Multiobjective reliability-based optimization for design of a vehicle door Jianguang Fang a , Yunkai Gao a,n , Guangyong Sun b,nn , Qing Li c a School of Automotive Studies, Tongji University, Shanghai 201804, China b State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha 410082, China c School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia article info Article history: Received 27 April 2012 Received in revised form 19 November 2012 Accepted 20 November 2012 Available online 2 January 2013 Keywords: Multiobjective reliability-based design optimization Automotive door Response surface method Probabilistic sufficiency factor Uncertainty abstract Structural optimization for vehicle door signifies one of the key topics of research to continuously improve its performances. However, majority of the studies to date have not considered uncertainties whilst it has been known that a deterministic optimization may lead to an unreliable design in practice. In this paper, a multiobjective reliability-based design optimization (MORBDO) procedure is proposed to explore the design of vehicle door. To improve the efficiency of optimization, response surface method (RSM) is used to replace the time-consuming finite element simulations. In conjunction with Monte Carlo simulation and descriptive sampling technique, probabilistic sufficiency factor is adopted as a design constraint. The multiobjective particle swarm optimization (MOPSO) algorithm is employed to perform the optimization. The results demonstrate that the proposed optimization procedure is capable of generating a well-distributed Pareto frontier of reliable solutions, and it is suggested to select an optimum from relative insensitive regions. Moreover, the influence of varying the uncertainty and increasing the target reliability level in the optimization results is analyzed, which provided decision- makers with insightful design information. & 2012 Elsevier B.V. All rights reserved. 1. Introduction Being an indispensable assembly of an automotive body, vehicle door serves as a key supporting component for functional accessories and sound insulation in occupant compartment. Poor performances of a door would lead to lots of functional problems such as bad sealing, abnormal sounds and severe intrusion in crashing. For this reason, structural optimization of a vehicle door has become one of the major concerns in automotive engineering. For example, Shin et al. [1] suggested a design procedure by integrating topology, shape and size optimization and design of experiments (DoE) to develop door structure for better stiffness and natural frequency. Song and Park [2] employed multi- disciplinary optimization (MDO) with multicriteria for weight reduction of a door made of a tailored blank. Lee and Kang [3] combined the Kriging interpolation method with a simulated annealing algorithm for design of frontal door. Zhu et al. [4] integrated finite element analysis (FEA), artificial neural network, and genetic algorithm for optimal design of an inner door panel. Cui et al. [5] developed multi-material configuration for light- weight design by combining a multiobjective genetic algorithm with an artificial neural network. These above-mentioned studies on structural optimization of vehicle doors are restricted to deterministic optimization, in which all design variables and parameters involved are regarded certain. In real world, a design optimization may not afford to neglect uncertainties, which can largely exist in material proper- ties, geometries and manufacturing precision, etc. In effect, the optimum obtained from a deterministic optimization could easily violate the imposed constraints and cause unreliable solutions [6–8]. In order to take into account various uncertainties, reliability-based design optimization (RBDO) was introduced and has drawn increasing attention recently. Compared with deterministic optimization, RBDO aims to seek a reliable opti- mum by converting the deterministic constraints into probabil- istic counterparts, in which failure probability is restricted to a pre-defined level. In this regard, Zhang and Liu [9] employed the second moment and reliability-based design theory to present a practical and effective method for the reliability-based design of automobile components. Acar and Solanki [10] performed vehicle RBDO for crashworthiness and analyzed the effect of reliability allocation on different failure modes. Song et al. [11] generated RBDO for an automotive knuckle component under different conditions, where a constraint-feasible moving least square method was used for modelling functional inequality constraint. d’Ippolito et al. [12] utilized the RBDO methodology to optimize the reliability of a vehicle knuckle for fatigue life, where the variability in the material parameters were considered. Ju and Lee [13] developed a Kriging metamodel for an active constraint Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design 0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.11.007 n Corresponding author. Tel./fax: þ86 21 6958 9845. nn Corresponding author. Tel.: þ86 731 8882 2051; fax: þ86 731 8881 1445. E-mail addresses: gaoyunkai@tongji.edu.cn (Y. Gao), sgy800@126.com (G. Sun). Finite Elements in Analysis and Design 67 (2013) 13–21