Aerodynamic shape optimization by variable-fidelity computational fluid dynamics models: A review of recent progress
Introduction
Aerodynamic (or hydrodynamic) shapes and surfaces are encountered in numerous engineering systems, such as aircraft, automobiles, ships, rockets, bicycles, turbines, and pumps; just to name a few. The task of the aerodynamic engineer is to find a shape (or adjust the existing one) that improves a given aerodynamic measure of merit while adhering to appropriate constraints. The complexity of engineering systems is growing and computer simulations are needed to provide a reliable evaluation of system performance. Given the nonlinear behavior of fluid systems it is often an impossible task to improve a given design by using a hands-on approach. Numerical design techniques are, therefore, essential to assist the engineer in solving the challenging task. Aerodynamic shape optimization (ASO) involves the use of search algorithms for the design of aerodynamic surfaces. This paper provides a review of recent progress in this field. In particular, several variable-fidelity optimization algorithms, which have been shown to be very efficient, will be described and compared with benchmark techniques.
Hicks et al. [1] are generally credited for the first practical application of ASO. They used a conjugate-gradient method to design two-dimensional airfoil shapes in transonic flow. Later, Hicks and Henne [2] extend the work to three-dimensional transonic wing design with a steepest-descent gradient method. Nowadays, gradient-based methods are considered the state-of-the-art in ASO and are the most widely used approaches; see for example [3], [4], [5], [6]. The key to using gradient-based ASO is the adjoint approach, first introduced by Pironneau [7], and later developed for aerodynamic design by Jameson [8]. The main advantage is that the cost of a gradient calculation can be made nearly independent of the number of design variables. This opens the gateway for applying ASO to problems with a large design space.
Various other types of algorithms are used for ASO, such as derivative-free methods, one-shot methods, and surrogate-based methods. Evolutionary algorithms, such as genetic algorithms, are the most popular derivative-free methods for ASO; see for example Holst and Pulliam [9], and Epstein and Peigin [10]. The fundamental advantage of evolutionary algorithms (or, more broadly, population-based metaheuristics) over gradient-based ones is their ability to perform global search. However, this comes at a price since a large number of model evaluations are needed, especially for a large design space. One-shot methods are based on the same Lagrangian formulation as the gradient-based methods, but the flow equations and the first-order optimality conditions are solved simultaneously, and, thereby, avoiding repeated flow and gradient evaluations. An overview of the approach can be found in Gunzburger [11] and applications can be found in Gatsis and Zingg [12], and Iollo et al. [13].
In surrogate-based optimization (SBO), a computationally expensive model is replaced by a cheap surrogate model [14], [15]. The main objective is to accelerate the optimization process and obtain an optimized design by using fewer evaluations of the expensive model. Typically, the surrogates are functional ones, i.e., constructed by using design of experiments and data fitting. A variety of techniques are available to create function-approximation surrogate models. These include polynomial regression [14], radial basis function interpolation [15], kriging [16], and support vector regression [17]. Function-approximation models are versatile, however, they normally require substantial amount of data samples to ensure good accuracy. Examples of SBO with various function–approximation models related to aerodynamic design can be found in Forrester et al. [18], Jouhoud et al. [19], and Brooks et al. [20].
Variable-fidelity optimization (VFO) refers to a certain type of SBO where the surrogate models are constructed using corrected physics-based low-fidelity models [21], [22], [23], [24], [25], [26], [27], [28], [29]. The low-fidelity models can be obtained using one of, or a combination of the following: simplified physics models (also called variable-fidelity physics models), the high-fidelity model with a coarser computational mesh discretization (called variable-resolution models), and relaxed flow solver convergence criteria (called variable-accuracy models). The surrogate model needs to be a reliable representation of the high-fidelity model. This is typically achieved by correcting the low-fidelity model. Examples of correction techniques for aerodynamic models include space mapping (SM) [24], [27], shape-preserving response prediction (SPRP) [25], [28], and adaptive response correction (ARC) [29]. The key benefit of the VFO approach is that compared to function-approximation surrogates, less high-fidelity model data may be needed to construct a physics-based one to obtain a given accuracy level, which will lead to improved algorithm efficiency.
In this paper, we provide a brief summary of recently developed VFO algorithms for the design of aerodynamic surfaces. In particular, we describe the multi-level optimization (MLO) algorithm [23], and the SM [24] and SPRP [28] correction techniques. The algorithms are applied to aerodynamic shape optimization of transonic airfoils.
Section snippets
Aerodynamic shape optimization
This section provides a discussion on the basic properties and characteristics of aerodynamic surfaces pertaining to the geometry and the performance measures, as well as an example of design objectives. A mathematical formulation of the ASO problem is given.
Variable-fidelity solution approaches
In this section, we first discuss the SBO concept, which is the basis of VFO, and then we describe the three different VFO algorithms.
High-fidelity model
The flow is assumed to be steady and inviscid. The compressible Euler equations are taken to be the governing fluid flow equations.
Case descriptions
The three SBO algorithms described in Section 3 are applied to the same design cases. We consider three cases at transonic flow conditions. Case 1 involves lift maximization of an airfoil at a Mach number M∞ = 0.75, and an angle of attack α = 0°. The maximum allowable drag coefficient is set Cdw.max = 0.005, and the minimum non-dimensional cross sectional area is Amin = 0.075. Case 2 involves drag minimization of an airfoil at M∞ = 0.70 and α = 1°, with minimum lift coefficient Cl.min = 0.60 and Amin = 0.075.
Conclusion
A review of aerodynamic design using variable-fidelity optimization algorithms and computational fluid dynamic (CFD) models has been presented. We have discussed a multi-level design optimization algorithm, as well as space mapping and shape-reserving response prediction techniques. All three methods employ the same set of CFD models (from high- to low-fidelity), but differ in how the low-fidelity models are exploited and/or corrected. For algorithm performance illustration, two design cases of
Leifur Leifsson received a Ph.D. degree in Aerospace Engineering from Virginia Tech, USA, in 2006. He is currently an Associate Professor with the School of Science and Engineering at Reykjavik University (RU), Iceland. Leifur is the director of the Laboratory for Unmanned Vehicles at RU. His research interests include applied aerodynamics, surrogate-based modeling and optimization, multidisciplinary design optimization, and unmanned vehicle design. Prior coming to RU, Leifur held positions at
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Leifur Leifsson received a Ph.D. degree in Aerospace Engineering from Virginia Tech, USA, in 2006. He is currently an Associate Professor with the School of Science and Engineering at Reykjavik University (RU), Iceland. Leifur is the director of the Laboratory for Unmanned Vehicles at RU. His research interests include applied aerodynamics, surrogate-based modeling and optimization, multidisciplinary design optimization, and unmanned vehicle design. Prior coming to RU, Leifur held positions at Airbus UK Ltd., and Hafmynd Ltd.
Slawomir Koziel received the M.Sc. and Ph.D. degrees in electronic engineering from Gdansk University of Technology, Poland, in 1995 and 2000, respectively. He also received the M.Sc. degrees in theoretical physics and in mathematics, in 2000 and 2002, respectively, as well as the Ph.D in mathematics in 2003, from the University of Gdansk, Poland. He is currently a Professor with the School of Science and Engineering, Reykjavik University (RU), Iceland. He is the director of the Engineering Optimization & Modeling Center, RU. His research interests include CAD of microwave circuits, surrogate-based modeling and optimization, circuit theory, and numerical analysis.