The Continuous Adjoint Turbulent Solver for Topology ...hani/kurser/OS_CFD_2019/Luis...Final...

The Continuous Adjoint Turbulent Solver for Topology Optimization “adjointOptimisationFoam”Industrial Approach

MSc. Ing. Luis Fernando Garcia Rodriguez

University of São Paulo (USP) Brazil

CFD with OpenSource Software course, Final Presentation

Chalmers University, Sweden

November 27th, 2019

The Continuous Adjoint Turbulent Solver for Topology Optimization “adjointOptimisationFoam”Industrial Approach

Sponsors

CFD with OpenSource Software course, Final Presentation

University of Chalmers, Sweden

November 27th, 2019

RESEARCH CENTRE FOR GAS INNOVATION

Contents

• Research Group

• Topology Optimization (TO)

• TO Formulation

• Frozen Turbulence TO solver

• Continous Adjoint Turbulent solver: “adjointOptimizationFoam”

• Turbulent Sensitivity Calculation

• How to use the solver?

• Modifications to the adjoint Turbulent sensitivity

3


Contents

• Research Group


• TO Formulation

• “adjointOptimizationFoam” solver

• Sensitivity Calculation

• Modifications

4


Research Group

5


Research Group

6

• >300 researchers

• Research programmes: Natural Gas, Biogas, Hydrogen, Gas Transport and Monitoring, CO2 Abatement and CarbonCapture and Storage

• >45 Projects

RESEARCH CENTRE FOR GAS INNOVATION 7

Power consumed by aCO2 compressor ~ 5%of the plant rating, e.g.a 600 MW plant wouldrequire 30 MW or40.000 [hp]


Contents

• Research Group


• TO Formulation



• Modifications

8


Topology Optimization Method (TOM)

• Goal:

– Distribute material in a domain to minimize/maximize an objective function while keeping constraints satisfied

• In flows:

– Uses porosity field to represent solid regions

9

High permeability

Low permeability


Contents

• Research Group


• TO Formulation



• Modifications

10


Topology Optimization Formulation*

11

In compressible Navier Stokes equations in non-conservative form

𝛻 ∙ 𝒖 = 0

𝒖 ∙ 𝛻 𝒖 +1

𝜌𝛻𝑝 − 𝛻 ∙ 𝜈 𝛻𝒖 + 𝛻𝒖𝑻 = 0

Navier Stokes Modified Equations

𝛻 ∙ 𝒖 = 0

(𝒖 ∙ 𝛻)𝒖 +1

𝜌𝛻𝑝 − 𝛻 ∙ 𝜈 𝛻𝒖 + 𝛻𝒖𝑻 + 𝛼𝒖 = 0


Topology Optimization Formulation*

12

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐽 = −න𝛤

𝑝 +1

2𝑢2 𝒖 ∙ 𝒏 𝑑𝛤

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜𝑅𝑢 = (𝒖 ∙ 𝛻)𝒖 +

1

𝜌𝛻𝑝 − 𝛻 ∙ 𝜈 𝛻𝒖 + 𝛻𝒖𝑻 + 𝛼𝒖 = 0

𝑅𝑝 = 𝛻 ∙ 𝒖 = 0𝛼𝑚𝑖𝑛 ≤ 𝛼 ≤ 𝛼𝑚𝑎𝑥

Objective Function

J. S. Romero and E. C. N. Silva, “A topology optimization approach applied to laminar flow machine rotor design,” Comput. Methods Appl. Mech. Eng., vol. 279, pp. 268–300, 2014.


Continuous Adjoint Differentiation – Frozen Turbulence

13

𝐿 = 𝐽 − 𝜆𝑣𝑜𝑙𝑐𝑣𝑜𝑙 + 𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2 ; 𝑐𝑣𝑜𝑙 ∗=

𝜌𝑑Ω

𝑑Ω

1

𝑉𝑡𝑎𝑟𝑔𝑒𝑡− 1

2

𝐿𝑎𝑢𝑔 = 𝐽 − 𝜆𝑣𝑜𝑙𝑐𝑣𝑜𝑙 +𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2 +න

Ω

𝑞𝑅𝑝 𝑑Ω +නΩ

𝒗𝑅𝒖 𝑑Ω

;𝜆𝑣𝑜𝑙 , 𝑞 and 𝑣 are Lagrange multipliers

*J. M. Verboom, “Department of Precision and Microsystems Engineering Design and Additive Manufacturing of Manifolds for Navier-Stokes Flow :,” 2017.

𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟𝑠

𝐷𝑒𝑟𝑖𝑣𝑖𝑛𝑔 𝑤. 𝑟. 𝑡 𝑒𝑎𝑐ℎ 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒

𝛿𝐿 = 𝛿𝑢𝐿 + 𝛿𝑝𝐿 + 𝛿𝛼𝐿 ∴ 𝛿𝛼𝐿 = 0 ⋀ 𝛿𝑢𝐿 + 𝛿𝑝𝐿 = 0


Contents

• Research Group


• TO Formulation




• Modifications

14


Flowchart for TOM

15

Initial guess

Solve Navier Stokes

Solve Adjoint Navier Stokes

Calculate sensitivities

Converged? End

Update material distribution

Yes

No


adjointShapeOptimizationFoam.C

16

(𝒖 ∙ 𝛻)𝒖 +1

𝜌𝛻𝑝 − 𝛻 ∙ 𝜈 𝛻𝒖 + 𝛻𝒖𝑻

+ 𝛼𝒖 = 0

State equations are included// Pressure-velocity SIMPLE

corrector

116 {

117 // Momentum predictor

118

119 tmp<fvVectorMatrix> tUEqn

120 (

121 fvm::div(phi, U)

122 + turbulence->divDevReff(U)

123 + fvm::Sp(alpha, U)

124 ==

125 fvOptions(U)

126 );

127 fvVectorMatrix& UEqn =

tUEqn.ref();

128

129 UEqn.relax();

130

131 fvOptions.constrain(UEqn);

https://www.openfoam.com/documentation/guides/latest/api/combustion_2reactingFoam_2UEqn_8H.html#a22a7f38a81e139953f1aa5df6560ff81

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam_1_1fac.html#a7f47cf8d21bf98bf234866e4efeab057

https://www.openfoam.com/documentation/guides/latest/api/chtMultiRegionSimpleFoam_2fluid_2setRegionFluidFields_8H.html#a1a97abcf6c141f539994c8a21c8daca7

https://www.openfoam.com/documentation/guides/latest/api/combustion_2PDRFoam_2pEqn_8H.html#a81cf6107131a3583e2b0b762cb9c2862

https://www.openfoam.com/documentation/guides/latest/api/solvers_2compressible_2rhoCentralFoam_2createFields_8H.html#a1b05071e3ebddbaee19d4e54729872a0


https://www.openfoam.com/documentation/guides/latest/api/interFoam_2alphaSuSp_8H.html#ae52fb7b78e597bb766e9f6861c127ab0

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam_1_1constant_1_1atomic.html#a97fb79d7ca4d61fd4a89879cf7f990ae


https://www.openfoam.com/documentation/guides/latest/api/chtMultiRegionSimpleFoam_2fluid_2setRegionFluidFields_8H.html#a17798bb69ca5f9a5d28c2a824053b8a6


https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a8f2120ea7c2de50040801eb09674904d

https://www.openfoam.com/documentation/guides/latest/api/combustion_2reactingFoam_2UEqn_8H.html#aeeb518d883492fb109c93a7959833ffb

https://www.openfoam.com/documentation/guides/latest/api/combustion_2reactingFoam_2UEqn_8H.html#a22a7f38a81e139953f1aa5df6560ff81





Flowchart for TOM

17

Initial guess

Solve Navier Stokes



Converged? End


Yes

No


Continuous Adjoint Differentiation Frozen Turbulence

18

𝛿𝐽

𝛿𝑢+න

Ω

𝑞𝛿 𝛻 ∙ 𝒖

𝛿𝑢𝑑Ω + න

Ω

𝒗𝛿 𝒖 ∙ 𝛻 𝒖 − 𝛻 ∙ 𝜈 𝛻𝒖 + 𝛻𝒖𝑻 + 𝛼𝒖

𝛿𝑢𝑑Ω

+𝛿𝐽

𝛿𝑝+න

Ω

𝒗𝛿

1𝜌 𝛻𝑝

𝛿𝑝𝑑Ω = 0

𝛿𝑢𝐿 + 𝛿𝑝𝐿 = 0

𝛿𝐽

𝛿𝑢+ න

Ω

𝑞𝛿𝑅𝑝

𝛿𝑢𝑑Ω + න

Ω

𝒗𝛿𝑅𝑢

𝛿𝑢𝑑Ω +

𝛿𝐽

𝛿𝑝+ න

Ω

𝑞𝛿𝑅𝑝

𝛿𝑝𝑑Ω + න

Ω

𝒗𝛿𝑅𝑢

𝛿𝑝𝑑Ω = 0


Continuous Adjoint Differentiation Frozen Turbulence

19


Adjoint NS−2D 𝒗 𝒖 − 𝛻 ∙ 2ν𝐷 𝒗 + 𝛼𝒗 = −𝛻𝑞

𝛻 ∙ 𝒗 = 0

Inlet and Wall BC

𝒗 ∙ 𝒖 + 𝑣𝑛𝑢𝑛 + ν 𝒏 ∙ 𝛻 𝑣𝑛 − 𝑞 +𝜕𝐽Γ𝜕𝑢𝑛

= 0

𝑢𝑛𝑣𝑡 + ν 𝒏 ∙ 𝛻 𝑣𝑡 +𝜕𝐽Γ𝜕𝑢𝑡

= 0

𝑣𝑡 = 0

𝑣𝑛 = −𝜕𝐽Γ𝜕𝑝

𝛻𝑞 ∙ 𝒏 = 0

Outlet BC



20

−2D 𝒗 𝒖 − 𝛻 ∙ 2ν𝐷 𝒗 + 𝛼𝒗 = −𝛻𝑞

Adjoint equations are included

// Adjoint Pressure-velocity

SIMPLE corrector

175 {

187 zeroCells(adjointTransposeConv

ection, inletCells);

188

189 tmp<fvVectorMatrix> tUaEqn

190 (

191 fvm::div(-phi, Ua)

192 - adjointTransposeConvection

193 + turbulence->divDevReff(Ua)

194 + fvm::Sp(alpha, Ua)

195 ==

196 fvOptions(Ua)

197 );

198 fvVectorMatrix& UaEqn =

tUaEqn.ref();

199

200 UaEqn.relax();

https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#abfa8944ece10aa6b51af5a32d4ad2c1c

https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#a8e833408629144ec36a99bb4c2a8a4aa

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam_1_1fac.html#a7f47cf8d21bf98bf234866e4efeab057



https://www.openfoam.com/documentation/guides/latest/api/interFoam_2alphaSuSp_8H.html#ae52fb7b78e597bb766e9f6861c127ab0



https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a8f2120ea7c2de50040801eb09674904d


Flowchart for TOM

21

Initial guess

Solve Navier Stokes



Converged? End


Yes

No



22

𝑁𝑜 𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑖𝑜𝑛

}

242

243 laminarTransport.correct();

244 turbulence->correct();

245

246 runTime.write();

247

248 runTime.printExecutionTime(Inf

o);

249 }

250

251 Info<< "End\n" << endl;

252

253 return 0;

254 }

https://www.openfoam.com/documentation/guides/latest/api/solvers_2heatTransfer_2buoyantBoussinesqSimpleFoam_2readTransportProperties_8H.html#ad46be8ed110e710f467476d62ef7e533


https://www.openfoam.com/documentation/guides/latest/api/createEngineTime_8H.html#a981cce1bc13a71369d805a93bbdf7da9


https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a501e80d1e14c94973e82f1555e7eb711


https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a2db8fe02a0d3909e9351bb4275b23ce4



23

𝛿𝛼𝐿 = 𝒖 ∙ 𝒗 V

Include the sensitivity calculation*

U. Nilsson, “Description of AdjointShapeOptimizationFoam and how to implement new cost functions,” pp. 1–40, 2013.

}



245 sens=(Ua&U)*vol


247


o);

249 }

250


252

253 return 0;

254 }









Material Distribution

24

And 𝛼 represents material distribution* , 0 ≤ 𝜌 ≤ 1

𝛼 𝜌 = ത𝛼 + 𝛼 − ത𝛼 𝜌1 + 𝑞

𝜌 + 𝑞

Where 𝑞 controls grey level

𝜌 = 𝛾 the volumen fraction

And 𝛼 =2.5𝜇

1002and ത𝛼 =

2.5𝜇

0.012, are the mínimum and

máximum values of 𝛼


𝛻 ∙ 𝒖 = 0

(𝒖 ∙ 𝛻)𝒖 +1

𝜌𝛻𝑝 − 𝛻 ∙ 2𝜈𝑫 𝒖 + 𝛼𝒖 = 0

*T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003.



25

And 𝛼 represents material distribution* , 0 ≤ 𝜌 ≤ 1

𝛼 𝜌 = ത𝛼 + 𝛼 − ത𝛼 𝜌1 + 𝑞

𝜌 + 𝑞

Where 𝑞 controls grey level

𝜌 = 𝛾 the volumen fraction

And 𝛼 =2.5𝜇

1002and ത𝛼 =

2.5𝜇

0.012, are the mínimum and máximum

values of 𝛼


𝛻 ∙ 𝒖 = 0

(𝒖 ∙ 𝛻)𝒖 +1

𝜌𝛻𝑝 − 𝛻 ∙ 2𝜈𝑫(𝒖) + 𝛼𝒖 = 0

*T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003.


Sensitivity Formulation – Frozen Turbulence

26

𝛿𝛼𝐿𝑎𝑢𝑔 = 𝛿𝛼 𝐽 − 𝜆𝑣𝑜𝑙𝑐𝑣𝑜𝑙 + 𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2 +න

Ω


𝒗𝑅𝒖 𝑑Ω

𝛿𝑅𝑢

𝛿𝛼𝒖 ∙ 𝛻 𝒖 + 𝛻𝑝 − 𝛻 ∙ 2ν𝐷 𝒖 + 𝛼𝒖 = 𝒖

𝛿𝛼

𝛿𝜌

𝛿𝛼𝐿 = නΩ

𝒖 ∙ 𝒗 𝑑Ω𝜕𝛼

𝜕𝜌+ −𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙

𝛿𝑐𝑣𝑜𝑙𝛿𝛼

𝛿𝛼𝐿 = 𝒖 ∙ 𝒗 V 𝛼 − ത𝛼 𝑞1 + 𝑞

𝜌 + 𝑞 2 + −λ𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2

ത𝛼 Ω𝑑Ω

Ω

1 −𝛼ത𝛼𝑑Ω

Ω𝑑Ω

− V

𝛿𝐿

𝛿𝛼=𝛿𝐽

𝛿𝛼+න

Ω

𝑞𝛿𝑅𝑝

𝛿𝛼𝑑Ω +න

Ω

𝒗𝛿𝑅𝑢

𝛿𝛼𝑑Ω +න

Ω

−𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙𝛿𝑐𝑣𝑜𝑙𝛿𝛼

𝑑Ω

𝑤ℎ𝑒𝑟𝑒



27

𝛿𝛼𝐿

= 𝒖 ∙ 𝒗 V 𝛼 − ത𝛼 𝑞1 + 𝑞

𝜌 + 𝑞 2

+ (−λ𝑣𝑜𝑙

Update the sensitivity calculation

}



245 sens=(Ua&U)*vol*((alphaMin-

alphaMax)*q*(1+q)/sqr(vf+q))*cte +

(-lvol+2*wf*sqr(1-(alpha/alphaMax)-

vf))*(2/(alphaMax*vol))*(1-

(alpha/alphaMax)-vf)*cte2


247


o);

249 }

250










Flowchart for TOM

28

Initial guess

Solve Navier Stokes



Converged? End


State Equations “adjointOptimizationFoam”

Yes

No


adjointShapeOptimizationFoam.CMaterial Distribution

29

𝛼𝑛+1 = 𝛼𝑛 + 𝛿𝛼𝐿 ∗ 𝑙𝑎𝑚𝑏𝑑𝑎

𝛼𝑛+1 = 𝛼𝑛 − 𝛿𝛼𝐿 ∗ 𝑙𝑎𝑚𝑏𝑑𝑎

𝛼𝑛+1 = 𝛼𝑛 − 𝒖 ∙ 𝒗 V ∗ 𝒍𝒂𝒎𝒃𝒅𝒂

1. Modify the “steepest descent algorithm”*

U. Nilsson, “Description of AdjointShapeOptimizationFoam and how to implement new cost functions,” pp. 1–40, 2013.

Info<< "\nStarting time loop\n" <<

endl;

while (simple.loop())

{

Info<< "Time = " <<

runTime.timeName() << nl << endl;

alpha +=

mesh.fieldRelaxationFactor("alpha

")

*(min(max(alpha - lambda*(Ua &

U), zeroAlpha), alphaMax) - alpha);



https://www.openfoam.com/documentation/guides/latest/api/readFluidMultiRegionSIMPLEControls_8H.html#a10dc3474cdabfe88f9c0c3083be23b3d



https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#afbf3861e53bc13fd6c82d9eae8f97378



https://www.openfoam.com/documentation/guides/latest/api/createDynamicFvMesh_8H.html#a4c1de256df6b3d186a85745b3149394a

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a253e112ad2d56d96230ff39ea7f442dc

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#ac993e906cf2774ae77e666bc24e81733


https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#a9fc1d5833978dab511347cfb4c4de101


https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#a733fd37432d2bc9bc416bddda72f3f70

https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#ae4775010d980ce5f959f64fc5a3e2011




30

𝛼𝑛+1 = 𝛼𝑛 − 𝛿𝛼𝐿 ∗ 𝑙𝑎𝑚𝑏𝑑𝑎

𝛼𝑛+1

= 𝛼𝑛 − 𝒖 ∙ 𝒗 V 𝛼 − ത𝛼 𝑞1 + 𝑞

𝜌 + 𝑞 2


Update the “steepest descent algorithm”

Info<< "\nStarting time loop\n" <<

endl;

while (simple.loop())

{

Info<< "Time = " <<

runTime.timeName() << nl << endl;

alpha +=

mesh.fieldRelaxationFactor("alpha

")

*(min(max(alpha -

lambda*(Ua&U)*vol*((alphaMin-




(alpha/alphaMax)-vf)*cte2,

zeroAlpha), alphaMax) - alpha);



https://www.openfoam.com/documentation/guides/latest/api/readFluidMultiRegionSIMPLEControls_8H.html#a10dc3474cdabfe88f9c0c3083be23b3d



https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#afbf3861e53bc13fd6c82d9eae8f97378



https://www.openfoam.com/documentation/guides/latest/api/createDynamicFvMesh_8H.html#a4c1de256df6b3d186a85745b3149394a

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a253e112ad2d56d96230ff39ea7f442dc

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#ac993e906cf2774ae77e666bc24e81733


https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#a9fc1d5833978dab511347cfb4c4de101

https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#a733fd37432d2bc9bc416bddda72f3f70

https://www.openfoam.com/documentation/guides/latest/api/solvers_2incompressible_2adjointShapeOptimizationFoam_2createFields_8H.html#ae4775010d980ce5f959f64fc5a3e2011



createFields.H

31

𝛿𝛼𝐿 = 𝒖 ∙ 𝒗

Update the design variable sensitivity

volScalarField alpha

(

IOobject

(

"alpha",

runTime.timeName(),

mesh,

IOobject::READ_IF_PRESENT,

IOobject::AUTO_WRITE ),

lambda*max(Ua&U,zeroSensitivity)

);


createFields.H

32

𝛿𝛼𝐿

= 𝒖 ∙ 𝒗 𝛼 − ത𝛼 𝑞1 + 𝑞

𝜌 + 𝑞 2


Update the design variable sensitivity per cell definition

volScalarField alpha

(

IOobject

(

"alpha",

runTime.timeName(),

mesh,



lambda*max(Ua&U,zeroSensitivity)

);


createFields.H

33

Define the variables to be used at the sensitivity calculation

//sensitivities variables

//domain volume

dimensionedScalar vol

(

"vol",

dimless,

laminarTransport

);

//minimum value of porosity

dimensionedScalar alphaMin

(

"alphaMin",

dimless/dimTime,

laminarTransport

);

.


createFields.H

34

Define the sensitivity calculation

volScalarField sens

(

IOobject

(

"sensitivity",

runTime.timeName(),

mesh,



(Ua&U)*vol*((alphaMin-




(alpha/alphaMax)-vf)*cte2

);


fvSolution

35

Include the new variables values at constant/transportProperties

transportModel Newtonian ;

nu 1e-5;

lambda 1 e5 ;

alphaMax 2 0 0 . 0 ;

alphaMin 2 0 0 . 0 ;

Vol 0.25 ;

q 0.01;

.

.

.

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003.


Examples

36

Cases tested

𝐿𝑎𝑚𝑖𝑛𝑎𝑟𝑢𝑖𝑛 = 1 𝑚/𝑠𝜈 = 1 𝑚2/𝑠 𝜌𝑑Ω

𝑑Ω= 25%

𝐹𝑟𝑜𝑧𝑒𝑛 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑐𝑒𝑢𝑖𝑛 = 5 𝑚/𝑠𝜈 = 10−5 𝑚2/𝑠𝐼 = 1%𝑘𝜔 − 𝑆𝑆𝑇𝜌𝑑Ω

𝑑Ω= 25%

𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 𝑐𝑎𝑠𝑒 𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝑠𝑜𝑙𝑣𝑒𝑟


Examples

37

Cases tested

𝑏𝑒𝑛𝑐ℎ𝑚𝑎𝑟𝑘 case 𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝑠𝑜𝑙𝑣𝑒𝑟

𝐿𝑎𝑚𝑖𝑛𝑎𝑟𝑢𝑖𝑛 = 1 𝑚/𝑠𝜈 = 1 𝑚2/𝑠𝜌𝑑Ω

𝑑Ω= 50%

𝐹𝑟𝑜𝑧𝑒𝑛 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛ce𝑢𝑖𝑛 = 5 𝑚/𝑠𝜈 = 10−5 𝑚2/𝑠𝐼 = 1%𝑘𝜔 − 𝑆𝑆𝑇 𝜌𝑑Ω

𝑑Ω= 50%


Contents

• Research Group


• TO Formulation




• Modifications

38


adjointOptimizationFoam Solver

39

• It is a functionality targeting automated gradient-based optimisation loop assisted

by the continuous adjoint method.

• Supports topology and shape optimization.

$FOAM _SOLVERS/incompressible/adjointOptimizationFoam


adjointOptimizationFoam Solver Description

40


adjointOptimizationFoam.C Description

41

#include "fvCFD.H"

#include "optimisationManager.H“

#include "primalSolver.H"

#include "adjointSolverManager.H"

Fundamental tools for FV calculations



42

#include "fvCFD.H"




Abstract class for optimizationmethods. Declaration of

primalSolvers, adjointSolverManager,

updateDesignVariables, solverPrimalEquations, solveAdjointEquations,

computeSensitivities,updatePrimalBasedQuantities



43

#include "fvCFD.H"




Class declaration of theprimalSolver with

its member functions that

assigns the class use at each cellof the mesh domain.



44

#include "fvCFD.H"




Class for managing theadjoint solvers, which are

suposed to be more tan oneper operating point. Define

the update function and clases for solving the adjoint

equations, agregatesensitivites, compute

sensitivities and so on.



45

#include "fvCFD.H"



#include "adjointSolverManager.H“

// * * * * * * * * * * * * * * //

int main(int argc, char *argv[])

{

#include "setRootCase.H“

#include "createTime.H“

#include "createMesh.H“

#include "createFields.H"

The construction of anobject which setss the

correct path, créate thetime and the mesh.



46

#include "fvCFD.H"



#include "adjointSolverManager.H“

// * * * * * * * * * * * * * * //

int main(int argc, char *argv[])

{

#include "setRootCase.H“

#include "createTime.H“

#include "createMesh.H“

#include "createFields.H" The declaration of theoptimization manager is

made.



47

createFields.H:

autoPtr<optimisationManager>

optManagerPtr

(

optimisationManager::New(mesh)

);

optimisationManager& om =

optManagerPtr();



48

optimisationManager

* “OpenFOAM: API Guide: optimisationManager Class Reference.” [Online]. Available: https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1optimisationManager.html#ab753cbc2d71764526653cdc4eac7c397. [Accessed: 25-Nov-2019].

*



49

// * * * * * * * * * * * * * * //

Info<< "\nStarting time loop\n"

<< endl;

for (om++; !om.end(); om++)

{

Info<< "* * * * * *" << endl;

Info<< "Time = "

<<runTime.timeName() <<

endl;

Info<< "* * * * * * * * * * * * *

* * * * * *" << endl;

The time loop is defined, a for loop starts where three

statements aim to show thetimestep at the terminal



50

if (om.update())

{

om.updateDesignVariables();

}

else

{

om.solvePrimalEquations();

}

If update is stablished, thenthe update design variable

field is used, otherwise, theprimal equations solver is

called.



51

if (om.update())

{


}

else

{


}

Employ a line search to finda correction satisfying thestop convergence criteria

(not implemented yet)



52

if (om.update())

{


}

else

{


}

Solve primal Equations, defined at

optimisationManager.C



53

optimizationManager.C

.

void Foam::optimisationManager::solvePrimalEquations()

{

// Solve all primal equations

forAll(primalSolvers_, psI)

{

primalSolvers_[psI].solve();

}

}

Solve primal Equations, defined at optimisationManager.C



54

primalSolvers

Sends the solver to solve the primal (state equations)

Incompressible Navier Stokes equations in non-conservativeform

𝛻 ∙ 𝒖 = 0

𝒖 ∙ 𝛻 𝒖 +1

𝜌𝛻𝑝 − 𝛻 ∙ 𝜈 𝛻𝒖 + 𝛻𝒖𝑻 = 0



55

if (om.update())

{


}

else

{


}

om.updatePrimalBasedQuantities();

om.solveAdjointEquations();

om.computeSensitivities();

After that, an update of primal Based Quantities is

called.



56

void

Foam::optimisationManager::updatePrimalBasedQuantities()

{

forAll(adjointSolverManagers_, amI)

{

PtrList<adjointSolver>& adjointSolvers =

adjointSolverManagers_[amI].adjointSolvers();

forAll(adjointSolvers, asI)

{

adjointSolvers[asI].updatePrimalBasedQuantities();

}

}

}

The update of primal Based Quantities is defined at optimisationManager.C



57

if (om.update())

{


}

else

{


}




Then , the adjoint equationsare requested to be solved




58

void Foam::optimisationManager::solveAdjointEquations()

{

// Solve all adjoint solver equations


{

adjointSolverManagers_[amI].solveAdjointEquations();

}

}

the adjoint equations solver is defined at optimisationManager.C



59

if (om.update())

{


}

else

{


}




Finally the sensitivities are requested to be solved

𝛿𝛼𝐿 = 𝟎



60

void Foam::optimisationManager::computeSensitivities()

{

// Compute senstivities from all adjoint solvers


{

adjointSolverManagers_[amI].computeAllSensitivities();

}

}

the adjoint equations solver is defined at optimisationManager.C



61

if (om.update())

{


}

else

{


}




}

Info<< "End\n" << endl;

return(0);

Ends with the loop of thesolver..


adjointSolverManager Description

62

void Foam::adjointSolverManager::solveAdjointEquations()

{ for (adjointSolver& solver : adjointSolvers_)

{

objectiveManager& objManager = solver.getObjectiveManager();

objManager.updateAndWrite();

solver.solve(); }}

Class for managing the adjoint solvers.

For example for managing the adjoint equations resolutions:


adjointSolverManager Description

63

void Foam::adjointSolverManager::computeAllSensitivities()

{

for (adjointSolver& adjSolver : adjointSolvers_)

{

adjSolver.clearSensitivities();

adjSolver.computeObjectiveSensitivities();

}}

Class for managing the adjoint solvers.

For computing the sensitivities calculation:


Core Solver: adjointSimple

64

Path for getting the core Solver:

The solver is being constructed for managing multiple objective functions andsensitivities calculation instead of a straight line solver, but at the end, what hasbeen implemented is in the “adjointSimple.C” file.


adjointSimple.C Description

65

At “member functions” in solveIter() object frome the “adjointSimple” classcan be found the solver implementation:

void

Foam::adjointSimple::solveIter()

{

const Time& time = mesh_.time();

Info<< "Time = " <<

time.timeName() << "\n" << endl;

Starts by defining the time stepcalculation

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1adjointSimple.html#aec2e61252bd8a247204b902036e16169

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1Time.html

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1objectRegistry.html#a0d94096809fe3376b29a2a29ca11bb18


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1Time.html#a882e958c2f124b46fdb52dd241ec0ca3




66

At “memeber functions” in solveIter() object frome the “adjointSimple” classcan be found the solver implementation:

void

Foam::adjointSimple::solveIter()

{

const Time& time = mesh_.time();

Info<< "Time = " <<

time.timeName() << "\n" << endl;

// Grab primal references

const surfaceScalarField& phi =

primalVars_.phi();

Followed by taking the results ofthe state equations variablesand defined them as “phi”

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1adjointSimple.html#aec2e61252bd8a247204b902036e16169

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1Time.html

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1objectRegistry.html#a0d94096809fe3376b29a2a29ca11bb18


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1Time.html#a882e958c2f124b46fdb52dd241ec0ca3


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1GeometricField.html




67


// Grab primal references

const surfaceScalarField& phi =

primalVars_.phi();

// Grab adjoint references

incompressibleAdjointVars&

adjointVars = getAdjointVars();

volScalarField& pa =

adjointVars.paInst();

volVectorField& Ua =

adjointVars.UaInst();

surfaceScalarField& phia =

adjointVars.phiaInst();

After that, the adjoint variablesto be used are defined



https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjointVars.html


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjointMeanFlowVars.html#a80d74ea8efe135ba46201c99179d5652


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjointMeanFlowVars.html#abbaac89baa2c873a9d754534013fe52f


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjointMeanFlowVars.html#a743a6a831334ea6e29ec245da412d67c



68


.

.

.

autoPtr<incompressibleAdjoint::adj

ointRASModel>& adjointTurbulence =

adjointVars.adjointTurbulence();

const label& paRefCell =

solverControl_().pRefCell();

const scalar& paRefValue =

solverControl_().pRefValue();

The new term, the adjointTurbulence 𝑣𝑎 *

E. M. Papoutsis-Kiachagias and K. C. Giannakoglou, “Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications,” Arch. Comput. Methods Eng., vol. 23, no. 2, pp. 255–299, 2016.

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1autoPtr.html

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjointVars.html#ae32ce47793cf7d364b48d17a1bf69f79

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a22f45a3cb4f074e609f58ebaeef0ecf9


Continuous Turbulent Adjoint Equations

69

𝛿𝑢𝐿 + 𝛿𝑝𝐿 + 𝛿෦𝑣𝑎𝐿 + 𝛿∆𝑎𝐿 = 0

𝑅𝑞 = −𝜕𝑢𝑗

𝜕𝑥𝑗= 0

𝑅𝑖𝑢 = 𝑢𝑗

𝜕𝑣𝑗

𝜕𝑥𝑖−𝜕 𝑣𝑗𝑢𝑖

𝜕𝑥𝑗−

𝜕

𝜕𝑥𝑗𝜐 + 𝜐𝑡

𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖+𝜕𝑞

𝜕𝑥𝑖+ 𝜈𝑎

𝜕𝜈

𝜕𝑥𝑖−

𝜕

𝜕𝑥𝑖𝜈𝑎𝜈

𝐶𝑌𝑌𝑒𝑚𝑗𝑘

𝜕𝑣𝑘𝜕𝑥𝑗

𝑒𝑚𝑙𝑖 +ต𝛼𝑢𝑖𝑇𝑎,𝑢

= 0

𝑅 𝑣𝑎

= −𝜕 𝑣𝑗 𝑣𝑎

𝜕𝑥𝑗−

𝜕

𝜕𝑥𝑗𝜐 +

𝑣

𝜎

𝜕 𝑣𝑎𝜕𝑥𝑗

+1

𝜎

𝜕 𝑣𝑎𝜕𝑥𝑗

𝜕 𝑣

𝜕𝑥𝑗+2𝑐𝑏2𝜎

𝜕

𝜕𝑥𝑗𝑣𝑎𝜕 𝑣

𝜕𝑥𝑗+ 𝑣𝑎 𝑣𝐶𝑣 +

𝜕𝑣𝑡

𝜕 𝑣

𝜕𝑢𝑖

𝜕𝑥𝑗×

𝜕𝑣𝑖

𝜕𝑥𝑗+𝜕𝑣𝑗

𝜕𝑥𝑖

+ −𝑃 + 𝐷 𝑣𝑎 + ต𝛼 𝑣𝑎𝑇𝑎,𝑣𝑎

= 0

𝑅∆𝑎 = −2𝜕

𝜕𝑥𝑗∆𝑎

𝜕∆

𝜕𝑥𝑗+ 𝑣 𝑣𝑎𝐶∆ = 0

Adjoint NS



BC Continuous Turbulent Adjoint Equations

70

Inlet


𝛿𝐹𝑎𝑢𝑔

𝛿𝑎𝑛= න

𝑆

𝐵𝐶𝑖𝑢 𝜕𝑣𝑖𝜕𝑎𝑛

𝑑𝑆 + න𝑆

𝑢𝑗𝑛𝑗 +𝜕𝐹𝑆𝑖𝜕𝑝

𝑛𝑖𝜕𝑝

𝜕𝑎𝑛𝑑𝑆 + න

𝑆

−𝑢𝑖𝑛𝑗 +𝜕𝐹𝑆𝑘𝜕𝜏𝑖𝑗

𝑛𝑘𝜕𝜏𝑖𝑗

𝜕𝑎𝑛𝑑𝑆

𝑢 𝑛 = 𝑢𝑗𝑛𝑗 = −𝜕𝐹𝑆𝐼,𝑖𝜕𝑝

𝑛𝑖

𝑢 𝑡𝐼 =

𝜕𝐹𝑆𝐼,𝑘𝜕𝜏𝑖𝑗

𝑛𝑘𝑡𝑖𝐼𝑛𝑗 +


𝑛𝑘𝑡𝑗𝐼𝑛𝑖

𝑢 𝑡𝐼𝐼 =


𝑛𝑘𝑡𝑖𝐼𝐼𝑛𝑗 +


𝑛𝑘𝑡𝑗𝐼𝐼𝑛𝑖

𝜕𝑣𝑖𝜕𝑎𝑛

= 0Outlet

Figure 3. Finite volume adjacent to the wall, where 𝒏 is the outwards normal unit vector, 𝒕𝐼 is parallel to the velocity velocity vector at first cell center P and 𝒕𝑰𝑰 =

𝑒𝑖𝑗𝑘𝑛𝑗𝑡𝑘𝐼 .


Modifications for Threating Turbulence

71

𝒌 − 𝝎 Turbulence model

𝛻 · (𝒖 𝑘) = 𝑻𝑡: 𝛻𝒖 − 𝛽∗𝜔𝑘 + 𝛻 · 𝜈 + 𝜎∗𝑘

𝜔𝛻𝑘 − 𝛼𝑘

𝛻 · 𝒖𝜔 =𝛼𝜔

𝑘𝑻𝑡: 𝛻𝒖 − 𝛽𝜔2 +

𝜎𝑑𝜔𝛻𝑘 · 𝛻𝜔 + 𝛻 · 𝜈 + 𝜎

𝑘

𝜔𝛻𝜔 + 𝛼(𝜔𝑏 − 𝜔)

Spalart Allmaras Turbulence model*

𝑢𝑗𝜕 ǁ𝜈

𝜕𝑥𝑗−

𝜕

𝜕𝑥𝑗𝜈 +

ǁ𝜈

𝜎

𝜕 ǁ𝜈

𝜕𝑥𝑗−𝑐𝑏2𝜎

𝜕 ǁ𝜈

𝜕𝑥𝑗

2

− ǁ𝜈𝑃 ǁ𝜈 + ǁ𝜈𝐷 ǁ𝜈 + 𝛼 ǁ𝜈 = 0

𝑇ℎ𝑒𝑛𝛿𝑅 𝑣

𝛿𝛼=𝛿𝛼

𝛿𝛼𝑣 + 𝛼

𝛿 𝑣

𝛿𝛼

*E. M. Papoutsis-Kiachagias and K. C. Giannakoglou, “Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications,” Arch. Comput. Methods Eng., vol. 23, no. 2, pp. 255–299, 2016.


BC Continuous Turbulent Adjoint Equations

72


Wall

𝛿𝐹𝑎𝑢𝑔

𝛿𝑎𝑛= න

𝑆

𝐵𝐶𝑖𝑢 𝜕𝑣𝑖𝜕𝑎𝑛

𝑑𝑆 +න𝑆

𝑢𝑗𝑛𝑗 +𝜕𝐹𝑆𝑖𝜕𝑝

𝑛𝑖𝜕𝑝

𝜕𝑎𝑛𝑑𝑆 +න

𝑆

−𝑢𝑖𝑛𝑗 +𝜕𝐹𝑆𝑘𝜕𝜏𝑖𝑗

𝑛𝑘𝜕𝜏𝑖𝑗

𝜕𝑎𝑛𝑑𝑆

+න𝑆

𝐵𝐶෦𝑣𝑎𝜕 𝑣

𝜕𝑎𝑛𝑑𝑆 −න

𝑆

෦𝑣𝑎 𝑣 +𝑣

𝜎

𝜕

𝜕𝑎𝑛

𝜕 𝑣

𝜕𝑥𝑗𝑛𝑗 𝑑𝑆 + න

𝑆

2∆𝑎𝜕∆

𝜕𝑥𝑗𝑛𝑗

𝜕∆

𝜕𝑏𝑛𝑑𝑆


𝑢𝑗𝜕 ǁ𝜈

𝜕𝑥𝑗−

𝜕

𝜕𝑥𝑗𝜈 +

ǁ𝜈

𝜎

𝜕 ǁ𝜈


𝜕 ǁ𝜈

𝜕𝑥𝑗

2


𝑢 𝑛 = −𝜕𝐹𝑆𝑊,𝑖

𝜕𝑝𝑛𝑖 𝑎𝑛𝑑 𝑢𝜏

2 = 𝜈 + 𝜈𝑡𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖𝑛𝑗𝑡𝑖

𝐼 = 0



73


.

.

.

autoPtr<incompressibleAdjoint::adj

ointRASModel>& adjointTurbulence =

adjointVars.adjointTurbulence();

const label& paRefCell =

solverControl_().pRefCell();

const scalar& paRefValue =

solverControl_().pRefValue();

The new term, the adjointTurbulence 𝑣𝑎 *


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1autoPtr.html

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjointVars.html#ae32ce47793cf7d364b48d17a1bf69f79

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a22f45a3cb4f074e609f58ebaeef0ecf9



74

At “memeber functions” in solveIter() object from the “adjointSimple” classcan be found the solver implementation:

// Momentum predictor

//~~~~~~~~~~~~~~~~~~~

tmp<fvVectorMatrix> tUaEqn

(

fvm::div(-phi, Ua)

+ adjointTurbulence-

>divDevReff(Ua)

+ adjointTurbulence-

>adjointMeanFlowSource()

==

200 fvOptionsAdjoint_(Ua)

);

fvVectorMatrix& UaEqn =

tUaEqn.ref();

Appears due to the differentiation of the turbulence model defined as:

nuTilda()*fvc::grad(nuaTi

lda() -

conservativeMomentumSourc

e());

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1tmp.html

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam_1_1fvm.html#a058dd4757c11ae656c8d576632b33d25


https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjoint_1_1adjointRASModel.html#a2fe41bb94b1df0e86fe7a6c83a78c5ad

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1incompressibleAdjoint_1_1adjointRASModel.html#ac0a158779308be444229d7e3aae2f909

https://www.openfoam.com/documentation/guides/latest/api/classFoam_1_1fvMatrix.html



75

After adding source terms of the adjoint equation:

if

(solverControl_().momentumPredicto

r())

{

Foam::solve(UaEqn == -

fvc::grad(pa));

fvOptionsAdjoint_.correct(Ua);

}

𝑈𝑎 = −𝛻𝑞

https://www.openfoam.com/documentation/guides/latest/api/readFluidMultiRegionSIMPLEControls_8H.html#a9f9361a09e3baa185c06c84ee72659fd

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#ad82fbab56a75a330a844997630e6644b

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam_1_1fvc.html#a7c00ec90ecc15ed3005255e42c52827a



76

Ending up with a proposed“adjoint” Pressure-VelocitySIMPLE corrector*

// Pressure Eq

//~~~~~~~~~~~~

{

volScalarField rAUa(1.0/UaEqn.A());

volVectorField

HabyA(constrainHbyA(rAUa*UaEqn.H(),

Ua, pa));

surfaceScalarField

phiaHbyA("phiaHbyA",

fvc::flux(HabyA));

adjustPhi(phiaHbyA, Ua, pa);

.

.

.

*F. Moukalled, L. Mangani, and M. Darwish, The finite volume method in computational fluid dynamics : An Advanced Introduction with OpenFOAM and Matlab, vol. 113. 2016.



https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#abce55f11a96150ac0fb5e20d59da3cfd


https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam_1_1fvc.html#a8fc1922666ca94c5fd207b8b96f99467

https://www.openfoam.com/documentation/guides/latest/api/namespaceFoam.html#a0ea353dc88166de8e48fb02639977b39


Contents

• Research Group


• TO Formulation




• Modifications

77


Flowchart for TOM

78

Initial guess

Solve Navier Stokes



Converged? End



Yes

No


Sensitivity Formulation – Frozen Turbulence

79

𝛿𝛼𝐿𝑎𝑢𝑔 = 𝛿𝛼 𝐽 − 𝜆𝑣𝑜𝑙𝑐𝑣𝑜𝑙 + 𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2 +න

Ω


𝒗𝑅𝒖 𝑑Ω

𝛿𝑅𝑢

𝛿𝛼𝒖 ∙ 𝛻 𝒖 + 𝛻𝑝 − 𝛻 ∙ 2ν𝐷 𝒖 + 𝛼𝒖 = 𝒖

𝛿𝛼

𝛿𝜌= 𝒖

𝛿𝛼𝐿 = නΩ

𝒖 ∙ 𝒗 𝑑Ω + −𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙𝛿𝑐𝑣𝑜𝑙𝛿𝛼

𝛿𝛼𝐿 = 𝒖 ∙ 𝒗 V + −λ𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2

ത𝛼 Ω𝑑Ω

Ω


Ω𝑑Ω

− V

𝛿𝐿

𝛿𝛼=𝛿𝐽

𝛿𝛼+න

Ω

𝑞𝛿𝑅𝑝

𝛿𝛼𝑑Ω +න

Ω

𝒗𝛿𝑅𝑢

𝛿𝛼𝑑Ω +න

Ω

−𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙𝛿𝑐𝑣𝑜𝑙𝛿𝛼

𝑑Ω

𝑤ℎ𝑒𝑟𝑒


Turbulent Sensitivity Formulation

80

𝐿 = 𝐽 − 𝜆𝑣𝑜𝑙𝑐𝑣𝑜𝑙 + 𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2 ; 𝑐𝑣𝑜𝑙 =

𝜌𝑑Ω

𝑑Ω

1


2


Ω


𝒗𝑅𝒖 𝑑Ω + නΩ

෦𝑣𝑎 𝑅𝑣 𝑑Ω +න

Ω

∆𝑎𝑅∆ 𝑑Ω


𝛿𝐿𝑎𝑢𝑔

𝛿𝛼=𝛿𝐽

𝛿𝛼+ (−𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙)


+නΩ

𝑞𝛿𝑅𝑝

𝛿𝛼𝑑Ω𝛿𝑢 +න

Ω

𝒗𝛿𝑅𝑢

𝛿𝛼𝑑Ω +න

Ω

෦𝑣𝑎𝛿𝑅 𝑣

𝛿𝛼𝑑Ω

+නΩ

∆𝑎𝛿𝑅∆

𝛿𝛼𝑑Ω

𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑑𝑒𝑠𝑖𝑔𝑛 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝛼


*



81

𝐿 = 𝐽 − 𝜆𝑣𝑜𝑙𝑐𝑣𝑜𝑙 + 𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙2 ; 𝑐𝑣𝑜𝑙 =

𝜌𝑑Ω

𝑑Ω

1


2


Ω


𝒗𝑅𝒖 𝑑Ω + නΩ

෦𝑣𝑎 𝑅𝑣 𝑑Ω +න

Ω

∆𝑎𝑅∆ 𝑑Ω



𝛿𝛼=𝛿𝐽

𝛿𝛼+ (−𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙)


+නΩ

𝑞𝛿𝑅𝑝

𝛿𝛼𝑑Ω𝛿𝑢 +න

Ω

𝒗𝛿𝑅𝑢

𝛿𝛼𝑑Ω +න

Ω

෦𝑣𝑎𝛿𝑅 𝑣

𝛿𝛼𝑑Ω

+නΩ

∆𝑎𝛿𝑅∆

𝛿𝛼𝑑Ω




82


𝛻 · (𝒖 𝑘) = 𝑻𝑡: 𝛻𝒖 − 𝛽∗𝜔𝑘 + 𝛻 · 𝜈 + 𝜎∗𝑘




𝜎𝑑𝜔𝛻𝑘 · 𝛻𝜔 + 𝛻 · 𝜈 + 𝜎

𝑘



𝑢𝑗𝜕 ǁ𝜈

𝜕𝑥𝑗−

𝜕

𝜕𝑥𝑗𝜈 +

ǁ𝜈

𝜎

𝜕 ǁ𝜈


𝜕 ǁ𝜈

𝜕𝑥𝑗

2



𝛿𝛼= 𝑣 + 𝛼

𝛿 𝑣

𝛿𝛼




83


𝛿𝛼= (−𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙)


+නΩ

𝒗𝛿𝑅𝑢

𝛿𝛼𝑑Ω + න

Ω

෦𝑣𝑎𝛿𝛼

𝛿𝛼𝑣 + 𝛼

𝛿 𝑣

𝛿𝛼𝑑Ω



𝛿𝛼

= 𝒖 ∙ 𝒗 V 𝛼 − ത𝛼 𝑞1 + 𝑞


ത𝛼 Ω𝑑Ω

Ω


Ω𝑑Ω

− V

+නΩ

𝑣෦𝑣𝑎 𝑑Ω + නΩ

𝛼෦𝑣𝑎𝛿 𝑣

𝛿𝛼𝑑Ω


Sensitivity implementation: adjointSensitivityIncompressible.C

84

// Compute dxdb multiplier

flowTerm =

ATCModel->getFISensitivityTerm()

- fvc::grad(p) * Ua

- nuEff*(gradU & (gradUa +

T(gradUa)))

+ (- nuEff*(gradUa & (gradU +

T(gradU)))

+ fvc::grad(nuEff * Ua &

(gradU + T(gradU)))

+ (pa * gradU)

// from the adjoint turbulence model

+ turbulenceTerm.T()

// Term 7, term from objective

functions

+ objectiveContributions;

𝐴𝑇𝐶 + 𝛻𝑝 ∗ 𝑈𝑎 − 𝑣𝑒𝑓𝑓∗ 𝛻𝑢 ∙ 𝛻𝑢𝑎 + 𝛻𝑢𝑇 + (−𝑣𝑒𝑓𝑓∗ 𝛻𝑢𝑎 ∙ 𝛻𝑢𝑎 + 𝛻𝑢𝑇 + 𝛻(𝑣𝑒𝑓𝑓∗ 𝑈𝑎 ∙ 𝛻𝑢𝑎 + 𝛻𝑢𝑇 + 𝑝𝑎 ∗ 𝛻𝑢+ 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑐𝑒𝑇𝑒𝑟𝑚+ 𝑜𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠



85

Contribution of turbulenceadjoint model

objectiveManager_.adjointSolverName(

)

).getATCModel();


flowTerm =


- fvc::grad(p) * Ua


T(gradUa)))

+ term4

+ (pa * gradU)




functions





86

Contribution of objectivefunction

objectiveManager_.adjointSolverName(

)

).getATCModel();


flowTerm =


- fvc::grad(p) * Ua


T(gradUa)))

+ term4

+ (pa * gradU)




functions





87

𝐴𝑇𝐶 + 𝛻𝑝 ∗ 𝑈𝑎 − 𝑣𝑒𝑓𝑓 ∗ 𝛻𝑢 ∙ 𝛻𝑢𝑎 + 𝛻𝑢𝑇 + (−𝑣𝑒𝑓𝑓 ∗ 𝛻𝑢𝑎 ∙ 𝛻𝑢𝑎 + 𝛻𝑢𝑇

+ 𝛻(𝑣𝑒𝑓𝑓 ∗ 𝑈𝑎 ∙ 𝛻𝑢𝑎 + 𝛻𝑢𝑇 + 𝑝𝑎 ∗ 𝛻𝑢 + 𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑐𝑒𝑇𝑒𝑟𝑚

+ 𝑜𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠


𝛿𝛼= 𝒖 ∙ 𝒗 V 𝛼 − ത𝛼 𝑞

1 + 𝑞

𝜌 + 𝑞 2+ −λ𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙

2

ത𝛼 Ω𝑑Ω

Ω


Ω𝑑Ω

− V +නΩ

𝑣෦𝑣𝑎 𝑑Ω + නΩ


𝛿𝛼𝑑Ω


Flowchart for TOM

88

Initial guess

Solve Navier Stokes



Converged? End



Yes

No


Contents

• Research Group


• TO Formulation





• Modifications

89


adjointOptimisationFoam for TO

90

Tutorials path

$FOAM_TUTORIALS/incompressible/adjointOptimisationFoam



91

Tutorials path


goleman, daniel; boyatzis, Richard; Mckee, A. (2019). Manual adjointOptimisationFoam. Journal of ChemicalInformation and Modeling, 53(9), 1689–1699. https://doi.org/10.1017/CBO9781107415324.004

https://doi.org/10.1017/CBO9781107415324.004



92

No Optimization

$FOAM_TUTORIALS/incompressible/adjointOptimisationFoam/system/optimisationDict


optimisationManager singleRun;

https://doi.org/10.1017/CBO9781107415324.004



93




primalSolvers{

op1

{

active true;

type incompressible;

solver simple;

https://doi.org/10.1017/CBO9781107415324.004



94




primalSolvers{

op1

{

active true;


solver simple;

useSolverNameForFields false; false as default

https://doi.org/10.1017/CBO9781107415324.004



95



primalSolvers

.

solutionControls

{

nIters 3000;

residualControl

{

"p.*" 1.e-6;

"U.*" 1.e-6;

"nuTilda.*" 1.e-6;

} } }}

Entries to manage the solutionprocess of the primal equations.

https://doi.org/10.1017/CBO9781107415324.004



96



adjointManagers

{

adjManager1

{

There is the possibility ofdefine one adjoint Managerper primalSolver used.

https://doi.org/10.1017/CBO9781107415324.004



97



adjointManagers

{

adjManager1

{

primalSolver op1;

https://doi.org/10.1017/CBO9781107415324.004



98



adjointManagers

{

adjManager1

{

primalSolver op1;

operatingPointWeight 1;

Default value 1.

Weight factor of usingmultiple objective functions:

𝐽 =𝑖𝑤𝑖𝑜𝑝𝐽𝑗𝑜𝑝

https://doi.org/10.1017/CBO9781107415324.004



99



adjointManagers

{

adjManager1

{

primalSolver op1;


adjointSolvers

{

adjS1

List of dictionaries to set upthe adjointSolvers to be used.Multiple can be used.

https://doi.org/10.1017/CBO9781107415324.004



100



adjointManagers

{

adjManager1

{

primalSolver op1;


adjointSolvers

{

// choose adjoint solver --//

active true; true set up as default

https://doi.org/10.1017/CBO9781107415324.004



101



adjointManagers

{

adjManager1

{

primalSolver op1;


adjointSolvers

{

// choose adjoint solver --//

active true;


https://doi.org/10.1017/CBO9781107415324.004



102



adjointManagers

{

adjManager1

.

// manage objectives //----

objectives

{


objectiveNames

{ lift

Objective functionsimplemented: lift, drag andPtLosses

https://doi.org/10.1017/CBO9781107415324.004



103



adjointManagers

{

adjManager1

.

//--------ATCModel

{

ATCModel standard;

extraConvection 0;

nSmooth 0;

zeroATCPatchTypes ();

maskType faceCells;

}

Threatment of the Adjointtranspose convection term,which is in the adjointmomentum equations.

https://doi.org/10.1017/CBO9781107415324.004



104



adjointManagers

{

adjManager1

.

//--------ATCModel

{

ATCModel standard;

extraConvection 0;

nSmooth 0;


maskType faceCells;

}

Entries avilable:

Standard (non-conservative)

𝑢𝑗𝜕𝑣𝑗

𝜕𝑥𝑖

UaGradU (conservative)

−𝑣𝑗𝜕𝑢𝑗

𝜕𝑥𝑖

cancel

https://doi.org/10.1017/CBO9781107415324.004



105



adjointManagers

{

adjManager1

.

//--------ATCModel

{

ATCModel standard;

extraConvection 0;

nSmooth 0;


maskType faceCells;

}

Default is zero.

Add/subtract the adjointconvection term this manytimes to add numericaldissipation.

https://doi.org/10.1017/CBO9781107415324.004



106



adjointManagers

{

adjManager1

.

//--------ATCModel

{

ATCModel standard;

extraConvection 0;

nSmooth 0;


maskType faceCells;

}

Default value is zero.

Propagate the smoothing ofthe ATC

https://doi.org/10.1017/CBO9781107415324.004



107



adjointManagers

{

adjManager1

.

//--------ATCModel

{

ATCModel standard;

extraConvection 0;

nSmooth 0;


maskType faceCells;

}

Default is empty.

Zero the ATC next to patchesprovided. Available also forcell zones.

https://doi.org/10.1017/CBO9781107415324.004



108



adjointManagers

{

adjManager1

.

//--------ATCModel

{

ATCModel standard;

extraConvection 0;

nSmooth 0;


maskType faceCells;

}

Default is faceCells, can bepointCells.

Define how the smoothing willbe executed at the types,either faces or points.

https://doi.org/10.1017/CBO9781107415324.004



109



adjointManagers

{

adjManager1

.

// solution control // --------

solutionControls

{

nIters 3000;

residualControl {

"pa.*" 1.e-6;

"Ua.*" 1.e-6;

"nuaTilda.*" 1.e-6;

https://doi.org/10.1017/CBO9781107415324.004



110



Optimisation

{ sensitivities

{

type surfacePoints;

patches (pressure);

includeSurfaceArea false;

includeDistance true;

includeMeshMovement true;

includeObjectiveContribution true;

writeAllSurfaceFiles true;

Available options: “Surface”,“Surface points” and“sensitivtyMultiple”.

https://doi.org/10.1017/CBO9781107415324.004



111



Optimisation

{ sensitivities

{

type surfacePoints;

patches (lower );






https://doi.org/10.1017/CBO9781107415324.004



112



Optimisation

{ sensitivities

{

type surfacePoints;

patches (lower );






Normal displacement of theboundary points

Available options: “Surface”,“Surface points” and“sensitivtyMultiple”.

https://doi.org/10.1017/CBO9781107415324.004



113



Optimisation

{ sensitivities

{

type surfacePoints;

patches (lower );






True (adjoint Spalart Allmaras)

Variable set as: 𝑑𝑎

BC: Inlet=outlet=fixedValue

Wall=zeroGradient

fcSchemes/divSchemes:𝑑𝑖𝑣(−𝑖𝑃ℎ𝑖, 𝑑𝑎)

https://doi.org/10.1017/CBO9781107415324.004



114



Optimisation

{ sensitivities

{

type surfacePoints;

patches (lower );



includeMeshMovement false;



Set as false for topologyoptimization.

https://doi.org/10.1017/CBO9781107415324.004



115

Tutorials path



https://doi.org/10.1017/CBO9781107415324.004


BC of adjointOptimisationFoam for TO

116


https://doi.org/10.1017/CBO9781107415324.004



117


https://doi.org/10.1017/CBO9781107415324.004



118


https://doi.org/10.1017/CBO9781107415324.004



119


https://doi.org/10.1017/CBO9781107415324.004


Contents

• Research Group


• TO Formulation





• Modifications to the adjoint Turbulent sensitivity

120



121


𝛻 · (𝒖 𝑘) = 𝑻𝑡: 𝛻𝒖 − 𝛽∗𝜔𝑘 + 𝛻 · 𝜈 + 𝜎∗𝑘




𝜎𝑑𝜔𝛻𝑘 · 𝛻𝜔 + 𝛻 · 𝜈 + 𝜎

𝑘



𝑢𝑗𝜕 ǁ𝜈

𝜕𝑥𝑗−

𝜕

𝜕𝑥𝑗𝜈 +

ǁ𝜈

𝜎

𝜕 ǁ𝜈


𝜕 ǁ𝜈

𝜕𝑥𝑗

2



𝛿𝛼=𝛿𝛼

𝛿𝛼𝑣 + 𝛼

𝛿 𝑣

𝛿𝛼




122


𝛿𝛼= (−𝜆𝑣𝑜𝑙 + 2𝑤𝑣𝑜𝑙𝑐𝑣𝑜𝑙)


+නΩ

𝒗𝛿𝑅𝑢

𝛿𝛼𝑑Ω + න

Ω

෦𝑣𝑎𝛿𝛼

𝛿𝛼𝑣 + 𝛼

𝛿 𝑣

𝛿𝛼𝑑Ω



𝛿𝛼

= 𝒖 ∙ 𝒗 V 𝛼 − ത𝛼 𝑞1 + 𝑞


ത𝛼 Ω𝑑Ω

Ω


Ω𝑑Ω

− V

+ 𝛼 − ത𝛼 𝑞1 + 𝑞

𝜌 + 𝑞 2නΩ

𝑣෦𝑣𝑎 𝑑Ω +නΩ


𝛿𝛼𝑑Ω


References

[1] T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003.

[2] J. S. Romero and E. C. N. Silva, “A topology optimization approach applied to laminar flow machine rotor design,” Comput. Methods Appl. Mech. Eng., vol. 279, pp. 268–300, 2014.

[3] OpenFOAM® v1906: New and improved numerics. (n.d.). Retrieved October 27, 2019, from https://www.openfoam.com/releases/openfoam-v1906/numerics.php#numerics-adjoint

[4] Papoutsis-Kiachagias, E. M., & Giannakoglou, K. C. (2016). Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications. Archives of Computational Methods in Engineering, 23(2), 255–299. https://doi.org/10.1007/s11831-014-9141-9

[5] Zymaris, A. S., Papadimitriou, D. I., Giannakoglou, K. C., & Othmer, C. (2010). Adjoint wall functions: A new concept for use in aerodynamic shape optimization. Journal of Computational Physics, 229(13), 5228–5245. https://doi.org/10.1016/j.jcp.2010.03.037

[6] goleman, daniel; boyatzis, Richard; Mckee, A. (2019). Manual adjointOptimisationFoam. Journal of Chemical Information and Modeling, 53(9), 1689–1699. https://doi.org/10.1017/CBO9781107415324.004

[7] F. Moukalled, L. Mangani, and M. Darwish, The finite volume method in computational fluid dynamics : An Advanced Introduction with OpenFOAM and Matlab, vol. 113. 2016.

[8] U. Nilsson, “Description of AdjointShapeOptimizationFoam and how to implement new cost functions,” pp. 1–40, 2013.

123

THANK YOU

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The Continuous Adjoint Turbulent Solver for Topology ...hani/kurser/OS_CFD_2019/Luis...Final...

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