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Figure 3. The S-SBDO framework The S-SBDO toolbox (Fig. 3) is made of three essential parts: the design modification tool, the (deterministic and stochastic) analysis tools, and the optimization algorithm. The reduced-dimensionality design space obtained by using KLE is selected to retain 95% of the original design variability (reducing the design variables from 20 to 4 !). The analysis tool used is CFDShip Iowa (v4.5),a high - -fidelity URANS solver. UQ methods include Monte Carlo simulation, importance sampling, and regular wave/correction models. DPSO ([7]) is used as optimization algorithm, including synchronous/asynchronous formulations for single- and multi-objective problems. Static and dynamic metamodels are used to increase the efficiency of both the UQ and optimization procedure. The North Pacific Ocean is assumed as the operations environment, and the annual probability of occurrence of sea states is used as shown in [14]. Considering stochastic operations and environment, a stochastic form of the RBRDO type (see Eq. 5) is used where –ϕ2 is the reliability of the design with respect to the set of constraints hk. These constraints are based on subsystem’s seakeeping performance criteria and address mobility (MOB), anti-submarine warfare (ASW), surface warfare (SUW) and anti-air warfare (AAW), as per the standardization agreement NATO STANAG 4154. Constraints pertain to the single significant amplitude (SSA = 2 RMS) of motion-related variables. It is worth noting that in this particular case of design optimization, the design reliability coincides with the ship operability. Ship’s main particulars, design variables and bounds, and geometric constraints gk may be found in [12] and [15]. Finally, the single-objective RBRDO problem of Eq. 9 is formulated and solved as a multi-objective problem as: Ideally, mean resistance in wave T R and motion-related constraints hk are evaluated by irregular wave simulations at each sea state and speed. Considering numerical integration including 5 sea states (from 1 to 7) and at least 10 different speeds, each geometry would require >400k CPU hours. Herein, the following assumptions and approximations are used to reduce the computational cost and the overall wall clock time, making the overall S-SBDO feasible: (A) In order to compute eq. 11, the expected value of the mean resistance in wave is evaluated considering only sea state 5, as this is the expected sea state for operations in the North Pacific Ocean. A regular wave model is used for optimization, coupled with metamodel-based UQ [14], providing: 1ˆ ( ) ( , , ) ( ) TUx R x S U p U dU (15) With a training set of 6 speeds, 5k CPU hours are required to solve eq. (15) for each geometry. (B) At each sea state and speed, the functional constraints related to the motions are evaluated by a metamodel-based UQ procedure for stochastic wave frequency, based on RMS transfer functions [14].
These are assessed at Fr=0.425 (speed near to constraints violation) and sea state 5 (expected sea state) and translated over the whole range of speed and sea states using a correction model. This latter model has been specifically developed for optimization and presented in [14]: it is based on a training set of 272 URANS simulations in regular wave, over the sea state and speed range, interpolated by a thin-plate spline which provides exact prediction at training conditions. Details may be found in [14]. Herein, using a training set of 9 regular wave simulations with the correction model, for the solution of eq. 14, each geometry requires 8k CPU hours. The definition of the conditions for performance assessment follows the concept of the operability cube presented in [15]. Accordingly, the operational speed is assumed to be uniformly distributed from 20% to 100% of the maximum design speed (0.115 ≤ Fr ≤ 0.575); the stochastic sea state ranges from 1 to 7 and follows the probability of occurrence in the North Pacific Ocean. The current work focuses only on head waves with the aim of reducing the computational work.. The design optimization is conducted using the framework in Figure 3, by KLE-reduced dimensionality space, stochastic regular wave UQ models, sequential metamodel training and multi-objective extension of DPSO. An initial training and two refinements are used, for a total of 120 designs, evaluated at an overall computational cost greater than 1.5M CPU hours. The final Pareto front of trade-off solutions is shown in Figure 4. The selected design shows concurring improvements for resistance and operability. This achieves more than 10% reduction for the expected value of the resistance in wave and nearly 6% increase in operability. The original and optimized hulls are compared in Figure 5, showing a quite unconventional optimized design, stemming from the reduced-dimensionality modification method used. Fig. 6 shows an example of the analysis: the high speed catamaran in head waves, heave and pitch motions time history and the instantaneous pressure on the hull, and Fig. 7 refers to a verification in calm water, showing the comparison between original and optimal wave pattern. A more detailed overview of the S-SBDO results are provided in [16]. Additionally, the optimized geometry will be built and tested in the near future in the INSEAN towing tank, for validation of the current achievements. Figure 4. Pareto front of trade-off designs. Figure 5. Original and optimized hulls. 5. CONCLUSIONS High-fidelity, deterministic SBDO is extended to a Stochastic-SBDO, to address the uncertainty that stems mainly from stochastic environment and operations, with the objective of producing designs with robust and reliable performance. This approach requires Uncertainty Quantification (UQ) analysis of the simulation tools and leads to Reliability-Based Robust Design Optimization (RBRDO) formulations. Each of the essential elements of the S-SBDO framework (geometry manipulation, stochastic UQ and optimization algorithms) needs to be accurate and efficient. Recent advancements on design-space dimensionality reduction, efficient metamodel-based UQ methodologies, and optimization methods are synthetically presented. A complex application is solved: the RBRDO of a high-speed catamaran, with user-defined probability density functions for the stochastic inputs of interest. The optimized design achieves more than 10% reduction for the expected value of the resistance in wave and nearly 6% increase in operability, considering realistic environment and operations in the North Pacific Ocean. The S-SBDO needed more than 1.5M CPU hours.