Research Article Reliability-Based Topology Optimization Using Stochastic Response Surface Method with Sparse Grid Design Qinghai Zhao, 1 Xiaokai Chen, 1 Zheng-Dong Ma, 2 and Yi Lin 3 1 Collaborative Innovation Center of Electric Vehicles in Beijing, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China 2 Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105, USA 3 Beijing Automotive Technology Center, Beijing 100081, China Correspondence should be addressed to Xiaokai Chen; chenxiaokai@263.net Received 18 July 2014; Revised 11 November 2014; Accepted 27 November 2014 Academic Editor: Chao Jiang Copyright © 2015 Qinghai Zhao et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A mathematical framework is developed which integrates the reliability concept into topology optimization to solve reliability-based topology optimization (RBTO) problems under uncertainty. Two typical methodologies have been presented and implemented, including the performance measure approach (PMA) and the sequential optimization and reliability assessment (SORA). To enhance the computational efciency of reliability analysis, stochastic response surface method (SRSM) is applied to approximate the true limit state function with respect to the normalized random variables, combined with the reasonable design of experiments generated by sparse grid design, which was proven to be an efective and special discretization technique. Te uncertainties such as material property and external loads are considered on three numerical examples: a cantilever beam, a loaded knee structure, and a heat conduction problem. Monte-Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach. Based on the results, it is demonstrated that application of SRSM with SGD can produce an efcient reliability analysis in RBTO which enables a more reliable design than that obtained by DTO. It is also found that, under identical accuracy, SORA is superior to PMA in view of computational efciency. 1. Introduction Te subject for optimal structural topologies under uncer- tainty is very important, challenging, and attractive for researchers. Te feld of structural topology optimization has become matured since the pioneering work by Bendsøe and Kikuchi [1]. Details of various proposed methodologies can be found in comprehensive reviews and text books [2–4]. Te aim of the optimization process is to obtain a material distribution within a fxed design domain, so as to optimize the specifed structural response subjected to prescribed constraints. However, most of the optimization problems are established in deterministic manner where the designs are carried out without considering the variations observed in geometry and material property as well as external loads due to inherent uncertainty. Consequently, optimum design obtained by the so-called deterministic topology optimiza- tion (DTO) may represent unreasonable reliability level. Hence, reliability-based topology optimization (RBTO) has emerged which achieves optimal topologies while quanti- tatively measuring the efects of uncertainty by means of probability constraints. Te state-of-the-art topology optimization methodolo- gies have been applied for obtaining optimized reliable designs of structures and mechanisms considering uncertain- ties. Density-based method taking into account loading and material uncertainties has been demonstrated for microelec- tromechanical systems (MEMS) [5], for geometrically non- linear structures [6]. A level set based approach for compliant mechanisms under uncertainties exhibited by loads, material properties, and member geometries has been reported by Zhang and Ouyang [7]. Recently, topology optimization with uncertainties has been demonstrated for the bidirectional evolutionary structural optimization (BESO) in the design for electrothermal compliant mechanisms [8] and for a vehicle’s hood reinforcement [9]. Although the RBTO is a rapidly Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 487686, 13 pages http://dx.doi.org/10.1155/2015/487686