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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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 2 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 3 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 4 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 5 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 6 / 21
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.
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 7 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 8 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 9 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 10 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 11 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 12 / 21
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)
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 13 / 21
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 .
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 14 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 15 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 16 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 17 / 21
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.
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 18 / 21
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)
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 19 / 21
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
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 20 / 21
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.
Fidkowski and Asher (UM) SIAM CSE 2011 March 2, 2011 21 / 21