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:

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):

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.

The basic elements of the method (presented in [6]) are easily summarized. The shape design problem (before starting the optimization phase) is considered affected by uncertainty: i.e. the optimal design in, obviously, unknown and a uniform probability of occurrence in the design space is assumed. The design space is then sampled randomly: a number S of geometries or are generated (but no objective function evaluation is performed!) by using    any arbitrary geometry deformation techniques (we have been using a Free Form Deformation technique, preferred since it allows high design flexibility and it is independent of grid topology) to sample the space.

摘要：本文描述一种在真实海况下对船舶的不确定性进行量化与设计优化的先进有效工具的发展过程。本文对比用于评估计算目标函数和约束条件的不同方法：用昂贵且高精度的方法比用现有方法区别在前者需要一个优化问题和解决问题的关键数学表达式，而用非定常雷诺兹平均Navier Stokes方程求解器（URANS）改进目标函数和约束条件的方法，也需要一个优化问题和解决问题的关键数学表达式。为了降低随机优化过程中的成本，作者和他们的同事已经开发一些改进措施：1）对高精度动态模型求解和随机模拟输出进行不确定性的量化；2）发展研究全局优化的进化型无导数算法；3）用Karhunen-Loève Expansion (KLE)方法提前找出减少大型设计空间维度方法从而减少对解决问题意义不大的基础功能（即设计变量）。最后给出了真实海况下船舶水动力设计优化的实例。

1.引言

近几十年内，人们不仅在海洋上进行广泛的运输和贸易，同时还在海洋里获取海洋食品、利用可再生能源发电、进行矿产开采。为了应对这种发展趋势，需要更自动化的设计过程。为了提高精度和稳定性，系统也要有能进行精确模拟的工具。在船舶流体力学中，为了应对这一趋势，最新应用工程是发展确定性模拟设计（SBD）。所有大型造船厂不断发展中大型计算模拟系统，想通过设计一个先验方法，提前评价替代设计的性能和相对优点。