Robust optimization for ship conceptual design Matteo Diez à , Daniele Peri INSEAN—The Italian Ship Model Basin, Resistance and Optimization Department, Via di Vallerano 139, 00128 Rome, Italy article info Article history: Received 31 December 2009 Accepted 22 March 2010 Keywords: Robust design optimization (RDO) Ship conceptual design Particle swarm optimization (PSO) abstract The paper presents an approach for the robust optimization of a bulk carrier conceptual design, subject to uncertain operating and environmental conditions. The uncertainties involved in the optimization process are addressed and a general formulation for robust design is given. Specifically, the uncertainties involved in the decision making process are taken into account by means of their probabilistic distributions. The expected values and the standard deviations of the relevant quantities are assessed and included in the optimization objectives, whereas the constraints are evaluated in the worst case. This leads to a robust design able to keep a good performance in the whole probabilistic operating scenario. A particle swarm optimization algorithm is used for the global minimization process, minimizing the expectation and the standard deviation of the unit transportation cost. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Over the years, optimization has been playing an increas- ingly important role in engineering. Advanced modeling and algorithms in optimization constitute now an essential part in the design and in the operations of complex aerospace (Hicks and Henne, 1978; Sobieszczanski-Sobieski and Haftka, 1997; Alexan- drov and Lewis, 2002; Willcox and Wakayama, 2003; Morino et al., 2006; Iemma and Diez, 2006) and automotive (Baumal et al., 1998; Kodiyalam and Sobieszczanski-Sobieski, 2001) applications, when, for example, it is by all means important to reduce costs and shorten time of development. In the design of large and complex systems, the use of efficient optimization tools leads to better product quality and improved functionality. The challenge is typically to model complex and frequently non- linear systems and structures to emulate and simulate possible operating conditions under a wide range of scenarios in order to allow for more intelligent design and optimization of these technologies. The recent years have seen some progress in optimization for ships too (Ray et al., 1995; Peri and Campana, 2003, 2005; Parsons and Scott, 2004; Pinto et al., 2004; Campana et al., 2007, 2009; Papanikolaou, 2009). However, arguably, the initial excitement that accompanied the emergence of these techniques has diminished somewhat over the years, due to the fact that these methods are not as generally accepted or widely used in practical ship design as the optimization community initially hoped. The explanation is not straightforward. It is certainly true that there are fundamental analytical and computational obstacles that must be overcome before optimization can make a widespread impact on the practice of ship design. Furthermore, robust and automated grid generation and manipulation has proved to be a serious challenge as well as the need to account for complex, real- industrial geometrical and functional constraints, together with the difficulty of generating the objective functions and their derivatives automatically and robustly when these functions are computed by solving systems of partial differential equations (PDE). The potential benefits and pay-offs of the impact of the optimization on the ship design process are so great, however, that despite the damping effects of reality on the immediate expectations, research on optimization has continued, yielding promising results and revealing specific new challenges and directions of research. The present paper presents an application of optimization for ship conceptual design. The standard deterministic formulation for design optimization is extended to take into account the uncertainty related to the design variables, to the operating conditions, and to the computational results of the simulations. In the standard approach to the conceptual optimal design, all the relevant quantities and models involved in the process are taken into account from a deterministic viewpoint. The operating conditions are statically specified, a mathematical programming problem is defined, and the final solution is obtained within this rigid framework. This means that objectives and constraints are evaluated through a ‘‘static’’ or deterministic assessment of the problem. As a consequence, no information is available about the performance in off-design conditions. Moreover, the design tools are not considered as a potential source of uncertainty, and the accuracy of the analysis is not assessed during the optimization process. Furthermore, the effects of small variations of the design ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/oceaneng Ocean Engineering 0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.03.010 à Corresponding author. Tel.: + 39 06 50299314; fax: + 39 06 5070619. E-mail addresses: m.diez@insean.it (M. Diez), d.peri@insean.it (D. Peri). Ocean Engineering 37 (2010) 966–977