Gaussian-Process based optimization methods have become very popular in recent years for the global optimization of complex systems with high computational costs. These methods rely on the sequential construction of a statistical surrogate model, using a training set of computed objective function values, which is refined according to a prescribed infilling strategy. However, this sequential optimization procedure can stop prematurely if the objective function cannot be computed at a proposed point. Such a situation can occur when the search space encompasses design points corresponding to an unphysical configuration, an ill-posed problem, or a non-computable problem due to the limitation of numerical solvers. To avoid such a premature stop in the optimization procedure, we propose to use a classification model to learn non-computable areas and to adapt the infilling strategy accordingly. Specifically, the proposed method splits the training set into two subsets composed of computable and non-computable points. A surrogate model for the objective function is built using the training set of computable points, only, whereas a probabilistic classification model is built using the union of the computable and non-computable training sets. The classifier is then incorporated in the surrogate-based optimization procedure to avoid proposing new points in the noncomputable domain while improving the classification uncertainty if needed. The method has the advantage to automatically adapt both the surrogate of the objective function and the classifier during the iterative optimization process. Therefore, non-computable areas do not need to be a priori known. The proposed method is applied to several analytical problems presenting different types of difficulty, and to the optimization of a fully nonlinear fluid-structure interaction system. The latter problem concerns the drag minimization of a flexible hydrofoil with cavitation constraints. The efficiency of the proposed method compared favorably to a reference evolutionary algorithm, except for situations where the feasible domain is a small portion of the design space.
This paper investigates the use of constrained surrogate models to solve the multi-design optimization problem of a flexible hydrofoil. The surrogate-based optimization (EGO) substitutes the complex objective function of the problem by an easily evaluable model, constructed from a limited number of computations at carefully selected design points. Associated with ad-hoc statistical strategies to propose optimum candidates within the estimated feasible domain, EGO enables the resolution of complex optimization problems. In this work, we rely on Gaussian processes (GP) to model the objective function and adopt a probabilistic classification method to treat non-explicit inequality constraints and non-explicit representation of the feasible domain. This procedure is applied to the design of the shape and the elastic characteristics of a hydrofoil equipped with deformable elements providing flexibility to the trailing edge. The optimization concerns the minimization of the hydrofoil drag while ensuring a non-cavitating flow, at selected sailing conditions (boat speed and lifting force). The drag value and cavitation criterion are determined by solving a two-dimensional nonlinear fluid-structure interaction problem, based on a static vortex lattice method with viscous boundary layer equations, for the flow, and a nonlinear elasticity solver for the deformations of the elastic components of the foil. We compare the optimized flexible hydrofoil with a rigid foil geometrically optimized for the same sailing conditions. This comparison highlights the hydrodynamical advantages brought by the flexibility: a reduction of the drag over a large range of boat speeds, less susceptibility to cavitation and a smaller angle of attack tuning range.
The precise control of geometric models plays an important role in many domains such as Computer Aided geometric Design and numerical simulation. For shape optimisation in Computational Fluid Dynamics, the choice of control parameters and the way to deform a shape are critical. In this paper, we describe a skeleton-based representation of shapes adapted for CFD simulation and automatic shape optimisation. Instead of using the control points of a classical B-spline representation, we control the geometry in terms of architectural parameters. We assure valid shapes with a strong shape consistency control. Deformations of the geometry are performed by solving optimisation problems on the skeleton. Finally, a surface reconstruction method is proposed to evaluate the shape's performances with CFD solvers. We illustrate the approach on two problems: the foil of an AC45 racing sail boat and the bulbous bow of a fishing trawler. For each case, we obtained a set of shape deformations and then we evaluated and analysed the performances of the different shapes with CFD computations.
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