Crashworthiness optimization with uncertainty from surrogate model and numerical error

Due to the expensive cost of full-scale tests, more and more designs rely on simulation. For highly nonlinear crash simulation, numerical uncertainty is an inherent by-product, which refers to the oscillation of results when the simulation is repeated at the same design or the design variables are slightly changed. This oscillation directly influences the quality and reliability of the optimal design. This paper shows how these issues can be addressed by proposing a simple uncertainty quantification method for numerical uncertainty (noise) and surrogate model uncertainty (error) in the optimization process. Three engineering problems, a tube crush example, an automotive front-rail crush example and a multi-cell structure crush example, are used to illustrate this method. Firstly, the level of numerical uncertainty is quantified in terms of noise frequency and amplitude, and the convergence study of these two criteria is employed to determine an appropriate data size to describe numerical noise. Secondly, an estimation method considering both numerical noise and surrogate model error is proposed based on the prediction variance of the polynomial response surface. Finally, the tube and front rail structures are optimized according to the proposed uncertainty quantification method. It was found that by considering the two sources of uncertainty, the optimal designs are more reliable than the deterministic solutions.