High-ordernumericalmethodsforLargeEddySimulation

INRIASeminarPalaiseau,France,November24,2016

J.Vanharen

JointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

INTRODUCTION

2

High-ordernumericalmethodsforLargeEddySimulation

Introduction

3

Folie 8 > AIAA/CEAS-2008-AFN > Dobrzynski

Major Sources of Airframe Noise

◆ Trackvorticesoverlongdistance◆ Turbulence◆ Generationandpropagationofsoundwaves◆ Complexgeometries

Creation

Propagation

Interaction

DoorruptureseemstobecausedbytheNLGwake/MLGinterac:on.

High-ordernumericalmethodsforLargeEddySimulation

Industrialneeds

4

1)Gotowardscomplexgeometries, includingmoreandmoretechnologyeffect.

2)Captureunsteadyflowphysics.

3)Needforimprovementinhighperformancecomputingcapability.

High-ordernumericalmethodsforLargeEddySimulation

Standardmeshforindustrialproblems

5

◆ Structuredapproach(S):

▪ Standardtechniquetoalignmeshlinesandflowphysics,especiallyintheboundarylayertocaptureliftanddragcoefficients.

▪ Initialtechniqueintroducedinindustry.

▪ Directdataaddressing.▪ Efficientalgorithms(multigrid,linear

solvers…)usingadirectionalapproachfollowingmeshlines(i,j,k).

▪ Timeconsumingmeshgenerationprocess:

• Fromacoupleofhourstomanyweeks.• Dedicatedtoolsforspecific

configurations.

High-ordernumericalmethodsforLargeEddySimulation

Advancedmeshforindustrialproblems

6

◆ Chimera/Oversetgrid:▪ Assemblemeshesgeneratedseparately.▪ Quickermeshgeneration:independentmeshes.▪ AppliedtoexternalaerodynamicswithRANS/URANS.

▪ Dataexchangebetweengridsbyinterpolation:notconservative.▪ Inaccurateforinternalaerodynamics:massflowrateloss.

High-ordernumericalmethodsforLargeEddySimulation

Advancedmeshforindustrialproblems

7

◆ Nonconforminggridinterface(NGI)betweenstructuredblocks:▪ Meshesgeneratedseparatelyshareageometricinterface.▪ Geometricinterfacediscretizationsdiffer.▪ Conservativeapproachforplaneinterface.

▪ Usedformanyapplicationswithexternalandinternalflows.

High-ordernumericalmethodsforLargeEddySimulation

Advancedmeshforindustrialproblems

8

◆ Unstructuredapproach(U):▪ Quickgenerationofthemeshonregularcomputers,

evenforacomplexgeometry.▪ Less control on mesh quality and mesh can be inaccurate in some

specificpartsofthedomain.▪ IndirectdataaddressinginducesanincreaseofCPUcost.

◆ Hybridapproach(H):▪ Blend(S)and(U)blocksinasinglemesh.▪ Blend(S)and(U)algorithms

inasinglesolver.

▪ Takebenefitofanyapproachinits bestdomainofaccuracy.

▪ IntroduceNGIat(S)/(U)interface. Onera

High-ordernumericalmethodsforLargeEddySimulation

Overview

Introduction

I)Nonconforminggridinterfaceforunsteadyflows

II)LimitsinaFiniteVolumeFormalism

III)Alternative:theSpectralDifferenceMethod

Conclusion&perpectives

9

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

I)NONCONFORMINGGRIDINTERFACEFORUNSTEADYFLOWS

10

High-ordernumericalmethodsforLargeEddySimulation

Spectralanalysis

11

◆ Mathematicalframework:cell-centeredfinitevolumeapproximation

◆ Modelproblem:linearadvectionequationonaregularstructuredgridwithfluxdensity

◆ In1D:

d(fV)dt

+ (cSf)i+1/2 � (cSf)i�1/2 = 0

d

dtfi + c ·

fi+1/2 � fi�1/2

�x= 0

High-ordernumericalmethodsforLargeEddySimulation 12

◆ In,needtodefineaninterfacevalue.Introduceacenteredsecond-orderapproximation:

◆ Andfinally:

◆ Thesameapproximationofthedivergencetermisconsideredinthefollowing.

d

dtfi + c ·

fi+1/2 � fi�1/2

�x= 0

fi+1/2 � fi�1/2

�x' fi+1 � fi�1

2�x

@f

@x

����i

' fi+1 � fi�1

2�x= 0

Spectralanalysis

High-ordernumericalmethodsforLargeEddySimulation 13

◆ Lastpoint:performaVonNeumann(orFourier)localanalysis.▪ Assumethatwithissolutionof

thelinearadvectionequation.▪ Injectthesolutionintheadvectionequationandconsider

auniformgrid:thestandardequivalentwavenumberisrecovered:

j2 = �1f = exp(jkx)

f 0i ' exp[jk(i+ 1)�x]� exp[jk(i� 1)�x]

2�x

' jk exp(jk · i�x) exp(jk�x)�exp(�jk�x)2jk�x

' jkfisin(k�x)

k�x

kmk

=sin(k�x)

k�x

Spectralanalysis

km =f 0ij fi

High-ordernumericalmethodsforLargeEddySimulation 14

Spectralanalysisforsecond-ordercenteredscheme.Thisschemeisnotdissipacve(centered)butdispersive.

Theoretical

SpectralanalysisR

e(k

mD

x )

Im(k

mD

x )

High-ordernumericalmethodsforLargeEddySimulation

Spectralanalysis

15

High-ordernumericalmethodsforLargeEddySimulation 16

◆ Ourgoal:▪ Performasimilaranalysisforan

unsteadyflowinthepresenceofaNGI.

▪ UnderstandtheunsteadybehaviouroftheNGI.

▪ Correctspuriouseffects.

▪ ThisanalysiswaspublishedintheJournalofComputationalPhysics[1].

Spectralanalysis

[1] J. Vanharen, G. Puigt,M.Montagnac. Theoreccal and numerical analysis of nonconforming grid interface forunsteadyflows.JournalofComputaconalPhysics,285:111–132,2015.

High-ordernumericalmethodsforLargeEddySimulation 17

Toyproblem

◆ In2D,linearadvectionequationisconsideredasin1D,withasecond-ordercentered(structured)approximation.

◆ Meshcomposedofrectangularelementsparametrisedby,andonsides.

◆ meansstandardjoininterface.

◆ Keypoint:computingtherightcontributionforthefluxincell.

�zh

h = 0, �zR = �zL

(m,n)

h = 0, �zR = �zL

High-ordernumericalmethodsforLargeEddySimulation

Toyproblem

18

◆ Toensureconservation,thefluxonisdecomposedintotwocontributions:onand.with

◆ Asaconsequence,theprincipleis:▪ Todefineintersectionfacets.▪ Tobuildtheinterfacefluxonfacets

usinganaverageofRandLcontributions.

[AB][AM ] [MB]

fAM =fm,n + fm0,n0+1

2, fMB =

fm,n + fm0,n0+2

2

fm+1/2,n =AM

AB· fAM +

MB

AB· fMB

High-ordernumericalmethodsforLargeEddySimulation

2Dspectralanalysis

Butthenonconforminggirdinterfacerequiresatwo-dimensionalstudy.Otherquanccesshouldbeploiedtoachievespectralanalysis.

characterisesdissipaRon(if<1)oramplificaRon(if>1).

characterisesdispersion(=0foranon-dispersivescheme).

tocomplytheNyquist-Shannonsamplingtheorem.

19

A (�x)

A(�x) = exp [=m [kxm + kym ]�x]

�(�x) =kx + ky �<e [kxm + kym ]

⇡�x

� (�x)

k�x 2 [0,⇡]

High-ordernumericalmethodsforLargeEddySimulation

2Dspectralanalysis

20

◆ OnaNGI,theapproachisvalidonLandRblocks.

◆ Breakinguptheanalytictime-integrationintotheleftandrightparts sequentially to show the combined effects of a wavetravellingtheNGI,theoveralleffectoftheNGIisintroduced:

◆ Amplitudeandphasedependonlocalmetrics.

f(t, x, z) = AL exp⇥j⇡�L

⇤| {z }

Left contribution

· AR exp⇥j⇡�R

⇤| {z }

Right contribution

· exp [j (kxx+ kzz � !t)

High-ordernumericalmethodsforLargeEddySimulation

Coarseningalongthez-axis

21

Left Block200 x 400

Right Block200 x 103

High-ordernumericalmethodsforLargeEddySimulation 22

0.0 0.5 1.0 1.5 2.0 2.5 3.0k · �x [�]

0.0

0.2

0.4

0.6

0.8

1.0

�L,R

( �x)

[�]

u = 1 Left Block

u = 1 Right Block

u = 2 Left Block

u = 2 Right Block

u = 4 Left Block

u = 4 Right Block

0.0 0.5 1.0 1.5 2.0 2.5 3.0k · �x [�]

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

AL,R

( �x)

[�]

Coarsening along the z-axis. Propagation non-orthogonal to the interface.

Coarseningalongthez-axis

High-ordernumericalmethodsforLargeEddySimulation

NGI-COVO

23

0.0 0.5 1.0 1.5 2.0x [m]

0.0

0.5

1.0

1.5

2.0

z[m

]

Initial V ortex Convected V ortex

Boundary conditions

Subsonic inlet

Subsonic outlet

Periodic

Join

-1.400

-1.145

-0.891

-0.636

-0.382

-0.127

0.127

0.382

0.636

0.891

1.145

1.400

u�

U0

[m/s

]

COnvection of a compressible and isentropic

VOrtex (CO-VO): exact solution of Euler equations.

An analytical solution is available.

Error analysis.

⇠ = kfanalytical � fcomputedk1

⌘ =kfanalytical � fcomputedk1

kfanalyticalk1

High-ordernumericalmethodsforLargeEddySimulation

Spuriousreflection

24

Aspectratiointhex-&z-direction:32

Thisspuriousreflectioncanbe correctedbyametric-dependentinterpolationforRiemannsolver

High-ordernumericalmethodsforLargeEddySimulation

Spuriousreflection

25

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

II)LIMITINAFINITEVOLUMEFORMALISM

26

High-ordernumericalmethodsforLargeEddySimulation

elsAHworkflow

27

Hybridmesh

Pre-

elsAH

Post-processing

Hybridsplitter

1. Hybridmeshgeneration.▪ Notooltomanagehybridgrids.▪ Basealgebra:A+B=C,eitherbygeometric

reconstructionorbyfamilies.2. Hybridsplitterforparallelcomputation.

▪ Couplingofunstructuredandstructuredsplitter3. Pre-processing4. elsaH&co-processing.

▪ Co-processingismandatoryforLEScomputations.5. Post-processing.

▪ Antareslibrary.

High-ordernumericalmethodsforLargeEddySimulation

elsAHworkflow

28

Unstructured

Structured

Hybridnonconforminggridinterface

High-ordernumericalmethodsforLargeEddySimulation

Necessityofhigh-ordermethods

Convecconofanisentropicvortexinaboxwithperiodicboundarycondicons.Highorderversusloworderapproach,same#DOF.

Forthesameaccuracy,mul:plythe#DOFby64fora2ndorderscheme.

29

High-ordernumericalmethodsforLargeEddySimulation

Extensiontohigh-ordermethods

Forstructuredblocks,thehigh-orderschemeisasixth-ordercompactFiniteVolumescheme.

30

High-ordernumericalmethodsforLargeEddySimulation

Forunstructuredblocks,thesituationismuchmorecomplicated.

31

Extensiontohigh-ordermethods

ENO/WENO,k-exact&Least-Squares:definitionofalocalpolynomialrepresentationofquantitiestocomputethefluxesainterface.

DifficulttoimplementinelsAforgeneralunstructuredgrids.StenciltoolargeforHPC.

5thorder:104cells

6thorder:167cells

High-ordernumericalmethodsforLargeEddySimulation

Forunstructuredblocks,thesituationismuchmorecomplicated.

32

Extensiontohigh-ordermethods

INRIAapproachintroducedbyDervieuxetal.foracell-vertexcode.Schemesfromthesecond-ordertothesixth-order(forCartesiangrid).Reverttoasecond-orderontetrahedrabutwithlimiteddissipation&dispersion.

A PhD at Cerfacs investigated these schemes to adapt them in a cell-centered FiniteVolumeformalismbutwithoutsuccess.

Theaccuracyisdrivenbythegradientsonthecompactstencil.Asecond-ordergradientneedsathird-orderextrapolationofthesolution.Athird-orderofthesolutionneedsextendedstencilonunstructuredgrid.

AndthisleadstoHPC-inefficientschemes.[3]P.Cayot,G.Puigt,J.Vanharen,J.-F.Boussuge,P.Sagaut.Towardssecond-orderfinite-volumecell-centereddiffusionschemeonhybridunstructuredmeshes.Inpreparacon,AIAAJournal.

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

III)ALTERNATIVE:THESPECTRALDIFFERENCEMETHOD

33

High-ordernumericalmethodsforLargeEddySimulation 34

◆ TheSpectralDifferenceMethodconsistsinprojectingequationsonthebasisofLagrange polynomials.

◆ Thesolutioniscontinuousinside acellbutcanbediscontinuous atcellinterfaces.

◆ Letusdefinep+1solutionpoints andp+2fluxpoints.

SpacetimediscretisationforSpectralDifferenceMethod

�x

High-ordernumericalmethodsforLargeEddySimulation 35

TheSpectralDifferenceMethoddiscretisestheadvectionequationwiththreesteps:

1)Extrapolationfromthesolutionpointstothefluxpoints(matrixE),

2)Fluxcomputationinfluxpoints(matrixF),

SpacetimediscretisationforSpectralDifferenceMethod

High-ordernumericalmethodsforLargeEddySimulation 36

3)Fluxpolynomialderivationtoobtainthefluxderivativeatsolutionpoints(matrixD),

Combiningallthesematrices,oneobtains:

whereisthevectorofsolutionspoints.

SpacetimediscretisationforSpectralDifferenceMethod

@Ui

@t= DFE · Ui

Ui

High-ordernumericalmethodsforLargeEddySimulation 37

Thegeneratingpatternforaone-dimensionalproblemisgivenbyonecell.Tointroduceanormalmode,onehastomultiplytheDFEmatrixbytheLmatrix:toobtain:

whereM=DFEL.Thissystemoflinearequationsissolvedwiththelow-storagesecond-ordersix-stageRunge-KuttaschemeofBogeyandBailly[5]tosolvetheadvectionequation.

SpacetimediscretisationforSpectralDifferenceMethod

[5]C.BogeyandC.Bailly.Afamilyoflowdispersiveandlowdissipacveexplicitschemesforflowandnoisecomputacons,J.Comput.Phys.194(1)(2004)194–214.

High-ordernumericalmethodsforLargeEddySimulation 38

◆ Thisequationcanbeseenasageometricprogression.

◆ Consideringtheeigenvalueproblem:

◆ Onefindstheeigenvalues:

◆ Onefindstheassociatedeigenvectorsspanning:

TheMatrixPowerMethod

Figure 6: Illustration of the spectral analysis. The mesh is composed of four cells. Each cell is delimited by the dotted lines.The black crosses + represent the solution point for each cell. The symbols • represent the initial solution at each solutionpoint and � the computed solution at each solution point.

4.5. Spectral analysis: the Matrix Power Method215

Using properties of geometric progessions and (34), one obtains:

Uni = Gn · U0

i . (38)

Carrying out an unsteady computation with the Spectral Di↵erence Method simply consists in computingthe matrix G to the nth power to obtain the solution at the nth iteration from the initial solution. Itis somewhat reminiscent of the Matrix Power Method. For estimating and computing eigenvalues, thisMethod generally gives the eigenvalue with the greatest absolute value of a given matrix. It consists incomputing the nth power of this matrix. The proof of this theorem can be found in [33]. The followingproof is considerably inspired from the previous reference but it will allow us to establish a general resultfor the spectral analysis of the Spectral Di↵erence Method. Note that this method could be applied apriori for other high-order methods which consider several degrees of freedom inside each cell. Considerthe eigenvalue problem G · Ui = �Ui, Ui 6= 0, G 2 Mp+1 (C), Ui 2 Cp+1, � 2 C, which we assume hasa complete normalized eigenvector space (v1,v2, ...,vp+1) spanning Cp+1, with corresponding eigenvalues

(�1,�2, ...,�p+1) where |�1| < |�2| < ... < |�p+1|. We assume that U0i is a given vector, for which

U0i =

p+1X

j=1

↵(0)j · vj , (39)

where ↵(0)n 6= 0. Let

8n 2 {1, 2, ...} , Uni = G · Un�1

i .

To analyze the convergence of the sequence Uni , one obtains

Uni =

p+1X

j=1

↵(0)j Gn · vj .

15

Un+1i = G · Un

i

Figure 6: Illustration of the spectral analysis. The mesh is composed of four cells. Each cell is delimited by the dotted lines.The black crosses + represent the solution point for each cell. The symbols • represent the initial solution at each solutionpoint and � the computed solution at each solution point.

4.5. Spectral analysis: the Matrix Power Method215

Using properties of geometric progessions and (34), one obtains:

Uni = Gn · U0

i . (38)

Carrying out an unsteady computation with the Spectral Di↵erence Method simply consists in computingthe matrix G to the nth power to obtain the solution at the nth iteration from the initial solution. Itis somewhat reminiscent of the Matrix Power Method. For estimating and computing eigenvalues, thisMethod generally gives the eigenvalue with the greatest absolute value of a given matrix. It consists incomputing the nth power of this matrix. The proof of this theorem can be found in [33]. The followingproof is considerably inspired from the previous reference but it will allow us to establish a general resultfor the spectral analysis of the Spectral Di↵erence Method. Note that this method could be applied apriori for other high-order methods which consider several degrees of freedom inside each cell. Considerthe eigenvalue problem G · Ui = �Ui, Ui 6= 0, G 2 Mp+1 (C), Ui 2 Cp+1, � 2 C, which we assume hasa complete normalized eigenvector space (v1,v2, ...,vp+1) spanning Cp+1, with corresponding eigenvalues

(�1,�2, ...,�p+1) where |�1| < |�2| < ... < |�p+1|. We assume that U0i is a given vector, for which

U0i =

p+1X

j=1

↵(0)j · vj , (39)

where ↵(0)n 6= 0. Let

8n 2 {1, 2, ...} , Uni = G · Un�1

i .

To analyze the convergence of the sequence Uni , one obtains

Uni =

p+1X

j=1

↵(0)j Gn · vj .

15

Figure 6: Illustration of the spectral analysis. The mesh is composed of four cells. Each cell is delimited by the dotted lines.The black crosses + represent the solution point for each cell. The symbols • represent the initial solution at each solutionpoint and � the computed solution at each solution point.

4.5. Spectral analysis: the Matrix Power Method215

Using properties of geometric progessions and (34), one obtains:

Uni = Gn · U0

i . (38)

Carrying out an unsteady computation with the Spectral Di↵erence Method simply consists in computingthe matrix G to the nth power to obtain the solution at the nth iteration from the initial solution. Itis somewhat reminiscent of the Matrix Power Method. For estimating and computing eigenvalues, thisMethod generally gives the eigenvalue with the greatest absolute value of a given matrix. It consists incomputing the nth power of this matrix. The proof of this theorem can be found in [33]. The followingproof is considerably inspired from the previous reference but it will allow us to establish a general resultfor the spectral analysis of the Spectral Di↵erence Method. Note that this method could be applied apriori for other high-order methods which consider several degrees of freedom inside each cell. Considerthe eigenvalue problem G · Ui = �Ui, Ui 6= 0, G 2 Mp+1 (C), Ui 2 Cp+1, � 2 C, which we assume hasa complete normalized eigenvector space (v1,v2, ...,vp+1) spanning Cp+1, with corresponding eigenvalues

(�1,�2, ...,�p+1) where |�1| < |�2| < ... < |�p+1|. We assume that U0i is a given vector, for which

U0i =

p+1X

j=1

↵(0)j · vj , (39)

where ↵(0)n 6= 0. Let

8n 2 {1, 2, ...} , Uni = G · Un�1

i .

To analyze the convergence of the sequence Uni , one obtains

Uni =

p+1X

j=1

↵(0)j Gn · vj .

15

Figure 6: Illustration of the spectral analysis. The mesh is composed of four cells. Each cell is delimited by the dotted lines.The black crosses + represent the solution point for each cell. The symbols • represent the initial solution at each solutionpoint and � the computed solution at each solution point.

4.5. Spectral analysis: the Matrix Power Method215

Using properties of geometric progessions and (34), one obtains:

Uni = Gn · U0

i . (38)

Carrying out an unsteady computation with the Spectral Di↵erence Method simply consists in computingthe matrix G to the nth power to obtain the solution at the nth iteration from the initial solution. Itis somewhat reminiscent of the Matrix Power Method. For estimating and computing eigenvalues, thisMethod generally gives the eigenvalue with the greatest absolute value of a given matrix. It consists incomputing the nth power of this matrix. The proof of this theorem can be found in [33]. The followingproof is considerably inspired from the previous reference but it will allow us to establish a general resultfor the spectral analysis of the Spectral Di↵erence Method. Note that this method could be applied apriori for other high-order methods which consider several degrees of freedom inside each cell. Considerthe eigenvalue problem G · Ui = �Ui, Ui 6= 0, G 2 Mp+1 (C), Ui 2 Cp+1, � 2 C, which we assume hasa complete normalized eigenvector space (v1,v2, ...,vp+1) spanning Cp+1, with corresponding eigenvalues

(�1,�2, ...,�p+1) where |�1| < |�2| < ... < |�p+1|. We assume that U0i is a given vector, for which

U0i =

p+1X

j=1

↵(0)j · vj , (39)

where ↵(0)n 6= 0. Let

8n 2 {1, 2, ...} , Uni = G · Un�1

i .

To analyze the convergence of the sequence Uni , one obtains

Uni =

p+1X

j=1

↵(0)j Gn · vj .

15

High-ordernumericalmethodsforLargeEddySimulation 39

◆ Projectingtheinitialsolutionontheeigenvectorsbasis:

◆ Usingthediagonalizationproperties:

◆ Factorising:

TheMatrixPowerMethod

n ! +10

Figure 6: Illustration of the spectral analysis. The mesh is composed of four cells. Each cell is delimited by the dotted lines.The black crosses + represent the solution point for each cell. The symbols • represent the initial solution at each solutionpoint and � the computed solution at each solution point.

4.5. Spectral analysis: the Matrix Power Method215

Using properties of geometric progessions and (34), one obtains:

Uni = Gn · U0

i . (38)

Carrying out an unsteady computation with the Spectral Di↵erence Method simply consists in computingthe matrix G to the nth power to obtain the solution at the nth iteration from the initial solution. Itis somewhat reminiscent of the Matrix Power Method. For estimating and computing eigenvalues, thisMethod generally gives the eigenvalue with the greatest absolute value of a given matrix. It consists incomputing the nth power of this matrix. The proof of this theorem can be found in [33]. The followingproof is considerably inspired from the previous reference but it will allow us to establish a general resultfor the spectral analysis of the Spectral Di↵erence Method. Note that this method could be applied apriori for other high-order methods which consider several degrees of freedom inside each cell. Considerthe eigenvalue problem G · Ui = �Ui, Ui 6= 0, G 2 Mp+1 (C), Ui 2 Cp+1, � 2 C, which we assume hasa complete normalized eigenvector space (v1,v2, ...,vp+1) spanning Cp+1, with corresponding eigenvalues

(�1,�2, ...,�p+1) where |�1| < |�2| < ... < |�p+1|. We assume that U0i is a given vector, for which

U0i =

p+1X

j=1

↵(0)j · vj , (39)

where ↵(0)n 6= 0. Let

8n 2 {1, 2, ...} , Uni = G · Un�1

i .

To analyze the convergence of the sequence Uni , one obtains

Uni =

p+1X

j=1

↵(0)j Gn · vj .

15

Figure 6: Illustration of the spectral analysis. The mesh is composed of four cells. Each cell is delimited by the dotted lines.The black crosses + represent the solution point for each cell. The symbols • represent the initial solution at each solutionpoint and � the computed solution at each solution point.

4.5. Spectral analysis: the Matrix Power Method215

Using properties of geometric progessions and (34), one obtains:

Uni = Gn · U0

i . (38)

Carrying out an unsteady computation with the Spectral Di↵erence Method simply consists in computingthe matrix G to the nth power to obtain the solution at the nth iteration from the initial solution. Itis somewhat reminiscent of the Matrix Power Method. For estimating and computing eigenvalues, thisMethod generally gives the eigenvalue with the greatest absolute value of a given matrix. It consists incomputing the nth power of this matrix. The proof of this theorem can be found in [33]. The followingproof is considerably inspired from the previous reference but it will allow us to establish a general resultfor the spectral analysis of the Spectral Di↵erence Method. Note that this method could be applied apriori for other high-order methods which consider several degrees of freedom inside each cell. Considerthe eigenvalue problem G · Ui = �Ui, Ui 6= 0, G 2 Mp+1 (C), Ui 2 Cp+1, � 2 C, which we assume hasa complete normalized eigenvector space (v1,v2, ...,vp+1) spanning Cp+1, with corresponding eigenvalues

(�1,�2, ...,�p+1) where |�1| < |�2| < ... < |�p+1|. We assume that U0i is a given vector, for which

U0i =

p+1X

j=1

↵(0)j · vj , (39)

where ↵(0)n 6= 0. Let

8n 2 {1, 2, ...} , Uni = G · Un�1

i .

To analyze the convergence of the sequence Uni , one obtains

Uni =

p+1X

j=1

↵(0)j Gn · vj .

15

Furthermore, 8j 2 {1, 2, ..., p+ 1}, Gn · vj = �nj vj yields

Uni =

p+1X

j=1

↵(0)j �n

j vj

or

Uni = ↵(0)

p+1�np+1

2

4vp+1 +pX

j=1

↵(0)j

↵(0)p+1

✓�j

�p+1

◆n

vj

3

5 .

Hence, using��� �j

�p+1

��� = 1� |�p+1|�|�j ||�p+1| , we find

�����

�����vp+1 �1

↵(0)p+1�

np+1

Uni

�����

�����1

6pX

j=1

�����↵(0)j

↵(0)p+1

����� (1� ap+1)k ,

where ap+1 = minj 6=p+1

��� |�p+1|�|�j |�p+1

���, that is, Uni is approximated by ↵(0)

p+1�np+1vp+1. The rate of convergence

depends on the gap between the absolute magnitude of eigenvalues. However, it is an exponential decreaseand during an unsteady computation, several thousands of iterations can be performed. More precisely, oneobtains:

Uni ⇠

n!+1↵(0)p+1�

np+1vp+1 (40)

This means that for n enough high, the solution at the iteration n� 1 is simply multiply by �p+1 to obtainthe solution at the iteration n. We will see that in practice that n has no need to be so high since the rate ofconvergence is exponentionally decreasing. This gives a way to perform the spectral analysis of the SpectralDi↵erence Method. The dispersion and the amplification are simply given by the spectral radius �p+1 of thematrix G. Between the iterations n and n+ 1, the dispersion is given by the argument ' = � arg (�p+1) /⌫220

and the amplification is given by the absolute value ⇢ = |�p+1|. The Fig. 7a represents the absolute value ofthe di↵erence between k�x and '. The more these quantity tends to zero, the less dispersive the numericalscheme is. This quantity is characterisitc of the dispersion error. The Fig. 7b represents the di↵erencebetween 1 and ⇢. The more these quantity tends to zero, the less there is dissipation. This quantity ischaracteristic of the dissipation error.225

(a) Dispersion error. (b) Dissipation error.

Figure 7: Spectral analysis for ⌫ = 0.1: e↵ect of the order of the solution reconstruction.

16

Furthermore, 8j 2 {1, 2, ..., p+ 1}, Gn · vj = �nj vj yields

Uni =

p+1X

j=1

↵(0)j �n

j vj

or

Uni = ↵(0)

p+1�np+1

2

4vp+1 +pX

j=1

↵(0)j

↵(0)p+1

✓�j

�p+1

◆n

vj

3

5 .

Hence, using��� �j

�p+1

��� = 1� |�p+1|�|�j ||�p+1| , we find

�����

�����vp+1 �1

↵(0)p+1�

np+1

Uni

�����

�����1

6pX

j=1

�����↵(0)j

↵(0)p+1

����� (1� ap+1)k ,

where ap+1 = minj 6=p+1

��� |�p+1|�|�j |�p+1

���, that is, Uni is approximated by ↵(0)

p+1�np+1vp+1. The rate of convergence

depends on the gap between the absolute magnitude of eigenvalues. However, it is an exponential decreaseand during an unsteady computation, several thousands of iterations can be performed. More precisely, oneobtains:

Uni ⇠

n!+1↵(0)p+1�

np+1vp+1 (40)

This means that for n enough high, the solution at the iteration n� 1 is simply multiply by �p+1 to obtainthe solution at the iteration n. We will see that in practice that n has no need to be so high since the rate ofconvergence is exponentionally decreasing. This gives a way to perform the spectral analysis of the SpectralDi↵erence Method. The dispersion and the amplification are simply given by the spectral radius �p+1 of thematrix G. Between the iterations n and n+ 1, the dispersion is given by the argument ' = � arg (�p+1) /⌫220

and the amplification is given by the absolute value ⇢ = |�p+1|. The Fig. 7a represents the absolute value ofthe di↵erence between k�x and '. The more these quantity tends to zero, the less dispersive the numericalscheme is. This quantity is characterisitc of the dispersion error. The Fig. 7b represents the di↵erencebetween 1 and ⇢. The more these quantity tends to zero, the less there is dissipation. This quantity ischaracteristic of the dissipation error.225

(a) Dispersion error. (b) Dissipation error.

Figure 7: Spectral analysis for ⌫ = 0.1: e↵ect of the order of the solution reconstruction.

16

Un+1i = G · Un

i

High-ordernumericalmethodsforLargeEddySimulation 40

◆ Onecouldfindtheasymptoticbehaviour:

◆ Thisasymptoticbehaviourisdrivenbythespectralradius.

◆ Wedefinecharacterisationsofdispersionanddissipation:

TheMatrixPowerMethod

Un+1i = G · Un

i

Furthermore, 8j 2 {1, 2, ..., p+ 1}, Gn · vj = �nj vj yields

Uni =

p+1X

j=1

↵(0)j �n

j vj

or

Uni = ↵(0)

p+1�np+1

2

4vp+1 +pX

j=1

↵(0)j

↵(0)p+1

✓�j

�p+1

◆n

vj

3

5 .

Hence, using��� �j

�p+1

��� = 1� |�p+1|�|�j ||�p+1| , we find

�����

�����vp+1 �1

↵(0)p+1�

np+1

Uni

�����

�����1

6pX

j=1

�����↵(0)j

↵(0)p+1

����� (1� ap+1)k ,

where ap+1 = minj 6=p+1

��� |�p+1|�|�j |�p+1

���, that is, Uni is approximated by ↵(0)

p+1�np+1vp+1. The rate of convergence

depends on the gap between the absolute magnitude of eigenvalues. However, it is an exponential decreaseand during an unsteady computation, several thousands of iterations can be performed. More precisely, oneobtains:

Uni ⇠

n!+1↵(0)p+1�

np+1vp+1 (40)

This means that for n enough high, the solution at the iteration n� 1 is simply multiply by �p+1 to obtainthe solution at the iteration n. We will see that in practice that n has no need to be so high since the rate ofconvergence is exponentionally decreasing. This gives a way to perform the spectral analysis of the SpectralDi↵erence Method. The dispersion and the amplification are simply given by the spectral radius �p+1 of thematrix G. Between the iterations n and n+ 1, the dispersion is given by the argument ' = � arg (�p+1) /⌫220

and the amplification is given by the absolute value ⇢ = |�p+1|. The Fig. 7a represents the absolute value ofthe di↵erence between k�x and '. The more these quantity tends to zero, the less dispersive the numericalscheme is. This quantity is characterisitc of the dispersion error. The Fig. 7b represents the di↵erencebetween 1 and ⇢. The more these quantity tends to zero, the less there is dissipation. This quantity ischaracteristic of the dissipation error.225

(a) Dispersion error. (b) Dissipation error.

Figure 7: Spectral analysis for ⌫ = 0.1: e↵ect of the order of the solution reconstruction.

16

' ⇠ arg (�p+1)

⇢ ⇠ |�p+1|

High-ordernumericalmethodsforLargeEddySimulation 41

TheMatrixPowerMethod

10−7

10−6

10−5

10−4

10−3

10−2

10−1

π/4 π/2 3π/4 π

|k·∆

x−ϕ|[−

]

k ·∆x [−]

SD2SD3SD4SD5

DispersionerrorCFL=0.1

High-ordernumericalmethodsforLargeEddySimulation 42

TheMatrixPowerMethod

10−12

10−10

10−8

10−6

10−4

10−2

π/4 π/2 3π/4 π

1−

ρ[−

]

k ·∆x [−]

SD2SD3SD4SD5

DissipationerrorCFL=0.1

High-ordernumericalmethodsforLargeEddySimulation 43

ExtensiontohighwavenumbersfortheSpectralDifferenceMethod

◆ Energylossestimationbetweentheiterationnandm(n>m):

◆ Phaseshiftestimationbetweentheiterationnandm(n>m):

High-ordernumericalmethodsforLargeEddySimulation 44

ComparisonwithstandardFiniteDifferencemethods

Dissipationfor100000iterationsatCFL=0.7.

RKo6s-SD5schemeisasaccurateasRKo6s-CF8-CS6scheme.

High-ordernumericalmethodsforLargeEddySimulation 45

ComparisonwithstandardFiniteDifferencemethods

Dispersionfor100000iterationsatCFL=0.7.

RKo6s-SD5schemeisasaccurateasRKo6s-CF8-CS6scheme.[2] J.Vanharen,G.Puigt,X.Vasseur, J.-F.Boussuge,P. Sagaut.Revisicng the spectral analysis forhigh-order spectraldisconcnuousmethods.Underreview,JournalofComputaconalPhysics,2016.

High-ordernumericalmethodsforLargeEddySimulation

HPCefficiency

46

High-ordernumericalmethodsforLargeEddySimulation 47

JAGUARCPUperformance(Euler)

GFLOPSobtainedarealmosttwiceasbigas standardCFDsolvers.

MeantimebyiterationandbyDoF:2.2µsin2DEuler(about8µsforNSin3D)

whichislowerthanstandardCFDsolvers.

High-ordernumericalmethodsforLargeEddySimulation

Partialconclusion

48

JAGUARisanalternativesolvertostandardFiniteVolumeones:

Unstructuredgridforcomplexgeometry.

Highperformancecapabilitywithverygoodscaling.

AsaccurateasstandardFiniteVolumeschemes.

Butonlyhexahedralmeshes…

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

CONCLUSION&PERPECTIVES

49

High-ordernumericalmethodsforLargeEddySimulation

Conclusion&perpectives

Extensionofthenonconforminggridinterfaceforhybridmeshes.

Second-orderaccurateevenfornonconforminggridinterface.

Workflowcreationforhybridmeshes.

RelativeHPCefficiency.

Theextensiontohigh-orderschemesisimpossible forunstructuredmeshes.

50

High-ordernumericalmethodsforLargeEddySimulation

JAGUARisanalternativetostandardFiniteVolumesolvers.

Anyarbitraryorderofaccuracy.HPCefficient.

Butonlyforhexahedralmeshes…

Splittetrahedraintohexahedra…ExtensionoftheSpectralDifference

Methodontetrahedra.

51

Conclusion&perpectives

High-ordernumericalmethodsforLargeEddySimulation

LaBS:solvetheBoltzmannequation.Goodaccuracy.

(likeasixth-orderschemesfordissipation).HPCefficient.

Canhandleverycomplexgeometries withnonbody-fittedoctreemeshes.

Transitionresolution.Weaklycompressibleformulation.

52

Conclusion&perpectives

High-ordernumericalmethodsforLargeEddySimulation 53

Conclusion&perpectives

High-ordernumericalmethodsforLargeEddySimulation

◆ High-orderschemesbibliographyforLES.◆ elsAH,ANTARES&JAGUARdevelopmentsinPython,C/C++&Fortran90.

◆ Parallelprogramming:ArgonneTrainingProgramonExtreme-ScaleComputing(ATPESC2016).

◆ Airbusinternship:– fluide/structureinteraction,– gustresponse.

◆ AvailableinMay2017.

54

Conclusion&perpectives

High-ordernumericalmethodsforLargeEddySimulation 55

Thankyouforyourattention.JulienVanharen

PhDstudentAirbus/Cerfacsvanharen@cerfacs.fr

References:

[1] J. Vanharen, G. Puigt, M. Montagnac. Theoreccal and numerical analysis ofnonconforming grid interface for unsteady flows. Journal of Computaconal Physics,285:111–132,2015.[2] J.Vanharen,G.Puigt, X.Vasseur, J.-F.Boussuge,P. Sagaut.Revisicng the spectralanalysis for high-order spectral disconcnuous methods. Under review, Journal ofComputaconalPhysics,2016.[3] P. Cayot, G. Puigt, J. Vanharen, J.-F. Boussuge, P. Sagaut. Towards second-orderfinite-volume cell-centered diffusion scheme on hybrid unstructured meshes. Inpreparacon,AIAAJournal.[4] J. Vanharen, G. Puigt, J.-F. Boussuge, P. Sagaut. Opcmized Runge-Kuia cmeintegracon forSpectralDifferenceMethod. Inpreparacon, JournalofComputaconalPhysics.

[A] J. Vanharen, G. Puigt, M. Montagnac. Theoreccal and numerical analysis ofnonconforminggridinterfaceforunsteadyflows.AERO2015,Toulouse,France,March30th-31st,April1st2015.[B] J.Vanharen,G.Puigt,and J.-F.Boussuge.Revisicngthespectralanalysis forhigh-order spectral disconcnuous methods. In TILDA - Symposium, Toulouse, France,November21st-23rd2016.

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

A)FLUXCONSERVATION

56

High-ordernumericalmethodsforLargeEddySimulation

A)Fluxconservation

57

Propertyoffluxconservationlostwhen:1.Noncoplanarfaces2. Curvedinterfaces

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

B)SHOCK-CAPTURING

58

High-ordernumericalmethodsforLargeEddySimulation

B)Shock-capturing

59

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

C)INTRODUCTION

60

High-ordernumericalmethodsforLargeEddySimulation 61

C)Industrialneeds

Complex geometries Accuracy

HPCPhysicalmodelling

High-ordernumericalmethodsforLargeEddySimulation

C)Introduction

Unsteady flow phenomena can occur on a large number of aircraqcomponents: -Landinggearsdoors, -Jet-flapinteraccons, -Airbrakesoutduringemergencydescent.

Theresulcngunsteadyaerodynamicforcesincreasethestructuralloadsthat may lead to vibracons and/or structural damage (facgue, limitloads).

The predicRon of these phenomena is tradiRonally based onexperiments,whichprovide resultsof appreciableaccuracy,but lateinthedesignprocess.

62

High-ordernumericalmethodsforLargeEddySimulation

C)Introduction

63

Trackingvorticeso

verlongdistancea

nd

propagatingacous

ticwavesoncomplex

geometriesarestillacha

llengeforCFDand

requireveryaccur

atenumericalmethods

intermsofdissipationand

dispersion.

High-ordernumericalmethodsforLargeEddySimulation 64

C)IndustrialCFDsimulationsatAirbus

Threeactiveresearchactivities:

Cabinnoise

Externalsourcenoise

Landinggear

High-ordernumericalmethodsforLargeEddySimulation

C)Summary

65

Statusandfutureneeds:◆ RANS/URANS, geometry not too complex, structured grid with mesh

movement(ALE).◆ Tendency to treat off-design configurations with complex geometry and

unsteadyflows(withURANSorLES).◆ Threekeypoints:complexgeometries,unsteadyflow&HPC.

Goalofthistalk:◆ Extend theNonconforming Grid Interface (NGI) inhybrid framework for

unsteadyflow.◆ FocusattentiononthelimitsofthisapproachinaFiniteVolumeformalism.◆ Exploreanewparadigms:theSpectralDifferenceMethodandtheLattice-

BoltzmannMethod.

High-ordernumericalmethodsforLargeEddySimulationINRIASeminarPalaiseau,France,November24,2016

J.VanharenJointworkwithG.Puigt,M.Montagnac,X.Vasseur,J.-F.BoussugeandP.Sagaut.

D)THESPECTRALDIFFERENCEMETHOD

66

High-ordernumericalmethodsforLargeEddySimulation 67

Thisexpressionwasobtainedbyaspace-timeFouriertransformbasedonthefollowingnormalmode:

Itis2periodic:

Inordertoavoidaliasing,shouldbelongto:thisistheNyquist-Shannontheorem.

p

kDx [0, p]

Un+1i = G · Un

i

D)SpacetimediscretisationforSpectralDifferenceMethod

High-ordernumericalmethodsforLargeEddySimulation 68

Finally,oneobtains:

D)SpacetimediscretisationforSpectralDifferenceMethod

Un+1i = G · Un

i

Firstremark:allthepropertiesofthespacetimediscretisationareinthismatrixG.

Secondremark:thisavectorexpressionsincethereareseveralsolutionpointspercell.