adjoint-based error estimation and mesh adaptation for ... · 20.01.2015 · context and motivation...

Adjoint-Based Error Estimation and MeshAdaptation for Problems with Output

Constraints

Marco CezeNASA - ORAU

Ben Rothacker and Krzysztof FidkowskiUniversity of Michigan

AMS SeminarJanuary 20, 2015

AMS Seminar, Jan. 20, 2015 Constrained Adjoint-Based Error Estimation and Mesh Adaptation 1/33

Context and MotivationOutput-based error estimation and mesh adaptation

Demonstrated applicability to a wide range of aerospaceengineering problems.Present in production-level codes, e.g.: Cart3D, FUN3D.Many challenges have been addressed: turbulence modeling,unsteady flows, mesh optimization, hp-adaptation, and complexgeometries.

Constrained problems are ubiquitousOutput prediction under trimmed conditions, e.g.: DPW.Current multi-output adaptive strategies consider staticcombinations of outputs.Various outputs have different domains of dependence.Most interesting optimization problems are constrained – how toincorporate constraint errors in the objective function?


Outline of this work

Discontinuous Galerkin spatial discretization.CPTC nonlinear solver with relaxed line-search.Exact Jacobian with element-line-Jacobi preconditioner andGMRES linear solver.Roe solver for inviscid flux and BR2 for viscous discretization.MPI parallelization node-edge weighted mesh partitioning.ICCFD7 version of the SA turbulence model.ALE mesh deformation for trimming.

In this talkDerive an error correction procedure for output-constrainedproblems.Demonstrate adaptive benefits of including constraint-relatederror.


Problem StatementSolve the residual equation:

R(U,α) = 0,

where:R ∈ RN : vector of N residuals that must be driven to zeroU ∈ RN : state vector that encodes the flow stateα ∈ RNα : trimming parameter vector (trimming "knobs")Jadapt(U,α): scalar output on which we want to adaptJtrim(U,α) ∈ RNα : vector of outputs used to define trimmingconstraints

We want to predict Jadapt(U,α) to εadapt accuracy, subject to flowequations and the following Nα constraints:

Jtrim(U,α)− Jtrim= 0

Jtrim ∈ RNα is a set of user-specified trim outputs/constraints.AMS Seminar, Jan. 20, 2015 Constrained Adjoint-Based Error Estimation and Mesh Adaptation 4/33

Problem Statement: Example

Consider drag prediction under fixed lift:Nα = 1α: single trimming parameter (angle of attack, for example)

Lift = Lift =

Drag =

AoA =


Key ingredient: an adjointSensitivity of the output w.r.t. residuals in the physics.Input perturbations are converted into residual perturbations.Advantage: residuals are generally cheap to compute.

Flow simula+on

Post-‐processing e.g.: drag, li7.

J(U)

Input parameters ↵

�R�J

26664

R1(U)R2(U)

...Rn(U)

37775 =

26664

00...0

37775

Adjoint variables

U

“Mesh” +



Flow simula+on


J(U)


�R�J

26664

R1(U)R2(U)

...Rn(U)

37775 =

26664

00...0

37775

Adjoint variables

U

�↵

“Mesh” +


Sensitivity analysis


Flow simula+on


J(U)


�R�J

26664

R1(U)R2(U)

...Rn(U)

37775 =

26664

00...0

37775

Adjoint variables

U

“Mesh” +

Residual evalua+on

in finer space


Output errorestimation

A derivation of the adjoint equation

Consider an output J ((α),u) where u satisfies R((α),u) = 0 fora fixed input parameter set α.We form a Lagrangian L((α),u,ψ) to incorporate the constraint:

L((α),u,ψ) = J (u) +ψTR((α),u).

We take the variation of the Lagrangian assuming R((α),u) = 0:

δL((α),u,ψ) =

(∂J∂u

T+ψT ∂R

∂u

)

︸︷︷︸= 0

Adjoint equation

δu +

(∂J∂α

T+ψT ∂R

∂α

)δα.

︸︷︷︸= 0


Making δL = 0 selects only realizable δJ ’s.

Linearequation!

Single output error estimation and adaptationuH,p will generally not satisfy the original PDE: R(uH,p,w) 6= 0Instead, it satisfies the weak form:

R(u,w) + δR(w) = 0 where δR(w) = −R(uH,p,w).

ψ ∈ V relates the residual perturbation to an output perturbation:

δJ = J(uH,p)− J(u) ≈ −R(uH,p,ψ)

We approximate ψ in a higher order space VH,p+1 ⊃ VH,p andestimate the error as:

δJ ≈ −∑

κH∈T H

RκH (uH,p,ψH,p+1 − ψH,p),

We assign an adaptive indicator to κH based on its contribution tothe error estimate:

ηκH =∣∣∣RκH (uH,p,ψH,p+1 − ψH,p)

∣∣∣.


Coupled flow-trim solve

In an output-constrained run we solve:{

R(U,α) = 0 ← N flow equationsRtrim(U,α) = 0 ← Nα trim conditions

where Rtrim(U,α) = Jtrim(U,α)− Jtrim.We seek variations of Jadapt that satisfy the constraints above, so:

δL = δJadapt +ψT δR︸︷︷︸(a)

+φT δRtrim︸︷︷︸

(b)

= 0.

“a” is the output error estimate for fixed α, “b” is the influence ofthe trim error in the output error due to inexact constraintsatisfaction.


Coupled adjoint solutionWe have an adjoint contribution from both the flow and trimmingresiduals: [

∂R∂U

T ∂Rtrim

∂UT

∂R∂α

T ∂Rtrim

∂α

T

] [ψφ

]+

[∂Jadapt

∂UT

∂Jadapt

∂α

T

]=

[00

]

ψ = −(∂R∂U

)−T[∂Jadapt

∂U

T

+∂Rtrim

∂U

T

φ

]

= −(∂R∂U

)−T ∂Jadapt

∂U

T

︸︷︷︸ψadapt

−(∂R∂U

)−T ∂Jtrim

∂U

T

︸︷︷︸Ψtrim

φ

= ψadapt +Ψtrimφ

Finally, the sensitivity of Jadapt to trim residuals is:

φ = −[

dJtrim

dα

]−T dJadapt

dα

T

= −

[dJadapt

dα

(dJtrim

dα

)−1]T


Constrained error estimate and adapt indicator

We get the output error estimate for the constrained case via a changeto the total adjoint:

δJ ≈ −[ψ

adapth +Ψtrim

h φ]T

Rh(UHh ,αH) = δJadapt + φT δJtrim

and the modified adaptive indicator is given by:

ηκ = ηadaptκ︸︷︷︸

Unconstrained

+ |φ|Tηtrimκ

︸︷︷︸Constrained

"Unconstrained" - adaptation uses ηk = ηadaptk

"Constrained" - adaptation uses ηk = ηadaptk + |φ|Tηtrim

k


Trim-constraint solver

At each adaptation iteration, we trim parameters to satisfy constraints:

Trimming implementationCurrent meshCurrent state, UCurrent αTrim tolerance, εtrim

Solve R(U,α) = 0Yes‖Jtrim(U,α)− J

trim‖ < εtrim Done

No

Compute output sensitivity matrix: dJtrim

dαUpdate parameters:

α← α+ dJtrim

dα

−1∆Jtrim

∆Jtrim ≡ Jtrim− Jtrim

1. Solve for Nα adjoints:

3. Form sensitivity matrix:

∂R∂U

Tψtrimi +

∂Jtrimi

∂U

T

= 0

∂R∂α (cheap finite differences)

dJtrimi

dα =(ψtrimi

)T∂R∂α +

∂Jtrimi

∂α

2. Compute Nα residual linearizations:

One can also solve flow and trimming equations simultaneously


Mesh adaptation mechanics

Select a fraction f frac of elements with largest error indicators foradaptation.

Isotropic refinement performed in reference space.

New boundary nodes are projected to the geometry.

One level of refinement is kept between adjacent cells.


Arbitrary Lagrangian-Eulerian mapping

We consider geometric parameters.Solve transformed PDE on an undeformed reference domain.Near-field rigid body motion blended into static farfield mesh.Quintic polynomial (in radial coordinate) blending functions.


Supersonic biplaneTrim on total biplane lift via changes to global angle of attack.Output of interest is drag on lower airfoil.Inviscid flow at M∞ = 1.5 with p = 2 approximation order.

Initial mesh Mach contours


Supersonic biplane

c`,total = 0.25 trimming constraint.cd ,lower adapt output.Different converged outputs.

0 2 4 6 8 101.61

1.615

1.62

1.625

1.63

1.635

Adaptive Iteration

An

gle

of

Att

ack (

o)

Unconstrained

Constrained

Angle of Attack Convergence

0 2 4 6 8 100.0426

0.0426

0.0426

0.0426

0.0427

0.0427

0.0427

0.0427

0.0427

0.0428

Adaptive Iteration

Dra

g C

oe

ffic

ien

t o

n L

ow

er

Airfo

il

Unconstrained: J

Unconstrained: J + δ Jadapt

Constrained: J

Constrained: J + δ Jadapt

Constrained: J + δ Jadapt

+ δ Jtrim

Drag Convergence


Supersonic biplaneTrim and adapt outputs have different domains of dependence.Unconstrained adaptation does not target the upper airfoil.

Unconstrained adaptation Constrained adaptation


NACA 0012

M∞ = 0.5, Re = 5000, p = 1.Lift-constrained drag prediction.Trimming parameter is the angle of attack.

StrategyCompute "exact" lift at α = 1◦

⇒ cexact` = 0.018250.

Set ctarget` = 0.018250.

Use cexact` and cexact

d to evaluateconstrained vs. unconstrainedadaptation.


NACA 0012

Trimming history:

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

Adaptation Iteration

An

gle

of

Att

ack (

o)

Unconstrained

Constrained

Ref. value

6 8 10 120.9

0.95

1

1.05

1.1


0 2 4 6 8 10 12 1410

−4

10−3

10−2

10−1

100

101


Angle

of A

ttack E

rror

(o)

Unconstrained

Constrained

Angle of Attack Error

Trimming parameter converges faster for constrained adaptation.


NACA 0012

Output history:

0 2 4 6 8 10 12 140.0545

0.055

0.0555

0.056

0.0565

0.057

0.0575

0.058

0.0585

Dra

g C

oeffic

ient


6 8 10 12 14

0.0553

0.0553

0.0554

0.0554

0.0554

0.0554

Drag Convergence

0 2 4 6 8 10 12 1410

−8

10−7

10−6

10−5

10−4

10−3

10−2

Adaptation IterationD

rag

Co

eff

icie

nt

Err

or

Unconstrained: J

Unconstrained: J+δ Jadapt

Constrained: J

Constrained: J+δ Jadapt


+δ Jtrim

Drag Error

Unconstrained adaptation lags behind constrained adaptation.


NACA 0012Difference in the final lift adjoints between constrained and unconstrained:

Density component of lift adjoint


High-lift configurationMDA 30P-30N main airfoil with NACA 0012 elevator.M∞ = 0.2, ReCw = 9x106 with p = 1.Freestream at 10◦.ctarget` = 3.0, ctarget

m = 0.0

Case ParametersTrim lift and momentconstraints via changes toangle of attack.Output of interest is total drag.Wing chord, Cw = 0.5588Tail chord, Ct = 0.3 ∗ Cw

Separation = 4.0 ∗ Cw


High-lift configurationFlow/trim solve for first adaptive iteration:

0 100 200 300 400 500 600 700 80010

−10

10−8

10−6

10−4

10−2

100

102

Nonlinear iteration

Resid

uals

Flow residual

Lift−constraint residual

Moment−constraint residual

Note: trimming succeeds despite flow solution stall.


High-lift configuration

Output and trimming history:

1 2 3 4 5 6 7 8

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

Tota

l D

rag C

oeffic

ient

Adaptation iteration

Unconstrained: J

Unconstrained: J+δ Jadapt

Constrained: J


+δ Jtrim

2 4 6 80.039

0.0395

0.04

0.0405

0.041

0.0415

Drag Convergence

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Angle

of attack (

o)

Adaptation iteration

Unconstrained: αwing

Unconstrained: αtail

Constrained: αwing

Constrained: αtail



High-Lift configuration - why the similarity?Adaptation targets same area with and without constraint:

Unconstrained Mesh


High-lift configuration - why the similarity?Adaptation targets same area with and without constraint:

Constrained Mesh


DPW5 caseNo-tail CRM geometry at M∞ = 0.85, Re = 5× 106, and C target

L = 0.5.

Approximation order is p = 1.

The output of interest is the total drag.

Initial cubic (q = 3) mesh generated by agglomerating 3 linear cells ineach direction.

Trimming diagram. 6th adapted mesh (155661 elements).AMS Seminar, Jan. 20, 2015 Constrained Adjoint-Based Error Estimation and Mesh Adaptation 27/33

DPW5 case

Adaptation proceeds with under-converged constraint residual due toexcessively coarse initial mesh.


DPW5 case

Initial Mach contours Final Mach contoursAMS Seminar, Jan. 20, 2015 Constrained Adjoint-Based Error Estimation and Mesh Adaptation 29/33

DPW5 case

Initial Mach contours Final Mach contoursAMS Seminar, Jan. 20, 2015 Constrained Adjoint-Based Error Estimation and Mesh Adaptation 30/33

DPW5 case

Including the constraint error smoothes the error correction.

Note: we are not fully solving the fine-space adjoints.


Concluding remarks

We extended the adjoint-based error estimation method to problemswith output constraints.

We demonstrated this extension in several problems with varyingcomplexity.

In cases where the domains of dependence largely overlap, the benefitsof the constrained adaptation is marginal.

In cases with disjoint outputs, the constrained adaptation method offersa clear advantage – most evident in the convergence of the trimmingparameter.

Future developments: simultaneous flow-trim solve, extension to errorsampling adaptive strategies e.g.: optimization-based hp (Ceze andFidkowski) and MOESS (Yano and Darmofal).


Acknowledgments

University of Michigan

NASA Ames computational resources

AFOSR FA9550-10-C-0040

Thank you!


adjoint-based error estimation and mesh adaptation for ... · 20.01.2015 · context and motivation...

Documents