description of adjointshapeoptimizationfoam and how to...

Overview Introduction Theory Implementation Modifications Results

Description of AdjointShapeOptimizationFoamand how to implement new cost functions

Ulf Nilsson

A project work in the courseCFD with OpenSource software,

taught byHakan Nilsson.

Chalmers University of Technology,Gothenburg, Sweden

December 10, 2013

Ulf Nilsson adjointShapeOptimizationFoam December 10, 2013 1 / 40


Overview

I Introduction

I Theory

I Implementation

I Modifications

I Results



Introduction

Optimization in CFD

I Increased importanceI Optimization algorithm

- Simplex- Evolution Strategies- Gradient based- ...

I Demanding

- Number of variables- Constraints



Introduction

Adjoint approach

I Gradient based method

I Tool to compute sensitivity w.r.t. design variables

I Well behaved objective function

I Large number of design variables

I Constraints, number of functions



Optimization problem

Objective function

minimize J = J(v, p, α) (1)

such that

(v · ∇)v +∇p−∇ · (2νD(v)) + αv = 0, (2)

∇ · v = 0. (3)

I Gerneral cost function

I Incompressible, steady state RANS equation

I Porosity α

I Topology optimization




Lagrangian relaxation

minimize L := J +

∫Ω

(u, q)RdΩ. (4)

I Lagrange multipliers, (u, q)

I A penalty to violate the constraints

I Adjoint velocity, u, and pressure, q

I No physical meaning




Variations w.r.t. flow and design variables

δL = δαL+ δvL+ δpL, (5)

δvL+ δpL = 0. (6)

I Total variation.I Choose adjoint variables so that the variation with respect to flow

variables vanishes.I Equation of the adjoint variables and an expression for the sensitivity

of L with respect to the porosity of each cell, derivations out of thescope of the project.

I Approximations needed to arrive at the adjoint equations.- Forzen turbulence.- Ducted flow specialization.

I Integration by parts to arrive at volume contribution and boundarycontribution.



Resulting equations

Sensitivity

∂L

∂αi= ui · viVi, (7)

I Expression for cell i

I Vi the cell volume.

I Equation of the adjoint variables and an expression for the sensitivityof L with respect to the porosity of each cell



Resulting equations

Adjoint equations

−2D(u)v = −∇q +∇ · (2νD(u))− αu− ∂JΩ

∂v, (8)

∇ · u =∂JΩ

∂p. (9)

I Contribution from volumetric part of the cost function, JΩ.



Resulting equations

Adjoint boundary conditions

Wall and inlet boundary conditions

ut = 0, (10)

un = −∂JΩ

∂p, (11)

n · ∇q = 0. (12)

Outlet boundary conditions

q = u · v + unvn + ν(n · ∇)un +∂JΓ

∂vn, (13)

0 = vnut + ν(n · ∇)ut +∂JΓ

∂vt. (14)

Ulf Nilsson adjointShapeOptimizationFoamDecember 10, 2013 10 /

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Resulting equations


Wall Inlet Outlet

v No-slip Prescribed value Zero gradient

p Zero gradient Zero gradient p = 0

Table: Boundary conditions for the primal quantities v and p.

I Only valid for specific primal boundary condition.

I Contribution from both the volume and boundary part of the costfunction, JΩ and JΓ respectively.

I Eq. 13 and 14 used to calculate the adjoint pressure and tangentialpart of the adjoint velocity at the outlet.


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Resulting equations

Summary of the theory

I Introducing a Lagrange relaxation, with multipliers u and q.

I Choice of adjoint variables gives adjoint equations and an expressionfor the sensititivity.

I Lengthy derivations.

I Approximations and requirements to ensure a valid method.

I If the cost function satisfies Jω = 0 the dependence of J enters thesolver only through the boundary conditions.

I The optimization problem reduces to solving two flow equations, theprimal and the adjoint.

I When the sensitivity is calculated an algorithm is needed to updatethe porosity.


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Steepest descent algorithm

Steepest descent

pk = −∇f(xk), (15)

αn+1 = αn − ui · viViδ. (16)

I General search direction in the opposite direction of the gradient ofthe objective function.

I Considering a linear problem a minimum will be found for smallenough steps in the search direction.

I Using the sensitivity, ∂L/∂α to update.

I The step length δ. An optimization problem of its own.

I Under relaxation and limits in the final implementation.


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Solver

Solution procedure

Main-loop

Solve theprimal system

(v, p)

Solve theadjoint system

(u, q)

Update theporosity

Steepest descent

ConvergenceassessmentEnd time or

convergence condition

EndConverged

“Frozen tubulence”

Figure 1: Block scheme of the solution procedure used in adjointShapeOptimizationFoam.

1


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Solver

$FOAM SOLVERS/incompressible/

adjointShapeOptimizationFoam/

adjointShapeOptimizationFoam.C

Let us study the implementation together, trying to identify the underlyingtheory we have covered so far. In the order it is implemented in the code.

I Porosity update

- Implementation of the steepest descent using under relaxation.- Correct?

I The primal pressure-velocity SIMPLE corrector

- How is the sink term implemented?

I The adjoint pressure-velocity SIMPLE corrector

- Is the current implementation done for objective functions withvolumteric contribution (JΩ 6= 0)?


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Solver

Porosity update

94 a l p h a +=95 mesh . f i e l d R e l a x a t i o n F a c t o r ("alpha" )96 ∗( min ( max ( a l p h a + lambda ∗(Ua & U) , z e r o A l p h a ) , alphaMax ) − a l p h a ) ;

Listing 1: file adjointShapeOptimizationFoam.C

αn+1 = αn(1− γ) + γmin (max ((αn + ui · viViδ) .0) , αmax) , (17)


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Solver

Porosity update

However the there seems to be an incorrect sign in the steepest descentimplementation. The correct eq. and implementation should be

αn+1 = αn(1− γ) + γmin (max ((αn − ui · viViδ) , 0) , αmax) . (18)

94 a l p h a +=95 mesh . f i e l d R e l a x a t i o n F a c t o r ("alpha" )96 ∗( min ( max ( a l p h a − lambda ∗(Ua & U) , z e r o A l p h a ) , alphaMax ) − a l p h a ) ;



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Solver

Primal SIMPLE loop

103 // Momentum p r e d i c t o r104105 tmp<f v V e c t o r M a t r i x> UEqn106 (107 fvm : : d i v ( phi , U)108 + t u r b u l e n c e−>d i v D e v R e f f (U)109 + fvm : : Sp ( a lpha , U)110 ) ;111112 UEqn ( ) . r e l a x ( ) ;113114 s o l v e ( UEqn ( ) == −f v c : : g rad ( p ) ) ;



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Solver

Primal SIMPLE loop

I fvm::Sp(alpha, U) Implicit implementation of the extrasource term.


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Solver

Adjoint SIMPLE loop

156 // A d j o i n t Momentum p r e d i c t o r157158 v o l V e c t o r F i e l d a d j o i n t T r a n s p o s e C o n v e c t i o n ( ( f v c : : grad (Ua) & U ) ) ;159 // v o l V e c t o r F i e l d a d j o i n t T r a n s p o s e C o n v e c t i o n160 // (161 // f v c : : r e c o n s t r u c t162 // (163 // mesh . magSf ( )∗ ( f v c : : snGrad (Ua) & f v c : : i n t e r p o l a t e (U) )164 // )165 // ) ;166167 z e r o C e l l s ( a d j o i n t T r a n s p o s e C o n v e c t i o n , i n l e t C e l l s ) ;168169 tmp<f v V e c t o r M a t r i x> UaEqn170 (171 fvm : : d i v (−phi , Ua)172 − a d j o i n t T r a n s p o s e C o n v e c t i o n173 + t u r b u l e n c e−>d i v D e v R e f f (Ua)174 + fvm : : Sp ( a lpha , Ua)175 ) ;176177 UaEqn ( ) . r e l a x ( ) ;178179 s o l v e ( UaEqn ( ) == −f v c : : g rad ( pa ) ) ;



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Solver

Adjoint SIMPLE loop

I Similar to the primal system.

I No additional terms containing information about the costfunction, i.e. current implementation treats cost functions ofthe type JΓ = 0


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Boundary conditions of the adjoint variables


The bad results when the sign in the steepest descent algorithm ischanged indicates additional problems with the current solver. Since theactual solver seems to agree with the theory, what remains to be examinedis the boundary conditions.First the boundary condition for a specific cost function need to becalculated.


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Power dissipation

J := −∫Γ

(p+1

2v2)v · n dΓ. (19)

I Only boundary contribution.

I Energy loss, due to pressure drop and kinetic energy.


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Inlet boundary conditions

ut = 0, (20)

un = −∂JΩ

∂p, (21)

n · ∇q = 0. (22)

ut = 0,

un =

0 at wall,

vn at inlet,

n · ∇q = 0.

(23)


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Inlet boundary conditions

I Use existing conditions

- Prescribed value- No slip- Zero-gradient

I Could be useful to implement a inlet boundary condition forthe adjoint velocity, if the velocity profile of the primal velocityat the inlet is not easily described.


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q = u · v + unvn + ν(n · ∇)un +∂JΓ

∂vn, (24)

0 = vnut + ν(n · ∇)ut +∂JΓ

∂vt. (25)

q = u · v + unvn + ν(n · ∇)un − 1

2v2 − v2

n,

0 = vn(ut − vt) + ν(n · ∇)ut.(26)


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I Use eq. 26 to prescribe a value to the adjoint pressure

I Use eq. 26 to prescribe a value to the tangential componentof the adjoint velocity

ut =− ν

∆uneigh,t − vp,nvp,tvp,n + ν

∆

, (27)


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Implementation of pressure condition

86 // ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Member F u n c t i o n s ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ //8788 void Foam : : a d j o i n t O u t l e t P r e s s u r e F v P a t c h S c a l a r F i e l d : : u p d a t e C o e f f s ( )89 90 i f ( updated ( ) )91 92 return ;93 9495 const f v s P a t c h F i e l d<s c a l a r>& p h i p =96 patch ( ) . l o o k u p P a t c h F i e l d<s u r f a c e S c a l a r F i e l d , s c a l a r >("phi" ) ;9798 const f v s P a t c h F i e l d<s c a l a r>& p h i a p =99 patch ( ) . l o o k u p P a t c h F i e l d<s u r f a c e S c a l a r F i e l d , s c a l a r >("phia" ) ;

100101 const f v P a t c h F i e l d<v e c t o r>& Up =102 patch ( ) . l o o k u p P a t c h F i e l d<v o l V e c t o r F i e l d , v e c t o r >("U" ) ;103104 const f v P a t c h F i e l d<v e c t o r>& Uap =105 patch ( ) . l o o k u p P a t c h F i e l d<v o l V e c t o r F i e l d , v e c t o r >("Ua" ) ;106107 operator==(( p h i a p / patch ( ) . magSf ( ) − 1 . 0 )∗ p h i p / patch ( ) . magSf ( ) + (Up & Uap ) ) ;108109 f i x e d V a l u e F v P a t c h S c a l a r F i e l d : : u p d a t e C o e f f s ( ) ;110

Listing 5: file adjointOutletPressureFvPatchScalarField.C


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Implementation of velocity condition

83 // ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Member F u n c t i o n s ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ //8485 // Update th e c o e f f i c i e n t s a s s o c i a t e d w i t h th e patch f i e l d86 void Foam : : a d j o i n t O u t l e t V e l o c i t y F v P a t c h V e c t o r F i e l d : : u p d a t e C o e f f s ( )87 88 i f ( updated ( ) )89 90 return ;91 9293 const f v s P a t c h F i e l d<s c a l a r>& p h i a p =94 patch ( ) . l o o k u p P a t c h F i e l d<s u r f a c e S c a l a r F i e l d , s c a l a r >("phia" ) ;9596 const f v P a t c h F i e l d<v e c t o r>& Up =97 patch ( ) . l o o k u p P a t c h F i e l d<v o l V e c t o r F i e l d , v e c t o r >("U" ) ;9899 s c a l a r F i e l d Un(mag( patch ( ) . n f ( ) & Up ) ) ;

100 v e c t o r F i e l d UtHat ( ( Up − patch ( ) . n f ( )∗Un ) / ( Un + SMALL ) ) ;101102 v e c t o r F i e l d Uan ( patch ( ) . n f ( )∗ ( patch ( ) . n f ( ) & p a t c h I n t e r n a l F i e l d ( ) ) ) ;103104 v e c t o r F i e l d : : operator=( p h i a p ∗ patch ( ) . Sf ( ) / s q r ( patch ( ) . magSf ( ) ) + UtHat ) ;105 // v e c t o r F i e l d : : o p e r a t o r =(Uan + UtHat ) ;106107 f i x e d V a l u e F v P a t c h V e c t o r F i e l d : : u p d a t e C o e f f s ( ) ;108

Listing 6: file adjointOutletVelocityFvPatchVectorField.C


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Analyzing the implementation

Listing 5 gives an implementation of the pressure condition according to

q = (un − 1)vn + u · v. (28)

compare to

q = u · v + unvn + ν(n · ∇)un −1

2v2 − v2

n, (29)

Listing 6 gives an implementation of the velocity condition according to

up,t =vp − vp,n

up,n + SMALL. (30)

compare to

ut =− ν

∆uneigh,t − vp,nvp,tvp,n + ν

∆

, (31)


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Analyzing the theory

I Not equal to the theory.

I Another cost function, total pressure loss.

I Approximations or derivations done makes the cost functionhard to identify.


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Analyzing the theory

I Changing the sign of the steepest descent implementation.

I Implementing the cost functions according to derivedequations.


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Adjoint pressure boundary condition

I Similar to the existing implementationI A few additional terms

- Velocity in the neighbouring node- Effective viscosity

const s c a l a r F i e l d& d e l t a i n v = patch ( ) . d e l t a C o e f f s ( ) ; // d i s t a n c e ˆ(−1) between patch and n e i g h b o u r i n g node .

// c r e a t e t he o b j e c t needed to g e t th e v i s c o u s i t y :const i n c o m p r e s s i b l e : : RASModel& rasMode l =

db ( ) . look upOb ject<i n c o m p r e s s i b l e : : RASModel>("RASProperties" ) ;

// n u e f f from t he t u r u l e n c e model , rasMode l :s c a l a r F i e l d n u e f f = rasMode l . n u E f f ( ) ( ) . b o u n d a r y F i e l d ( ) [ patch ( ) . i n d e x ( ) ] ;

// N e i g h b o u r i n g node ’ s v e l o c i t y ( normal component ) :s c a l a r F i e l d Uane igh n = ( Uap . p a t c h I n t e r n a l F i e l d ( ) & patch ( ) . n f ( ) ) ;

Listing 7: file myAdjointOutletVelocityFvPatchVectorField.C


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Adjoint velocity boundary condition

I Vector fields instead of scalar fields

// p a t c h I n t e r n a l F i e l d to g e t th e a d j o i n t v e l o c i t y o f n e i g h b o u r i n g node .v e c t o r F i e l d Uaneigh = Uap . p a t c h I n t e r n a l F i e l d ( ) ;// I t s t a n g e n t i a l and normal componentsv e c t o r F i e l d Uane igh n = ( Uaneigh & normal )∗ normal ;v e c t o r F i e l d U a n e i g h t = Uaneigh − Uane igh n ;

Listing 8: file myAdjointOutletVelocityFvPatchVectorField.C


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Case settings

The boundary conditions of the primal and adjoint variables are setaccording to the table below. More detailed information and m4 script ofthe “box” example case can be found in the provided files.

Wall Inlet Outlet

u Prescribed value, 0 Prescribed value, un = vn adjointOutletVelocityPower

q Zero gradient Zero gradient adjointOutletPressurePower


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Power dissipation

pitzDaily porosity field


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Power dissipation

pitzDaily value of the objective function

0

0.001

0.002

0.003

0.004

0.005

0.006

0 500 1000 1500 2000 2500 3000

J

Iteration step

”valueObjFunc.xy”


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Power dissipation

pitzDaily value of the objective function, without

porosity update

0

0.001

0.002

0.003

0.004

0.005

0.006

0 500 1000 1500 2000 2500 3000

J

Iteration step

”valueObjFunc.xy”


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Power dissipation

Pipe bend example

A movie can be found in the case file.


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Power dissipation

Thank you for listening!

Questions


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