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(9) After each DPSO iteration, the local search is performed if the swarm has not find a new global minimum. The initial step size for the local search is set equal to 0.25 times the variable domain range, and it is reduced by 0.5 at each local search iteration. Local searches continue in each direction until the step size is greater than 10-3. If the local search stops without providing a new Figure 1. Example of PSSs global minimum, the actual global minimum is declared as a stationary point. The line search method is not allowed to violate the box constraints. 3.3 Static and Dynamic Metamodels In order to reduce the computational cost of the stochastic SBDO, metamodel techniques have been widely applied in several engineering fields, including naval applications [9,10]. The objective functions f and the constraints g in eqs. (1-5) are approximated by surrogate models (metamodels), versus both the design variables x and the environmental/stochastic parameters y. This is usually done by interpolating a set of CFD-computed values, which form the metamodel training set. In the static approach, the training set is defined a priori and the training points are usually spread in the whole x and y domains, without exploiting any knowledge of the desired functions. Although the use of a metamodel usually saves computational time, the static approach may result in a lack of information in noteworthy domain regions, along with a waste of simulations in regions of no concern (because far from the optimum or showing linear trends, etc.). (a) (b) (c) Figure 2. Uncertainty-driven sequential sampling using dynamic RBF: (a) M=7, (b) M=8, (c) M=9 In order to improve the metamodel-training efficiency, adaptive approach have been developed, based on sequential sampling methodologies exploiting the information that becomes available during the sampling process. An adaptive dynamic radial basis function (RBF) metamodel for UQ and optimization applications in ship hydrodynamics has been presented in [11]. The method uses a stochastic sample of RBF. Specifically, given a training set of M points, 1 {}Miix , with the associated simulation values () iiy f x the standard RBF gives the following approximation for every point in the domain: 1ˆ ()Miiif x w x x (10) where is the kernel function and wi are the coefficients of the combination. These are solution of the linear system that satisfies the condition of exact predictions at the training points, ˆ ( ) ( ) iif x f x . If one considers a kernel function of the type iix x x x with a stochastic exponent , distributed arbitrarily, then the expected prediction and its uncertainty may be easily evaluated by the Monte Carlo method over , for every x. It is worth noting that, since RBF is an interpolation method, the uncertainty at the training point is zero. Therefore, if UQ is performed, then the uncertainty in the prediction may be used to drive the sequential sampling process, in order to minimize the overall approximation uncertainty (as shown in Figure 2). If SBDO is performed, then a combination of prediction uncertainty and objective function value may be used for sequential sampling, in order to minimize both the approximation uncertainty and the objective function value at the same time. Details of the method and comparison with other dynamic and static metamodels may be found in [11]. 4. HYDRODYNAMIC DESIGN OF A HIGH-SPEED CATAMARAN SAILING IN REALISTIC, STOCHASTIC ENVIRONMENT In this section, the design optimization of the high-speed Delft catamaran is presented, considering a realistic stochastic ocean environment. The background for deterministic optimization is given by [12] and [13], where barehull and waterjet ducting system were respectively optimized, for calm water performance at Fr=0.5. In the references [14] and [15] the authors provide the extension to stochastic optimization, considering a 100 m ship length. Specifically, a multi-objective reliability-based robust design optimization (RBRDO) for the catamaran hull is presented, for (a) the reduction of the expected value of the total resistance in wave and (b) the increase of operability in the North Pacific Ocean, considering head waves and variable speed and sea state.