discrete adjoint approach to cfd-based shape...

Discrete adjoint approach to CFD-based shapeoptimization

Praveen. [email protected]

Tata Institute of Fundamental ResearchCenter for Applicable Mathematics

Bangalore 560065http://math.tifrbng.res.in

Indo-German Conference onPDE, Sc. Comp. and Opt. in Appl.

IIT Kanpur7-9 October, 2009

Praveen. C (TIFR-CAM) Shape Optimization IITK, Oct 2009 1 / 40

[email protected]

http://math.tifrbng.res.in

Objectives and controls

• Objective function I(β)mathematical representation of system performance

I(β) = I(β,Q(β))

• Control variables β

β represents the shape

• State variable Q: solution of an ODE or PDE

R(β,Q) = 0 =⇒ Q = Q(β)


Example: Aerodynamics

http://www.aviation-history.com

Lift = WeightDrag = Thrust

minβD(β), such that L(β) = W


Gradient-based minimization

minβ∈RN

I(β,Q(β))

• Initialize β0, n = 0• For n = 0, . . . , Niter

I Solve R(βn, Qn) = 0I For j = 1, . . . , N

βn(j) = [βn1 , . . . , βnj + ∆βj , . . . , β

nN ]>

Solve R(βn(j), Qn(j)) = 0

dIdβj

≈I(βn

(j),Qn(j))−I(βn,Qn)

∆βj

I Steepest descent step

βn+1 = βn − sn dI

dβ(βn)

Cost of FD-based steepest-descent

Cost = O(N + 1)Niter = O(N + 1)O(N) = O(N2)Praveen. C (TIFR-CAM) Shape Optimization IITK, Oct 2009 4 / 40

Elements of shape optimization

1 Shape parameterization2 Surface grid generation/deformation3 Domain grid generation/deformation4 Flow solution (Euler/Navier-Stokes solver)5 Adjoint flow solution6 Optimization method

Shape parametersβ

Surface gridXs

Volume grid

XCFD solution

Q I

dI

dβ=

dI

dXdXdXs

dXs

dβ


Adjoint approach: dIdX

• For shape optimization: I = I(X,Q)

dI

dX=

∂I

∂X+∂I

∂Q

∂Q

∂X

• Differentiate state equation R(X,Q) = 0

∂R

∂X+∂R

∂Q

∂Q

∂X= 0

• Flow sensitivity ∂Q∂X ; costly to evaluate

• Introducing an adjoint variable Ψ, we can write

dI

dX=[∂I

∂X+∂I

∂Q

∂Q

∂X

]+ Ψ>

[∂R

∂X+∂R

∂Q

∂Q

∂X

]


Adjoint approach

• Collect terms involving the flow sensitivity

dI

dX=[∂I

∂X+ Ψ>

∂R

∂X

]+[∂I

∂Q+ Ψ>

∂R

∂Q

]∂Q

∂X

• Choose Ψ so that flow sensitivity vanishes

∂I

∂Q+ Ψ>

∂R

∂Q= 0 =⇒

(∂R

∂Q

)>Ψ +

(∂I

∂Q

)>= 0

• GradientdI

dX=

∂I

∂X+ Ψ>

∂R

∂X

Cost of AD-based steepest-descent

Cost = O(1)Niter = O(N)


Optimization steps

• β =⇒ Xs =⇒ X

• Solve the flow (primal) equations to steady-state

X =⇒ dQdt

+R(X,Q) = 0 =⇒ Q, I

• Solve adjoint equations to steady-state

X,Q =⇒ dΨdt

+(∂R

∂Q

)>Ψ +

(∂I

∂Q

)>= 0 =⇒ Ψ

• Compute gradient wrt grid X

dI

dX=

∂I

∂X+ Ψ>

∂R

∂X

dI

dβ=

dI

dXdXdXs

dXs

dβ=⇒ β ←− β − εdI

dβ


Continuous and discrete approaches

• Continuous approach (differentiate and discretize)PDE −→ Adjoint PDE −→ Discrete adjoint

• Discrete approach (discretize and differentiate)PDE −→ Discrete PDE −→ Discrete adjoint

• We use the discrete approachI R(X,Q) = 0 finite volume equations which are algebraic equationsI Use ordinary calculus to differentiateI Need to compute

∂I

∂Q,

∂I

∂X,

(∂R

∂Q

)>Ψ,

(∂R

∂X

)>Ψ

• Computer code available to compute

I(X,Q), R(X,Q)


Automatic differentiation

• Code is made of composition of elementary functions

X −→ Y = F (X)T0 = X

Tr = Fr(Tr−1)Y = F (X) = Fp ◦ Fp−1 ◦ . . . ◦ F1(T0)

• Use differentiation by parts formula: (X, X) −→ Y

Y = F ′(X)X = F ′p(Tp−1)F ′p−1(Tp−2) . . . F ′1(T0)X

• Automated using AD tools

Computer code

PAutomatic

DifferentiationNew code

P

• Note: Computation of Y goes from first line to last line of code


Reverse differentiation

• Reverse mode computes transpose: (X, Y ) −→ X

X = [F ′(X)]>Y = [F ′1(T0)]>[F ′2(T1)]> . . . [F ′p(Tp−1)]>Y

• Forward sweep and then reverse sweep

Func: T0 −→ F1T1−→ F2

T2−→ . . .Tp−1−→ Fp −→ F

Grad: Tp−1, Y −→ [F ′p]T Tp−2−→ [F ′p−1]T

Tp−3−→ . . .T0−→ [F ′1]T −→ X

Forward variables Tj required in reverse order: store or recompute• Reverse mode useful to compute(

∂I

∂Q

)>,

(∂I

∂X

)>,

(∂R

∂Q

)>Ψ,

(∂R

∂X

)>Ψ


Differentiation: Example

• A simple example

f = (xy + sinx+ 4)(3y2 + 6)

• Computer code, f = t10

t1 = x

t2 = y

t3 = t1t2

t4 = sin t1t5 = t3 + t4

t6 = t5 + 4t7 = t22

t8 = 3t7t9 = t8 + 6t10 = t6t9


F77 code: costfunc.f

subroutine costfunc(x, y, f) t1 = x t2 = y t3 = t1*t2 t4 = sin(t1) t5 = t3 + t4 t6 = t5 + 4 t7 = t2**2 t8 = 3.0*t7 t9 = t8 + 6.0 t10 = t6*t9 f = t10 end


AD Tool: TAPENADE

• TAPENADE: http://www-sop.inria.fr/tropics• Source transformation tool• Fortran and C• Free software• http://www.autodiff.org

Use of tapenade

$ tapenade -backward -vars "x y" -outvars "f" costfunc.fTapenade 3.1 (r2757) - 01/12/09 17:50 - Java 1.5.0_20 Mac OS XCommand: Took subroutine costfunc as default differentiation root.@@ Creating ./costfunc_b.f@@ Creating ./costfunc_b.msg


Automatic Differentiation: Reverse mode

SUBROUTINE COSTFUNC_B(x, xb, y, yb, f, fb) t1 = x t2 = y t3 = t1*t2 t4 = SIN(t1) t5 = t3 + t4 t6 = t5 + 4 t7 = t2**2 t8 = 3.0*t7 t9 = t8 + 6.0 t10b = fb t6b = t9*t10b t9b = t6*t10b t8b = t9b t7b = 3.0*t8b t5b = t6b t3b = t5b t2b = t1*t3b + 2*t2*t7b t4b = t5b t1b = t2*t3b + COS(t1)*t4b yb = t2b xb = t1b fb = 0.0 END

Z =[xy

]

Z =[xbyb

]=[∂f

∂Z

]>f

If f = 1, then

xb =∂f

∂x

yb =∂f

∂y


Implementation of AD (steady-state problems)

• Do not differentiate iterative schemes• CFD code written with many subroutines• Subroutines differentiated individually• Assembled together in reverse order to form adjoint solver• Only non-linear portions differentiated with AD

I numerical flux (Roe)I Limiters

• Linear portions differentiated manually• Automate using make utility• Leads to an efficient code with less memory requirements


Shape parameterization

• Parameterize thedeformations[xsys

]=

[x

(0)s

y(0)s

]+[nxny

]h(ξ)

h(ξ) =m∑k=1

βkBk(ξ)

• Hicks-Henne bump functions

Bk(ξ) = sinp(πξqk), qk =log(0.5)log(ξk)

• Move points along normal toreference line AB

A B

~n

ξ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ξ

h(ξ)

Exact derivatives dXsdβ can be

computed


Grid deformation

• Interpolate displacement ofsurface points to interiorpoints using RBF

f(x, y) = a0 + a1x+ a2y +N∑j=1

bj |~r − ~rj |2 log |~r − ~rj |

where ~r = (x, y)

• Results in smooth grids• Exact derivatives dX

dXscan be

computed

Initial grid

Deformed grid


NUWTUN flow solver

Based on the ISAAC code of Joseph Morrisonhttp://isaac-cfd.sourceforge.net

• Finite volume scheme• Structured, multi-block grids• Roe flux• MUSCL reconstruction with Hemker-Koren limiter• Implicit scheme

Source code of NUWTUN available onlinehttp://nuwtun.berlios.de


Validation of adjoint gradients

• Dot-product test

R := R′(Q)Q

≈ R(Q+ εQ)−R(Q)ε

Q := [R′(Q)]>R

R>R?= Q>Q

• Upwind schemes and limiterscan cause problems due tonon-differentiability

• Check adjoint derivativesagainst finite difference

I NACA0012: Cd/Cl

I RAE2822: Cd

5 10 15 20Hicks-Henne parameter

-40

-20

0

20

40

60

80

Gra

dien

t

ADFD

5 10 15 20Hicks-Henne parameter

-100

-50

0

50

100

150

Gra

dien

t

ADFD


Convergence and limiter: RAE2822 airfoil

L(a, b) =a(b2 + 2ε) + b(2a2 + ε)

2a2 − ab+ 2b2 + 3ε, 0 < ε� 1

ε = 10−8 ε = 10−4

1e-05

0.0001

0.001

0.01

0.1

1

0 200 400 600 800 1000 1200 1400 1600

Res

idue

Number of iterations

Densityx-momeny-momenz-momen

Energy

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 200 400 600 800 1000 1200 1400 1600

Res

idue

Number of iterations

Densityx-momeny-momenz-momen

Energy

Adjoint iterations blow-up No blow-up


Iterative convergence tests

Primal residual Adjoint residualRn = ||R(Qn)|| Rn = ||[R′(Q∞)]>Ψn + I′(Q∞)||

0 500 1000 1500 2000 2500Number of iterations

1x10-151x10-141x10-131x10-121x10-111x10-101x10-91x10-81x10-71x10-6

0.000010.0001

0.0010.01

0.11

10100

Resid

ue

Flow residual

Adjoint residual

0 500 1000 1500 2000 2500 3000Number of iterations

0.2

0.4

0.6

0.8

Cl

0

0.02

0.04

0.06

0.08

0.1

Cd

ClCd

Convergence characteristics for the flow and adjoint solutions, andconvergence of lift and drag coefficients,

for RAE2822 airfoil at M∞ = 0.73


Test cases

• NACA0012: M∞ = 0.8, α = 1.25o

I =CdCl

Cl0Cd0

• RAE2822: M∞ = 0.729, α = 2.31oI Penalty approach

I =Cd

Cd0

+ ω

∣∣∣∣1− Cl

Cl0

∣∣∣∣I Constrained minimization

min I =Cd

Cd0

s.t. Cl = Cl0


NACA0012: Maximize L/D

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1

-Cp

Initialconmin_frcgoptpp_q_newtonsteep

0 0.2 0.4 0.6 0.8 1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06


Method I 100Cd Cl Nfun Ngrad

Initial 1.000 2.072 0.295 - -conmin frcg 0.170 0.389 0.325 50 13

optpp q newton 0.142 0.329 0.329 50 39steep 0.166 0.335 0.287 50 45


RAE2822: Drag minimization, penalty approach

0 0.2 0.4 0.6 0.8 1x

-1

-0.5

0

0.5

1

1.5

-Cp


0 0.2 0.4 0.6 0.8 1x

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

y



Initial 1.000 1.150 0.887 - -conmin frcg 0.355 0.405 0.890 50 13

optpp q newton 0.351 0.400 0.884 50 51steep 0.341 0.388 0.884 50 47


RAE2822: Lift-constrained drag minimization

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1

1.5

-Cp

Initialconmin_mfdfsqpipopt

0 0.2 0.4 0.6 0.8 1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Initialconmin_mfdfsqpipopt


Initial 1.000 1.150 0.887 - -conmin mfd 0.342 0.394 0.881 50 22

fsqp 0.325 0.374 0.887 50 32ipopt 0.335 0.385 0.887 45 62


RAE2822: Lift-constrained drag minimization

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1

1.5-Cp

ipoptInitial

0 0.2 0.4 0.6 0.8 1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

ipoptInitial


RANS computation

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1

1.5

-Cp

RAE2822Optimized

M=0.729, Re=6.5 million, Cl=0.88, k-w turbulence model

Shape Inviscid RANSAOA Cl 100Cd AOA Cl 100Cd

RAE2822 2.31 0.887 1.150 3.28 0.887 2.558Optimized 2.31 0.887 0.386 3.22 0.885 1.692


Sensitivity to perturbations

0.7 0.71 0.72 0.73 0.74 0.75 0.76Mach number

0

0.01

0.02

0.03

0.04

Dra

g co

effi

cien

t

InitialOptimized

0.7 0.71 0.72 0.73 0.74 0.75 0.76Mach number

0

50

100

150

200

Lif

t/Dra

g

InitialOptimized

(a) (b)Variation of (a) drag coefficient and (b) L/D with Mach number for

RAE2822 airfoil and optimized airfoil

Need for robust aerodynamic optimization


ATR


Thick airfoil

• NACA0040 airfoil: 40% thickness• M∞ = 0.5, α = 0 deg.• Inviscid optimization• A posteriori RANS verification:

Re = 0.5 million, k − ω turbulence model


NACA0040: inviscid and RANS Cp

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1

1.5

2-Cp

InviscidRANS


Mitigating adverse pressure gradients

• ~q = velocity vector, q = |~q|• θ = momentum thickness• Separation of boundary layer occurs when (Nash and Macdonald)

θ

q

dqds≤ −0.004

• Objective functional

I =∫A

1q2

[~q

q· ∇p

]+

dA

Minimizing I should reduce tendency for BL separation


Inviscid: Initial and optimized pressure

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1

1.5

2-C

p

InviscidInviscid opt


Optimized airfoil: inviscid and RANS solution

0 0.2 0.4 0.6 0.8 1x/c

-1

-0.5

0

0.5

1-C

p Inviscid optRANS opt


RANS: Mach number


Drag coefficient

Shape Inviscid RANSInitial 7.0426e-4 4.0229e-2

Optimized 7.5306e-4 2.6779e-2


Initial and optimized shapes

0 0.2 0.4 0.6 0.8 1

-0.2

-0.1

0

0.1

0.2

InitialOptimized


Summary

• Adjoint approach in CFD for steady-state problems• AD promises to ease generation/maintenance of adjoint solver

I Differentiation of entire design chain

• True gradient of numerical function• But, is AD gradient consistent with analytical gradient ?

Shocks, non-smooth schemes, turbulence models, etc.• Whenever you need gradients, think of AD

I OptimizationI Adjoint-based error estimation and grid adaptationI Exact linearization for implicit schemes, GMRESI Sensitivity analysisI Uncertainty quantification

• AD tools available for fortran, C, matlab• AD for MPI programs


Homi Bhabha Centenary Year

http://www.tifr.res.in/~hbbccc


http://www.tifr.res.in/~hbbccc

discrete adjoint approach to cfd-based shape...

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