Adjoint-Based Numerical Error Estimation for the Unsteady ...Fidkowski and Asher (UM) SIAM CSE 2011...

Adjoint-Based Numerical Error Estimation for theUnsteady Compressible Navier-Stokes Equations

Krzysztof J. Fidkowski and Isaac M. AsherUniversity of Michigan

SIAM Conference on Computational Science and EngineeringReno, Nevada

March 2, 2011

Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 1 / 21

Outline

1 Introduction

2 Discretization

3 Output Error Estimation

4 Space-Time Mesh Adaptation

5 Conclusions


Introduction

Output-based adaptive methods

Provide accurate answers with efficient use of resources

Return error estimates that improve calculation robustness

Have been and are being studied in-depth for steady problems

Unsteady simulations

Application of output-based methods has been more limited

Cost of adjoint solve increases (e.g. storage for nonlinear problems)

Often contain a wide-range of spatial and temporal scales whoseresolution is not known in advance

Risk of unquantified discretization errors in outputs is high

As is the potential benefit of output-based adaptive methods


Goal and Approach of this Work

GoalTo present the benefits of adjoint-based space and time error estimation andmesh adaptation for the unsteady compressible Navier-Stokes equations.

Approach

High-order discontinuous Galerkin finite elements in space and time

Tensor-product space-time mesh structure

Unsteady discrete adjoint solution

Approximation of the fine-space adjoint

Adjoint-weighted residual error estimation

Combined spatial and temporal adaptation with a space-time anisotropymeasure


Discretization: Solution Approximation

Model problem∂u∂t

+ r(u) = 0

u(x, t) ∈ Rs = state vector

r(u) = spatial operator

space

time

ϕnH

φH,j

Finite element approximation

uH(x, t) =∑

n

∑j

unH,j φH,j(x)ϕn

H(t)

n = time degree of freedom index

j = space degree of freedom index

φH,j (x) = j th spatial basis function

ϕnH(t) = nth temporal basis function

Basis functions are discontinuous inspace and time (DGFEM)

UnH =

un

H,j

∀j = vector of

unknowns at time node n

Order is p in space, r in time


Discretization: Space-Time Mesh

Time is discretized in slabs (all elements advance the same ∆t)

Each space-time element is prismatic (tensor product: T He ⊗ IH

k )

The spatial mesh is assumed to be invariant in time

−+−+

time slab IH,k

element TH,e

x

y

t

tktk−1Ωtime


Discretization: Primal and Adjoint Solution

Primal weighted-residual statementr + 1 DGFEM unsteady residual vectors per time slab

RmH ≡ am,n

r MHUnH − ϕm

H (t0)MHUprevH +

∫ t1

t0ϕm

H (t)RH(UH(t)) dt = 0

where MH is the spatial mass matrix and for a Lagrange temporal basis,

am,n1 =

[1/2 1/2−1/2 1/2

]am,n

2 =

1/2 2/3 −1/6−2/3 0 2/3

1/6 −2/3 1/2

Discrete adjoint equationFor an output JH(uH(x , t)), the adjoint residual is

Rψ,nH (Ψm

H ) =

(∂R

mH

∂UnH

)T

ΨmH +

(∂JH

∂UnH

)T

= 0

Primal and adjoint are solved via Newton with an approximate factorization.


Output Error Estimation

The adjoint-weighted residual

uH ∈ VH will generally not be exact

The output JH(uH) will be affected by discretization errors

We estimate the output error using an adjoint-weighted residual

δJ = output error ≈ −(Ψm

h)T R

mh (UH,n

h )

Ψmh is a fine space adjoint solution

UH,nh is the injection of the coarse solution into the fine space

Two metrics: effectivity and cost

ηeff ≡δJ

JH(uH)− JexactηCPU ≡

CPU time of adjoint solve and error estimationCPU time of forward solve


Output Error Estimation: Approximations

Investigated six methods ofapproximating the adjoint:

A p + 1, r + 1B p + 1,∆t/2C Reconstruct(p + 1), r + 1D p + 1, Reconstruct(r + 1)E Reconstruct(p + 1, r + 1)F Smooth(p + 1, r + 1)

p + 1 = spatial order increaser + 1 = temporal order increase“Smooth” = Jacobi smoothing

patch

p

p

ppp+ 1

Spatial reconstruction using patch ofadjacent elements

tktk−1 tR

reconstructedr=2 adjointsolution

IH,k IH,k+1

= superconvergent nodes

t

r=1 adjoint solution

Temporal reconstruction usingsuper-convergent Radau points


Output Error Estimation: Space/Time Decomposition

Semi-refined spacesVHh = order p in space, r + 1 in timeVhH = order p + 1 in space, r in time

Output error due to spatial resolution

δJspaceH = J(uH)− J(uhH)

≈ (ΨmhH)T Rm

hH(UH,nhH )

Output error due to temporal resolution

δJ timeH = J (uH)− J (uHh)

≈ (ΨmHh)T Rm

Hh(UH,nHh )

δJ timeh

δJ spaceH

δJ spaceh uh

uHh

uhH

spatial resolution

temporal

resolution

uH

δJ timeH

Adjoints on semi-refinedspaces are computed usingleast-squares projection

ψHh = Πspace,hH ψh

ψhH = Πtime,hH ψh


Impulsively-Started Airfoil in Viscous Flow

Coarse mesh Initial Mach contours Final time entropy contours

Compressible Navier-Stokes equations

At t = 0 the velocity is blended smoothlyto zero in a circular disk around the airfoil

Freestream conditions: M∞ = 0.2,α = 20, Re = 1000

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

3

TimeLift coeffic

ient

output

Lift coefficient history


Impulsively-Started Airfoil: Error Convergence

Convergence of |δJ| under uniform refinement in space and time.

418 1672 6688

10−3

10−2

10−1

Spatial elements

Outp

ut err

or

3

actual

p+1 estimate

Reconstruct(p+1) estimate

Smooth(p+1) estimate

Spatial convergence, p = 2

32 64 128 256 51210

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Time slabs

Outp

ut err

or

3

5actual, r=1

r+1 estimate, r=1

Reconstruct(r+1) estimate, r=1

Smooth(r+1) estimate, r=1

actual, r=2

Temporal convergence


Impulsively-Started Airfoil: Error Effectivity

0 2 4 6 8 10 12−0.5

0

0.5

1

1.5

Forw

ard

solv

e c

ost

Ideal

CPU Factor

Err

or

Effectivity

Nelem

= 418, Nslab

= 32

Nelem

= 418, Nslab

= 128

Nelem

= 6688, Nslab

= 32

Nelem

= 6688, Nslab

= 128

D: p+1,Recon(r+1)

F: Smooth(p+1,r+1)

C: Recon(p+1),r+1

B: p+1,∆ t/2

A: p+1,r+1

E: Recon(p+1,r+1)


Impulsively-Started Airfoil: Error Decomposition

ftemporal =|δJ time

H ||δJspace

H |+ |δJ timeH |

418 1672 66880

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nslab

Tem

pora

l err

or

fraction

Nelem

= 418

418 1672 66880

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nslab

Nelem

= 1672

418 1672 66880

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nslab

Nelem

= 6688

actual

A: p+1,r+1

C: Recon(p+1),r+1

D: p+1, Recon(r+1)

E: Recon(p+1,r+1)

F: Smooth(p+1,r+1)

“Actual” temporal error fractions are computed from using very fine spatialand temporal discretization spaces in the expressions for δJspace

H and δJ timeH .


Adaptive Strategy

We use the output error estimate to adapt the space-time mesh:

Solution steps

yesDone

no

Initial space−time mesh and error tolerance

Solve primal by marching forward in time

indicators in a loop over time slabs backwards in time

Error tolerance met?

Identify elements and time slabs for refinement

Adapt space-time mesh

Solve adjoint, estimate the output error, and localize to adaptive


Error LocalizationThe output error estimate can be written as a sum over space-timeelements

δJ =

Nslab,H∑k=1

Nelem,H∑e=1

εk,e, error indicator = εk,e =∣∣εk,e

∣∣.For each space-time element e, k , we define:

βspacee,k = fraction of error attributable to the spatial discretizationβtime

e,k = fraction of error attributable to the temporal discretization

based on an output error calculation using the fine adjoint projected ontothe semi-coarsened spaces.

Aggregate adaptive indicators:

spatial indicator on element e = εe =

Nslab,H∑k=1

εk,eβspacee,k

temporal indicator on time slab k = εk =

Nelem,H∑e=1

εk,eβtimee,k


Adaptation Mechanics

Adaptation preserves tensor-product mesh structure

Aggregate indicators εe and εk identify elements, slabs for refinement

Decision between additional elements versus slabs is made in agreedy fashionFigure of merit is error addressed per degree of freedom addedIt is possible to target only spatial elements or time slabs

A fixed growth factor, f growth, is prescribed for the degrees of freedom

Mechanics of adaptation are hanging-node refinement of spatialelements and bisection of time slabs

x

ty

x

ty


Impulsively-Started Airfoil: Output Convergence

Output = lift coefficient integral late in the simulation

105

106

107

108

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

degrees of freedom

Outp

ut

Actual

Output error

Approximation error

Residual

Uniform adaptation

Convergence of output using various adaptiveindicators. Error estimates shown.

104

105

106

107

108

109

10−2

10−1

100

101

Degrees of freedomL

2 T

ime h

isto

ry e

rror

Output error

Approximation error

Residual

Uniform adaptation

Convergence of the L2 time history error forvarious adaptive indicators

Output-based adaptation yields not only accurate scalar outputs, butalso accurate lift coefficient time histories.


Impulsively-Started Airfoil: Adapted Spatial Meshes

Meshes shown at iterations with similar total degrees of freedomSpatially-marginalized output error estimate εe is shown on theelements of the output-adapted mesh

Adapted on output error (5956elements)

Adapted on approximationerror (4585 elements)

Adapted on residual (7929elements)


Impulsively-Started Airfoil: Adapted Temporal Meshes

0 1 2 3 4 5 6 7 8 9 10

Output error: 141 time slabs

Approximation error: 420 time slabs

Residual: 211 time slabs

Time

0

0.5

1x 10

−3

|Temporally−marginalized output error| Output indicator yields afairly-uniform temporalrefinement

Approximation errorfocuses on the initialtime (dynamics of theIC) and the latter 1/3 ofthe time, when the shedvortices develop

Residual creates amostly-uniform temporalmesh as it tracksacoustic waves


Conclusions and Ongoing Work

ConclusionsPresented an adjoint-based output error estimation and mesh adaptationalgorithm for unsteady simulations.

Context was a space-time discontinuous Galerkin method but ideasextend to other discretizations.

Spatial and temporal reconstruction of the fine adjoint solution wasshown to yield efficient error estimation.

Space-time error decomposition measured through semi-refined spaces.

Adjoint-based adaptation was demonstrated advantageous in degrees offreedom when compared to uniform refinement and cheaper heuristics.

Ongoing Work

Dynamically-refined spatial meshes and grid motion.

Coupling with uncertainty quantification methods.


Adjoint-Based Numerical Error Estimation for the Unsteady ...Fidkowski and Asher (UM) SIAM CSE 2011...

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