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This paper will present and discuss some of the fundamental aspects in building a S-SBDO framework designed specifically for ships, reporting results collected from author’s recent publications supported by a series of national and international projects. 2. INCLUDING UNCERTAINTY IN THE SIMULATION-BASED HYDRODYNAMIC DESIGN OPTIMIZATION Nowadays, the process of designing complex engineering systems - such as ships and off-shore platforms - has been substantially modified with the advent of simulation tools, driven by two major elements: (i) an increased robustness and accuracy of the numerical algorithms on which the simulations are based and (ii) an exponential development of the hardware, including the fast development of parallel architectures. Below the surface of this design revolution, there is the constant search for improvements - even marginal - imposed by the global market competition: forced by the need of finding better designs one is prone to accept larger design spaces, more design variables, and more alternatives have to be explored and compared. Despite the increased computational power and robustness of numerical algorithms, high-fidelity SBDO for shape optimization still remains a challenging process, from theoretical, algorithmic and technological viewpoints: searching high-dimensional, large design spaces when using high-fidelity computationally-expensive black-box functions trying to solve a stiff optimization problem in which the computation of an objective function has been transformed into the evaluation of an integral, whose kernel is the product of the objective function with some probability density function to include uncertainty. 2.1 Mathematical Formulation of the Optimization Problem under Uncertainty The general formulation of a robust optimization problem starts from a deterministic one: ˆ Minimize ( , ), for a given ˆ Subject to ( , ) 0, 1,...,xAnf x y y y Bg x y n N (1) where x is the design variables vector (intended as the designer choice) and y is the design parameters vector collecting those quantities that are independent of the designer choice (e.g., environmental and operational conditions). It is then possible to introduce several sources of uncertainty: (a) x is affected by a stochastic error (e.g. tolerance of the design variables); (b) y is an intrinsic stochastic random process (i.e. the environmental and operational conditions are given in terms of probability density); (c) the evaluation of objective f and constraints g is affected by an error due to inaccuracy in modeling or computing. To formulate a stochastic optimization problem, the probability density function (PDF) of y, p(y), has to be evaluated or given somehow. The UQ consist in evaluating the PDFs and the related moments of the functions f and g. The first two moments of f (and, similarly, of g) are: 2 2( ) : ( , ) ( )( ) : ( , ) ( )BBf f f x y p y dyf f x y f p y dy
(2) The task of computing the integrals in (2) is usually computational highly expensive due to the computational cost of the high-fidelity solvers used. The RDO problem If the objectives are defined in terms of the first two moments of the original objective function f and the constraints are still given in terms of deterministic inequalities constraints (y is a user specified deterministic design condition), the RDO problem can be simply stated as: 2nMinimize ( , ) and ( , )ˆ Subject to g ( , ) 0, 1,...,xAf x y f x yx y n N (3) The RBDO problem The task here is handling the constraints, which are now defined in terms of probabilistic inequalities . The RBDO problem can be formulated as (P0 is a target probability or reliability): 0x AMinimize ( , )subject to ( , ) 0 , 1,...,nf x yP g x y P n N (4) The RBRDO problem Finally, both the constraints and the objective function are defined in terms of stochastic variables. The RBRDO problem can be formulated as a combination of eqs. 3 and 4 as: 3. MAKING HIGH-FIDELITY, STOCHASTIC SBD OPTIMIZATION AFFORDABLE The solution of S-SBDO involves the integration of expensive simulation outputs, for the evaluation of mean, variance, and distribution. To enhance the computational efficiency some methods have been developed by the authors and are illustrated in the following (geometry and grid modification methods will not be illustrated here, see [1] for these techniques). 3.1 Reducing the Design Space Dimensionality with KLE (Karhunen–Lòeve Expansion). When the number of design variables is large (because of the complexity of the design, or because one is searching for large final improvements), the solution of the optimization problem becomes quickly extremely expensive. In a nutshell, what KLE provides is a tool for reducing this complexity by selecting a reduced number of design variables, and at the same time, giving the guarantee that a desired maximum geometrical variance is maintained. Even more important is to understand that results is provided by KLE without computing the objective function(s). The entire procedure is an “a priori” analysis of the geometries populating the original design space. The design space is populated by random geometries, and an eigenvalue problem is defined to analyze the statistical properties of these random designs, with focus on their geometrical variance.