Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Full length article Computational efficiency and accuracy of multi-step design optimization method for variable stiffness composite structures Mohammad Rouhi ⁎ , Hossein Ghayoor, Suong V. Hoa, Mehdi Hojjati Department of Mechanical and Industrial Engineering, Concordia Center for Composites, Concordia University, Montreal, Quebec, Canada, H3G 1M8 ARTICLE INFO Keywords: Design optimization Metamodeling Variable stiffness composites Fiber steering Buckling ABSTRACT A multi-step metamodel based optimization method is presented for the buckling design of a variable stiffness composite cylinder. Followed by sampling, metamodeling and optimization at the first step, the design domain is shrunk by narrowing down the side constraints of the design variables around the optimum points resulted from the previous steps. This procedure is repeated until a convergence is reached for both the design variables and the objective function. The structural response of the optimum variable stiffness composite cylinder resulting from this method is shown to significantly improved compared with its constant stiffness counterpart. The computational efficiency along with the accuracy of this method is also assessed by analyzing the metamodeling and evaluation errors in different optimization steps and different sample sizes. The results show that the multi- step method is considerably more efficient than a single-step optimization method. 1. Introduction Variable stiffness (VS) composites have attracted intensive attention in the past two decades due to the tailorability of stiffness and strength in structures made by these materials. The fibers/tows are steered by automated fiber placement (AFP) machines in the layup process to produce plies with continuously variable fiber orientation angle. The resulting VS laminates gives the designers more room to extend the design space and fully exploit the directional properties of composite materials [1]. As a result, for structures made by these materials a more efficient load path between the loading points and the supports can be created offering a significantly improved performance compared with their constant stiffness (CS) counterparts at the same weight [2–6]. Design optimization of VS composite cylinders for buckling was first studied by Tatting [7]. It was followed by many researchers who investigated the potential structural improvement of VS composite cylinders with cylindrical [5,6,8,9] and elliptical [10,8] cross-sections. Stiffness tailoring resulting from the design optimization process, helps the sectional resultant forces to be redistributed over the area of the cylinders such that a larger portion of the circumference becomes involved in carrying the buckling load. In such a way, the directional properties of the composite material is used more efficiently and the buckling load is improved as a result. There are different methods for optimization of VS structures that were reviewed in [11]. Due to the complexity and computational costs involved in the design optimization of VS composite structures, it usually includes metamodeling. Due to the iterative nature of the design optimization processes, it is extremely important for function calls to be inexpensive in each iteration. Metamodels, i.e. simple analytical functions that approximate a high fidelity model, can be used as surrogate models for expensive high fidelity functions in the optimiza- tion process to search for the optimum design, which is referred as metamodel-based design optimization (MBDO) [12]. Approximation- based optimization methods have intensively attracted the designers' attention in manufacturing companies, as a tool, to produce high quality and low cost products more quickly. In a number of reviews [13,12,14] the accuracy, robustness and computational performance of different metamodeling techniques including response surface method (RSM), radial basis functions (RBF), Kriging (KRG) and Multivariate Adaptive Regression Splines (MARS) were compared to each other and it was shown that the RBF is the best choice for metamodeling in terms of the overall performance and efficiency. Metamodeling has been shown to be very efficient for computationally intensive design optimization problems in different engineering applications including, but not limited to, design of multiscale composite materials for buckling [15] or energy absorption [16], design for crashworthiness [17] and design of variable stiffness composite structures [6,9,18–20]. In a comparative study, the performance of different metamodeling meth- ods in the design optimization of variable stiffness composite structures were compared [21]. Their results also showed that radial basis functions (RBF) and Kriging (KRG) metamodels give the best results in terms of accuracy and suitability for designing VS composite http://dx.doi.org/10.1016/j.tws.2017.01.019 Received 16 June 2016; Received in revised form 23 December 2016; Accepted 13 January 2017 ⁎ Corresponding author. E-mail address: m_rouh@encs.concordia.ca (M. Rouhi). Thin-Walled Structures 113 (2017) 136–143 0263-8231/ © 2017 Elsevier Ltd. All rights reserved. MARK