A two-stage approach to the optimization design of multi-cell square tubal structures

Multi-cell tubal structures have widely been used in the automobile industry due to proven superior crashworthiness performance than single-cell and foam-filled tubes. This superior performance is attributed to the number of corners within the cross-sectional profile of the tube. In this paper, a two-stage optimization design of a multi-cell tubal structure is presented to address an important design problem by combining a discrete and continuous optimization process into a sequential optimization that generates an overall optimum. The first stage entails a configurational optimization which is realized by formulating a discrete topological optimization problem where the webs within the tube configuration are taken as the topological design variables. Each topological configuration is represented using a binary scheme that shows the presence or not of an edge to create different combinations of the corners. The constraints in the first stage are connectivity, mass ratio, and peak crushing force (PCF). The binary genetic algorithm (BGA) is utilized in searching for the optimal configuration in the first stage. The second stage entails parameter optimization where the cell sizes are the design variables. The objective functions in the second stage are defined using meta-models. Multi-objective particle swarm optimization (MOPSO) is employed for Pareto searching and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is used to find the optimal point for each of the mass ratios considered. Compared with the baseline configuration, the optimized tubal structures demonstrated superior crashworthiness performance. The two-stage discrete and continuous optimization approach has demonstrated that it not only provides a systematic approach to searching optimal structure but also creates a series of novel multi-cell topological configurations with enhanced crashworthiness.