ADJOINT METHODS IN A HIGHER-ORDER SPACE-TIME DISCONTINUOUS-GALERKIN SOLVER FOR TURBULENT FLOWS

Laslo DiosadyScience & Technology Corp.

Patrick BloniganUSRA

Scott MurmanNASA ARC

Anirban GaraiScience & Technology Corp.

https://ntrs.nasa.gov/search.jsp?R=20170002294 2020-02-21T14:56:34+00:00Z

L. Diosady 03-01-17

Motivation

• Perform detailed time-accurate scale-resolving simulations of practical, complex, compressible flows

• High-Reynolds-number separated flows involving large-scale unsteadiness, where RANS models are unreliable

2

L. Diosady 03-01-17

Approach

• Turbulent flows involve a large range of spatial and temporal scales which need to be resolved

• Efficient algorithms and implementation necessary for wall-resolved high Reynolds number flows

• Numerical method must be capable of handling complex geometry

• Numerical method must be “robust”

3

Developed higher-order space-time discontinuous-Galerkin spectral element framework

L. Diosady 03-01-17

• Gradient computation needed for error-estimation, adaptation, design, sensitivity analysis, etc.

• Tangent and adjoint methods have been successfully applied to a variety of steady and unsteady flows

• High-fidelity simulations we are targeting are chaotic • Can traditional tangent/adjoint methods work? • Develop efficient implementation of tangent and adjoint method in space-time discontinuous solver

• Assess the applicability of traditional adjoint and tangent methods for chaotic flows

4

Approach

L. Diosady 03-01-17

Outline

• Space-time DG formulation

• Discrete primal formulation

• Discrete tangent and adjoint formulations

• NACA0012

• Flow sensitivity with increasing Reynolds number

• Properties of adjoint for chaotic flows

• T106a LPT

• Adjoint solutions corresponding to a practical simulation

• Summary/Outlook

5

L. Diosady 03-01-17

Space-Time Discontinuous-Galerkin (DG) Formulation

• Compressible Navier-Stokes Equations: • Space-Time Discontinuous-Galerkin Discretization:

• Entropy variables: Hughes (1986)

• Inviscid Flux: Ismail & Roe (2009) • Viscous Flux: Interior penalty method with penalty parameter

given by 2nd method of Bassi and Rebay (2007) • Integrals evaluated using numerical quadrature with 2N points • Discrete entropy stability: Barth (1995)

6

@u

@t+r · F (u,ru) = 0

Ao

v,t

+Ai

v,i

� (Kij

v,j

),i

= 0

SPD Sym SPSDv =

2

64� s

��1 + �+1��1 � ⇢E

p⇢ui

p

�⇢p

3

75

(⇢s),t +

✓qicvT

◆

,i

= vT,iKijv,j � 0

L. Diosady 03-01-17

Discrete Formulation• Discrete system solved each time-slab:

• where:

• beginning/end of time-slab

7

w(tn+) = Iswn

w(tn+1� ) = Iew

n

Rn(un, wn) +Gn(un�1, wn) = 0

Rn(un, wn) =X

⇢Z

I

Z

�✓@w

@tu+rw · F

◆+

Z

I

Z

@w[F · n+

Z

w(tn+1

� )u(tn+1� )

�

Gn(un�1, wn) =X

⇢Z

�w(tn+)u(t

n�)

�

L. Diosady 03-01-17

Discrete Tangent formulation

• Output: • where α is a parameter (i.e. angle of attack, Reynolds number etc.)

• Compute sensitivity of output to parameters:

• Tangent equation:

• Matrix form:

8

J(u;↵)

@Rn

@un (�un, wn) + @Gn

@un�1 (�un�1, wn) = �@Rn

@↵

2

64

@Rn�1

@un�1 0 0@Gn

@un�1@Rn

@un 0

0 @Gn+1

@un@Rn+1

@un+1

3

75

2

4�un�1

�un

�un+1

3

5 = �

2

64

@Rn�1

@un�1

@Rn

@un

@Rn+1

@un+1

3

75

dJ

d↵=

@J

@↵+

@J

@u

@u

@↵=

@J

@↵+

@J

@R̄

@R̄

@↵

L. Diosady 03-01-17

Discrete Adjoint Formulation• Lagrangian:

• Stationarity of Lagrangian:

• Adjoint equation:

• Matrix Form:

9

Adjoint Sensitivity

L(u, ;↵) = J(u;↵) + T R̄(u;↵)

�L =

@J

@u

T�����↵

+ T @R̄

@u

����↵

!�u+

@J

@↵

T�����u

+ T @R̄

@↵

����u

!�↵ = 0

@Rn

@un (wn, n) + @Gn

@un�1 (wn�1, n) = �@Jn

@un ( n)

2

664

@Rn�1

@un�1

T@Gn

@un�1

T0

0 @Rn

@un

T @Gn+1

@un

T

0 0 @Rn+1

@un+1

T

3

775

2

4 n�1

n

n+1

3

5 = �

2

64

@Jn�1

@un�1

@Jn

@un

@Jn+1

@un+1

3

75

L. Diosady 03-01-17

Primal Solver: Implementation Details• Efficient implementation of higher order DG

• Tensor-product basis

• Take advantage of hardware (SIMD/optimized kernels)

• Jacobian-free Approximate Newton-Krylov solver

• Tensor-product based ADI-preconditioner

• Primal Residual Evaluation:

1. Evaluate state/gradient at quadrature points

2. Evaluate flux at quadrature points

3. Weight fluxes with gradient of test functions

10

Optimizedsum-factorization

Vectorized Kernels

L. Diosady 03-01-17

Tangent/Adjoint: Implementation Details• Reuse optimization from primal for tangent and adjoint

• Tangent Residual Evaluation:

1. Evaluate state/gradient and update/gradient at quadrature points

2. Evaluate linearized flux at quadrature points

3. Weight fluxes with gradient of test functions

• Adjoint Residual Evaluation:

4. Evaluate state/gradient and adjoint/gradient at quadrature points

5. Evaluate adjoint flux at quadrature points

6. Weight fluxes with gradient of test functions

11

Optimizedsum-factorization

Vectorized Kernels

L. Diosady 03-01-17 12

• At high angle of attack, flow over NACA0012 airfoil exhibits vortex shedding which become chaotic with increasing Reynolds number. (Pulliam 1993)

• Examine primal and adjoint solutions

NACA0012, α = 10

L. Diosady 03-01-17

NACA0012, Re = 800, α = 10

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0 50 100-Time

10 -2

10 -1

10 0

Adjo

int M

agni

tude

0 50 100Time

0

0.1

0.2

0.3

0.4

0.5

Force

Directional Force Output Adjoint Magnitude

• Unsteady shedding gives periodic output signal • Adjoint solution also periodic (after initial transient)

L. Diosady 03-01-17

NACA0012, Re = 1600, α = 10

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Directional Force Output Adjoint Magnitude

• With increasing Reynolds number force has multiple frequencies • Adjoint solution still essentially appears periodic

0 50 100-Time

10 -2

10 -1

10 0

10 1

Adjo

int M

agni

tude

0 50 100Time

0

0.1

0.2

0.3

0.4

0.5

Force

L. Diosady 03-01-17

NACA0012, Re =2400, α = 10

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Directional Force Output Adjoint Magnitude

• With increasing Reynolds number simulated flow become chaotic • Adjoint solution begins to grow unboundedly

0 50 100Time

0

0.1

0.2

0.3

0.4

0.5

Force

0 50 100-Time

10 -2

10 0

10 2

10 4

Adjo

int M

agni

tude

L. Diosady 03-01-17

NACA0012, Re =800-2400, α = 10

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Adjoint Magnitude• With increasing Reynolds number simulated flow become chaotic • Adjoint solution begins to grow unboundedly

0 50 100-Time

10 -2

10 0

10 2

10 4

Adjo

int M

agni

tude

Re = 800Re = 1600Re = 2400

L. Diosady 03-01-17

NACA0012, Re =800-2400, α = 10

17

Adjoint Magnitude

• Solution is in fact chaotic at Re = 1600, but growth rate is much slower than at Re = 2400

• Windowing approaches may be successful at Re = 800, 1600

0 50 100 150 200-Time

10 -5

10 0

10 5

10 10

Adjo

int M

agni

tude

Re = 800Re = 1600Re = 2400

L. Diosady 03-01-17

T106c Low Pressure Turbine

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• Re = 80,000, Minflow = 0.243, alpha = 32.7, Mexit = 0.65

• Periodic BCs in span-wise and pitch-wise directions • No free-stream turbulence • Spanwise domain is 20% of chord

L. Diosady 03-01-17

Primal Solution

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• Nearly steady flow upstream and over first 2/3 of blade

• Separation leading to transition/vortex shedding on suction-side of blade

• Fully turbulent wake

L. Diosady 03-01-17

0 0.5 1 1.5Time

1

1.1

1.2

1.3

1.4

1.5Ax

ial F

orce

, = 32.7, = 32.701

Sensitivity to Inflow Boundary Condition

20

• Modify inlet flow angle from α = 32.7 to α = 32.701

Domain flow-through time

Convective disturbancehits leading edgeAcoustic disturbance

hits leading edge

L. Diosady 03-01-17

0 0.5 1 1.5Time

-0.05

0

0.05

Axia

l For

ce D

iffer

ence

0 0.5 1 1.5Time

10 -7

10 -5

10 -3

10 -1

10 1

|Axi

al F

orce

Diff

eren

ce|

Sensitivity to Inflow Boundary Condition

21

Domain flow-through time

Convective disturbancehits leading edge

Acoustic disturbance hits leading edge

L. Diosady 03-01-17

Adjoint of mean Axial Force

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• Output is integrated axial force

• Also define output without temporal normalization

J̄ =1

T

ZT

0Fx

(u(⌧))d⌧

J(t) =

Zt

0Fx

(u(⌧))d⌧

Range: [-1e6, 1e6]

L. Diosady 03-01-17

Sensitivity computed using adjoint

23

0 0.5 1 1.5Time

10 -8

10 -4

10 0

10 4

10 8

|J(,

+"

,)-J

(,)|

Finite DifferenceAdjoint

�J(t) = J(t;↵+�↵)� J(t;↵) =

Zt

0Fx

(u(⌧ ;↵+�↵))� Fx

(u(⌧ ;↵))d⌧

⇡Z

t

0 (⌧ ; t,↵)TR(u(⌧);↵+�↵)

L. Diosady 03-01-17

Sensitivity computed using adjoint

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• Approximation holds since flow upstream of blade is essentially time-independent

• Adjoint correctly captures sensitivity in part of flow upstream of separation

�Fx

(⌧) = Fx

(u(⌧ ;↵+�↵))� Fx

(u(⌧ ;↵))

?? ⇡ (t� ⌧ ; t,↵)TR(u;↵+�↵)

0 0.5 1 1.5Time

10 -11

10 -9

10 -7

10 -5

10 -3

10 -1

10 1

|Axi

al F

orce

Diff

eren

ce|

Finite DifferenceAdjoint T=0.1Adjoint T=0.2Adjoint T=0.3Adjoint T=0.4Adjoint T=0.7

L. Diosady 03-01-17

Sensitivity computed using adjoint

25

• Sensitivity computed using adjoint only valid for very short time windows

• Adjoint computed using long time window blows up

• Sensitivity computed using short time window, not representative long time behaviour

0 0.5 1 1.5Time

10 -810 -610 -410 -210 010 210 410 610 8

|" J

- *

T R|

L. Diosady 03-01-17

Graphical demonstration of concepts

26

• Adjoint correctly captures sensitivity in part of flow upstream of separation

• Sensitivity computed using adjoint only valid for very short time windows

• Sensitivity computed using short time window, not representative long time behaviour

27

Adjoint-based Error Estimation

Element-based error-indicator

S. Murman 20-oct-15

• Estimate error using dual-weighted residual method (Becker & Rannacher 1995)

• Localize error

• Flag elements with largest error for refinement

Primal Solution

✏ = J(u)� J(uH) ⇡ RH(uH , h)

✏ ⌘ RH(uH , h|)

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Adjoint-based error indicator

S. Murman 20-oct-15

• Unbounded adjoint not useful for error estimation

• Estimate is orders of magnitude larger than actual signal

• Error localization simply flags regions where adjoint is large

L. Diosady 03-01-17

Adjoint growth with mesh resolution

29

• Refined mesh has essentially double mesh resolution near separation region

• Increase mesh resolution results in faster growth of adjoint (i.e. larger Lyapunov exponent)

• Adaptation mechanism is not convergent

0 0.5 1 1.5Time

10 -110 110 310 510 710 9

|*|

CoarseFine

29

L. Diosady 03-01-17

Summary

• Presented space-time adjoint solver for turbulent compressible flows

• Confirmed failure of traditional sensitivity methods for chaotic flows

• Assessed rate of exponential growth of adjoint for practical 3D turbulent simulation

• Demonstrated failure of short-window sensitivity approximations.

30

L. Diosady 03-01-17

Outlook/Future Work:• Lyapunov exponents, least-square shadowing and beyond…

31

Questions??