numerical shape optimization and adjoint equations - tifr...

Numerical shape optimization and adjoint equations

Praveen. [email protected]

Tata Institute of Fundamental ResearchCenter for Applicable Mathematics

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

TIFR-CAM18 August, 2009

Praveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 1 / 37

Effect of shape on flow

http://www.aerospaceweb.org


Airfoils

http://www.centennialofflight.gov


Compressible flow and shocks

Mach number, M =speed of air

speed of sound

Range, R = Ma

cT

CLCD

logWi

Wf


Golf ball



Effect of shape on flow



Objectives and controls

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

• Control variables βI Parametric controls β ∈ Rn

I Infinite dimensional controls β : X → YI Shape β ∈ set of admissible shapes

• State variable Q: solution of an ODE or PDE

R(β,Q) = 0 =⇒ Q = Q(β)


Mathematical formulation

• Constrained minimization problem

minβ

I(β,Q) subject to R(β,Q) = 0

• Find δβ such that δI < 0 (to first order)

δI =∂I

∂βδβ +

∂I

∂QδQ =

[∂I

∂β+∂I

∂Q

∂Q

∂β

]︸︷︷︸

G

δβ

• Steepest descentδβ = −εG>, ε > 0

δI = −εGG> = −ε‖G‖2 < 0

How to compute gradient G cheaply and accurately ?


Elements of shape optimization

Shape parametersβ

Surface gridXs

Volume grid

XCFD solution

Q I

dI

dβ=

dI

dXdXdXs

dXs

dβPraveen. C (TIFR-CAM) Shape Optimization TIFR, 18 Aug 2009 9 / 37

Adjoint approach

• For shape optimization: I = I(X,Q)

dI

dX=

∂I

∂X+∂I

∂Q

∂Q

∂X

• Flow sensitivity ∂Q∂X ; costly to evaluate

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

∂R

∂X+∂R

∂Q

∂Q

∂X= 0

• 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 or

(∂R

∂Q

)>Ψ +

(∂I

∂Q

)>= 0

• GradientdI

dX=

∂I

∂X+ Ψ>

∂R

∂X


Optimization steps

• β =⇒ Xs =⇒ X

• Solve the flow equations to steady-state

dQdt

+R(X,Q) = 0 =⇒ Q, I(X,Q)

• Solve adjoint equations to steady-state

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 represent the 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


Automatic differentiation

• Computer code available to compute

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

• Code is made of composition of elementary functions

T0 = X

r′th line of program: Tr = Fr(Tr−1)Y = F (X) = Fp ◦ Fp−1 ◦ . . . ◦ F1(T0)

I Use differentiation by parts formula

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

I Automated using AD tools: Algorithmic Differentiation

Computer code

PAutomatic

DifferentiationNew code

P


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


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


Test cases: airfoil shape optimization

RAE2822: M∞ = 0.729, α = 2.31o

min I =CdCd0

s.t. Cl = Cl0


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


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


Wing optimization

Initial shape


Wing optimization

Optimized shape


ATR


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Pressure surface

(Rakshith, JNCASR)


Unmanned air vehicle


Adjoint-based aposteriori error estimation

• uh = numerical solution on grid h

• Numerical analysis

‖u− uh‖ = O(hp), p > 0 as h→ 0+

How to choose the grid size ?• Functional of interest J(u)

J(u)− J(uh) = O(hp)

Choose grid so that error in J is small


Cartesian Points

NACA-0012 - coarseM∞ = 1.2, α = 0o

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Points Mach number(Mohan Varma)


PDE with homogeneous BC

• Linear PDELu = f in D

• Adjoint operator(v, Lu) = (L∗v, u)

• Functional of interestJ = (g, u)

• Adjoint PDEL∗v = g

• Primal-dual equivalence

J = (g, u)= (L∗v, u)= (v, Lu)= (v, f)


Error estimation: homogeneous BC

fh := Luh, gh := L∗vh

The functional can be written as

J = (g, u)= (g, uh)− (gh, uh − u) + (gh − g, uh − u)= (g, uh)− (L∗vh, uh − u) + (gh − g, uh − u)= (g, uh)− (vh, L(uh − u)) + (gh − g, uh − u)= (g, uh)− (vh, fh − f) + (gh − g, uh − u)

J − Jh = −Computable error︷︸︸︷(vh, fh − f) +(gh − g︸︷︷︸

O(hp)

, uh − u︸︷︷︸O(hp)

)

J − [Jh + (vh, fh − f)] = O(h2p)


PDE with non-homogeneous BC

• Linear PDE

Lu = f in D

Bu = e on ∂D

• Functional of interest

J = (g, u)D + (h,Cu)∂D

• Adjoint PDE

L∗v = g in D

B∗v = h on ∂D

• Primal-dual equivalence

J = (g, u)D + (h,Cu)∂D= (v, f)D + (C∗v, e)∂D


PDE with non-homogeneous BC

Given approximate solutions uh, vh we define

fh := Luh, gh := L∗vh

eh := Buh, hh := B∗vh

The error in output functional is

J − Jh =

Computable error︷︸︸︷− (vh, fh − f))D − (C∗vh, eh − e))∂D+ (gh − g, uh − u)D + (hh − h,C(uh − u))∂D


Goal-based grid adaptation

D = ∪kDk

J − Jh = −(vh, fh − f)D = −∑k

(vh, fh − f)Dk

Divide element Dk if|(vh, fh − f)Dk

| > e0

where e0 is user-specified error level.


Example: Linear scalar convection

y∂u

∂x− x∂u

∂y= 0

IsoValue-0.05263160.02631580.07894740.1315790.1842110.2368420.2894740.3421050.3947370.4473680.50.5526320.6052630.6578950.7105260.7631580.8157890.8684210.9210531.05263


Example: Linear scalar convection

y∂u

∂x− x∂u

∂y= 0

IsoValue-0.194481-0.1701-0.153846-0.137591-0.121337-0.105083-0.0888287-0.0725745-0.0563203-0.040066-0.0238118-0.007557580.008696640.02495090.04120510.05745930.07371360.08996780.1062220.146858

Error from LTMIsoValue-0.003321130.001660560.004981690.008302820.01162390.01494510.01826620.02158730.02490840.02822960.03155070.03487180.03819290.04151410.04483520.04815630.05147750.05479860.05811970.0664225

Error indicator


Application to compressible flows

• H = grid on which computations are performed• h = finer grid, perhaps h = H/2• We try to estimate

J(uh)− J(uH)

by using discrete adjoint method.

Not on that, it will tell us how to adapt the grid, in an optimalway, to control errors in J .


Application to compressible flows

0 2000 4000 6000 8000 100000.3

0.305

0.31

0.315

0.32

N

CL

JH

Jh

JhH+J

cc

(a) M∞ = 0.63, α = 2◦

0 2000 4000 6000 8000 100000.108

0.11

0.112

0.114

0.116

0.118

0.12

NC

D

JH

Jh

JhH+J

cc

(b) M∞ = 0.95, α = 0◦

Figure: Adjoint-based Lift and drag correction for NACA 0012 airfoil section


numerical shape optimization and adjoint equations - tifr...

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