Crash worthiness design optimization using multipoint sequential linear programming

1 I n t r o d u c t i o n Safety measures like airbags and safety belts are nowadays commonly applied to improve the crash worthiness of road vehicles. Many governments stimulate or oblige by law car producers to include these devices in the vehicle design. Maximum values are imposed on parameters that quantitatively describe the severity of injuries. Examples are the Head Injury Criterion (HIC) and the chest 3 MilliSeconds criterion (chest 3MS). Physical crash tests of newly designed vehicles are used to determine the vehicle's crash worthiness. However, these physical tests are very cost expensive, and therefore car producers generally use crash simulation software in the design stage instead. One of the leading simulation packages is the combined multibody finite element program MADYMO (TNO 1994), which has been developed at TNO Crash-Safety Research Center in The Netherlands. Crash worthiness design is characterized by a variety of design variables and several conflicting injury parameter demands. Computer simulation tools like MADYMO enable systematic improvements early in the design stage. One can for example analyse some different designs and select the best set of design variable values found. Although much cheaper than physical tests, crash worthiness simulations are computationally expensive. As a result, the number of designs that * Currently: Department of Mathematics, Delft University of Technology, P.O. Box 5033 Delft, The Netherlands can be analysed is limited. Here, optimization tools can aid the designer in his search for the optimum design using as few crash worthiness analyses as possible. Design optimization has been hardly applied to crash worthiness analysis. Due to the impact situation and numerical inaccuracies, the behaviour of injury parameters as a function of the design variables may not be smooth. Sensitivity based optimization strategies will then fail to find the correct optimum solution. Bennett and Park (1995) also noticed this problem, and used sequential linear programming with somewhat larger finite difference steps to calculate the sensitivities, as well as Vanderplaats' multipoint optimization method (Vanderplaats 1979). Starting from MADYMO simulations, Bosio and Lnpker (1991) used Taguchi's method to deal with the noisy functional behaviour of the injury parameters. They built a global linear response surface model including interaction terms, and studied the influence of several seat belt and airbag design variables on the HIC in a frontal crash situation. Schoofs et al. (1992) constructed global response surface models (linear and quadratic) from a set of MADYMO analyses using design of experiments and regression techniques. An objective function built from several neck injury parameters was formulated to optimize a child seat. A new design optimization tool has been developed for the crash worthiness simulation package MADYMO. A sequential approximate optimization approach has been adopted. This yields a more automated optimization compared with global response-surface modeling. Linear model functions are built based upon the responses calculated for a multipoint experimental design in a restricted part of the design space. The multipoint approximations are able to deal with noisy responses since no local sensitivity information is required. A linear programming problem follows that can be solved within the search subregion, separately from the numerical analysis. The optimum solution is used as starting point of a new cycle of approximation and optimization, which results in a series of design cycles to reach the final optimum design. The effectiveness of the approach will be illustrated for an analytical test problem and a full-scale frontal impact design