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IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Mathematical details of adjoint-based shapeoptimization for the Euler and
Reynolds-Averaged Navier-Stokes equations(With thanks to Francisco Palacios, Jeff Fike, and Joaquim Martins)
Juan J. Alonso
Department of Aeronautics and Astronautics, Stanford University
OPTPDE 2011July 5th, 2011 - Bilbao, Spain
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Outline
1 IntroductionOptimal design in aerodynamicsAlternatives for sensitivity calculations
2 Design Using the Euler Equations
3 Design Using the RANS EquationsSpalart-Allmaras Turbulence Model
4 Continuous adjoint for the Spalart–Allmaras modelAnalytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
5 Conclusions
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Introduction to Optimization
Non-linear program
minimize I(~x)
~x ∈ RNx
subject to gm(~x) ≥ 0, m = 1,2, . . . ,Ng
I: objective function, output (e.g. structural weight).xn: vector of design variables, inputs (e.g. aerodynamic shape);bounds can be set on these variables.gm: vector of constraints (e.g. element von Mises stresses); ingeneral these are nonlinear functions of the design variables.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Optimization Methods
Intuition: decreases with increasing dimensionality.
Grid or random search: the cost of searching thedesign space increases rapidly with the number ofdesign variables.Evolutionary/Genetic algorithms: good for discre-te design variables and very robust; are they feasiblewhen using a large number of design variables?
Nonlinear simplex: simple and robust but inefficientfor more than a few design variables.
Gradient-based: the most efficient for a large num-ber of design variables; assumes the objective fun-ction is “well-behaved”. Convergence only guaran-teed to a local minimum.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Aerodynamic shape optimization
Shape optimization problem
Find Smin ∈ Sad such that
J(Smin) = minS∈SadJ(S),
where J(S) =R
S j(U,~n, ~x) ds.
Cost function j : drag/lift coefficients,deviation from pressure distribution, sonicboom intensity measures, total pressureloss, entropy increase. . .
Surface parameterized by a suitablenumber of shape functions.
Figure 1: Generic wing with parameterizing control points1.
Figure 2: Flow diagram in aerodynamic optimization2.
1 J.E. Hicken, “Efficient algorithms for future aircraft design”.2 M. Khurana, “Airfoil geometry parameterization throughshape optimizer and computational fluid dynamics”.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Gradient-Based Optimization: Design Cycle
Analysis computes objectivefunction and constraints (e.g.aero-structural solver)Optimizer uses the sensitivityinformation to search for theoptimum solution(e.g. sequential quadraticprogramming)Sensitivity calculation is usuallythe bottleneck in the designcycle, particularly for largedimensional design spaces.Accuracy of the sensitivities isimportant, specially near theoptimum.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Sensitivity Analysis Methods
Finite Differences: very popular; easy, but lacks robustness andaccuracy; run solver Nx times.
dfdxn≈ f (xn + h)− f (x)
h+O(h)
Complex-Step Method: relatively new; accurate and robust;easy to implement and maintain; run solver Nx times.
dfdxn≈ Im [f (xn + ih)]
h+O(h2)
Algorithmic/Automatic/Computational Differentiation:accurate; ease of implementation and cost varies.(Semi)-Analytic Methods: efficient and accurate; longdevelopment time; cost can be independent of Nx .
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Finite-Difference Derivative Approximations
From Taylor series expansion,
f (x + h) = f (x) + hf ′(x) + h2 f ′′(x)
2!+ h3 f ′′′(x)
3!+ . . . .
Forward-difference approximation:
⇒ df (x)
dx=
f (x + h)− f (x)
h+O(h).
f (x) 1,234567890123484f (x + h) 1,234567890123456
∆f 0,000000000000028
x x+h
f(x)
f(x+h)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Complex-Step Derivative Approximation
Can also be derived from a Taylor series expansion about x with acomplex step ih:
f (x + ih) = f (x) + ihf ′(x)− h2 f ′′(x)
2!− ih3 f ′′′(x)
3!+ . . .
⇒ f ′(x) =Im [f (x + ih)]
h+ h2 f ′′′(x)
3!+ . . .
⇒ f ′(x) ≈ Im [f (x + ih)]
h
No subtraction! Second order approximation. (Martins, Sturdza,Alonso, ACM Trans. Math. Soft., 2003)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Simple Numerical Example
Step Size, h
Norm
aliz
ed
Err
or,
Complex-StepForward-DifferenceCentral-Difference
Estimate derivative atx = 1,5 of the function,
f (x) =ex
√sin3x + cos3x
Relative error defined as:
ε =
∣∣f ′ − f ′ref
∣∣∣∣f ′ref
∣∣
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Would You Like Second Derivatives?
Unfortunately, complex step formulations are also subject tosubtractive cancellation when used for second derivatives? What canyou do? If you are interested, we have recently developed a methodbased on hyper-dual numbers that gives exact second derivatives,independently of the step, h!Hyper-dual numbers have one real part and three non-real parts:
x = x0 + x1ε1 + x2ε2 + x3ε1ε2
ε21 = ε22 = 0ε1 6= ε2 6= 0
ε1ε2 = ε2ε1 6= 0
With these definitions, the Taylor series expansion truncates exactlyat the second-derivative term.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Hyper-Dual NumbersIn other words:
f (x +h1ε1 +h2ε2 +0ε1ε2) = f (x)+h1f ′(x)ε1 +h2f ′(x)ε2 +h1h2f ′′(x)ε1ε2
There is no truncation error and no subtractive cancellation error(because of the definition of the hyper-dual numbers, see Fike andAlonso, AIAA-2011-3847). Evaluate a function with a hyper-dual step:
f (~x + h1ε1~ei + h2ε2~ej + ~0ε1ε2)
Derivative information can be found by examining the non-real parts:
∂f (~x)
∂xi=ε1part [f (~x + h1ε1~ei + h2ε2~ej + ~0ε1ε2)]
h1
∂f (~x)
∂xj=ε2part [f (~x + h1ε1~ei + h2ε2~ej + ~0ε1ε2)]
h2
∂2f (~x)
∂xi∂xj=ε1ε2part [f (~x + h1ε1~ei + h2ε2~ej + ~0ε1ε2)]
h1h2Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Challenges in Large-Scale Sensitivity Analysis
There are efficient methods to obtain sensitivities of manyfunctions with respect to a few design variables - Direct Method.There are efficient methods to obtain sensitivities of a fewfunctions with respect to many design variables - Adjointmethod.Unfortunately, there are no known methods to obtain sensitivitiesof many functions with respect to many design variables.This is the curse of dimensionality.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Symbolic Development of the Adjoint MethodLet I be the cost (or objective) function
I = I(w ,F)
where
w = flow field variablesF = grid variables
The first variation of the cost function is
δI =∂I∂w
T
δw +∂I∂F
T
δF
The flow field equation and its first variation are
R(w ,F) = 0
δR = 0 =
[∂R∂w
]δw +
[∂R∂F
]δF
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Symbolic Development of the Adjoint MethodIntroducing a Lagrange Multiplier, ψ, and using the flow field equationas a constraint
δI =∂I∂w
T
δw +∂I∂F
T
δF − ψT[
∂R∂w
]δw +
[∂R∂F
]δF
=
∂I∂w
T
− ψT[∂R∂w
]δw +
∂I∂F
T
− ψT[∂R∂F
]δF
By choosing ψ such that it satisfies the adjoint equation[∂R∂w
]T
ψ =∂I∂w
,
we have
δI =
∂I∂F
T
− ψT[∂R∂F
]δF
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Symbolic Development of the Adjoint Method
The expression for each component of the gradient no longerdepends on δw and, therefore, no flow re-evaluation is need (as is thecase in finite-difference methods). Variations with respect to theshape δF can be computed with relatively little computational effort.
This reduces the gradient calculation for an arbitrarily large number ofdesign variables at a single design point to
One Flow Solution+ One Adjoint Solution
independently of the number of design parameters.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Optimal design in aerodynamicsAlternatives for sensitivity calculations
Design Cycle
sectionsplanform
Shape & GridModification
repeated untilConvergence
Design Cycle
Flow Solver
Adjoint Solver
Gradient CalculationAerodynamics
Structure
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Outline
1 IntroductionOptimal design in aerodynamicsAlternatives for sensitivity calculations
2 Design Using the Euler Equations
3 Design Using the RANS EquationsSpalart-Allmaras Turbulence Model
4 Continuous adjoint for the Spalart–Allmaras modelAnalytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
5 Conclusions
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Design Using the Euler EquationsThe following is a simplified version of the derivation of the adjointequations and gradient computation formulae. In a body-fittedcoordinate system, the Euler equations can be written in conservationlaw form as
∂W∂t
+∂Fi
∂ξi= 0 in D, (1)
whereW = Jw ,
andFi = Sij fj .
The vector of conserved variables is typically given by:
w =
ρρu1ρu2ρu3ρE
fj =
ρuj
ρu1uj + pδ1jρu2uj + pδ2jρu3uj + pδ3j
ρujH
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Design Using the Euler EquationsAssuming that the surface being designed, BW , conforms to thecomputational plane ξ2 = 0, the flow tangency condition can bewritten as
U2 = 0 on BW . (2)Introduce the cost function
I =12
∫ ∫BW
(p − pd )2 dξ1dξ3.
A variation in the shape will cause a variation δp in the pressure andconsequently a variation in the cost function
δI =
∫ ∫BW
(p − pd ) δp dξ1dξ3. (3)
Since p depends on w through the equation of state the variation δpcan be determined from the variation δw . Define the Jacobians
Ai =∂fi∂w
, Ci = SijAj . (4)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Formulation of the Design ProblemThe weak form of the equation for δw in the steady state becomes∫
D
∂ψT
∂ξiδFidD =
∫B
(niψT δFi )dB,
where we have integrated the governing equations by parts and
δFi = Ciδw + δSij fj .
Adding to the variation of the cost function
δI =
∫ ∫BW
(p − pd ) δp dξ1dξ3
−∫D
(∂ψT
∂ξiδFi
)dD
+
∫B
(niψ
T δFi)
dB, (5)
which should hold for an arbitrary choice of ψ.Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Formulation of the Design ProblemIn particular, the choice that satisfies the adjoint equation
∂ψ
∂t− CT
i∂ψ
∂ξi= 0 in D, (6)
subject to far field boundary conditions
niψT Ciδw = 0,
and solid wall conditions
S21ψ2 + S22ψ3 + S23ψ4 = (p − pd ) on BW , (7)
yields and expression for the gradient that is independent of thevariation in the flow solution δw :
δI = −∫D
∂ψT
∂ξiδSij fjdD
−∫ ∫
BW
(δS21ψ2 + δS22ψ3 + S23ψ4) p dξ1dξ3. (8)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Formulation of the Design Problem
The volume integral in blue can be evaluated with ease (however, oneneeds to compute δSij using mesh perturbations). The surfaceintegral in red is also easily evaluated. Note that there are otherformulations where the volume integral can be converted to a surfaceintegral (see next lecture) and the gradient evaluation is simpifiedconsiderably.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Why use the adjoint approach?
Figure 3: CFD as a design tool3.
3 P. Castonguay, S. Nadarajah, “Effect of shape parameterization on aerodynamic shape optimization”.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Spalart-Allmaras Turbulence Model
Outline
1 IntroductionOptimal design in aerodynamicsAlternatives for sensitivity calculations
2 Design Using the Euler Equations
3 Design Using the RANS EquationsSpalart-Allmaras Turbulence Model
4 Continuous adjoint for the Spalart–Allmaras modelAnalytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
5 Conclusions
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Spalart-Allmaras Turbulence Model
Viscous turbulent flows
Large Reynolds numbers: the laminarmotion becomes unstable and the fluidturns turbulent (most applications ofindustrial interest).
Figure 4: Flow transitions (experimental observations).
Turbulent flows are computationallychallenging because:
– Fluid properties exhibit random spatialfluctuations.
– Strong dependence from initialconditions.
– Contain a wide range of scales (eddies).
Figure 5: DNS simulation of a turbulent flow.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Spalart-Allmaras Turbulence Model
Reynolds Averaged Navier–Stokes equations
Flow quantities are expressed as the sum of time fluctuations over small timesscales about a steady or slowly varying mean flow:
ui = ui + u′i , ρ = ρ+ ρ′, p = p + p′, T = T + T ′, . . .
Averaging of the Navier–Stokes equations yields for the mean flow:
Navier–Stokes equations
• ∂∂t ρ+ ∂
∂xi(ρui ) = 0
• ∂∂t (ρui ) + ∂
∂xj(ρuj ui ) = − ∂
∂xip
+ ∂∂xj
tji
• ∂∂t (ρE) + ∂
∂xj
`ρuj H
´=
∂∂xj
(ui tij )
− ∂∂xj
qj
RANS equations
• ∂∂t ρ+ ∂
∂xi(ρui ) = 0
• ∂∂t (ρui ) + ∂
∂xj(ρuj ui ) = − ∂
∂xip
+ ∂∂xj
ht ji − ρu′i u
′j
i• ∂∂t (ρE) + ∂
∂xj
“ρuj H + 1
2ρu′i u′i
”=
∂∂xj
hui
“t ji − ρu′i u
′j
”i− ∂∂xj
hqj + ρu′j H
′ − tji u′i + 12ρu′j u
′i u′i
iNew terms require additional modeling to close the RANS equations.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Spalart-Allmaras Turbulence Model
Turbulent Spalart–Allmaras modelThe Spalart–Allmaras model solves an addition convection-diffusion equation(with a source term):8<: ∂t ν +∇ · ~T cv − T s = 0 in Ω,
ν = 0 on S,ν∞ = σ∞ν∞ on Γ∞.
~T cv (U, ν) = − ν+νσ∇ν + ~v ν
T s(U, ν, dS) = cb1Sν − cw1fw“νdS
”2+
cb2σ|∇ν|2
Coupling to the main stream flow:
µtur = ρνfv1 → µ1tot = µdyn + µtur µ2
tot =µdyn
Prd+µtur
Prt
Figure 8: Ratio µtur/µdyn for a RAE-2822 profile in transonic conditions.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Outline
1 IntroductionOptimal design in aerodynamicsAlternatives for sensitivity calculations
2 Design Using the Euler Equations
3 Design Using the RANS EquationsSpalart-Allmaras Turbulence Model
4 Continuous adjoint for the Spalart–Allmaras modelAnalytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
5 Conclusions
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete system of equations and boundaryconditions
Navier–Stokes equations:
8>>><>>>:RU (U, ν) = ∇ · ~Fc −∇ ·
“µ1
tot~Fv1 + µ2
tot~Fv2
”= 0 in Ω
~v = 0 on S∂nT = 0 on S(W )+ = W∞ on Γ∞
~Fci =
0BBBB@ρvi
ρvi v1 + Pδi1ρvi v2 + Pδi2ρvi v3 + Pδi3
ρvi H
1CCCCA , ~Fv1i =
0BBBB@·τi1τi2τi3
vjτij
1CCCCA , ~Fv2i =
0BBBB@····
Cp∂i T
1CCCCA , i = 1, . . . , 3
τij = ∂j vi + ∂i vj −2
3δij∇ ·~v, µdyn =
µ1T 3/2
T + µ2, µ
1tot = µdyn + µtur , µ
2tot =
µdyn
Prd+µtur
Prt
Spalart–Allmaras model:8<: Rν (U, ν, dS ) = ∇ · ~T cv − T s = 0 in Ω
ν = 0 on Sν∞ = σ∞ν∞ on Γ∞
~T cv (U, ν) = −ν + ν
σ∇ν +~vν, T s (U, ν, dS ) = cb1Sν − cw1 fw
0@ ν
dS
1A2
+cb2
σ|∇ν|2.
µtur = ρνfv1, fv1 =χ3
χ3 + c3v1
, χ =ν
ν, ν =
µdyn
ρ, S = |Ω| +
ν
κ2d2S
fv2
fv2 = 1 −χ
1 + χfv1, fw = g
24 1 + c6w3
g6 + c6w3
351/6
, g = r + cw2(r6 − r), r =ν
Sκ2d2S
Eikonal equation:(
Rd (dS ) = |∇dS |2 − 1 = 0 in Ω
dS = 0 on S
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete system of equations and boundaryconditions
Navier–Stokes equations:
8>>><>>>:RU (U, ν) = ∇ · ~Fc −∇ ·
“µ1
tot~Fv1 + µ2
tot~Fv2
”= 0 in Ω
~v = 0 on S∂nT = 0 on S(W )+ = W∞ on Γ∞
~Fci =
0BBBB@ρvi
ρvi v1 + Pδi1ρvi v2 + Pδi2ρvi v3 + Pδi3
ρvi H
1CCCCA , ~Fv1i =
0BBBB@·τi1τi2τi3
vjτij
1CCCCA , ~Fv2i =
0BBBB@····
Cp∂i T
1CCCCA , i = 1, . . . , 3
τij = ∂j vi + ∂i vj −2
3δij∇ ·~v, µdyn =
µ1T 3/2
T + µ2, µ
1tot = µdyn + µtur , µ
2tot =
µdyn
Prd+µtur
Prt
Spalart–Allmaras model:8<: Rν (U, ν, dS ) = ∇ · ~T cv − T s = 0 in Ω
ν = 0 on Sν∞ = σ∞ν∞ on Γ∞
~T cv (U, ν) = −ν + ν
σ∇ν +~vν, T s (U, ν, dS ) = cb1Sν − cw1 fw
0@ ν
dS
1A2
+cb2
σ|∇ν|2.
µtur = ρνfv1, fv1 =χ3
χ3 + c3v1
, χ =ν
ν, ν =
µdyn
ρ, S = |Ω| +
ν
κ2d2S
fv2
fv2 = 1 −χ
1 + χfv1, fw = g
24 1 + c6w3
g6 + c6w3
351/6
, g = r + cw2(r6 − r), r =ν
Sκ2d2S
Eikonal equation:(
Rd (dS ) = |∇dS |2 − 1 = 0 in Ω
dS = 0 on S
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete system of equations and boundaryconditions
Navier–Stokes equations:
8>>><>>>:RU (U, ν) = ∇ · ~Fc −∇ ·
“µ1
tot~Fv1 + µ2
tot~Fv2
”= 0 in Ω
~v = 0 on S∂nT = 0 on S(W )+ = W∞ on Γ∞
~Fci =
0BBBB@ρvi
ρvi v1 + Pδi1ρvi v2 + Pδi2ρvi v3 + Pδi3
ρvi H
1CCCCA , ~Fv1i =
0BBBB@·τi1τi2τi3
vjτij
1CCCCA , ~Fv2i =
0BBBB@····
Cp∂i T
1CCCCA , i = 1, . . . , 3
τij = ∂j vi + ∂i vj −2
3δij∇ ·~v, µdyn =
µ1T 3/2
T + µ2, µ
1tot = µdyn + µtur , µ
2tot =
µdyn
Prd+µtur
Prt
Spalart–Allmaras model:8<: Rν (U, ν, dS ) = ∇ · ~T cv − T s = 0 in Ω
ν = 0 on Sν∞ = σ∞ν∞ on Γ∞
~T cv (U, ν) = −ν + ν
σ∇ν +~vν, T s (U, ν, dS ) = cb1Sν − cw1 fw
0@ ν
dS
1A2
+cb2
σ|∇ν|2.
µtur = ρνfv1, fv1 =χ3
χ3 + c3v1
, χ =ν
ν, ν =
µdyn
ρ, S = |Ω| +
ν
κ2d2S
fv2
fv2 = 1 −χ
1 + χfv1, fw = g
24 1 + c6w3
g6 + c6w3
351/6
, g = r + cw2(r6 − r), r =ν
Sκ2d2S
Eikonal equation:(
Rd (dS ) = |∇dS |2 − 1 = 0 in Ω
dS = 0 on S
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete system of equations and boundaryconditions
Navier–Stokes equations:
8>>><>>>:RU (U, ν) = ∇ · ~Fc −∇ ·
“µ1
tot~Fv1 + µ2
tot~Fv2
”= 0 in Ω
~v = 0 on S∂nT = 0 on S(W )+ = W∞ on Γ∞
~Fci =
0BBBB@ρvi
ρvi v1 + Pδi1ρvi v2 + Pδi2ρvi v3 + Pδi3
ρvi H
1CCCCA , ~Fv1i =
0BBBB@·τi1τi2τi3
vjτij
1CCCCA , ~Fv2i =
0BBBB@····
Cp∂i T
1CCCCA , i = 1, . . . , 3
τij = ∂j vi + ∂i vj −2
3δij∇ ·~v, µdyn =
µ1T 3/2
T + µ2, µ
1tot = µdyn + µtur , µ
2tot =
µdyn
Prd+µtur
Prt
Spalart–Allmaras model:8<: Rν (U, ν, dS ) = ∇ · ~T cv − T s = 0 in Ω
ν = 0 on Sν∞ = σ∞ν∞ on Γ∞
~T cv (U, ν) = −ν + ν
σ∇ν +~vν, T s (U, ν, dS ) = cb1Sν − cw1 fw
0@ ν
dS
1A2
+cb2
σ|∇ν|2.
µtur = ρνfv1, fv1 =χ3
χ3 + c3v1
, χ =ν
ν, ν =
µdyn
ρ, S = |Ω| +
ν
κ2d2S
fv2
fv2 = 1 −χ
1 + χfv1, fw = g
24 1 + c6w3
g6 + c6w3
351/6
, g = r + cw2(r6 − r), r =ν
Sκ2d2S
Eikonal equation:(
Rd (dS ) = |∇dS |2 − 1 = 0 in Ω
dS = 0 on S
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Objective function and surface deformation
We consider the following choice of objective function:
J(S) =
ZS
j(~f ,T , ∂nν,~n) ds, ~f = P~n − σ · ~n.
Incorporating flow equations as constraints to the cost functional by means ofLagrange multipliers, the Lagrangian reads:
J (S) =
ZS
j(~f ,T , ∂nν,~n) ds
+
ZΩ
“ΨT
URU (U, ν) + ψνRν(U, ν, dS) + ψd Rd (dS)”
dx .
We consider deformations of size δS along the normal direction to the surfaceS′ =
˘~x + δS ~n, ~x ∈ S
¯. So, the following holds:
δ~n = −∇S(δS)δ(ds) = −2HmδS ds
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Variation of the functional under the deformation
Total variation of the functional
δJ =
ZSδj(~f ,T , ∂nν,~n) ds +
ZδS
j(~f ,T , ∂nν,~n) ds
+
ZΩ
“ΨT
UδRU (U, ν) + ψνδRν(U, ν, dS) + ψdδRd (dS)”
dx
Linearization of the system of equations:
N.S.:
8>><>>:∂RU∂U δU = ∇(~AcδU)−∇ ·
„~F vk ∂µ
ktot
∂U δU + µktot~AvkδU + µk
tot Dvk∇δU
«∂RU∂ν δν = −∇ ·
„~F vk ∂µ
ktot
∂ν δν
«
S.A.:
8>><>>:∂Rν∂U δU = ∇ · (~F cvδU)− F sδU − Ms∇δU∂Rν∂ν δν = ∇ ·
“~Bcvδν + Ecv∇δν
”− Bsδν − Es∇δν
∂Rν∂dS
δdS = −K sδdS
Eikonal:n
∂Rd∂dS
δdS = ∇dS · ∇δdS = 0
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Variation of the functional under the deformation
Total variation of the functional
δJ =
ZSδj(~f ,T , ∂nν,~n) ds +
ZδS
j(~f ,T , ∂nν,~n) ds
+
ZΩ
“ΨT
UδRU (U, ν) + ψνδRν(U, ν, dS) + ψdδRd (dS)”
dx
Linearization of the system of equations:
N.S.:
8>><>>:∂RU∂U δU = ∇(~AcδU)−∇ ·
„~F vk ∂µ
ktot
∂U δU + µktot~AvkδU + µk
tot Dvk∇δU
«∂RU∂ν δν = −∇ ·
„~F vk ∂µ
ktot
∂ν δν
«
S.A.:
8>><>>:∂Rν∂U δU = ∇ · (~F cvδU)− F sδU − Ms∇δU∂Rν∂ν δν = ∇ ·
“~Bcvδν + Ecv∇δν
”− Bsδν − Es∇δν
∂Rν∂dS
δdS = −K sδdS
Eikonal:n
∂Rd∂dS
δdS = ∇dS · ∇δdS = 0
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Variation of the functional under the deformation
Total variation of the functional
δJ =
ZSδj(~f ,T , ∂nν,~n) ds +
ZδS
j(~f ,T , ∂nν,~n) ds
+
ZΩ
“ΨT
UδRU (U, ν) + ψνδRν(U, ν, dS) + ψdδRd (dS)”
dx
Linearization of the system of equations:
N.S.:
8>><>>:∂RU∂U δU = ∇(~AcδU)−∇ ·
„~F vk ∂µ
ktot
∂U δU + µktot~AvkδU + µk
tot Dvk∇δU
«∂RU∂ν δν = −∇ ·
„~F vk ∂µ
ktot
∂ν δν
«
S.A.:
8>><>>:∂Rν∂U δU = ∇ · (~F cvδU)− F sδU − Ms∇δU∂Rν∂ν δν = ∇ ·
“~Bcvδν + Ecv∇δν
”− Bsδν − Es∇δν
∂Rν∂dS
δdS = −K sδdS
Eikonal:n
∂Rd∂dS
δdS = ∇dS · ∇δdS = 0
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete continuous adjoint method
System of adjoint equations8<:0 = AU
U ΨU + AUνψν
0 = AνU ΨU + Aννψν0 = Ad
νψν + Addψd
BCs :
8>>>><>>>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ+ ψνg5
”ψν = − 1
g3
∂j∂(∂n ν)
ψd = 0
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Adνψν = −K s
ψν
Addψd = −∇ · (ψd∇dS)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete continuous adjoint method
System of adjoint equations8<:0 = AU
U ΨU + AUνψν
0 = AνU ΨU + Aννψν0 = Ad
νψν + Addψd
BCs :
8>>>><>>>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ+ ψνg5
”ψν = − 1
g3
∂j∂(∂n ν)
ψd = 0
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Adνψν = −K s
ψν
Addψd = −∇ · (ψd∇dS)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete continuous adjoint method
System of adjoint equations8<:0 = AU
U ΨU + AUνψν
0 = AνU ΨU + Aννψν0 = Ad
νψν + Addψd
BCs :
8>>>><>>>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ+ ψνg5
”ψν = − 1
g3
∂j∂(∂n ν)
ψd = 0
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Adνψν = −K s
ψν
Addψd = −∇ · (ψd∇dS)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete continuous adjoint method
System of adjoint equations8<:0 = AU
U ΨU + AUνψν
0 = AνU ΨU + Aννψν0 = Ad
νψν + Addψd
BCs :
8>>>><>>>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ+ ψνg5
”ψν = − 1
g3
∂j∂(∂n ν)
ψd = 0
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Adνψν = −K s
ψν
Addψd = −∇ · (ψd∇dS)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete continuous adjoint method
System of adjoint equations8<:0 = AU
U ΨU + AUνψν
0 = AνU ΨU + Aννψν0 = Ad
νψν + Addψd
BCs :
8>>>><>>>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ+ ψνg5
”ψν = − 1
g3
∂j∂(∂n ν)
ψd = 0
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Adνψν = −K s
ψν
Addψd = −∇ · (ψd∇dS)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete adjoint vs. Frozen viscosity
System of adjoint equations0 = AU
U ΨU + AUνψν
0 = AνU ΨU + AννψνBCs :
8>><>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ
”− g5
g2ψν
ψν = − 1g3
∂j∂(∂n ν)
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Complete adjoint vs. Frozen viscosity
System of adjoint equations0 = AU
U ΨU + AUνψν
0 = AνU ΨU + AννψνBCs :
8>><>>:ϕi = − ∂j
∂fi− ψνg4ni
∂nψ5 = − 1g2
“∂j∂T − ~g1 · ~ϕ
”− g5
g2ψν
ψν = − 1g3
∂j∂(∂n ν)
Adjoint operators:
AUU ΨU = −∇ΨT
U · ~Ac −∇ ·
“∇ΨT
U · µktot D
vk”
+∇ΨTU · µ
ktot~Avk +∇ΨT
U · ~Fvk ∂µ
ktot
∂U
AUνψν = −∇ψν · ~F cv − ψνF s +∇ · (ψνMs)
AνU ΨU = ∇ΨTU · ~F
vk ∂µktot
∂ν
Aννψν = −∇ψν · ~Bcv +∇ · (∇ψν · Ecv )− ψνBs +∇ · (ψνEs)
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Evaluation of surface sensitivities
Solving the adjoint system and introducing in the variation of the functional:
δJ =
ZS
„∂j∂fi∂nfi +
∂j∂T
∂nT +∂j
∂ (∂nν)∂2
n ν
«δS ds
−Z
S
„∂j∂~n
+∂j
∂~fP −
∂j
∂~f· σ«· ∇S(δS) ds −
ZS
(g + 2Hm j)δS ds
−2Z
Sψνg4(P~n − ~n · σ) · ∇S(δS) ds.
Usual objective functions are of the form j(~f ) = ~f · ~d . Then:
Sensitivity computation
δJ = −Z
Sh3δS ds
whereh3 = −~n · Σϕ · ∂n~v + µ2
tot Cp∇Sψ5 · ∇ST
Σϕ = µ1tot
„∇~ϕ+∇~ϕT − Id
23∇ · ~ϕ
«
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Evaluation of surface sensitivities
Solving the adjoint system and introducing in the variation of the functional:
δJ =
ZS
„∂j∂fi∂nfi +
∂j∂T
∂nT +∂j
∂ (∂nν)∂2
n ν
«δS ds
−Z
S
„∂j∂~n
+∂j
∂~fP −
∂j
∂~f· σ«· ∇S(δS) ds −
ZS
(g + 2Hm j)δS ds
−2Z
Sψνg4(P~n − ~n · σ) · ∇S(δS) ds.
Usual objective functions are of the form j(~f ) = ~f · ~d . Then:
Sensitivity computation
δJ = −Z
Sh3δS ds
whereh3 = −~n · Σϕ · ∂n~v + µ2
tot Cp∇Sψ5 · ∇ST
Σϕ = µ1tot
„∇~ϕ+∇~ϕT − Id
23∇ · ~ϕ
«
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Evaluation of surface sensitivities
Solving the adjoint system and introducing in the variation of the functional:
δJ =
ZS
„∂j∂fi∂nfi +
∂j∂T
∂nT +∂j
∂ (∂nν)∂2
n ν
«δS ds
−Z
S
„∂j∂~n
+∂j
∂~fP −
∂j
∂~f· σ«· ∇S(δS) ds −
ZS
(g + 2Hm j)δS ds
−2Z
Sψνg4(P~n − ~n · σ) · ∇S(δS) ds.
Usual objective functions are of the form j(~f ) = ~f · ~d . Then:
Sensitivity computation
δJ = −Z
Sh3δS ds
whereh3 = −~n · Σϕ · ∂n~v + µ2
tot Cp∇Sψ5 · ∇ST
Σϕ = µ1tot
„∇~ϕ+∇~ϕT − Id
23∇ · ~ϕ
«
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical experimentsTransonic RAE-2822
(M∞ = 0,734; Re = 6,5× 106; α = 2,54o)
Figure 9: RAE-2822: Density profile and mesh.
Transonic NACA-0012(M∞ = 0,8; Re = 6,5× 106; α = 1,25o)
Figure 10: NACA-0012: Density profile and mesh.
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 1: Transonic RAE-2822
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 1: Transonic RAE-2822
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 1: Transonic RAE-2822
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 2: Transonic NACA-0012
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 2: Transonic NACA-0012
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 2: Transonic NACA-0012
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Analytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
Numerical test 3: 2D unconstrained drag minimizationusing adjoint RANS
The goal of this academic problem is to reduce the drag of aRAE-2822 profile, by means of modifications of its surface.A total of 38 Hicks–Henne bump functions have been used as designvariables.
Figure 11: Optimization convergence history, adjoint method vs. frozen viscosity (left). Pressure coefficient distribution, original configurationand final design (right).
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Outline
1 IntroductionOptimal design in aerodynamicsAlternatives for sensitivity calculations
2 Design Using the Euler Equations
3 Design Using the RANS EquationsSpalart-Allmaras Turbulence Model
4 Continuous adjoint for the Spalart–Allmaras modelAnalytical formulationTransonic RAE-2822Transonic NACA-0012Optimization of a transonic airfoil
5 Conclusions
Juan J. Alonso Mathematical details of adjoint solvers
IntroductionDesign Using the Euler Equations
Design Using the RANS EquationsContinuous turbulent adjoint
Conclusions
Conclusions
Continuous adjoint formulations for both the Euler and Reynolds-AveragedNavier-Stokes equations can be derived.
Significant care required to obtain highly-accurate gradients (cannot freeze theeddy viscosity in RANS models).
Automating the derivation and enabling the use of arbitrary cost functions isdesirable (see next lecture).
Powerful tool for shape optimization.
Juan J. Alonso Mathematical details of adjoint solvers