the importance of mesh adaptation for higher-order...

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mmThe Importance of Mesh Adaptation forHigher-Order Discretizations of Aerodynamic Flows

Masayuki Yano, James Modisette, and David Darmofal

Acknowledgements: Dr. Steve Allmaras, Bob Haimes, ProjectX Team,The Boeing Company

Aerospace Computational Design LaboratoryMassachusetts Institute of Technology

June 29, 2011

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Motivation: Next Generation of CFD

Take advantage of ever-increasing computational powerEnable higher-fidelity simulationsProvide automated error control

Simulate complex flows beyond user’s knowledgeDesign exploration, database generation, optimization, etc

Mach numberM∞ = 0.2, Rec = 9× 106, α = 16.21

Mach numberM∞ = 0.775, Rec = 2× 107, α = −0.7

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Motivation: Grid Dependent Solutions

Two mesh families generated using best practicesWith 10 times dof used in practice a notable difference exists

0 0.2 0.4 0.6 0.8 1 1.2

x 10−4

0.0194

0.0196

0.0198

0.0200

0.0202

0.0204

0.0206

0.0208

0.0210

0.0212

h2=N

−2/3

CD

Topology 1

Topology 2

1M nodes

36M nodes

Typical grid size used∼ 4 counts

AIAA DPW3 Case II, wing only: M∞ = 0.76, Rec = 5× 106 (Mavriplis 2007)3 / 23

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Objective and Approach

Our choice of technologies:High-order discontinuous Galerkin (DG) methodSimplex mesh adaptation

Complex geometriesArbitrarily oriented anisotropy

Adjoint-based error estimateError estimate for outputs of interest (lift, drag, etc)Error localization

Robust nonlinear solverPseudo-time continuation and line search

Objective: Realize the full potential of high order methods throughautomated mesh adaptation and error control

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Outline

1 Motivation

2 Impact of Adaptation on High-order Efficiency

3 Adaptive Solution Strategy

4 Adaptive High-order RANS

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Example 1: Subsonic EulerNACA0012: M∞ = 0.5, α = 2.0

Optimal meshes produced at 2.5k and 5k DOFsUniform vs. adaptive refinement to 10k and 20k

Mesh p = 3, dof = 5k

2.5k 5k 10k 20k

10−3

10−2

10−1

100

101

degrees of freedom

cd e

rror

estim

ate

(counts

)

−1.50

−3.38

−0.67

0.1 drag count

p=1 (uniform)

p=3 (uniform)

p=1 (adapt)

p=3 (adapt)

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Error Distribution near Trailing EdgeNACA0012: M∞ = 0.5, α = 2.0

p = 3, dof = 20k distribution of elemental error log10(ηK )

Trailing edge singularity limits convergence rate

Uniform refinement Adaptive refinement

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Example 2: Subsonic RANS-SAM∞ = 0.2, Rec = 6.5× 106, α = 2.31

Adaptive refinement performed at 20k and 40kUniform vs. adaptive refinement to 80k and 160k

Mach number

20k 40k 80k 160k

10−3

10−2

10−1

100

101

degrees of freedom

cd e

rror

estim

ate

(counts

)

−0.98

−3.28

−1.71

0.1 drag count

p=1 (uniform)

p=3 (uniform)

p=1 (adapt)

p=3 (adapt)

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RANS-SA Boundary Layer Edge SingularityM∞ = 0.2, Rec = 6.5× 106, α = 2.31

p = 3, dof = 160k distribution of elemental error log10(ηK )

Irregularity in SA equation limits convergence rate

Uniform refinement

Adaptive refinement

Challenge: Design adaptation algorithm that produces propermeshes for complex flows in automated manner

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Outline

1 Motivation




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Adaptive Solution StrategyProblem: Find output J(u) that satisfies conservation lawDG-FEM: Find Jh,p(uh,p) with uh,p ∈ Vh,p s.t.

Rh,p(uh,p, vh,p) = 0 ∀vh,p ∈ Vh,p

Output-based adaptation: estimate and control

E ≡ J(u)− Jh,p(uh,p)

ProblemOutputMax errorMax time

Calculateflow andoutputs

Estimateoutputerror

?Flow solutionOutputsError estimate

Adapt gridto control

error

Error metTime over

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Error Estimation

Dual-weighted residual method of Becker and Rannacher [2001]

E ≡ Jh,p(uh,p)− J(u) = Rh,p(uh,p, ψ)

Adjoint, ψ, is a transfer function from δR to EElemental error indicator, ηK ≡ Rh,p(uh,p, ψh,p′ |K )

Error indicator, ηKPrimal uh,p (Mach)

Adjoint ψh,p (mass)

RAE2822 RANS-SA: M∞ = 0.729, Rec = 6.5× 106, α = 2.31, J = cd

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Adaptation Mechanics

1 Size redistribution: Fixed-fraction marking– Refine/coarsen elements with large/small errors

2 Shape selection: Mach-based anisotropy detection– Anisotropy from p + 1 derivative of Mach, M(p+1)

δx3 DOF control: Metric-field scaling

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Metric Field M(x)x∈Ω

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Mesh Th

Mesh Gen.

Implied Metric

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High-Order Mesh Generation

4 Recover high-order geometry representationOption 1: Elastically-curved boundary-conforming (bc) meshesOption 2: Cut-cell (cc) meshes

Valid linear mesh

Curving onlyboundary surface

Curvingwith

elasticity

Cut-cellIntersection

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Subsonic RANS: Fixed-DOF AdaptationRAE2822: M∞ = 0.2, Rec = 6.5× 106, α = 2.31

p = 3, dof = 40k adaptationError decreased via size redistribution and anisotropic resolution

Initial Mesh Adapted Mesh

0 5 10 15 20 2560

70

80

90

100

110

120

130

140

adaptation iteration

cd (

co

un

ts)

drag

drag est. range

true drag

0 5 10 15 20 2510

−2

10−1

100

101

102


cd e

rro

r (c

ou

nts

)

error estimate

true error

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Outline

1 Motivation




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Transonic RANSRAE2822: M∞ = 0.729, Rec = 6.5× 106, α = 2.31

Assessment of adaptive DG for transonic RANSKey features: boundary layers, wake, shock

Mach number

Initial mesh

p = 3, dof = 40k mesh

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Transonic RANS Drag ErrorRAE2822: M∞ = 0.729, Rec = 6.5× 106, α = 2.31

Efficient BL resolution more than offset shock inefficiencyThere is a limit to which effectiveness of HO can be extended

20k 40k 80k 160k

10−2

10−1

100

101

102

103

degrees of freedom

cd e

rror

estim

ate

(counts

)

−1.09

−2.27

−2.98

0.1 drag count p=1 (cc)

p=2 (cc)

p=3 (cc)

p=1 (bc)

p=2 (bc)

p=3 (bc)

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Supersonic RANSNACA0006: M∞ = 2.0, Rec = 1.0× 106, α = 2.0

Assessment of adaptive DG for supersonic RANSKey features: boundary layers, wake, oblique shocks

Mach number p = 2, dof = 80k mesh

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Supersonic RANS Drag ErrorNACA0006: M∞ = 2.0, Rec = 1.0× 106, α = 2.0

p = 2 outperforms p = 1 for high-fidelity simulations

40k 80k 160k10

−2

10−1

100

101

102

degrees of freedom

cd e

rror

estim

ate

(counts

)

−1.56

−3.00

0.1 drag count

p=1 (cc)

p=2 (cc)

p=3 (cc)

p=1 (bc)

p=2 (bc)

p=3 (bc)

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Parameter SweepMDA Three-Element: M∞ = 0.2, Rec = 6.5× 106, α = 0.0 − 24.5

Application of fixed-dof adaptation to parameter sweep

α = 8.1

α = 16.21

α = 21.34

α = 23.28

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Lift CurveMDA Three-Element: M∞ = 0.2, Rec = 6.5× 106, α = 0.0 − 24.5; p = 2, dof = 90k

α sweep using fixed (8.1 adapted) vs. adaptive meshesSingle adapted mesh not suited for entire α rangeError estimate captures solution quality

Lift curve cl error estimate

0 5 10 15 20 252

2.5

3

3.5

4

4.5

5

angle of attack

cl

fixed mesh

adaptive

0 5 10 15 20 2510

−3

10−2

10−1

100

101

102

angle of attack

cl e

rror

estim

ate

fixed mesh

adaptive

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Fixed vs. Adapted MeshesMDA Three-Element: M∞ = 0.2, Rec = 6.5× 106, α = 0.0 − 24.5; p = 2, dof = 90k

Mach number for α = 23.28

Fixed mesh (α = 8.1 adapted)

Adapted mesh

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Adapted MeshesMDA Three-Element: M∞ = 0.2, Rec = 6.5× 106, α = 0.0 − 24.5; p = 2, dof = 90k

α=0.0

α=8.1

α=23.28

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Conclusions

Highlighted the importance of proper mesh selection forhigh-order methodsPresented mesh adaptation strategy that works towarddof-optimal meshesDemonstrated competitiveness of adaptive high-order methods forsubsonic, transonic, and supersonic RANS flowsShowed capability of elastic-curving and cut-cell techniques forhigh-order mesh generationApplied fixed-dof adaptation to parameter sweep

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Backup Slides

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Mesh Grading near Trailing EdgeNACA0012: M∞ = 0.5, α = 2.0

T.E. mesh grading for p = 1 and p = 3 adapted meshes with2,000 elementsHigh-order discretization requires stronger grading toward T.E.

10−4

10−3

10−2

10−1

10−4

10−3

10−2

10−1

distance from trailing edge, r

ele

me

nt

siz

e,

h

h=0.16r0.66

h=0.27r0.86

p=1

p=3

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Option 1 – Isotropic subdivision of elements

Option 2 – Anisotropic subdivision of quadrilateral elements

Ceze [2010]

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Option 3 – Metric-based remeshingElement size and shape encoded in metric field, M(x)x∈Ω

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Metric Field M(x)x∈Ω

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Mesh Th

Mesh Gen.

Implied Metric

Adaptation philosophySize: equidistribute elemental errorsShape: capture directional flow featuresDOF: control computational cost

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Overview of Adaptation Strategy

1 Size redistribution: Fixed-fraction marking– Refine/coarsen elements with large/small errors

2 Shape selection: Mach-based anisotropy detection– Anisotropy from p + 1 derivative of Mach, M(p+1)

δx3 DOF control: Metric-field scaling4 HO mesh generation: Elastic curving or cut cells

Size redistribution Shape selection

Element-wise Error

Num

bero

fEle

men

ts

RefinefR · nelem

CoarsenfC · nelem

Element-wise Error

Num

bero

fEle

men

ts

M(p+1)δx

Mani

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Subsonic RANS Drag Error

High-order more efficient than 2nd-orderBoth boundary-conforming and cut-cell benefit from HO

cd error estimate cd

20k 40k 80k10

−2

10−1

100

101

degrees of freedom

cd e

rro

r e

stim

ate

(co

un

ts)

−1.03

−2.16

−2.82


p=2 (cc)

p=3 (cc)

p=1 (bc)

p=2 (bc)

p=3 (bc)

20k 40k 80k

83.8

83.9

84

84.1

84.2

84.3

84.4

84.5

84.6

84.7

degrees of freedom

cd (

co

un

ts)

+0.1 drag count

p=1 (bc)

p=2 (bc)

p=3 (bc)

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Transonic RANS Drag ErrorRAE2822: M∞ = 0.729, Rec = 6.5× 106, α = 2.31

Efficient BL resolution more than offset shock inefficiencyThere is a limit to which effectiveness of HO can be extended


20k 40k 80k 160k

10−2

10−1

100

101

102

103

degrees of freedom

cd e

rro

r e

stim

ate

(co

un

ts)

−1.09

−2.27

−2.98


p=2 (cc)

p=3 (cc)

p=1 (bc)

p=2 (bc)

p=3 (bc)

20k 40k 80k 160k

118.5

118.6

118.7

118.8

118.9

119

119.1

119.2

119.3

119.4

degrees of freedom

cd (

counts

)

+0.1 drag count

p=1 (bc)

p=2 (bc)

p=3 (bc)

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Supersonic RANS Drag ErrorNACA0006: M∞ = 2.0, Rec = 1.0× 106, α = 2.0

p = 2 outperforms p = 1 for high-fidelity simulations


40k 80k 160k10

−2

10−1

100

101

102

degrees of freedom

cd e

rro

r e

stim

ate

(co

un

ts)

−1.56

−3.00

0.1 drag count

p=1 (cc)

p=2 (cc)

p=3 (cc)

p=1 (bc)

p=2 (bc)

p=3 (bc)

40k 80k 160k403.9

404

404.1

404.2

404.3

404.4

degrees of freedom

cd (

counts

)

+0.1 drag count

p=1 (bc)

p=2 (bc)

p=3 (bc)

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Subsonic RANSRAE2822: M∞ = 0.2, Rec = 6.5× 106, α = 2.31

Assessment of adaptive DG for subsonic RANSKey features: boundary layers (including SA BL edges), wake

Mach numberInitial mesh

Adapted mesh

p = 3, dof = 40k

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Subsonic RANS: Fixed-DOF AdaptationRAE2822: M∞ = 0.2, Rec = 6.5× 106, α = 2.31

p = 3, dof = 40k adaptationError decreased via size redistribution and anisotropic resolutionLess iterations required for better initial mesh

cd

0 5 10 15 20 2560

70

80

90

100

110

120

130

140


cd (

counts

)

drag

drag est. range

true drag

cd error

0 5 10 15 20 2510

−2

10−1

100

101

102


cd e

rror

(counts

)

error estimate

true error

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Subsonic RANS Drag Error

High-order more efficient than 2nd-orderBoth boundary-conforming and cut-cell benefit from HO


20k 40k 80k10

−2

10−1

100

101

degrees of freedom

cd e

rro

r e

stim

ate

(co

un

ts)

−1.03

−2.16

−2.82


p=2 (cc)

p=3 (cc)

p=1 (bc)

p=2 (bc)

p=3 (bc)

20k 40k 80k

83.8

83.9

84

84.1

84.2

84.3

84.4

84.5

84.6

84.7

degrees of freedom

cd (

co

un

ts)

+0.1 drag count

p=1 (bc)

p=2 (bc)

p=3 (bc)

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What are Cut Cells?

Meshes do not conform to higher-order geometrySolution only exists within computational domainKey idea: metric driven mesh adaptation within a box/cube is robustand autonomous

Boundary conforming Cut cell

Selected previous work:1979 – Purvis and Burkhalter: FV for 2D Full Potential Equations1986 – Boeing’s TRANAIR: FEM for 3D Full Potential Equations1999 – Cart3D: Finite Volume for 3D Euler Equations2007 – Fidkowski: DG with simplex cut cells

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Why Simplex Cut Cells

Cartesian cut-cell methodRobust and automated mesh generationInability to adapt anisotropically

Euler M = 3.0

Lahur and Nakamura [2000]

Simplex cut-cell methodRobust and automated mesh generationAbility to generate meshes withanisotropy of arbitrary orientation

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Intersection Problem

Geometry definition:Cubic splices used to define curved boundariesProvide slope and continuity at spline knots

s = 0

s = smax

Embeddededge

Cut edge

Spline-edgeintersection

Spline geometry

1

2

Implementation:Solve a cubic equation for intersections between splines and edgesTreat multiply-cut elements as separate cut cells

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Integration Rules

DG Residual: R(u, v) =∑κ∈T

∫κ(·)dA +

∑e∈EI

∫e(·)ds +

∑e∈EB

∫e(·)ds

Edge integrationGauss points used on each edge

Area integration

Canonical shapesRecognition of triangles andquadrilaterals when possible

Arbitrarily cut elements, Fidkowski [2007]Uses a set of uniformly distributedsampling pointsQuadrature-like weights are computedfor each point

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Solution Efficiency: Boundary-Conforming vs. Cut-Cell

Solution Efficiency =Accuracy

DOF

A reduction in solution efficiency is expected when using cut-cellmeshes compared to elastically-curved meshesBoundary layers of thickness δ can be more efficiently resolved withcurved elements

δ δ

Elastically curved Cut cell

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the importance of mesh adaptation for higher-order...

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