidelity FE models is still too time-consuming to perform direct design optimization. As a result, response surface models (RSMs) are commonly used as surrogates of the FE models to reduce the turn-around time. However, RSM may introduce additional sources of uncertainty, such as model bias, and so on. The uncertainty and model bias will affect the trustworthiness of design decisions in design processes. This calls for the development of stochastic model interpolation and extrapolation methods that can address the discrepancy between the RSM and the FE results, and provide prediction intervals of model responses under uncertainty.

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.

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.

Computer Aided Engineering (CAE) has become a vital tool for product development in automotive industry. Various computer models for occupant restraint systems are developed. The models simulate the vehicle interior, restraint system, and occupants in different crash scenarios. In order to improve the efficiency during the product development process, the model quality and its predictive capabilities must be ensured. In this research, an objective model validation metric is developed to evaluate the model validity and its predictive capabilities when multiple occupant injury responses are simultaneously compared with test curves. This validation metric is based on the probabilistic principal component analysis method and Bayesian statistics approach for multivariate model assessment. It first quantifies the uncertainties in both test and simulation results, extracts key features, and then evaluates the model quality.

Response surface methods have been widely used in robust design for reducing turn-around time and improving quality. That is, from a given set of CAE data (design-of-experiments results), many different robust optimization studies can be performed with different constraints and objectives without large, recurring, computation costs. However, due to the highly nonlinear and non-convex nature of occupant injury responses, it is difficult to generate high quality response surface models from them. In this paper, we apply a cross validation technique to estimate the accuracy of response surface models, particularly in the context of robustness assessment. We then decompose selected occupant injury responses into more fundamental signals before fitting surfaces to improve the predictivity of the response surface models. Real-world case studies on an occupant restraint system robust design problem are used to demonstrate the methodology.

In an attempt to make vehicle restraint systems more effective in protecting occupants, many advanced safety technologies have been introduced. These advanced technologies are mostly adaptive technologies. The ability of a restraint system to adapt itself to crash parameters like crash speed and type, occupant size, and belt-usage status, offers possible enhancements in occupant protection. Designing a restraint system boils down to the determination of the design variables of either the restraint technologies or vehicle interiors. A restraint system of adaptive technologies involves much more design variables than a restraint system containing only load-limited belts and dual stage inflators, possibly posing a challenge to safety engineers. In this paper, a genetic algorithm (GA) tailored for restraint system optimization will be presented.

Response surface methodologies (RSMs) have been proposed as surrogate models in vehicle design processes to gain insight and improve turnaround time for optimization and robust design. However, when studying the vehicle occupants during crash events, nonlinearities in responses, coupled with the relatively high dimensionality of vehicle design, can yield misleading results with little or no warning from the response surface algorithms. To ensure the accuracy and reliability of RSMs, fast and dependable error estimation procedures are essential for enlightening how well a response surface predicts highly nonlinear phenomena, given a limited number of model simulations. Such error estimation methods are also useful for providing guidance on how many simulation runs are needed for reliable RSM construction. In this paper, a fast cross validation error estimate procedure is first presented, applied to the multivariable adaptive regression spline (MARS) response surface method.

This paper focuses on the methodology development and application of reliability-based design optimization to a vehicle exhaust system under noise, vibration and harshness constraints with uncertainties. Reliability-based design optimization provides a systematic way for considering uncertainties in product development process. As traditional reliability analysis itself is a design optimization problem that requires many function evaluations, it often requires tremendous computational resources and efficient optimization methodologies. Multiple functional response constraints and large number of design variables add further complexity to the problem. This paper investigates an integrated approach by taking advantages of variable screening, design of experiments, response surface model, and reliability-based design optimization for problems with functional responses. A typical vehicle exhaust system is used as an example to demonstrate the methodology.

With the continuous improvement of powerful computers, vehicle structural designs have been addressed using computational methods, resulting in more efficient development of new vehicles. Most simulation-based optimization generates deterministic optimal designs without considering variability effects in modeling, simulation, and/or manufacturing. This paper presents a new hybrid stochastic optimization method for vehicle side impact design. Nonlinear response surface models are employed as the ’real’ models for the side impact related performance functions to conduct this study. The main goal is to maintain or enhance the vehicle side impact performance while minimizing the vehicle weight under various uncertainties. The new method alleviates the computational burden of excessive model evaluations by estimating the objective and constraint functions during the optimization process through a reweighting approach.