二维可压缩流场下气动外形优化的Newton-Krylov方法

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"《最优气动布局设计：牛顿-克利夫算法》是一篇关于复杂气动外形优化设计的研究论文。作者Marjan Nemec针对非线性效应强烈的气流特性，如冲击波、边界层和分离，提出了一个基于梯度数值优化的牛顿-克利夫算法。研究的核心是将二维可压缩纳维-斯托克斯方程与一阶湍流模型结合，采用多块结构网格进行离散化处理。在计算目标函数梯度时，采用了离散雅可比方法，并利用预条件广义最小残差（GMRES）求解雅可比方程。论文创新地引入了一种新颖的预条件器，结合对离散纳维-斯托克斯方程和湍流模型的完整导数计算，极大地提高了梯度评估的精度和效率。通过这种方法，可以在达到收敛所需的五分之一到一半时间内得到梯度。在实际应用方面，该算法展示了在提高升力、在多个跨音速工作点下实现升阻最小化、以及根据竞争目标计算帕累托前沿等设计问题上的性能。通过惩罚形式强加优化约束，将有约束问题转化为无约束问题，再利用拟牛顿方法求解。结果表明，新算法显著降低了梯度值，说明找到了局部最优解，从而证明了其在气动外形优化领域的高效性。这项研究提出的方法对于当前的气动形状优化来说具有很高的速度优势，为实际气动设计提供了一种有效且实用的工具。它能够在保证设计精度的同时，显著缩短计算时间，使得复杂气动外形的设计过程更加便捷和高效。"

2 Chapter 1. INTRODUCTION

ﬂow features include regions of separated ﬂow, conﬂuent boundary layers and wakes, and even

regions of supercritical ﬂow where compressibility eﬀects are important. Such ﬂow features have

a strong inﬂuence on the aerodynamic performance of a conﬁguration due to their nonlinear

eﬀects. Consequently, the nonlinear eﬀects must be carefully controlled by the shape of the

conﬁguration in order to realize optimal performance. Note that the design problem inherently

involves multiple operating conditions.

The design of practical conﬁgurations with optimal aerodynamic properties requires the

consideration of not only aerodynamics, but also several other disciplines, namely, structures,

acoustics, and controls [161]. The aero-structural coupling is perhaps most signiﬁcant, since

an optimal aerodynamic design may incur excessive weight penalties [109, 129]. Furthermore,

the design is constrained by numerous engineering requirements [60], such as the desirable fuel

tank volume within the wing. Jameson and Vassberg [88] present an insightful example of

multidisciplinary design based on requirements for a small race plane.

Although aerodynamic design is a complex task, the incentive for the development of ef-

fective design strategies is substantial. The potential beneﬁts include safer and more eﬃcient

aircraft and shorter design-cycle times. For example, an eﬃcient high-lift conﬁguration can sig-

niﬁcantly improve the aerodynamic performance of an aircraft, as well as provide weight savings

and reductions in mechanical complexity of the high-lift system [172]. For cruise conﬁgurations,

the beneﬁts include low drag and improved fuel burn, an increase in the drag-divergence Mach

number, and also sonic-boom reductions for the design of quiet supersonic platforms [85, 4].

Experimental methods, using wind tunnel and ﬂight testing, provide the basis of the tradi-

tional “cut and try” approach for the design of new aerodynamic shapes. This approach alone,

however, is too expensive, which has motivated the continual development of sophisticated com-

putational methods for ﬂow simulation [104] and established the ﬁeld of computational ﬂuid

dynamics. These methods, along with performance gains in computer technology, complement

and in some instances even replace the use of the wind tunnel during the design process. In this

setting, the experimental and computational techniques are analysis to ols that provide reliable

estimates of aerodynamic performance for given conﬁgurations and operating conditions. A

good physical insight of the designer is required to select and evolve the candidate aerodynamic

shapes and provide an overall control for the design process.

The fundamental question that the design process seeks to answer is the following:

What is the aerodynamic shape that attains the design objectives?

The “cut and try” approach does not address this question directly, resulting in an ineﬃcient

design process. Consequently, a signiﬁcant research eﬀort has been devoted to the development

of computational methods for the solution of the design problem. The resulting methods can

1.1 MOTIVATION 3

be divided into two categories, namely, inverse design methods and numerical optimization

methods.

Inverse design methods, ﬁrst introduced in 1945 by Lighthill [102], are an established ap-

proach for the determination of an airfoil shape that attains a given pressure distribution. For

example, an experienced designer is able to specify a lift-preserving pressure distribution that is

shock-free for transonic ﬂow conditions, thereby achieving signiﬁcant drag reductions. Inverse

methods were also used to design the well-known Liebeck high-lift airfoils [101]. An advantage

of this approach is its low computational cost. Giles and Drela [61] developed an inverse de-

sign method using the two-dimensional, coupled Euler and boundary layer equations with a

computational cost equivalent to the solution of just one analysis problem. Although inverse

design methods have been applied to three-dimensional problems [55], their primary limitation

is the speciﬁcation of desirable pressure distributions that lead to optimal designs, especially

for problems with multiple operating conditions and turbulent and separated ﬂow.

Numerical optimization metho ds provide a more general approach for solving design prob-

lems. The creativity and insight of an experienced designer are required to reduce the design

problem to a well-posed optimization problem. This involves the deﬁnition of objective func-

tions that specify the goals of the optimization, design variables that determine the aerodynamic

shape, as well as constraints that qualify a feasible region of the design space. Note that for

practical problems, it is very likely that the objectives are competing and that changes in the

speciﬁcation of the optimization problem occur as the design evolves. Typically, the problem is

cast as a minimization, where the objective functions include lift, drag, and moment functionals.

Inverse design can be considered a subset of numerical optimization by deﬁning an objective

function that represents the diﬀerence between the target and the actual pressure distributions.

Once an aerodynamic shape optimization problem is deﬁned, a numerical optimization

method coupled with a suitable ﬂow analysis tool (ﬂow solver) is used to ﬁnd a solution. The

goal of the optimization is to determine a set of design variables that satisﬁes the objectives

within the feasible region of the design space. This approach provides an eﬀective design

strategy, since the selection of an optimal conﬁguration is based on a systematic and potentially

fast evaluation of candidate designs. Furthermore, the eﬀects of nonlinear ﬂow features, multiple

operating conditions, and multidisciplinary interactions can be intrinsic in the formulation of

the optimization problem.

The main diﬃculty in the realization of such general optimization algorithms is computa-

tional cost, which is due to the need for repeated evaluation of the objective function, and hence

the ﬂow equations. Objective function gradients can be used to accelerate the convergence of

the optimization problem; however, fast evaluation of accurate gradients is a challenging task.

The goal of this thesis is to develop an accurate and robust algorithm for complex aerodynamic

4 Chapter 1. INTRODUCTION

shape optimization problems that provides an eﬃcient evaluation of the objective function and

gradient.

An overview of algorithms for aerodynamic shape optimization problems is given in the

following section, which is divided into two parts. In the ﬁrst part, Subsection 1.2.1, we provide

an outline of the techniques used for the essential components of the optimization problem. In

Subsection 1.2.2, we summarize methods used for the evaluation of the gradient. The objectives

of this thesis are stated in Section 1.3, which also provides an outline of the document.

1.2 Review of Aerodynamic Shape Optimization Algorithms

and Applications

1.2.1 Essential Components

The aforementioned, essential components of an aerodynamic shape optimization problem are

objective functions and constraints, design variables

, a ﬂow solver, and a numerical optimiza-

tion method. These components should be selected carefully, since they have a direct impact

on the accuracy and eﬃciency of the optimization. Jou et al. [89], Giles [60], and Drela [38]

present important factors that inﬂuence the selection of the objective functions and constraints

for practical problems. These factors include limitations of the ﬂow solver that can be exploited

by the optimization procedure and the consideration of multiple operating conditions.

Samareh [154, 153], Oyama [136], and Reuther et al. [146] provide excellent summaries of

shape parameterization techniques that can be used to deﬁne design variables. Overall, it is

important that the selected parameterization technique provides suﬃcient ﬂexibility in order to

realize truly optimal designs; yet, it is also desirable that the number of parameters required to

deﬁne the shape is small in order to ensure a reasonable convergence rate of the optimization.

Additional desirable qualities include local shape control, as well as smooth shape perturbations.

The simplest choice of design variables, which does not require an explicit parameterization

technique, is the location of surface nodes in the computational grid. Jameson [84] and Mo-

hammadi [113] have successfully used this approach in conjunction with smoothing operators.

Unfortunately, the number of design variables becomes prohibitively large for three-dimensional,

turbulent ﬂow problems.

The most promising shape parameterization approaches are the shape-function, polynomial

and spline approaches. The shape-function approach was introduced by Hicks and Henne [75].

For recent publications that contain a detailed description of this approach see [48, 146]. An

Also referred to as design parameters or controls in the context of control theory or genes in the context of

genetic algorithms.

1.2 REVIEW OF AERODYNAMIC SHAPE OPTIMIZATION ALGORITHMS AND APPLICATIONS 5

example of the polynomial approach is PARSEC [160, 59, 136], where practical airfoil parame-

ters such as the trailing-edge angle, airfoil thickness and curvature are used to deﬁne the airfoil

shape. Spline approaches, such as basis splines (B-splines) and nonuniform rational B-splines

(NURBS) [32, 53], have received considerable attention [20, 8, 98] since they can be tailored

to provide local and smooth airfoil shape control and are easily incorporated into the CAD

environment.

Closely associated with the design variables is a grid-perturbation algorithm, which adjusts

the computational grid relative to the surface shape deﬁned by the design variables. Although

this task could be accomplished by a grid generator, present-day grid generation tools for com-

plex geometries are computationally expensive and typically not fully automated. Therefore,

grid-perturbation algorithms are often used to modify a baseline grid, which is generated only

once at the start of the optimization. For two- and simple three-dimensional problems using

structured grids, algebraic grid-perturbation strategies work well. Burgreen and Baysal [19]

use an algebraic strategy that preserves the distance to the outer grid boundary and relocates

the nodes in the normal direction proportional to the distance from the surface. A similar

strategy is used by Reuther et al. [145]. However, they note that this strategy fails for three-

dimensional problems where multiple surfaces require simultaneous adjustment. A new strategy

is introduced based on transﬁnite interpolation [167]. Grid-perturbation algorithms for unstruc-

tured grids generally use a force equilibrium of springs approach, but safeguards are required

for viscous-grid perturbations [8, 47, 42]. A promising strategy for large surface deformations

based on a modiﬁed linear elasticity theory is proposed by Nielsen and Anderson [132].

The accuracy of the optimization ultimately depends on the modelling of the ﬂow, and

hence, the ﬂow solver. Accurate modelling of the nonlinear ﬂow eﬀects described in Section 1.1

requires the solution of the compressible, Reynolds-averaged, Navier–Stokes equations combined

with a suitable turbulence model. Two-dimensional ﬂow problems for single and multi-element

airfoils are considered in this work. Detailed reviews of two-dimensional ﬂow solvers and re-

sults for transonic cases are provided by Holst [78] and for high-lift cases see Rumsey and

Ying [149], Klausmeyer and Lin [95], and Fejtek [54]. With the exception of stall and post-

stall conditions, the ﬂow solvers provide accurate estimates of aerodynamic performance. The

algorithm developed in this work is based on two established solvers, namely, ARC2D [143]

and TORNADO [121, 67]. TORNADO is a generalization of the approximate-factorization

algorithm of ARC2D for multi-block structured grids. Both solvers have been a subject of

detailed accuracy studies for a wide range of ﬂow conditions and airfoil conﬁgurations, see for

example [107, 78, 182, 122, 54, 120, 34, 149].

The ﬂow solver has also a signiﬁcant inﬂuence on the eﬃciency of the optimization. The

repeated evaluations of the objective functions required during the optimization demand fast

6 Chapter 1. INTRODUCTION

ﬂow solutions. Detailed reviews of eﬀective ﬂow solvers for the Navier–Stokes equations are

given by Nelson [119], Pueyo [141], and Walsh [176]. Among the fastest algorithms are the

Newton–Krylov solvers [175, 9, 7, 142, 183, 22]. For example, promising results are presented

by Pueyo and Zingg [142], who used the preconditioned generalized minimal residual (GMRES)

Krylov subspace method [152] in conjunction with an inexact-Newton strategy. A critical com-

ponent in this approach is a fast solution of the linear system at each Newton iteration, which

is provided by the preconditioned GMRES method. For the aerodynamic shape optimization

problem, such Newton–Krylov solvers are very appealing, since they not only provide fast so-

lutions to the ﬂow equations, but the preconditioned GMRES method can also be used to

compute objective function gradients.

Numerical optimization methods, which control the solution of the optimization problem,

can be divided into the following four categories: 1) classical direct search methods, 2) stochas-

tic methods, 3) gradient-based methods, and 4) fully-coupled methods. An insightful summary

of direct search methods is provided by Lewis et al. [99]. Among the most well-known di-

rect search methods is the simplex method of Nelder and Mead. Duvigneau and Visonneau [42]

investigate the performance of this method for aerodynamic and hydrodynamic shape optimiza-

tion problems, including the optimization of a high-lift conﬁguration. The ﬂow is governed by

the two-dimensional incompressible Navier–Stokes equations. Although this approach provides

a robust optimization method, its convergence to the optimal solution is slow and requires a

large number of ﬂow evaluations.

Simulated annealing [94] and genetic algorithms [68] are good examples of stochastic meth-

ods. The latter, in particular, have received considerable attention for application in aero-

dynamic shape optimization problems, see for example Giannakoglou [59], Marco et al. [108],

Obayashi [134], and Oyama [136]. The fundamental concepts of genetic algorithms are based

on the process of natural selection. These algorithms are capable of ﬁnding a global optimum,

are well-suited for problems with multiple and non-smooth objectives, and can be used with

categorical design variables. The primary disadvantage of genetic algorithms is high compu-

tational cost. Tse and Chan [170] and Holst and Pulliam [79] provide optimization examples

where up to 10,000 ﬂow solutions are required to reach convergence.

Gradient-based methods [66, 36, 133] are potentially the most eﬀective methods for aero-

dynamic shape optimization problems, since signiﬁcant design improvements can be obtained

in a relatively few evaluations of the objective function and gradient. However, these methods

require a smooth design space

and inherently converge to a local optimum. The gradient can

be used directly to determine a search direction for the optimization process, which is the case

Recently, Moreau and Aeyels [115] examined the optimization of discontinuous objective functions using the

concept of generalized gradients.

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