Comput Mech (2011) 47:251–263 DOI 10.1007/s00466-010-0532-y ORIGINAL PAPER Probability-based least square support vector regression metamodeling technique for crashworthiness optimization problems Hu Wang · Enying Li · G. Y. Li Received: 21 May 2009 / Accepted: 20 August 2010 / Published online: 7 October 2010 © Springer-Verlag 2010 Abstract This paper presents a crashworthiness design optimization method based on a metamodeling technique. The crashworthiness optimization is a highly nonlinear and large scale problem, which is composed various nonlinear- ities, such as geometry, material and contact and needs a large number expensive evaluations. In order to obtain a robust approximation efficiently, a probability-based least square support vector regression is suggested to construct metamodels by considering structure risk minimization. Fur- ther, to save the computational cost, an intelligent sampling strategy is applied to generate sample points at the stage of design of experiment (DOE). In this paper, a cylinder, a full vehicle frontal collision is involved. The results demonstrate that the proposed metamodel-based optimization is efficient and effective in solving crashworthiness, design optimization problems. Keywords Metamodeling technique · Support vector regression · Intelligent sampling · Crashworthiness · Optimization H. Wang (B ) · G. Y. Li (B ) Key Laboratory of Advanced Technology for Vehicle Body Design and Manufacture, College of Mechanical and Automobile Engineering, Hunan University, Changsha 410082, People’s Republic of China e-mail: wanghuenying@hotmail.com G. Y. Li e-mail: gyli@hnu.cn E. Li School of Logistics, Central South University of Forestry and Teleology, Changsha 41004, People’s Republic of China e-mail: enyasteven@hotmail.com Abbreviations AIMS Adaptive and interactive modeling system BBNS Boundary and best neighbor sampling D-OPT D-optimum DOE Design of experiment ERM Empirical risk minimization FE Finite element FF Full factorial KKT Karush–Kuhn–Tucker LHS Latin hypercube sampling LS-SVR Least square support vector regression PR-RSM Polynomial regression response surface methodology PSO Particle swarm optimization RBF Radial basis function PLS-SVR Probability-based least square support vector regression SM Space mapping SVR Support vector regression SSE Sum-squared error SRM Structure risk minimization 1 Introduction Crashworthiness optimization is of special interest in the automotive industry to archive a lightweight vehicle and insure the occupant safety in the event of the crash. How- ever, the crashworthiness optimization commonly requires many evolutions to obtain optimum results. With increas- ing of complexity and scale of finite element (FE) model, the computational cost becomes extremely expensive and unusable for application for real-world problems. Therefore, metamodeling techniques are often applied to these kinds of nonlinear design problems. In the past 20 years, several 123