High Order Discontinuous Galerkin Methods for...

High Order Discontinuous Galerkin Methods

for Aerodynamics

Per-Olof Persson

Massachusetts Institute of Technology

Collaborators: J. Peraire, J. Bonet, N. C. Nguyen, A. Mohnot

Symposium on Recent Developments and Emerging Trends

in Computational Design, Analysis and Optimization

WPAFB, OH

January 30, 2007

1

Outline

• Background

• Introduction to High-Order DG

• Cost and efficiency issues

• Robustness, RANS and shocks

• Deformable domains

2

Application Areas

• DNS/LES/DES applications

• Accurate RANS for engineering applications

– Drag prediction

– Rotor dynamics

– Hypersonics

– Fluid/Structure Interaction, Flapping flight

• Computational aeroacoustics

– Direct simulation using the compressible Navier-Stokes equations

– Accurate computation of noise sources

3

Numerical Schemes

Three fundamental requirements:

High-order/Low dispersion

Unstructured meshes

Stability for convervation laws

4

Numerical Schemes


FVM


Unstructured meshes


5

Numerical Schemes


FVM FDM


Unstructured meshes


6

Numerical Schemes


FVM FDM FEM


Unstructured meshes


7

Numerical Schemes


FVM FDM FEM DG


Unstructured meshes


8

Discontinuous Galerkin: Problems

• High-order geometry representation and mesh generation

• High CPU/memory requirements (compared to FVM or high-order FDM)

• Low tolerance to under-resolved features (no robustness)

The challenge is to make DG methods competitive for real-world problems

9

Example: Aeroacoustics and K-H Instability

• Inspired by calculations of Munz et al (2003) (linearized)

• Nonlinear behavior: Large scale acoustic wave interacts with small scale

flow features, leading to vorticity generation

• p = 7 (8th order accuracy), 100-by-20 square elements

10

Example: Flow around Wedge Box

• Simple but representative geometry

• Well-studied aeroacoustics test case

• Low Mach number flow, high order

methods required to capture and prop-

agate acoustic waves

11

Example: Solid Dynamics

• Cast governing equations as system of

first order conservation laws

• Account for involutions by appropriate

selection of approximating spaces

– Curl-free deformation tensor

• Use High-Order DG

• Lagrangian non-dissipative Neo-

Hookean material

12

Outline

• Background





13

The Discontinuous Galerkin Method

• (Reed/Hill 1973, Lesaint/Raviart 1974, Cockburn/Shu 1989-, etc)

• Consider non-linear hyperbolic system in conservative form:

ut +∇ · Fi(u) = 0

• Triangulate domain into simplicial elements

• Use space of element-wise polynomials of degree p

• Solve weighted residual formulation

• ErrorO(hp+1) for smooth problems

• Reduces to finite volume method for p = 0

∂κ

κ

n

uL

uR

14

Nodal Basis and Curved Boundaries

• Nodal basis within each element→ sparse element connectivities

• Isoparametric treatment of curved elements (not only at boundaries)

• Important research topic:

Meshing/Node placement for DG

W1 wing

Isotropic mesh

Anisotropic mesh

15

Viscous Discretization

• Treat the viscous terms by rewriting as a system of first order equations:

ut +∇ · Fi(u)−∇ · Fv(u,σ) = 0

σ −∇u = 0

• Discretize using DG, various formulations and numerical fluxes such as:

– The BR1 scheme (Bassi/Rebay 97):

Averaging of fluxes, connections between non-neighboring elements

– The BR2 scheme (Bassi/Rebay 98):

Different lifting operator for each edge, compact connectivities

– The LDG scheme (Cockburn/Shu 98):

Upwind/Downwind, sparse connectivities, some far connections

– The CDG scheme (Peraire/Persson 06):

Modification of LDG for local dependence – sparse and compact

16

Outline

• Background





17

Sparsity Patterns

and

and

CDG :

LDG :

BR2 :

1

23

4

1

1

2

2

3

3

4

4

18

Implicit DG Memory Requirements

• Example: Navier-Stokes + S-A in 3-D, p = 4, 10, 000 tetrahedra

– DOFs = 6 · 35 · 10, 000 = 2, 100, 000

– Jacobian: 1.20 · 109 matrix entries = 9.6GB

• Observations:

– For p ≥ 2, 6 tetrahedra

cheaper than 1 hex (!)

– p = 2 and p = 3 optimal for

tetrahedra

0 2 4 6 810

2

103

104

Polynomial order p

Sto

rage

Memory Requirements − Fixed resolution r, 3−D

MCDG

Simplex

MCDG

Block

19

Matrix Storage and Operations

• Block matrix representation fundamental for high performance

– Up to factors of 10 difference between optimized matrix libraries and

hand-coded for-loops

• Compromise between general purpose and application specific format

• The CDG scheme is well-suited for a compact block storage format

• With a general format (e.g. compressed column), it is hard to take

advantage of the block structure

20

Convergence of Iterative Solvers

• Convergence of various solvers with block-diagonal preconditioner

• Block Jacobi great when it works, but unreliable

• Krylov solvers comparable convergence

• Focus on GMRES(20)

0 10 20 30 40 50 6010

2

103

104

105

106

107

108

# Matrix−Vector products

Err

or

NACA, ∆ t=10−3

0 50 100 150 200 250 300 350 400 450 50010

1

102

103

104

105

106

107

108

# Matrix−Vector products

Err

or

NACA, ∆ t=10−1

Block JacobiQMRCGSGMRES(20)GMRES

21

Preconditioners for Krylov Methods

• Performance of Krylov methods greatly improved by preconditioning

• Various standard methods:

– Block-diagonal: A ≈ blockdiag(A)

– Block-Incomplete LU: A ≈ LU

– p-multigrid (low-degree) preconditioner

• Highly efficient combination (Persson/Peraire 06)

– Use block-ILU(0) as post-smoother for coarse scale correction

– In particular, convergence is almost independent of Mach number

22

Mach Number Dependence

• The ILU(0)/Coarse scale preconditioner appears to make convergence

almost independent of Mach number

• Individually, ILU(0) and coarse corrections perform poorly for low Mach

• Clearly, ILU(0) is an efficient

smoother and reduces errors

in high frequencies (within and

between the elements)

10−3

10−2

10−1

100

100

101

102

103

Mach Number

# G

MR

ES

Iter

atio

ns

NACA, ∆ t=1.0

Block diagonalILU(0)p0−correctionILU(0)/p0

23

Outline

• Background





24

Artificial Viscosity Idea

• High-order discretization might be unstable when features are unresolved

(Gibbs oscillations)

• Unrealistic to require the mesh to resolve all features

• (Persson/Peraire 06) Add a small amount of viscosity to original equations

so that unresolved features can be represented by approximating space

∂u

∂t+∇ · F (u,∇u) = ∇ · (ν∇u)

ε

2εε ∼ ν

25

Resolution – SA Turbulence Model

0 4 80.0

0.2

0.4

0.6

0.8

1.0

Velocity

viscosityEddy

Shearstress

1 10 100 1000 100000

5

10

15

20

25

30

Rθ = 104

(from Spalart-Allmaras 1992)

Eddy Viscosity Profile

• Benign at the wall

• High resolution required at boundary

layer edge

26


Eddy Viscosity Profile:

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

y/δ ∗

Coarse Mesh

hfill

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

y/δ ∗

Fine Mesh

27


Eddy Viscosity Profile:

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

y/ δ*�

ε

In the transition region ν ∼ ν

∂ν

∂y∼ (ν + ν)

∂2ν

∂y2+ C

(∂ν

∂y

)2

ν

ε∼ νν

ε2⇒ ε ∼ ν !!!

Detailed structure of ν in transition region likely to have small effect on overall solution

28

Artificial Viscosity vs. Limiting

Width δs to capture a discontinuity:

• Low order schemes:

– Modern Finite Volume Schemes: δs ∼ 2h

– Artificial Viscosity Methods: δs ∼ 4h

• High order schemes:

– Limiting: δs ∼ 2h (Not obvious how to exploit increased resolution)

– Artificial Viscosity Methods: δs ∼ 4h/p (Easy to extend to high order)

h

h/p

29

Example: Transonic Flow over NACA Airfoil

• NACA 0012, M = 0.8, p = 4

• Shock stabilization by artificial

viscosity with non-linear sensor

(Persson/Peraire 06)

• Implicit steady-state solution with

Newton solver

Pressure

Entropy Sensor

30

Example: Supersonic Flow over NACA Airfoil

• NACA 0012, M = 1.5, p = 4

• Shock stabilization by artificial

viscosity with non-linear sensor

(Persson/Peraire 06)

• Implicit steady-state solution with

Newton solver

Mach

Sensor

31

Proposed Approach

• Use Artificial Viscosity with a Non-Linear Sensor

• Select Parameters based on Available Resolution

• DG implementation

– Consistent Treatment of High Order Terms (CDG)

– Do not exploit nature of Discontinuous Approximation

• Artificial Viscosity Models:

Shocks

– Viscosity Model: Laplacian form∇ · (ν1∇U )

– Sensor: Density

Eddy Viscosity Equation

– Viscosity Model: Laplacian form∇ · (ν2∇ν)

– Sensor: ν32

Sensor

• Determine regularity of the solution based on rate of decay of expansion

coefficients

• Periodic Fourier case:

f(x) =∞∑

k=−∞

gkeikx

If f(x) has m continuous derivatives → |gk| ∼ k−(m+1)

• For simplices: Use orthonormal Koornwinder basis within each element

33

Example: Flat Plate

• M = 0.2, Re = 1.02 · 107

Eddy Viscosity Sensor

p = 4

34

Example: Flat Plate

Velocity Skin friction

100

101

102

103

104

0

5

10

15

20

25

30

y

u

Law of the wallExp. datap =4, grid Cp =2, grid Bp =1, grid A

+

+

0 2 4 6 8 10

x 106

0

1

2

3

4

5

6x 10

−3

Rex

Cf

Exp. datap =4, grid Cp =2, grid Bp =1, grid A

p = 1, 2, 4 (constant h/p)

35

Example: Flat Plate

Output: Drag on (0.0148 ≤ x ≤ 4.687)

A B C10

−8

10−7

10−6

10−5

10−4

CD

err

or

Grid

p =1

p =2

p =3

p =4

Convergence rate∼ hp

36

Example: RAE2822

• M = 0.675, α = 2.31o, Re = 6.5 · 106

p = 2 (constant h/p) p = 4

CL = 0.6144 CD = 0.0104 CL = 0.6131 CD = 0.0103

37

Example: RAE2822

• M = 0.675, α = 2.31o, Re = 6.5 · 106

p = 2 (constant h/p) p = 4

CL = 0.6144 CD = 0.0104 CL = 0.6131 CD = 0.0103

38

Example: RAE2822

Cp Cf

0 0.2 0.4 0.6 0.8 1

−1.5

−1

−0.5

0

0.5

1

x/c

Cp

p =4p =2

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

4

5

6

x 10−3

x/c

Cf

p =4p =2

39

Example: NACA0012

• M = 0.85, Re = 1.85 · 106

p = 3 (constant h) p = 4

CD = 0.05347 CD = 0.05346

40

Example: NACA0012

• M = 0.85, Re = 1.85 · 106

p = 3 (constant h) p = 4

CD = 0.05347 CD = 0.05346

41

Example: NACA0012

0 0.2 0.4 0.6 0.8 1

−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

x/c

Cp

p =4p =3p =2

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4 x 10−3

x/c

Cf

p =4p =3p =2

p = 2 CD = 0.05185

p = 3 CD = 0.05347

p = 4 CD = 0.05346

42

Example: NACA0012

• Sensor for ν

p = 3

p = 4

43

Example: NACA0012

Mach Number Shock Sensor

p = 4

44

Outline

• Background





45

Background and Applications

• Deformable domains important in many applications:

– Rotor-stator flows

– Flapping flight

– Fluid-structure interaction

• Several attempts using Arbitrary Lagrangian-Eulerian (ALE) formulations

– Equations discretized on a deforming grid, time-dependent metric

– At most third order accuracy in space and time demonstrated

• Alternative approach [Visbal/Gaitonde 02] in finite difference setting

– Map from fixed reference domain to real time-varying domain

• We use a mapping approach, however in a DG setting and with a

conservative formulation for free-stream preservation

46

ALE Formulation

• Map from fixed reference domain V to physical deformable domain v(t)

• A point X in V is mapped to a point x(t) = G(X, t) in v(t)

• Introduce the mapping deformation gradient G and the

mapping velocity vX as

G = ∇XG

vX =∂G∂t

∣∣∣∣X

and set g = det(G)

X1

X2

NdA

V

x1

x2

nda

vG , g, vX

47

Transformed Equations

• The system of conservation laws in the physical domain v(t)

∂Ux

∂t

∣∣∣∣x

+ ∇x · Fx(Ux, ∇xUx) = 0

can be written in the reference configuration V as

∂UX

∂t

∣∣∣∣X

+ ∇X · FX(UX , ∇XUX) = 0

where

UX = gUx , FX = gG−1Fx −UXG−1vX

and

∇xUx = ∇X(g−1UX)G−T = (g−1∇XUX −UX∇X(g−1))G−T

• Proof. See paper (only a few lines)

48

Geometric Conservation Law

• A constant solution in v(t) is not necessarily a solution of the discretized

equations in V , due to inexact integration of the Jacobian g

• The time evolution of g is

∂g

∂t

∣∣∣∣X

−∇X · (gG−1vX) = 0 ,

which in general is non-zero

• Visbal and Gaitonde added corrections to cancel the errors

• Our approach solves instead the convervative system

∂(gg−1UX)

∂t

∣∣∣∣X

−∇X · FX = 0

∂g

∂t

∣∣∣∣X

−∇X · (gG−1vX) = 0

49

Mappings for Deformable Domains

• Need mapping x = G(X, t) given the boundary deformations

• Common approach in ALE settings: Move boundary mesh nodes, update

interior nodes by mesh smoothing (Laplacian, spring analogy, etc)

– Would require interpolation in both space and time to obtain G, vX

– Stability and accuracy of scheme unclear

• Can also solve PDEs (usually elliptic) for

x in the interior

• In this work, we use a blending technique

which gives explicit expressions for x =

G(X, t)

– Easy to differentiate to obtain G, vX

50

Blending of Meshes

• Our approach forms a smooth weighted combination of two meshes,

displaced according to each boundary deformation

Original Mesh Rigidly Moved Mesh

Blended Mesh

51

Example: Euler Vortex

• Propagate an Euler vortex on a variable domain with

x(ξ, η, t) = ξ + 2.0 sin(2πξ/20) sin(πη/7.5) sin(1.0 · 2πt/t0)

y(ξ, η, t) = η + 1.5 sin(2πξ/20) sin(πη/7.5) sin(2.0 · 2πt/t0)

• Deformed meshes used for visualization only, everything is computed on

the reference mesh

52

Example: Euler Vortex, Convergence

• Optimal order of convergence O(hp+1) for mapped scheme

10−1

100

10−8

10−6

10−4

10−2

100

Element size h

L 2−er

ror

p=1

p=2

p=3

p=4

p=5

12

1

6

MappedUnmapped

53

Example: Free-stream Preservation

• Solve with initial free-stream solution u0(x, y) = u∞

• L2-error of error u(x, y, T )− u0 with and without g equation

10−1 10010−15

10−10

10−5

100

Element size h

L 2−erro

r

p=1p=1

p=2p=2

p=3 p=3

p=4

p=4p=5

p=5

With Free−stream PreservationWithout Free−stream Preservation

54

Example: Flow around Oscillating Cylinder

• Oscillating unit circle, yc(t) = A sin(2πft) with A = 4/3,f = 0.1.

Reynolds number 400, Mach 0.2, degree p = 4 approximation

• Compare rigid and blended mappings

Rigid Mapping Blended Mapping

55

Example: Flow around Oscillating Cylinder

• Good agreement between different mappings of the same domain

deformation

Rigid Mapping

0 5 10 15 20 25 30 35−5

0

5

10

Time

DragLift

Blended Mapping

0 5 10 15 20 25 30 35−5

0

5

10

Time

DragLift

56

Example: Heaving and Pitching Foil in Wake

• NACA 0012 foil heaving and pitching in wake of D-section cylinder

• Both oscillate y(t) = A sin(2πft), foil pitching θ = a sin(2πft + π/2)

• Based on experimental study [Gopalkrishnan et al 94]

57

Example: Heaving and Pitching Foil in Wake

• Thrust/Drag of combined cylinder/foil highly dependent on location/phase

• Future work: Parametric study to identify different modes of interaction

Lift

0 2 4 6 8 10 12 14 16 18−3

−2

−1

0

1

2

3

Time

Lift (cylinder)Lift (foil)

Drag

0 2 4 6 8 10 12 14 16 18−2

−1

0

1

2

3

Time

Drag (cylinder)Drag (foil)

58

Example: Pitching Airfoil

• HT13 airfoil attached to translating and heaving point by torsional spring

• Fluid properties: Re = 5000, M = 0.2

• Forced vertical motion rz(t) = r0 sin ωt (at leading edge)

• Moment equation: Iθ + Cθ − Srz(t) + Maero = 0

– I moment of inertia, C spring stiffness, S = mxc static unbalance

– Maero moment from fluid

rz(t) θ

59

Example: Pitching Airfoil

Low Amplitude

0 10 20 30 40 50 60 70 80 90 100

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time

rz

θMDL

High Amplitude

0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5

Time

rz

θMDL

60

Conclusions

• Important steps toward a practical DG solver for real-world problems

• Efficient discretizations and solvers

• Artificial viscosity approach for producing grid independent RANS

solutions and sub-cell shock capturing

• Effect of grid distortion not fully understood (grid generation ?)

• RANS is much harder than Euler/Laminar NS

• Optimal accuracy for deformable domains by mapping approach

• Current work: Extensions for LES/DES, parallelization for large scale 3-D

problems, shock capturing for solids, etc

61

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