An All-At-Once Approach to Airfoil Design

34 Pages·2000·0.37 MB·English

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Airfoil Design by an All–At–Once Method
ƒ
Ajit Shenoy § Matthias Heinkenschloss ¤ Eugene M. Cliff '
Abstract
The all–at–once approach is implemented to solve an optimum airfoil design problem.
Theairfoildesignproblemisformulatedasaconstrainedoptimizationprobleminwhichflow
variables and design variables are viewed as independent and the coupling steady state Eu-
ler equation is included as a constraint, along with geometry and other constraints. In this
formulation, the optimizer computes a sequence of points which tend toward feasiblility and
optimality at the same time (all–at–once). This decoupling of variables typically makes the
problem less nonlinear and can lead to more efficient solutions. In this paper an existing op-
timization algorithm is combined with an existing flow code. The problem formulation, its
discretization,and the underlyingsolvers are described. Implementationissues are presented
and numerical results are given which indicate that the cost of solving the design problem is
approximatelysixtimesthecostofsolvingasingleanalysisproblem.
Key words Airfoil design, optimization, computational fluid dynamics, Euler equa-
tions,nonlinearprogramming,optimaldesign.
1 Introduction
Optimum airfoil design is an active area of research. See, e.g., the recent papers [2], [4], [18],
[19],[20], [22],[23], [26],[27], [33],[34]. Abstractly,theoptimumairfoildesignproblemcan be
formulated as a constrained optimization problem, and many techniques have been applied to its
solution. Most of the recent approaches, in fact all of the references above, combineoptimization
andoptimalcontroltechniqueswithcomputationalfluiddynamics. Severalissueshavetobedealt
“
« ThisversionwasgeneratedNovember5,1997.
InterdisciplinaryCenterforAppliedMathematicsandAerospaceEngg. Dept.,VirginiaPolytechnicInstituteand
StateUniversity,Blacksburg,Virginia24061–0531.ThisauthorwassupportedbyAFOSRunderGrantsF49620–93–
1–0‹280andF49620–96–1–0329.
DepartmentofComputationalandAppliedMathematics,RiceUniversity,Houston,Texas77005–1892,USA,E-
Mail: [email protected]. ThisauthorwassupportedbyNSFunderGrantDMS–9403699,byDoEunderGrant
DE-›FG03-95ER25257,AFOSRunderGrantF49620–96–1–0329.
Interdisciplinary Center for Applied Mathematics and Aerospace Engg. Dept., Virginia Polytechnic Institute
and State University, Blacksburg, Virginia 24061–0531, USA, E-Mail: [email protected]. This author was
supportedbyAFOSRunderGrantsF49620–93–1–0280andF49620–96–1–0329.
1
2 A.R.SHENOY,M.HEINKENSCHLOSS,ANDE.M.CLIFF
with when one follows this path. Among these are the formulation of the optimization problem,
the discretization of the infinite dimensional airfoil design problem, and the differentiationof ob-
jectivefunctionandconstraints. Forthesolutionoftheairfoildesignproblemallissueshavetobe
dealt with simultaneously. Great caremust be taken to avoid, or at leastto controlinconsistencies
and to develop a robust and efficient solution method. While these techniques have already been
successfully applied to airfoil design problems, further investigations are needed to improve effi-
ciencyandrobustnessofthesolutiontechniquesandtoincreasethesetof(airfoil)designproblems
towhichthesetechniquescan beapplied.
Inthispaperweinvestigatetheapplicabilityoftheall–at–onceformulationoftheoptimization
problem to solve an airfoil design problem. In most papers on airfoil design, the flow variables
fi fi–(cid:176)
are viewed as functions of the design parameters fl . This function fl‡† is implicitly defined
(cid:176)(cid:181)fi•¶
by the governing equations · fl‡†C‚”„ , in our case the steady state 2-D Euler equations. The
optimizationformulationdescribingtheairfoildesignproblemisthenposedinthedesignvariables
fl . Thisis calledtheblack-boxapproach. TheEulerequationsarenotvisibleto theoptimizer,but
fi
hidden by eliminating the flow variables, i.e., by expressing the flow variables as functions of
the designs variables fl . An alternative to this approach is the all–at–once formulation in which
fi
one views flow variables and design variables fl as independent variables in the optimization
problem. The Euler equations coupling these two are included into the optimization formulation
asaconstraintalongwithotherconstraintssuchasgeometricconstraints,dragconstraints,etc. The
optimizerisnowresponsibleforcomputingapointwhichisfeasibleandoptimalatthesametime,
i.e., move towards feasibility and optimality at once, rather than moving along the manifold of
feasiblepointstowardsoptimality. Comparisonsbetweenthesetwoapproachesonotherproblems
have shown that the all–at–once approach can be substantially faster. The reason is that viewing
fi
and fl as independent variables, allows the optimizer to violate the Euler equations during the
iterations. These are only required to be satisfied at the solution. This makes the optimization
problem less nonlinear and often results in fewer iterations. For example, Iollo et al. [18] report
thattheirpseudo-timesteppingimplementationof theall–at–onceapproachrequiresonly threeto
four times as many iterations to solve the design problem as compared to the effort required for
the solution of a single analysis problem. Our results indicate a factor of five or six. However,
our optimization approach is different and our problem includes geometric constraints. It is also
important to note that an optimizer implementing the all–at–once approach requires roughly the
same problem informationas an optimizerapplied to the black-box approach, except that the all–
at–once approach does not require solutions to the nonlinear flow equations. We give a more
detailedpresentationoftherelationsinthenextsection.
Rather thanformulatingtheairfoil designproblem,its discretization,anda solutionalgorithm
and then implement all components from scratch, we decided to build upon existing codes. In
our implementation of the all–at–once method for our airfoil design problem we combine the
optimizer,TRICE,withtheflowcode,ErICA.Thisimposescertainlimitsonthechoiceofproblem
formulationanddiscretization,butwebelievethistobearealisticapproach. Aswehaveindicated
above, various issues have to be addressed when solving the airfoil design problem. We focus
on the optimization formulation. Our airfoil parameterization is obtained by choosing a set of
basis airfoils and computing an optimized airfoil as a linear combination of those. Moreover,
AIRFOILDESIGNBYANALL–AT–ONCEMETHOD 3
grid generationand discretizationof the Euler equationwas done to limit difficulties arisingfrom
nonsmoothness and inconsistencies. Further gains in efficiency and accuracy can be achieved by
using more refined discretization techniques and improving the coupling of the flow solver with
theoptimizer. Thiswasbeyondthescopeofthisstudyandisplannedforfutureinvestigations.
Thispaperisorganizedasfollows: InSection2wediscussoptimizationformulationsandtheir
relations. This section also provides further motivation for the all–at–once approach and reviews
some existing optimization approaches. The governing equations and the flow code ErICA are
discussed in Section 3. The design problem is formulated in Section 4 and Section 5 contains a
description of the optimizer TRICE. Section 6 contains some implementation issues that have to
be resolved when combining an optimizer with a flow code for our situation. Section 7 presents
somenumericalresultsandcontainsadiscussionofournumericalexperimentsandopenissues.
2 Optimization Problem
Thereareseveralwaystocastthedesignproblemoutlinedintheintroductionintoanoptimization
problem. Two formulations will be discussed in this section. The main purpose of this section is
to provide a background for the discussion of our approach to the airfoil design problem and for
a comparison with other approaches in the literature. In this section we proceed as follows: First,
we present the two formulationsand their relation in an abstract framework. Then we discuss the
applicabilitytotheairfoildesignproblem.
Thefirstformulationoftheairfoildesignproblemis givenby
»\…]‰ (cid:190)j(cid:176)(cid:181)fi•¶ ¶
fl‡† (2.1)
(cid:176)Ofi$¶ ¶
s.t. · fl¿†(cid:17)‚(cid:192)„ (2.2)
`
(cid:176)(cid:181)fi•¶
fl¿†9´ˆ„–˜ (2.3)
fi
Here represent the flow variables and fl are the design parameters. The constraint function ·
represents the Euler equations. The inequality constraints (2.3) represent geometric constraints,
`
dragconstraintsandthelike. Thespecialcaseinwhich doesnotdependontheflowparameters
fi (cid:190)(cid:2)¯s˘ ˘ ˘ ¯
deserves attention. In this section we assume that the functions ·’˙(cid:8)¨C(cid:201) ·’˙x˚(cid:204)¸ · , ·
`
˘ ˘ ˘ ¯3˘ ˘ ˘
·’˙(cid:8)¨˝(cid:201) ·’˙˛˚(cid:153)¸ ·’˙˛¨ ,and ·’˙(cid:8)¨˝(cid:201) ·’˙x˚V¸ ·’˙˛ˇ aretwicecontinuouslydifferentiableatthepoints
under consideration. However,we note thatfor the formulationand executionof the optimization
`
¶
algorithmappliedtoourairfoildesignproblem,weonlyneedfirstderivatives. For · wedenote
(cid:176)(cid:181)fi•¶
differentiationwithrespecttoavariablebyusingthevariableasasubscript,e.g., ·’— fl‡† denotes
fi
thepartialJacobianof · withrespectto . Inadditiontothedifferentiabilityassumption,wemake
(cid:176)Ofi$¶ (cid:176)(cid:181)fi•¶
theassumptionthat ·’— fl¿† isinvertibleatallpoints fl¿† underconsideration.
Under the assumption of the implicit function theorem (see, e.g., [24]), the constraint (2.2)
fi(cid:13)¯3˘ ˘
locallydefinesafunction ·’˙ ˚ ¸ ·’˙ ¨ asthesolutionof
(cid:176)Ofi–(cid:176) ¶
· fl‡† fl‡†N‚(cid:192)„–˜ (2.4)
fi"(cid:176) ˘
If the equation (2.2) has a unique solution fl¿† for all fl(cid:210)(cid:209) ·’˙ ˚ under consideration (typically,
˘
(2.3) represents an explicit restriction of the design space and therefore not the whole ·’˙ ˚ is
4 A.R.SHENOY,M.HEINKENSCHLOSS,ANDE.M.CLIFF
fi
relevant), then we can eliminate the flow variables and formulate (2.1)–(2.3) in the following
reducedform:
»\…]‰ (cid:190)j(cid:211) (cid:176) (cid:190)j(cid:176)Ofi–(cid:176) ¶ ¶
fl¿†s‚ fl‡† fl‡† (2.5)
` `
(cid:211) (cid:176) (cid:176)(cid:181)fi"(cid:176) ¶
s.t. fl¿†s‚ fl¿† fl¿†9´ˆ„•˜ (2.6)
The optimization formulation(2.1)–(2.3) correspondsto the all–at–once(AAO) approach [7],
[11](alsocalledthesimultaneousanalysisanddesign(SAND)approach[3],[26]). Theoptimiza-
tion formulation (2.5), (2.6) corresponds to the black–box approach [11] (also called the nested
analysisand design(NAND) [3], [26]) and it correspondsto the multidisciplinefeasibleand indi-
vidualdisciplinefeasibleapproachin [7].1
In the following, we present optimality conditions for (2.1)–(2.3) and we discuss the relation
between these two problems. These results are knownand can be found in a similar form, e.g.,in
[9],[13]. Let (cid:212)
`
(cid:176)(cid:181)fi•¶ ¶(cid:214)(cid:213)X¶(cid:216)(cid:215) (cid:190)j(cid:176)(cid:181)fi•¶ (cid:213)–(cid:219) (cid:176)Ofi$¶ (cid:215)@(cid:219) (cid:176)Ofi$¶
fl †*‚ fl‡†(cid:218)(cid:217) · fl‡†(cid:218)(cid:217) fl‡† (2.7)
be the Lagrangian corresponding to (2.1)–(2.3). If a constraint qualification is met, then for an
(cid:176)(cid:181)fi•¶ (cid:213) (cid:215)
optimalpoint fl‡† of(2.1)–(2.3)thereexist , suchthat
(cid:220) (cid:190)j(cid:176)Ofi$¶ (cid:176)Ofi$¶ (cid:219) (cid:213) ` (cid:176)(cid:181)fi•¶ (cid:219) (cid:215) ¶
— fl‡†(cid:218)(cid:217)(cid:204)·’— fl¿† (cid:217) — fl¿† ‚ „
(cid:220)(cid:222)(cid:221) (cid:190)j(cid:176)(cid:181)fi•¶ (cid:221) (cid:176)(cid:181)fi•¶ (cid:219) (cid:213) ` (cid:221) (cid:176)(cid:181)fi•¶ (cid:219) (cid:215) ¶
fl¿†(cid:218)(cid:217)T· fl¿† (cid:217) fl¿† ‚ „
(cid:176)Ofi$¶ ¶
· fl¿†(cid:223)‚ „
` (cid:176)Ofi$¶ ¶ (2.8)
fl¿†(cid:224)´ „
(cid:215) Æ ¶
„
`
(cid:176)(cid:181)fi•¶ (cid:219) (cid:215)
fl¿† ‚ „–˜
`
fi
If doesnotdependon ,thenthefirstequationin(2.8)reducesto
(cid:220) (cid:190)s(cid:176)Ofi$¶ (cid:176)(cid:181)fi•¶ (cid:219) (cid:213)
— fl¿†X(cid:217)T·’— fl‡† ‚(cid:192)„–˜ (2.9)
`
fi
Equation(2.9) is called theadjointequationand, if doesnot depend on , definesthe Lagrange
(cid:213) (cid:213) (cid:220)(cid:222)(cid:221) (cid:190)j(cid:176)(cid:181)fi•¶ (cid:221) (cid:176)(cid:181)fi•¶ (cid:219) (cid:213)
multiplier (or the co–state) . With given by (2.9), the term fl¿†s(cid:217)(cid:192)· fl‡† of the
secondequationin(2.8)iscalledthereducedgradient.
A commonly used constraint qualification is the linear independent constraint qualification
`1(cid:226) `
(cid:176)Ofi$¶ (cid:176)(cid:181)fi•¶ (cid:176)Ofi$¶
(LICQ): Let fl‡† denote the vector of functions of fl‡† which are active at fl¿† . Then
`ª(cid:226)
(cid:176)Ofi$¶ (cid:176)(cid:181)fi•¶
LICQ is satisfied if the gradient of the component functions in · fl¿† and fl¿† are linearly
` `1(cid:226)
fi (cid:176)
independent. If doesnotdependon andiftherowsof fl‡† arelinearindependent,whichis,
`
(cid:176) (cid:176)Ofi$¶
e.g., the case if fl¿†(cid:27)‚n(cid:228)¿fl , then our assumption that ·’— fl¿† is invertibleimplies that LICQ is
satisfied[9],[13].
Thesecondordernecessary[sufficient]optimalityconditionsaregivenby(2.8)and
(cid:229);(cid:230) (cid:219)’Ł (cid:229)s(cid:230) Æ
— (cid:176)Ofi$¶ ¶(cid:214)(cid:213)F¶5(cid:215) —
(cid:230) (cid:221)9(cid:231) fl † (cid:230) (cid:221);(cid:231) غŒ(cid:21)(cid:236) „ (2.10)
1Sinceweonlyhaveone discipline (cid:237)h(cid:238)](cid:239)x(cid:240)Oæ*(cid:242)s(cid:243)J(cid:244) thereis nodistinctionbetweenthe multidisciplinefeasibleand
individualdisciplinefeasibleformulationin[7].
AIRFOILDESIGNBYANALL–AT–ONCEMETHOD 5
(cid:230) ¶ (cid:230) (cid:221)
forall — satisfying
(cid:176)(cid:181)fi•¶ (cid:230) (cid:221) (cid:176)(cid:181)fi•¶ (cid:230) (cid:221) ¶
·’— fl‡† —F(cid:217)T· fl¿† ‚ „ (2.11)
` (cid:176)Ofi$¶ (cid:230) ` (cid:221) (cid:176)(cid:181)fi•¶ (cid:230) (cid:221)
— fl¿† —F(cid:217) fl¿† ‚ „–˜ (2.12)
` ` (cid:226)
(cid:176)(cid:181)fi•¶ (cid:176)(cid:181)fi•¶
UŁ nlessnotedotherwise, fl‡† wouldusuallyrefertothesetofactiveconstraints, fl¿† . Here
(cid:176)(cid:181)fi•¶ ¶(cid:214)(cid:213)X¶(cid:216)(cid:215)
fl † denotestheHessianoftheLagrangian
Ł (cid:212)
(cid:176)Ofi$¶ ¶(cid:214)(cid:213)X¶(cid:216)(cid:215) (cid:220)(cid:222)ı(cid:246) (cid:221)"ł (cid:176)(cid:181)fi•¶ ¶(cid:214)(cid:213)X¶(cid:216)(cid:215)
fl †*‚ —(cid:4)(cid:247) fl †(cid:214)˜
(cid:176)(cid:181)fi•¶ (cid:230) (cid:221)
Pointssatisfyingthehomogeneousstateequation(2.11)canbecharacterizedby ø(cid:18)œ ¨ ‚(cid:253)(cid:252) fl¿† ,
(cid:2) (cid:3) (cid:4) œ ˚(cid:218)ß
where
(cid:176)Ofi$¶ (cid:221) (cid:176)Ofi$¶
(cid:176)Ofi$¶ (cid:1)(cid:0) ·’— fl¿† · fl¿†
(cid:252) fl¿†N‚(cid:255)(cid:254) ˘ ˜ (2.13)
Withthis,(2.10),(2.11),(2.12)canberewrittenas
Ł
(cid:230) (cid:221) (cid:176)Ofi$¶ (cid:219) (cid:176)(cid:181)fi•¶ ¶(cid:214)(cid:213)X¶(cid:216)(cid:215) (cid:176)(cid:181)fi•¶ (cid:230) (cid:221) Æ Ø(cid:140)Œ(cid:9)(cid:236)
(cid:252) fl¿† fl †(cid:30)(cid:252) fl‡† „ (2.14)
(cid:230) (cid:221)
forall satisfying
(cid:3)
(cid:2)
Ø ` (cid:176)(cid:181)fi•¶ (cid:176)(cid:181)fi•¶ (cid:221) (cid:176)(cid:181)fi•¶ ` (cid:221) (cid:176)Ofi$¶ (cid:236) (cid:230) (cid:221)
(cid:0) — fl¿†(cid:4)·.— fl‡† · fl¿†(cid:218)(cid:217) fl¿† ‚%„–˜ (2.15)
Ł
(cid:176)Ofi$¶ (cid:219) (cid:176)Ofi$¶ ¶(cid:214)(cid:213)F¶5(cid:215) (cid:176)Ofi$¶ (cid:3)
Thematrix(cid:252) fl¿† fl †(cid:30)(cid:252) fl‡† iscalledthereducedHessian. Thet(cid:2)ermonthelefthand
(cid:176)(cid:181)fi•¶ (cid:221) (cid:176)(cid:181)fi•¶
sideof(2.15)caneitherbecomputedbycalculatingthesensitivities ·’— fl‡† · fl¿† orusing
(cid:5)
anadjointapproach. Ifwedefine
(cid:2)
` (cid:5)
(cid:176)(cid:181)fi"(cid:176) ¶ (cid:219) (cid:176)Ofi–(cid:176) ¶ (cid:219) ¶
‚ (cid:0) ·’— fl¿† fl¿† — fl‡† fl‡† (2.16)
(cid:219) (cid:221) (cid:176)Ofi–(cid:176) ¶ ` (cid:221) (cid:176)(cid:181)fi"(cid:176) ¶
then the left hand side of (2.15) can be written in the form · fl‡† fl‡†s(cid:217) fl¿† fl¿† . In
(cid:6)(cid:8)(cid:7) (cid:6) (cid:221)
particularif is smaller than the adjoint equation based approachseems more attractivethan
thesensitivityequationapproach.
Itisknown,see,e.g.,[9],[13],thatderivativesforthereducedproblem(2.5),(2.6)arerelatedto
(cid:220) (cid:190)j(cid:211) (cid:176)
thereducedquantitiesoftheproblem(2.1)–(2.3). Forexample,thegradient fl¿† ofthereduced
(cid:220)(cid:222)(cid:221) (cid:190)s(cid:176)Ofi$¶ (cid:221) (cid:176)Ofi$¶ (cid:219) (cid:213) (cid:213) fi
pfi–r(cid:176) oblem is equal to the reducŁ(cid:9)ed(cid:176) gr¶(cid:216)a(cid:215)dient(cid:220) (cid:221)ı (cid:212)(cid:211) (cid:176) ¶(cid:216)fl¿(cid:215) †F(cid:217) · fl¿† , wi(cid:212)(cid:211)th(cid:176) ¶5g(cid:215) iven(cid:190)sb(cid:211) y(cid:176) (2.9)`(cid:211), a(cid:176) t (cid:219) ‚(cid:215)
fl¿† . Moreover,theHessian fl †s‚ fl † oftheLagraŁngian fl †N‚ fl¿†(cid:11)(cid:217) fl¿†
(cid:176)Ofi$¶ (cid:219) (cid:176)(cid:181)fi•¶ ¶(cid:214)(cid:213)X¶(cid:216)(cid:215) (cid:176)(cid:181)fi•¶
ofthereducedproblemisequaltothereducedHessian(cid:252) fl‡† fl † (cid:252) fl¿† . Finally,the
`(cid:211) (cid:221) (cid:176) fi fi–(cid:176)
Jacobian fl¿† is equalto thematrixonthelefthandsideof(2.15)at ‚ fl‡† .
Comparisonsbetweentheall–at–once(SAND)approachandtheblack–box(NAND,discipline
feasible) approach can be found in, e.g., [7], [11], [26]. The all–at–once approach decouples
state and design variables. An optimizer for (2.1)–(2.3) can use this decoupling and is allowed
to violate constraints during the iteration. This can result in substantial gains in performance.
Significant reductions in solution times for problems related to the one considered in this paper
are reported in [11], [12]. However, the optimizer must achieve feasibility and optimality at the
same time. This requires carefully designed optimization codes to maintain robustness. In the
6 A.R.SHENOY,M.HEINKENSCHLOSS,ANDE.M.CLIFF
computational experiments in [11] the (particular) implementation of the black–box formulation
always performed more robustly than the implementation of the all–at–once approach for a one–
dimensionalductdesignproblem. Concerningtheapplicabilityoftheall–at–onceapproachandthe
black–box approach, it should be mentioned that most of the quantities needed to implement the
all–at–once approach also have to be provided for an implementation of the black–box approach.
This is indicated by the relations between derivativesfor the reduced problem (2.5), (2.6) and the
reducedquantitiesoftheproblem(2.1)–(2.3)summarizedabove.
In the context of airfoil design problems both formulations (2.1)–(2.3) and (2.5), (2.6) have
been used, however, currently the black–box formulation (2.5), (2.6) seems to be dominant [19],
[20], [23], [27], [33]. For airfoil design problems the all–at–once formulation (2.1)–(2.3) is con-
sideredin[18],[34]. Inbothcasesonlytheequalityconstrainedproblem(2.1),(2.2)isconsidered.
Theoptimizationmethodsarederivedfromtheoptimalitysystemfor(2.1),(2.2). In[18]theopti-
malitysystemissolvedbyapplyingafewpseudo–timestepsto theEulerequationandtheadjoint
equationtoimprovestateandco–stateestimatesandagradient–likesteptoupdatethedesignvari-
ables. The optimization methods in [34] are derived from the application of Newton’s method to
theoptimalitysystem;theseareparticularversionsofsequentialprogramming(SQP)methods.
We use the all–at–once formulation (2.1)–(2.3). For our particular design problem the con-
`
straints are simple constraints on the design variables. The exact problem formulation will be
introducedinthesubsequentsections. WeuseanSQPmethodfromtheclassofmethodsdescribed
in[9],[15]forthesolutionoftheall–at–onceformulation. ThisSQPmethodusesaninteriorpoint
strategy to handle the inequality constraints and employs a trust–region strategy for globalization
of convergence and to enhance robustness. See also Section 5. If only equality constraints are
present, then the Newton based methods in [34] are related to the SQP methods in [9], [15]. Be-
sidesthecapabilityofhandlinginequalitiesonthedesigns,othermaindifferencesarethattheSQP
methods in [9], [15] use a trust–region globalization and, in addition to exact second derivatives,
provide quasi–Newton approximations to the full and reduced Hessian of the Lagrangian. First
and second order convergence results are proven in [9] and the influence of inexact derivatives is
analyzedin[14].
For the formulation of the airfoil design problem as an optimization problem, several other
issues are of great importance. These are the issues of discretization, differentiability,and unique
solvability of state equations and linearized state equations. We give a more detailed description
below. For general airfoil design problems comprehensive, rigorous treatments of these issues
are still missing. In the case of a one-dimensional duct design problem, which is related to the
airfoil design problem, such a comprehensive,rigorous treatment can be found in [6]. It is shown
in [6] that an understanding of these issues can be used to improve robustness and efficiency of
the optimization code. These improvementsare based on the understandingof the problem, of its
discretization, and of the optimization method. They are achieved with very little programming
effortandalmostnoadditionalcomputingeffortperiteration.
The airfoil design problem originally is an infinite dimensional problem. Therefore, the opti-
mizationformulationandoptimizationalgorithmhavetobecombinedwithadiscretizationscheme.
Variousapproachesarepossible. Twoofthosearetheoptimize–then–discretizeapproachinwhich
theoptimizationalgorithmisformulatedintheinfinitedimensionalsettingandthendiscretization
AIRFOILDESIGNBYANALL–AT–ONCEMETHOD 7
are applied to the individual steps, and the discretize–then–optimize approach in which one first
discretizestheproblemandthenappliesanoptimizationalgorithmtothediscretizedproblem. The
processes of discretization and optimization are usually not interchangeable and therefore these
two approaches are different. The numerical solution of an infinite dimensional problem requires
a careful study of the problem at hand. Several issues have to be kept in mind. In the optimize–
then–discretize approach the derivatives after discretization are usually not the derivatives of the
discretized functions. Therefore optimization algorithms have to cope with inexact derivative in-
formation. See e.g., [5], [33]. The discretize–then–optimizeapproach often neglects the fact that
the infinite dimensional problem structure still influences the finite dimensional problem. If this
influenceisnotincorporatedproperly,thentheoptimizationproblemtypicallybecomesartificially
ill–conditioned and one observes a severe degradation in performance and robustness of the opti-
mizer,see[6],[28].
fi
Derivativesofconstraintfunctionsandsolutions tothestateequationsareusedintheformu-
lationofoptimalityconditionsandinefficientoptimizers. See,forexample,gradientcomputations
using sensitivities or adjoint equations. For problems governed by the Euler equations, differen-
tiability in the infinite dimensional context is problematic, due to the presence of shocks. This
might be differentfor the discretizedEuler equations. If smoothingprocedures (e.g., introduction
of artificial viscosity) are applied in discretization schemes for the Euler equations, the resulting
finite dimensional system may be differentiable. However, since the discretization schemes used
in CFD codes are very complex, ‘derivatives’ and ‘adjoint equations’ should be treated with care
andusuallymustbeunderstoodformally.
It is also important to keep in mind that the formulations (2.1)–(2.3) and (2.5), (2.6) are only
fi–(cid:176) ˘ (cid:176)Ofi$¶
equivalentif(2.2)hasauniquesolution fl¿† forallfl (cid:209) · ˙ ˚ underconsideration. If · fl¿†s‚ „
represents the (discretized) Euler equations, this assumption seems to be rather strong in view of
the non-uniqueness result presented in [21] for discretized Euler equations. The existence and
fi (cid:176)Ofi$¶
uniqueness of the solution of · fl¿†(cid:9)‚q„ for given fl is often also related to the existence and
(cid:230) (cid:176)(cid:181)fi•¶ (cid:230) (cid:221) (cid:176)(cid:181)fi•¶ (cid:230) (cid:221) (cid:176)(cid:181)fi•¶
uniquenessofthesolution — ofthelinearizedstateequations·’— fl‡† —|(cid:217)C· fl‡† (cid:217)C· fl¿†N‚
(cid:176)Ofi$¶ ¶ (cid:230) (cid:221)
„ forgiven fl¿† .
As we have noted before, for a one-dimensional duct design problem, which is related to the
airfoil design problem, the above issues have been rigorously discussed in [6]. For general air-
foil design problems these issues are subject of current research. In our approach to the airfoil
design problem we parameterize the airfoil using linear combinations of existing airfoils. This
can be viewed as a reduced basis approach (see e.g., [32, Sec.7.3]) leading to a low dimensional
(cid:6) (cid:221) (cid:11)(cid:10)
( ‚ ) design space. Our grid generationscheme leads to grids which depend smoothly on the
designparameters. OurapplicationprogramsarebasedonthepackageErICAforthesimulationof
flows over airfoils governed by the Euler equations. Among the discretization schemes available
in that package, we use the schemes with better smoothness properties. We use the discretize-
then-optimize approach. Since we have a low dimensional design space and a rather simple grid
generationscheme, we believethis is sensible. However, givenour experiencesin [6], we believe
this approach has to be rethought if more complex discretization schemes are used. More details
onthediscretizationschemesareprovidedin Sections3and6.
8 A.R.SHENOY,M.HEINKENSCHLOSS,ANDE.M.CLIFF
3 Analysis Problem, Discretization, and Flow Code
In this section we discuss the analysis problem underlying our design problem and its discretiza-
fi
tion. The analysis problem is the flow around the airfoil governed by the steady state Euler
equationsforaperfectgas. WealsooutlinetheflowcodeErICAusedforthesolutionoftheanaly-
sis problem. Althoughour optimizationformulationis basedontheall–at–onceapproachandour
optimizernever needs to solvethe Euler flow equations,we will extract severalsubtasks from the
flowcodeErICA.ThepresentationoftheErICAcodewillhelpto describethesetasks.
The unsteady Euler equations for a perfect gas, written in integral conservation law form is
(cid:12)
givenby
(cid:29)(cid:31)(cid:30)
(cid:12)(cid:14)(cid:13)(cid:16)(cid:15)(cid:18)(cid:17)(cid:20)(cid:19)(cid:22)(cid:21)(cid:24)(cid:23) (cid:15)(cid:18)(cid:25)(cid:26)(cid:17)(cid:28)(cid:27) (cid:21)
(cid:6)(cid:27) (cid:230)
(cid:217) ‚(cid:192)„ (3.1)
where,in Cartesiancoordinates, (cid:29) (cid:27) "(cid:27) %(cid:27)
! $#
‚ (cid:217)
and * 1 *,+ 1 *,- 1
( *,+ ( *,+ ( *,+6-
(cid:19) ı 54
‚ &’’’ *,- ’’’2 ¶ ‚ &’’’ *,+6(cid:217)- ’’’2 ¶ #J‚ &’’’ *,- ı 94 ’’’2
*/.(cid:18)0 3 7*(cid:24)8(cid:14)0 + 3 7*(cid:24)8(cid:14)0(cid:217) - 3
’’’) ’’’ ’’’) (cid:176) † ’’’ ’’’) (cid:176) † ’’’
+ - * .(cid:18)0 . 7+ - .
(cid:176) ı ı ;:=<
with velocitycomponents , , density , total energy per unit mass ‚ (cid:217) (cid:217) † , with
4
being the internal ene?r> gy per*/u. nit mass and pressure , which for a perfect ga8(cid:14)s0may be?e>xpressed
4 (cid:176) BA ı : (cid:176)
b+ y the r-elation,.(cid:18)Y0 ‚ * (cid:0)(cid:28)@ † . The to> tal*enthalpy per unit mass is giv(cid:19) en by ‚ (cid:0)(cid:28)@ †(cid:17)(cid:217)
ı :C< ı :C< D4E: A GF 4E:
((cid:217) 1‚ (cid:217) *H+I,- where (cid:13)‚ isthesonicvelocity. Here, representstheconserved
Ø 4 (cid:236) #
fi (cid:219)
variableswith ‚ denotingtheprimitivevariables,and, and representtheinviscid
J KLJ
fluxes. Theproblemdomainisdenotedby and representstheboundaryofthedomain. Fora
detailedtreatmentoftheEulerequationsrefer[10].
3.1 Discretization of the Euler Equations
For a given airfoil configuration, the shape of which is represented as a function of the design
variables fl , the analysis problem corresponds to the solution of the Euler equations of flow. The
flow is simulated numerically using the solver ErICA (EuleR Inviscid Code for Aerodynamics)
whichwasdevelopedbyNarducci[25].
Computational simulations were performed on a C-type grid, which is wrapped around the
airfoil. Weonlysketchthegridgenerationtofixsomenotation. Fortheomitteddetailswereferto
[29,Sec.4.2.2]. Thegridisgeneratedalgebraically,bythefollowingprocedure: Wefirstdistribute
points on bottom boundary, corresponding to the airfoil surface and the trailing edge wake, and
on the top boundary of the computational grid, corresponding to the far-field boundary. Once the
boundariesaredefined,weconnectcorrespondingpairsofpo"int%sontheto"pandbottomboundaries
using straight% lines. Grid cells and nodes are numbered by (cid:176) ¶ † , whe% re refers to the horizontal
positionand refers to the vertical position in the grid. Indices with ‚ @ refer to nodes or cells
AIRFOILDESIGNBYANALL–AT–ONCEMETHOD 9
< NM=O
onor atthe airfoil,respectively. A typical e„ @ (cid:201) gridis shownin Figures3.1and 3.2with 121
points on the airfoil surface. In practical CFD codes more sophisticated grid generation schemes
areused. Eventually,suchgrid generationshaveto be incorporated. However,in a first attemptto
apply the all–at–once methodology to airfoil design, we preferred this simple grid because of the
relativeeaseofgeneratingthegridandbecauseofitsguaranteedsmoothdependenceonthedesign
parameters(airfoil).
< 5M=O
Figure3.1: The e„ @ (cid:201) grid.
< 9M=O
Figure3.2: Close-upViewofthe e„ @ (cid:201) grid.
Giventhegrid,theErICAcode[25]isusedforthesolutionofthesteadystateEulerequations.