RESEARCH PAPER Model uncertainty approximation using a copula-based approach for reliability based design optimization Hao Pan 1 & Zhimin Xi 1 & Ren-Jye Yang 2 Received: 21 December 2015 /Revised: 12 May 2016 /Accepted: 21 June 2016 # Springer-Verlag Berlin Heidelberg 2016 Abstract Reliability-based design optimization (RBDO) has been widely used to design engineering products with mini- mum cost function while meeting reliability constraints. Although uncertainties, such as aleatory uncertainty and epi- stemic uncertainty, have been well considered in RBDO, they are mainly considered for model input parameters. Model un- certainty, i.e., the uncertainty of model bias indicating the inherent model inadequacy for representing the real physical system, is typically overlooked in RBDO. This paper ad- dresses model uncertainty approximation in a product design space and further integrates the model uncertainty into RBDO. In particular, a copula-based bias modeling approach is pro- posed and results are demonstrated by two vehicle design problems. Keywords Model uncertainty . Reliability-based design optimization . Copula modeling . Bias correction . Vehicle design 1 Introduction Reliability based design optimization (RBDO) has been wide- ly employed for engineering product design to minimize the cost function while satisfying the reliability constraints. Research on RBDO can be mainly categorized into three areas: i) optimization strategy development, ii) improvement of algorithm accuracy and efficiency, and iii) uncertainty modeling. The first area includes many efforts for optimiza- tion strategy evolvement from double loop (Youn et al. 2003, 2005; Yang and Gu 2004), decoupled double loop (Zou and Mahadevan 2006; Agarwal and Renaud 2006; Agarwal et al. 2007), to single loop RBDO (Liang et al. 2008; Shan and Wang 2008; Nguyen et al. 2011). The second area focuses on the development of reliability analysis algorithms such as the sampling approach (Bucher 1988; Melchers 1989; Au and Beck 1999), the most proper point (MPP)-based approach (Lee et al. 2008, 2010; Zhang and Du 2010), the classical response surface approach (Bucher and Bourgund 1990; Kaymaz 2005; Kaymaz and McMahon 2005), the dimension reduction method (Rahman and Xu 2004; Xu and Rahman 2004; Youn et al. 2008), the polynomial chaos expansion (Crestaux et al. 2009; Oladyshkin and Nowak 2012; Wei et al. 2008), etc. The third area deals with uncertainty model- ing of model input parameters such as aleatory and epistemic uncertainty modeling using probability estimation methods, Bayes statistics, and interval uncertainties (Zhang and Mahadevan 2000; Wang et al. 2009; Du et al. 2005; Zhang et al. 2010). Typically, RBDO is conducted based on a simulation mod- el (e.g., a finite element model, an analytical model, a com- puter fluid dynamics model, etc.) and the model is assumed to be an accurate representation of the real physical system. However, this assumption may not be valid in many engineer- ing design applications, which could result in significant * Zhimin Xi zxi@umich.edu Hao Pan hapan@umich.edu Ren-Jye Yang ryang@ford.com 1 Department of Industrial and Manufacturing Systems Engineering, University of Michigan – Dearborn, Dearborn, MI 48168, USA 2 Ford Research & Advanced Engineering, MD2115-RIC, 2101 Village Road, Dearborn, Michigan 48121, USA Struct Multidisc Optim DOI 10.1007/s00158-016-1530-2