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. SWATH的参数化设计和多目标优化英文文献和中文翻译(5):http://www.youerw.com/fanyi/lunwen_63562.html