January 6, 2016 Dr. Corentin Carton de Wiart corentin.carton@gmail.com Dear Dr. Carton de Wiart, As you know, administration of the NASA Postdoctoral Program is transitioning from Oak Ridge Associated Universities to the Universities Space Research Association. On January 31, 2016, your appointment under Oak Ridge Associated Universities will end. This letter is to inform you of your new appointment with USRA, which commences February 1, 2016. As a current NASA Postdoctoral Fellow, we want to welcome you to Universities Space Research Association and to introduce ourselves to you, and to provide you with guidance during this transition. Universities Space Research Association (USRA), founded in 1969, is an independent, nonprofit research corporation whose mission is to work with NASA, other federal agencies, and the University community to advance earth science, space science, and technology. USRA’s research expertise includes astrophysics, planetary science, earth science, engineering, biomedicine, and quantum computing. We are an association of 105 US and international universities, and we manage research institutes at NASA centers, including the Goddard Space Flight Center, the Ames Research Center, the Glenn Research Center, and the Johnson Space Center and we manage astrophysics activities at the Arecibo Observatory. We employ over 400 scientists and engineers at the postdoctoral and more senior level, and manage internship programs for NASA and the Air Force Research Lab. We are headquartered in Columbia, Maryland; our website is http://www.usra.edu. I’d like to provide you an overview of the transition process. The transition from ORAU to USRA began on October 15, 2015. As of February 1, 2016, USRA will be responsible for supporting all appointed Fellows (including stipend payments, healthcare, visas and scientific travel support) as well as managing the reviews of all applicants, announcing the results of award decisions, and providing reports on the status of the Program to NASA. I’ve provided some specific information on important, near-term issues regarding center access, stipends, taxes, travel, visas, healthcare, and the USRA NPP web portal. Center Access: If you work at a NASA Center, your NASA Center Representative will update your entry in IDMAX to provide physical access to your Center, and will update your TTCP if necessary to ensure your access to the NASA internet network. If you are a foreign national, the Center Representative will also upload a copy of your DS2019 to IDMAX. Stipends: By agreement, ORAU will pay your stipend through January 31, 2016. Your first stipend payment from USRA will be made on the last business day of February 2016, i.e. Monday, February 29, 2016. If you only work for part of the month, your stipend will be pro-rated. If you participate in the USRA Healthcare plan, your USRA stipend payment will include the NASA contribution to your healthcare premium; the NASA contribution to your healthcare premium, and your portion of the healthcare premium, will be automatically deducted from your monthly stipend by USRA.

UNIVERSITIES SPACE RESEARCH ASSOCIATION — NASA Postdoctoral Program —

7178 Columbia Gateway Drive • Columbia, MD 21046 • (410) 730-2656

February27-March3,2017SIAMCSE2017-Atlanta(GA)

Adjoint Sensitivity Analysis for Scale-Resolving Turbulent Flow Solvers

PatrickBlonigan,LasloDiosady,AnirbanGarai,andScottMurman

NASAAmesResearchCenter

https://ntrs.nasa.gov/search.jsp?R=20170010266 2020-07-10T22:17:32+00:00Z

• TypesofSensitivityAnalysis• Tangent:sensitivityofmanyobjectives

tooneinputparameter• Adjoint:sensitivityofoneobjectiveto

manyinputparameters• Gradient-basedDesignOptimization

• ErrorEstimation

• MeshAdaptation

• UncertaintyQuantification

• Systemswithunsteadyflowshavemanyimportantobjectivefunctionsthataretimeaveraged

2

AdjointSensitivityAnalysisofHighFidelitySimulations

Traditionalsensitivityanalysisfailsfortheseobjectivesinhighfidelitysimulations,whichexhibitchaoticdynamics

Failureofconventionalsensitivityanalysisforchaos

3

• Lorenz63System• Objectivez:rateofheattransfer• Inputρ:temperaturedifference

Input Input

Timeaveraged

objective

Deriv

ativeof

Timeaveraged

objective

Failureofconventionalsensitivityanalysisforchaos

4

InputParameter InputParameter

Time

TimeDependentOutput TimeAveragedOutput

1. EnsembleAdjointMethod• Leaetal.2000,Eyinketal.2004.

2. Fokker-PlanckMethods• Thuburnetal.2005.,Bloniganand

Wang2014

3. Fluctuation-DissipationTheorem• Leith1975,AbramovandMajda2007

4. LeastSquaresShadowing(LSS)• Wang,Hui,andBlonigan2014

5

Sensitivityanalysisapproachesforchaoticsystems

1.Ensembleadjointsensitivitiesforshort,medium,andlongtimesegments.

2.Fokker-Planckcomputedstationarydensity(left)anditsadjoint(right).

4.LSSreferenceandshadowtrajectories.

ShadowingObjectiveSurface

• Chooseinitialconditionforsmoothvariationofobjectivehistorywithinputparameter.

ConventionalObjectiveSurface

• Fixedinitialconditionforallinputparametervalues.

6

Sensitivityanalysiswithshadowing

InputParameter

Time

InputParameter

InPhaseSpace: ShadowingObjectiveSurface

• Chooseinitialconditionforsmoothvariationofobjectivehistorywithinputparameter.

7

Sensitivityanalysiswithshadowing

Time

InputParameter

ShadowTrajectory

ReferenceTrajectorydu

dt= f(u; s)

du

d⌧= f(u; s+ �s)

• Assumeergodicity,replaceinitialconditionforu(t)with

8

Leastsquaresshadowing

minu,⌧

1

2

Z T1

T0

W (t)ku(⌧(t))� ur(t)k2 dt

s.t.du

d⌧= f(u; s+ �s)

LinearizefortangentLSS:

v ⌘ @u

@s) min

v

1

2

Z T1

T0

W (t)kv(t)k2 dt

s.t.

⌧v,

du

dt

�= 0

s.t.dv

dt=

@f

@uv +

@f

@s+

✓1� d⌧

dt

◆

| {z }⌘

f

ShadowingObjectiveSurface

• Chooseinitialconditionforsmoothvariationofobjectivehistorywithinputparameter.

InputParameter

Time

LeastsquaresshadowingforLorenz63

9

• Lorenz63System• Objective(z-28):deviationofrateof

heattransfer• Inputρ:temperaturedifference

Input Input

Timeaveraged

objective

Timeaveraged

objective

T1-T0=5.0 T1-T0=5000.0

“Non-Intrusive”leastsquaresshadowing

10

• OriginallyproposedbyNietal.(AIAA2016-4399)• ReducessizeofLSSminimizationproblemconsiderablyby• Minimizingv(t)atKdiscretecheckpointsintime

• Expressingv(t)intermsofhomogeneousandinhomogeneouscomponents

minv(ti)

1

2

KX

i=0

kv(ti)k2 s.t.dv

dt=

@f

@uv +

@f

@s+ ⌘f,

⌧v,

du

dt

�= 0

v(t) =X

j

↵j vj(t) + v(t)

Chooseαthatsolvestheleastsquaresproblem

t0

||v(t)||

Segment 1 Segment 2 Segment 3

t1 t2 t3

dvjdt

=@f

@uvj ,

vj(ti) = V ji

dv

dt=

@f

@uv +

@f

@s+ ⌘f,

v(ti) = Vi

Homogeneous

Inhomogeneous

TangentNILSSAlgorithm

11

• Setand,arandomorthonormalmatrix.• Foreachsegmentstartingwith1:1.Computeprimalfromti-1toti2.Computeallmfrom3.ComputeQR-decomposition,where4.Set5.Computeforwith

• Solve

• Computesensitivitytoswithαi’sandsegmentsensitivitycontributionsand

V0 = 0 V0 = Q0

u(t)

vj(t) vj(ti�1) = V ji�1

v(t)

QiRi = V �i

V ji = Qj

i

v(ti�1) = Vi Vi = (I �Qi�1QTi�1)v(t

�i�1)

[V �i ]j = vj(ti)

dJ

ds=

1

tK � t0

KX

i=1

�gi

T↵i + hi

�+

@J

@s

gi hi

min

���������

↵1...

↵K

↵K+1

���������2

s.t.

2

64R1 �I

. . .. . .RK �I

3

75

2

6664

↵1...

↵K

↵K+1

3

7775=

2

64�QT

1 v(t�1 )

...�QT

K v(t�K)

3

75

AdjointNILSSAlgorithm

12

• Set,arandomorthonormalmatrix.• Foreachsegmentstartingwith1:1.Computeprimalfromti-1toti2.Computeallmfrom3.ComputeQR-decomposition,where4.Set

• Solvetheminimizationproblem

• Set.ForeachsegmentstartingwithKsolvetheadjointequationbackwardsfromtitoti-1,wherethematrixandvectorarerelatedtoτ(t).

• Computesensitivitieswith

V0 = Q0

u(t)

vj(t) vj(ti�1) = V ji�1

QiRi = V �i

V ji = Qj

i

[V �i ]j = vj(ti)

min

����������

2

66664

RT1

�I. . .. . . RT

K�I

3

77775

2

64 1... K

3

75�

2

64g1...gK

3

75

����������2

w(t+K) = 0

�dw

dt=

@f

@u

�Tw +

1

tK � t0

@J

@uw(t�i ) = Pti

�(I �QiQT

i )w(t+i )�Qi i

�+ xi

dJ

ds=

Z tK

t0

@f

@s

����t

w(t) dt+@J

@s

Pti xi

• 1costunit=primalsolutionforasinglesegment

• CostofAdjointNILSS:~(m+3)Kunits• PrimalonKsegmentscostsKunits• mtangentsolutionscost~mKunits• KQR-decompositions:• ParallelTSQR:2NDOFm2/P+2m3/3flops

• MinimizationProblem• Usuallyarelativelysmallcost

• AdjointonKsegmentscosts~2Kunits• FileI/Ocoulddrivecomputetime

• misatleastthenumberofpositiveLyapunovexponents.• Reτ=180channelflow,m≈1,500• T106Cturbineblade,m≈400

13

NILSScomputationalcost

Channelflow:Vorticitymagnitudeisosurfacescoloredbystreamwisevelocity

Turbineblade:Vorticitymagnitudeisocontourscoloredbymachnumber

• Smallestchannelthatcansustainturbulentflow(JimenezandMoin,1991).• Verygoodagreementwithturbulentchannel

statisticsbelowy+=40

• Currentstudyreplicatesacaseintheoriginalpaper• Re=3000,Reτ=140• Channelsize=πh×2h×0.34πh

• FlowSolver:• DiscontinuousGalerkinSpectralElementMethod

(DGSEM)framework• Space-timeDGdiscretization• EntropystablefluxofIsmailandRoe

• Mesh:32x128x16DegreesofFreedom• Roughly150positiveLyapunovexponents

14

MinimumTurbulentFlowUnit

Q-Criterionisosurfacescoloredbyx-momentum

NI-LSSSensitivity

15

• NI-LSSrunwith160modes• Objectivefunctionisvolume-integratedkineticenergy• SensitivitytoReτcomputed• Slowconvergenceofsensitivityduetolongtimescalespresentinflowunit

• Shadowingadjointdoesnotexhibitexponentialgrowth

• Adjointprovidesphysicalinsights• Largestadjointmagnitudesoccurbefore

“blooming”ofturbulenceindicatedbywallshearstressτ.

16

NI-LSSAdjoint

t=12.13 t=13.18

Q-Criterionisosurfacescoloredbyx-momentum

Turbulence“Blooming”

FlowUnitAdjointField

17

• Integratedkineticenergyadjointshowswhenandwhereflowismostsusceptibletoflowinstabilities

Q-Criterionisosurfacescoloredbyx-momentum

AdjointX-momentumisocontoursfor±2.0

WallscoloredbyshearforceMagnitude

AdjointFieldandZ-vorticity

18

• Time-averaged,volume-integratedkineticenergyissensitivetoperturbationsinthesheetsofZ-vorticitybeingtransportedawayfromthewalls.

Contourlines:adjointX-momentumColormap:Z-vorticity

Contourlines:adjointX-momentumColormap:X-mometum

Coloredisocontours:adjointX-momentumfor±2Whiteisosurfaces:Z-vorticitymagnitude=1.9

Flow

Flow

Flow

Snapshotsfromt=12.13

• AdjointX-momentumfieldforflowunitpriortoturbulence“blooming”:

• StreamwisevelocitymagnitudecontoursforaflowperturbationoptimizedtoincreasethekineticenergyofRe=610flowoveraflatplate(Cherubinietal.2010,JFM):

19

OptimalPerturbationforTransition

Contourlines:X-momentumadjointColormap:Z-vorticity

Solidlines:domainlength=400unitsDottedlines:domainlength=800units

X-momentumperturbationssuggestedbytheadjointaresimilartotheoptimalvelocityperturbationscomputedbyCherubinietal.

ConclusionsandFutureWork

20

• Conventionalsensitivityanalysisfailsforchaoticdynamicalsystemssuchasscale-resolvingturbulentflowsimulations

• Shadowing-basedsensitivityanalysisisapromisingapproachforchaoticsystems• Non-IntrusiveLSScancomputeusefulsensitivities- CostscaleswiththenumberofpositiveLyapunovexponents

• Shadowingadjointprovidesvaluablephysicalinsightsintoturbulentflows

• NextSteps:- Shadowingforothercanonicalturbulentflowsincludingaxis-symmetricjets- ExploreapproachestoreducecostofNILSS- Studyothershadowingalgorithmssuchasmultipleshootingshadowing

January 6, 2016 Dr. Corentin Carton de Wiart corentin.carton@gmail.com Dear Dr. Carton de Wiart, As you know, administration of the NASA Postdoctoral Program is transitioning from Oak Ridge Associated Universities to the Universities Space Research Association. On January 31, 2016, your appointment under Oak Ridge Associated Universities will end. This letter is to inform you of your new appointment with USRA, which commences February 1, 2016. As a current NASA Postdoctoral Fellow, we want to welcome you to Universities Space Research Association and to introduce ourselves to you, and to provide you with guidance during this transition. Universities Space Research Association (USRA), founded in 1969, is an independent, nonprofit research corporation whose mission is to work with NASA, other federal agencies, and the University community to advance earth science, space science, and technology. USRA’s research expertise includes astrophysics, planetary science, earth science, engineering, biomedicine, and quantum computing. We are an association of 105 US and international universities, and we manage research institutes at NASA centers, including the Goddard Space Flight Center, the Ames Research Center, the Glenn Research Center, and the Johnson Space Center and we manage astrophysics activities at the Arecibo Observatory. We employ over 400 scientists and engineers at the postdoctoral and more senior level, and manage internship programs for NASA and the Air Force Research Lab. We are headquartered in Columbia, Maryland; our website is http://www.usra.edu. I’d like to provide you an overview of the transition process. The transition from ORAU to USRA began on October 15, 2015. As of February 1, 2016, USRA will be responsible for supporting all appointed Fellows (including stipend payments, healthcare, visas and scientific travel support) as well as managing the reviews of all applicants, announcing the results of award decisions, and providing reports on the status of the Program to NASA. I’ve provided some specific information on important, near-term issues regarding center access, stipends, taxes, travel, visas, healthcare, and the USRA NPP web portal. Center Access: If you work at a NASA Center, your NASA Center Representative will update your entry in IDMAX to provide physical access to your Center, and will update your TTCP if necessary to ensure your access to the NASA internet network. If you are a foreign national, the Center Representative will also upload a copy of your DS2019 to IDMAX. Stipends: By agreement, ORAU will pay your stipend through January 31, 2016. Your first stipend payment from USRA will be made on the last business day of February 2016, i.e. Monday, February 29, 2016. If you only work for part of the month, your stipend will be pro-rated. If you participate in the USRA Healthcare plan, your USRA stipend payment will include the NASA contribution to your healthcare premium; the NASA contribution to your healthcare premium, and your portion of the healthcare premium, will be automatically deducted from your monthly stipend by USRA.

UNIVERSITIES SPACE RESEARCH ASSOCIATION — NASA Postdoctoral Program —

7178 Columbia Gateway Drive • Columbia, MD 21046 • (410) 730-2656

ScottMurmanNASAAmes

NicholasBurgessLasloDiosadyAnirbanGarai

Science&TechnologyCorp.

DirkEkelschotCorentinCartonDeWiart

NASA/USRANPP

Acknowledgments

ThisresearchwassponsoredbyNASA'sTransformationalToolsandTechnologies(TTT)ProjectoftheTransformativeAeronauticsConceptsProgramundertheAeronauticsResearchMissionDirectorate.

LyapunovAnalysis

22

• PhaseSpaceforsystem:

Lyapunov exponents: Avg. rate of deformation

Covariant vectors: direction of deformation

Uniform

Perturbations

Trajectory for @u/@t = f(u; s)

Exponentsignsindicatelong-timedynamics:

PositiveLyapunovexponentsresponsibleforthebutterflyeffect

Steady Periodic Chaotic

AllNegative Zero,Negative Positive,Zero,Negative

du

dt= f(u; s)

Considerasystemgovernedby

Foranyδ>0thereexistsε>0,suchthatforevery“ε-pseudo-solution”usatisfyingǁdu/dt−f(u)ǁ<ε,thereexistsatruesolutionusatisfyingdu/dτ−f(u)=0underatimetransformationτ(t),suchthatǁu(τ)−u(t)ǁ<δ,|1−dτ/dt|<δ

TheShadowingLemma

23

du

dt= f(u; s)

ShadowTrajectory ReferenceTrajectory

TimeTransformation

24

• Timetransformationisrequiredtokeepthetrajectoriescloseinphasespaceforalltime

d⌧

dt= 1

d⌧

dt6= 1

WithoutTransformation WithTransformation

AdditionalNILSSDefinitions

25

• Tangent:

• Adjoint:

• Definitions:

xi =1

tK � t0(J � J(u(ti)))

f(u(ti); s)

kf(u(ti); s)k22Pti = I � f(u(ti); s)

f(u(ti); s)T

kf(u(ti); s)k22

�dw

dt=

@f

@u

�Tw +

1

tK � t0

@J

@uw(t�i ) = Pti

�(I �QiQT

i )w(t+i )�Qi i

�+ xi

dJ

ds=

Z tK

t0

@f

@s

����t

w(t) dt+@J

@s

dJ

ds=

1

tK � t0

KX

i=1

�gi

T↵i + hi

�+

@J

@s

hi =1

tK � t0

Z ti

ti�1

@J

@u

����t

v(t) dt+ x

Ti v(t

�i )gi =

1

tK � t0

Z ti

ti�1

@J

@u

����t

V (t) dt+ x

Ti V

�i

Sensitivity:

Sensitivity:

dvjdt

=@f

@uvj , vj(ti) = V j

i

dv

dt=

@f

@uv +

@f

@s+ ⌘f, v(ti) = Vi

26

Developmentofacompressibleentropy-stablehigh-orderspace-timediscontinuousGalerkinspectralelementmethod(DGSEM)framework

• DGSEMtoefficientlyreachspectrallimitbothinspaceandtime(N≥8)• Lessdiscretizationerrorsandefficiency• Bettermatchforcurrent/futurehardware• Lowdependanceonmeshquality• h-padaptation

• Entropy-stableformulation• Entropyvariables• Space-timeDGdiscretization• EntropystablefluxofIsmailandRoe• “Exact”quadratureusinglocalde-aliasing

SOA1/4

1

4

8

16

1/8 1/2 2 8 16

Perf

orm

ance

(flop

s/cy

cle)

Operational Intensity (flops/byte)

Nehalem

Bandw

idth

Sandybridge

HaswellTarget

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 5 10 15 20

Dissipation

Rate

Time

Conservative

Entropy

FlowUnitAdjointField

27

• Adjointforintegratedkineticenergyshowswhenandwhereflowismostsusceptibletoflowinstabilities

Contourlines:X-momentumadjointColormap:X-mometum