Deterministic SBDO can be then augmented in an ad hoc formulation that include uncertainties, leading to the development of a Stochastic-SBD Optimization framework (S-SBDO), with the goal of producing optimal designs relatively insensible to the stochastic variations of environment and operations, and safe with respect to degradation of the performances in off- design conditions. This can be realized by introducing user-defined probability density functions for some stochastic inputs, coming from real-life or full scale trials, to better characterize the effective environmental and operational conditions encountered by the ship. From the mathematical standpoint, the application of statistical decision theories to deterministic analysis requires the Uncertainty Quantification (UQ) of the simulation tools as a pre-requisite to Robust Design Optimization. The difficulty with exploiting this approach is mostly computational, since the solution always involves the integration of expensive simulation outputs with respect to uncertain quantities for the evaluation of mean, variance and distribution. If high-fidelity simulations are used, the evaluation of these integrals is very expensive, and for this reason, the application of statistical decision theory to high-fidelity robust design has been infrequently attempted and lies at the frontier of current computational science.

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:

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: