an optimization-based framework for controlling...

An Optimization-based Frameworkfor Controlling Discretization Errorthrough Anisotropic h-Adaptation

Masayuki Yano and David Darmofal

Aerospace Computational Design LaboratoryMassachusetts Institute of Technology

Acknowledgments: Dr. Steven Allmaras, ProjectX Team, The Boeing Company

March 24, 2011

M. Yano (MIT) FEF 2011 March 24, 2011 1 / 22

Outline

1 Introduction

2 Optimization Framework : Anisotropic h-Adaptation

3 Results : L2 Error Control

4 Results : Functional Output Error Control

5 Conclusion


Objective and Approach

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

Our choice of technologies:Discontinuous Galerkin (DG) method

hp-flexible on unstructured meshesConvergence limited by regularity

- shocks, geometric singularities,turbulence models

Simplex mesh adaptationComplex geometriesArbitrarily oriented anisotropy

Adjoint-based error estimatorError estimate / localization foroutput of interest (e.g. lift, drag)


Functional Error Estimation

Problem: Find J(u) with u ∈ V s.t.

R(u, v) = 0 ∀v ∈ V

Dual-weighted residual (DWR) error representation [Becker,2001]

E ≡ Jh,p(uh,p)− J(u) = Rh,p(uh,p, ψ), ψ = adjoint

Local error indicator

ηK ≡ Rh,p(uh,p, ψh,p+1|K )

Error indicator assigns 1 value to each elementOnly sufficient for 1 parameter adaptation

type nparam parameters ηK sufficientisotropic h 1 size 4

isotropic hp 2 size, poly. order 8

anisotropic h 3 (6) size, shape 8

anisotropic hp 4 (7) size, shape, poly. order 8


Review of Anisotropic h-Adaptation

Need additional info to make anisotropy decisionPrevious work on output-based anisotropic h-adaptation

Method p > 1 ψ ani. simplexMach Hessian [Venditti, 2003] 8 8 4Mach “Hessian” [Fidkowski, 2007] 4 8 4Mach jump [Hartmann, 2008] 4 8 8Edge comp. ref [Park, 2008][Sun, 2009] 4∗ 4∗ 4∗

Comp. ref. [Georgoulis, 2009][Ceze, 2010] 4 4 8Flux reconstruction + C.V. [Loseille, 2010] 8 4 4Anisotropic error estimate [Richter, 2010] 8 4 8

Contribution: Anisotropic h-adaptation framework for simplexmeshes that implicitly synthesizes information about

primal and adjoint solution behaviorsinterpolation ordersolution regularity


Outline

1 Introduction




5 Conclusion


Problem Definition / Example

Objective: Develop “optimal” anisotropic meshExample problem: L2-projection error control of boundary layer

ED(Th) = ‖u − uh,p‖2L2(Ω) where u = exp (−x1/δ)

CD(Th) = DOF(Th)

u = exp(−x1/δ) optimal isotropic mesh optimal anisotropic mesh(δ = 0.01) E

12 = 1.6× 10−4 E

12 = 5.1× 10−8

*each mesh has ≈ 200 elements (p = 3)


Continuous Relaxation

(Intractable) discrete optimization problem

Th,opt = arg minThED(Th) s.t. CD(Th) = DOFtarget

Approximability of Th encoded in metric tensor field

M(x) ∈ Sym+d (d × d SPD matrix)

Continuous relaxation [cf. Loseille, 2009]

Mopt = arg minME(M) s.t. C(M) = DOFtarget

Metric Field M Mesh Th

Mesh Gen.

Implied Metric


Continuous Optimization

Problem Statement: Seek optimal metric field

Mopt = arg minME(M) s.t. C(M) = DOFtarget

Assume locality of error and cost functionals

E(M) =

∫Ω

e(x ,M(x))dx and C(M) =

∫Ω

c(M(x))dx

local error function : e(·, ·) : Rd × Sym+d → R+

local cost function : c(M) = Cp√

det(M)

First order optimality

∂e∂M

(x ,M(x))− λ ∂c∂M

(x ,M(x)) = 0 ∀x ∈ Ω

Need to estimate behavior of e(M)


Local Sampling

Split edges of K0 to form Ki ’s, i = 0, . . . ,mObtain ηKi on split config.

Freeze neighbor statesSolve local problem

Obtain metricMKi implied by split config.Associate ηKi ≈ e(MKi )|Ki |

original edge 1 split edge 2 split edge 3 split uniform splitMK0 , ηK0 MK1 , ηK1 MK2 , ηK2 MK3 , ηK3 MKu , ηKu


Manipulation of SPD Matrix Space

Original problem on Sym+d

∂e∂M

(x ,M)− λ ∂c∂M

(x ,M) = 0 M∈ Sym+d

SPD matrices define cone in Euclidean spaceHard to manipulate SPD matrices directly

Consider substitutionsM(Σ) =M1/2

0 exp(Σ)M1/20 ⇒ M(Σ) ∈ Sym+

d ∀Σ ∈ Symde(f ) = e0 exp(f ) ⇒ e(f ) ∈ R+ ∀f ∈ Rc(g) = c0 exp(g) ⇒ c(g) ∈ R+ ∀g ∈ R

New problem on Symd

e0∂f∂Σ

(x ,Σ)− λc0∂g∂Σ

(x ,Σ) = 0 Σ ∈ Symd

Manipulation of symmetric matricesApproximate f (Σ) and move toward optimality


Local Error Model / Parameter Identification

Construct linear approximation of f (Σ)

f (Σ) = tr(RΣ), R ∈ Symd = rate tensor

R is generalization of convergence rate to tensor spacetr(R) : sensitivity to uniform scalingR − tr(R)

d I : sensitivity to shape change

Identify R from local samples MKi , ηKimi=1 − ΣK1 −...

− ΣKm −

R11

2R12

R22

=

f (ΣK1 )...

f (ΣKm )

Σ(Ki ) = log(M−1/2

0 MKiM−1/20 ) and f (ΣKi ) = log(ηKi/ηK0 )


Discrete Optimality Condition

Discrete Optimality Conditions (Anisotropic)Global isotropic error balance : ∃λ ∈ R s.t.

ηK tr(RK )− λd2 dof(K ) = 0 ∀K ∈ Th

Local anisotropic optimality :RK ∝ I ∀K ∈ Th

e.g. R 6= I : more reduction by increasingM11 thanM22⇒ decrease h1 and increase h2

RK ,11 RK ,22


Algorithm

1 Obtain primal solution uh,p on T (i)h,p

2 Obtain adjoint solution ψh,p+1 on T (i)h,p+1 (for DWR)

3 Solve local problems and obtain MKi , ηKimi=0, ∀K ∈ T (i)h

4 Construct local error function e(M) ≈ e(M; MKi , ηKimi=0)

5 Update metric fieldM(i+1)

6 Generate new mesh T (i+1) fromM(i+1) and go to 1

Adapt 0 : E12 = 9.4× 10−3 Adapt 2 : E

12 = 1.8× 10−5 Adapt 7 : E

12 = 7.9× 10−8

*each mesh has ≈ 200 elements (p = 3)


Assessment of Boundary Layer L2 Error Control

Function: u = exp(−x1/δ) (δ = 0.01)Analytical optimal h⊥ vs. x1

h⊥,opt = C exp(k⊥,optx

), k⊥,opt =

1δ(p + 3/2)

Weaker grading for higher p

0 0.02 0.04 0.06 0.08 0.110

−4

10−3

10−2

10−1

kperp

=43.2

kperp

=44.1

kperp

=43.5

kperp

=42.8

x

h⊥

dof=500dof=1000

dof=2000dof=4000

0 0.02 0.04 0.06 0.08 0.110

−4

10−3

10−2

10−1

kperp

=24.9

kperp

=24.9

kperp

=22.9

kperp

=22.7

x

h⊥

dof=1000dof=2000

dof=4000dof=8000

p = 1 (k⊥,opt = 40.0) p = 3 (k⊥,opt = 22.2)M. Yano (MIT) FEF 2011 March 24, 2011 13 / 22

Assessment of Singular Corner L2 Error Control

Function: u = rα sin[α(θ + π

2

)](α = 2/3)

Analytical optimal h vs. r

hopt(r) = Cr kopt , kopt = 1− α + 1p + 2

Stronger grading for higher p

10−3

10−2

10−1

100

10−3

10−2

10−1

k=0.43

k=0.44

k=0.44

k=0.44

r

h

dof=500

dof=1000

dof=2000

dof=4000

10−3

10−2

10−1

100

10−3

10−2

10−1

k=0.68

k=0.67

k=0.67

k=0.67

r

h

dof=500

dof=1000

dof=2000

dof=4000

p = 1 (kopt = 0.44) p = 3 (kopt = 0.67)M. Yano (MIT) FEF 2011 March 24, 2011 14 / 22

Outline

1 Introduction




5 Conclusion


RAE2822 Transonic AirfoilRANS-SA : M∞ = 0.729, Rec = 6.5× 106, α = 2.31; p = 2

Objective: Application to transsonic turbulent flowFunctional : DragComparison : Mach-based anisotropy (∇p+1M)

Mach

Mass adjoint

Drag error

20000 30000 40000 60000

10−7

10−6

10−5

10−4

10−3

dof

|J −

Jre

f|

mach ani.

unified


RAE2822 Transonic AirfoilRANS-SA : M∞ = 0.729, Rec = 6.5× 106, α = 2.31; p = 2

Anisotropic resolution of featuresPrimal: BLs, wake, shockAdjoint: BLs, stagnation streamline, shock-induced perturbations

Unified (p = 2, dof = 60k) Mach anisotropy (p = 2, dof = 60k)


NACA0006 Shock PropagationEuler : M∞ = 2.0; p = 2

Objective: Application to sonic-boom predictionFunctional : Pressure 50c below airfoil, J =

∫l(p − p∞)2ds

Pressure

Mass adjoint

Pressure line output error

10000 20000 30000 40000

10−4

10−3

10−2

10−1

100

dof

|J −

Jre

f|/|J

ref|

mach ani.

unified


NACA0006 Shock PropagationEuler : M∞ = 2.0; p = 2

Unified (p = 2, dof = 40k) Mach anisotropy (p = 2, dof = 40k)


Three-Element MDA AirfoilRANS-SA : M∞ = 0.2, Rec = 9× 106, α = 8.10 ; p = 2

Objective: Application to complex turbulent flowFunctional : Drag

Mach

Mass adjoint

Drag error

30000 45000 60000 90000

10−5

10−4

10−3

dof

|J −

Jre

f|

mach ani.

unified


Three-Element MDA AirfoilM∞ = 0.2, Rec = 9× 106, α = 8.10; p = 2

Unified (p = 2, dof = 90k)

Mach anisotropy (p = 2, dof = 90k)


Outline

1 Introduction




5 Conclusion


Conclusion and Ongoing Work

Conclusion:Presented optimization framework for anisotropic h-adaptationthat implicitly synthesizes information about

primal and adjoint solution behaviorsinterpolation ordersolution regularity

Verified optimality of algorithm for L2 error controlCompared performance of algorithm against primal derivativebased anisotropy detection

Ongoing Work:Extension to hp-adaptationImproving robustness of error estimator for highly nonlinearproblems


an optimization-based framework for controlling...

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