Lightweight design of vehicle parameters under crashworthiness using conservative surrogates

Abstract

Lightweight design of vehicle structures parameters under crashworthiness is hard to accomplish because of the complexity of simulations required in crash analysis. To reduce the computation demand, surrogates (metamodels) are often used in place of the actual simulation models in design optimization to fit the mathematical relationship between design variables and responses. Each optimization cycle consists of analyzing a number of designs, fitting surrogates for the responses, performing optimization based on the surrogates for a candidate optimum, and finally analyzing that candidate. Even so, optimization using crash analysis codes is often allowed to run only for very few cycles. While traditional surrogate is unbiased which means prediction values at half region is lower than actual values, predicted candidate optimum usually is not feasible after validating by crash simulation. This paper explores the use of conservative surrogates for safe estimations of crashworthiness responses (e.g., intrusion and peak acceleration). We use safety margins to conservatively compensate for fitting errors associated with surrogates. Conservative surrogates minimize the risks associated with underestimation of the responses, which helps push optimization toward the feasible region of the design. We also propose an approach for sequential relaxation of the safety margins allowing for further weight minimization. The approach was tested on the lightweight design of a vehicle subjected to the full-overlap frontal crash. We compare this approach with the traditional use of unbiased surrogates (that is, without adding any safety margin). We find that conservative surrogates successfully drive optimization toward the feasible region of a design space, while that is not always the case with unbiased surrogates.

Introduction

Design optimization of vehicle structures under crashworthiness is a very computer intensive task. Because of that, we often use surrogate models, also called as approximations or metamodels in place of actual simulation models [1]. The popular surrogates including polynomial response surface, kriging neural network, radial basis function, support vector regression, etc. has been widely used in engineering application to alleviate the computation burden and conduct design space exploration and optimization [2], [3], [4], [5]. The fitting capability of surrogate is one of research topic and many publications indicate no surrogate can always be better than others in different samples and problems [6], [7], [8]. Besides computational cost, the literature has used surrogate models to build crashworthiness response for practical design optimization and has also pointed difficulties associated crashworthiness responses such as noise, nonlinearities, and limited number of samples [9], [10], [11], [12], [13], [14], [15]. Youn et al. [9] used polynomial response to fit side impact response and indicated that the optimum highly depended on surrogate accuracy. Same conclusions can also be reached that optimum solution depends on the selection of surrogate and sample points etc. Altogether, the consequence is that optimization is conducted for very few cycles at the risk of violating constraints [11], [15].

The present work focuses on the weight reduction of vehicle front end structures subject to frontal crash. The design has to satisfy constraints with respect to the peak acceleration of auto body and the toe-board intrusion for occupant safety [16], [17]. The front end structures have to balance compliance to reduce the acceleration level and stiffness to minimize the intrusion of passenger compartment. These responses are obtained via expensive finite element-based simulations. A single detailed crash computation on 8 CPUs requires about 3–4 h on a current computer platform (3.00 GHz Intel Core 2 Duo processors on a Linux cluster). In our model, the objective function, i.e. the mass, is relatively cheap to evaluate (geometry of mechanical components is simple enough) and constraints are expensive (so that we can only afford few simulations). We approximate the constraints with surrogates to alleviate the costs. However, we are aware that after running the optimization the solution might turn out to be infeasible due to surrogate errors. We propose pushing optimization toward the feasible domain by compensating surrogate errors with safety margins.

Viana et al. [18] have proposed an approach for estimating the safety margins using cross validation. Cross validation errors are used to estimate the cumulative density function of the prediction errors. That in turn is used for designing the safety margin. Here, we use the same scheme for estimating the safety margins. We also propose a strategy for sequentially updating the conservativeness level throughout the optimization task by monitoring the current best feasible sample. This will hopefully allow further weight savings, moving the solution close to the limit state.

The remainder of the paper is organized as follows. Section 2 describes the finite element modeling and formulation of the optimization problem. Section 3 briefly reviews the concept of conservative surrogates and describes the proposed updating strategy for the safety margins. Section 4 presents the results and discussions of our numerical experiment. Finally, the concluding remarks are presented in Section 5.