anadaptivediscontinuousgalerkin reducedbasiselementmethod...

An adaptive discontinuous Galerkinreduced basis element method:

application to aerodynamic flows

Masayuki Yano

University of Toronto

Acknowledgment: NSERC

AIAA Aviation 2017Denver, United States

9 June 2017

Problem statement

CFD problem:geometry +flow condition

⇒ state u ⇒ output J(u)

Goal: find J(u) as accurately, efficiently, and reliably as possible.

Claim: automatic error control is a key to achieve this goal.

This talk: focus on discretization (and not modeling) error1. error estimation2. efficient approximation spaces (i.e. mesh, order, . . . )

1

Example 1: adaptive finite element

start from a crude meshgenerate sequence of “optimal” spaces informed by error estimate

V0 → V1 → · · · → Vn.

⇒ ⇒

+ error estimate+ high efficiency relative to “best-practice” meshes− best suited for single-/few-query scenarios

⇒ no reuse of information across cases

2

Example 2: projection-based model reduction (e.g. POD)

designed for parametrized problems in many-query scenariosoffline: collect n snapshots u1, . . . , un

online: approximate solution in Vn ≡ spanuini=1

+ potential for significant online speedup− reduced model is case-specific (i.e. geometry & parametrization)⇒ significant offline training for each case (n ∼ 100+ runs)

Q.

can we eliminate case-specific training and use trainingknowledge across different cases?

3

Example 2: projection-based model reduction (e.g. POD)

designed for parametrized problems in many-query scenariosoffline: collect n snapshots u1, . . . , un

online: approximate solution in Vn ≡ spanuini=1

+ potential for significant online speedup− reduced model is case-specific (i.e. geometry & parametrization)⇒ significant offline training for each case (n ∼ 100+ runs)

Q. can we eliminate case-specific training and use trainingknowledge across different cases?

3

Library-based localized model reduction

Idea: forgo reduction of a specific case,and focus on reducible & reusable features.

Library of reducible featuresExample 1: boundary layers (BLs)− BLs require high resolution− BLs are similarly shaped (Blasius, . . .).

Example 2: trailing edge singularity...

Vision: an intelligent CFD solver for many-query scenarios thatlearns reducible features and accelerates computation.

4

Formulation

DiscretizationReduced basis element (RBE)StabilityError estimation and adaptationRelated work

Discontinuous Galerkin (DG) method

PDE: steady conservation laws

∇ · F inv(u) +∇ · F visc(u,∇u) = S(u,∇u) in Ω ; (+BCs).

DG: introduce a discontinuous FE space Vn;find un ∈ Vn s.t.

Rn(un, vn) = 0 ∀vn ∈ Vn,x

y

uh(x, y)

Thwhere

Rn(un, vn) = −∫

Ωh

∇vn · F inv(un)dx+

∫Σh

v+n F

inv(· · · )ds+ · · · .

Features:Flexible choice of FE spacesStability for conservation laws

5

Polynomial FE spaces: adaptivity

FE space:

Vn = Vh,p = v ∈ L2(Ω)︸︷︷︸discontinuous

: (v gq)|κ ∈ Pp(κref)︸︷︷︸polynomial in

each reference element

, κref ∈ Th,ref︸︷︷︸tessellation

.

h-adaptivity: uh → u as h→ 0

p-adaptivity: up → u as p→∞

Polynomial spaces with good general approximability in each κ. 6

Formulation


Reduced basis element (RBE) [Maday & Rønquist, 2005; . . .]

Idea. Construct spaces with feature-specific approximability.

Discontinuous non-polynomial FE space

Vn = v ∈ L2(Ω) : (v gq)|κ ∈ FEn(κref), κref ∈ Th,ref.

where

FEn(κref) =

RBEN(κref), κ ∈ reducible regionPp(κref), otherwise

.

Reduced basis elements (RBEN) are1. tailored for a specific reducible feature (e.g. BLs)2. hierarchical (N = 1, 2, . . . ).

· · ·7

RBE: offline training

Key: RBEs are not prescribed but are identified through training.

Step 1. Solve training cases

Step 2. Extract BL functions in reference element κref

· · · · · ·Step 3. Identify BL modes using POD → RBEN

· · ·8

RBE: offline training

Step 3 (alternate). Extract 1d wall-normal BL modes

i. Sample BL profile in wall-normal direction

ii. Apply POD to 1d functions → RBE1dN1d

iii. Tensorize RBE1dN1d

with polynomials

RBEN(κref) = RBE1dN1d

(κref,1d)⊗ Pp1d(κref,1d)

9

Formulation


Stability

For entropy variables, the discontinuous Galerkin method is stable.[Barth, 1999; Harten 1983; Hughes et al 1986]

Linear equations: energy stability

‖un(T )‖2M ≤ ‖un(0)‖2

M + (inflow data).

Nonlinear equations: (generalized) entropy stability

U(un(T )) ≤ U(un(0)) + (inflow data).

Remarks:RBE: test space does not have good general approximability⇒ entropy stability is crucial.

10

Formulation


Output error estimation: dual-weighted residual (DWR)

Adjoint ψ: quantifies sensitivity of output J(u) to disturbances:

R′n(un; vn, ψn) = J ′n(un; vn) ∀vn ∈ Vadjn .

Global error estimate: [Becker & Rannacher, 1996; Ainsworth & Oden, 1998]

E ≡ J(u)− Jn(un) ≈ −Rn(un, ψn).

Element-wise error indicator:

ηκ ≡ |Rn(un, ψn|κ)|, κ ∈ Th.

Note: we also train RBEs for the adjoint.11

Adaptation

EmploySolve︸︷︷︸DG-RBE

→ Estimate︸︷︷︸DWR

→ Mark︸︷︷︸fixed-fraction

→ Refine.

Refine: increase dof in a manner equivalent to isotropic p refinement

reducible: RBEN(κ); set N ≡ n21d ← (n1d + 1)2

otherwise: Pp(κ); set p← p+ 1.

12

Formulation


Related work

Viscous-inviscid couplingPotential/Euler-IBL [Le Balleur 1978–; Drela & Giles 1987; . . . ]

6= mathematical reduction; error estimation wrt full equation

FEM with non-polynomial, special basis functionsPUM, XFEM, GFEM, DEM, . . .

[Babuška 1994–; Belytschko 1999–; Farhat 2001–; . . . ]

6= analytical functions vs empirical training

Reduced basis element methodsRBE [Maday & Rønquist 2005–; . . . ]

Static-condensation RBE [Patera, Knezevic, & Huynh 2013–; . . . ]

Localized RB multiscale method [Ohlberger & Schindler 2015–]

13

Numerical results

Training setupEuler flow over airfoilLaminar airfoilFlat plate

Randomized training: NACA 4-digit family

parameter rangecamber 00, 14, 24thickness [0.06, 0.12]M∞ [0.2, 0.5]α [0, 3]Rec [3000, 6000]

boundary layer and trailing edge

Note: only 10 training solves; we train the features, not the case.

14

Numerical results


NACA 2410 Euler: setup

Case: M∞ = 0.3, α = 1

primal (Mach) adjoint (x-momentum)

RBE Library:1. Euler trailing edge elements (⇐ 10 random NACA cases)

Note: RBE regions are prescribed.

15

NACA 2410 Euler: convergence

103 104

dof

10-4

10-3c d

err

or

1 count

uniform (p=1)ani-hp adaptRBE-adapt

ani-hp adaptation: discrete optimization[Houston 2006; Ceze & Fidkowski 2012]

RBE-adaptation: <1% error using 1012 dof.16

NACA 2410 Euler: convergence with error estimate

103 104

dof

10-4

10-3c d

err

or

1 count


ani-hp adaptation: discrete optimization[Houston 2006; Ceze & Fidkowski 2012]

RBE-adaptation: <1% error using 1012 dof.16

NACA 2410 Euler: dof distribution (error level: 1 count)

p adapt (p ∈ [1, 8])

TE dof: 198

RBE adaptTE dof: 31

Significant TE DOF reduction ⇒ as if singularity does not exist

17

Numerical results


NACA 2410 laminar: setup

Case: M∞ = 0.3, α = 1, Rec = 5000


RBE Library:1. NS trailing edge elements (⇐ 10 NACA cases)2. NS boundary layer elements (⇐ 10 NACA cases)

18

NACA 2410 laminar: convergence

103 104

dof

10-4

10-3

10-2

10-1

c d e

rror

1% error


RBE-adaptation: <1% error using 908 dof.

19

NACA 2410 laminar: dof distribution (error level: 1%)

p adapt (p ∈ [1, 7])

BL dof: 693TE dof: 146

RBE adaptationBL dof: 162TE dof: 111

BL DOF reduction ⇒ as if singular perturbation does not exist

20

Numerical results


Flat plate: setup

Case: M∞ = 0.3, ReL = 5000


RBE Library:1. NS boundary layer elements (⇐ 10 NACA cases)

note: using existing library; no cases-specific training.

21

Flat plate: convergence

102 103 104

dof

10-4

10-3

10-2

10-1

c d e

rror

1% error


RBE on per with “state-of-the-art” ani-hp adaptation

22

Flat plate: dof distribution (error level: 1%)

p adaptationBL dof: 339LE dof: 164

RBE adaptationBL dof: 84LE dof: 89

+ significant reduction in boundary layer− leading/trailing edge singularities not efficiently resolved

23

Library update

Idea: “learn” from the flat plate (FP) casesidentify reducible and reusable featuresadd the features to the RBE library

RBE Library:1. NS boundary layer elements (⇐ 10 NACA cases)2. NS flat leading-edge elements (⇐ 5 FP cases)3. NS flat trailing-edge elements (⇐ 5 FP cases)

24

Flat plate, new library: dof distribution (error level: 1%)

Old library: NACA boundary layer only

RBE-adapt (BL only)BL dof: 84LE dof: 89TE dof: 74

Updated library: NACA boundary layer & FP leading/trailing edges

RBE-adapt (BL+LE+TE)BL dof: 120LE dof: 13TE dof: 20

+ significant reduction in LE (89→ 13) and TE (74→ 20) dofs.25

Perspectives and summary

DG-RBE method in CFD workflow

Classical adaptive FEM:

case 2

case 1...

case 3

...

...

solver solution 2...

solution 1...

solution 3...

26

DG-RBE method in CFD workflow

Library-based adaptive RBE method:

case 2

case 1...

case 3

...

...

acceleratedsolver

? solution 2...

solution 1...

solution 3...RBE library

Accelerated solve:known, reducible features: apply RBEunknown or irreducible features: apply hp-adaptation⇒ add reducible feature to library for future use

27

Summary

Idea: eliminate case-specific training anduse training knowledge across different cases.⇒ library of RBEs for reducible & reusable features

Ingredients:

RBE library + DG + error estimate + adaptivity.

Open questions: can wedynamically identify features?perform seamless hp/RBE switching?mine existing CFD database to construct a library?

...

28

Backup

POD eigenvalues and modes

Eigenvalues:primal adjoint

0 10 20 30 40n

10-10

10-8

10-6

10-4

10-2

100

6n

comp 1comp 2comp 3comp 4

0 10 20 30 40n

10-8

10-6

10-4

10-2

100

6n

comp 1comp 2comp 3comp 4

Modes: primal, component 2

· · ·29

Anisotropy detection

Discrete localized error sampling [Houston 2006-, Ceze & Fidkowski 2011-]

1. consider i = 1, . . . , nconfig split configurations

2. solve local problems (primal only)

find uin ∈ V in s.t. Rn(uin, vn) = 0 ∀vn ∈ V in

3. estimate local error (implicit local Galerkin orthogonality)

ηκi ≡ |Rn(uin, ψn|κ) = infvin∈Vi

n

|Rn(uin, (ψn − vin)|κ)

4. select configuration

i? ≡ maxi=1,...,nconfig

|ηκi/ηκ||dofκi |/|dofκ|

30

Perspective: general vs specialized spectrum

adaptive FEMglobal-snapshotmodel reduction

adaptive RBE

“bad” lib. “good” lib.

approximation space Vngeneral specific

offline/training costnone high

marginal evaluation costhigh low (?)

scenarios: # queriessingle many

31

Adjoint approximation

Adjoint approximability is critical for1. effective error estimate2. accurate prediction (i.e. implicit “output superconvergence”)

|J(u)− Jn(un)| = infvn∈Vn

|Rn(un, ψ − vn)|

Two optionsO1 Construct an adjoint-specific RBE spaces

Vadjn = v ∈ L2(Ω) : (v gq)|κ ∈ FEadj

n (κref), κref ∈ Th,ref.

O2 Include both primal and adjoint solution in POD processing

FEadjn = FE = POD(uBL

1 , . . . , uBLm ∪ ψBL

1 , . . . , ψBLm )

32

RBE: implementation

Note: evaluation of Rn(wn, vn) requires evaluation of basis functionsat quadrature points.

Approach:1. Represent RBE basis as piecewise high-order polynomials

φRBEi (x) =

∑j

wijφpolyj (x), x ∈ κsub ⊂ κref ;

2. Apply piecewise Gauss quadrature

Remark: Gauss quadrature is non-optimal for the specialized basis⇒ hyper-reduction can be applied [Ryckelynck 2005; . . . ]

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