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1 P. Ackerer, IMFS, Barcelona 2006 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides, Strasbourg, France [email protected]
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Page 1: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

1P. Ackerer, IMFS, Barcelona 2006

About Discontinuous Galerkin Finite Elements

P. Ackerer, A. Younès

Institut de Mécanique des Fluides et des Solides, Strasbourg, France

[email protected]

Page 2: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

2P. Ackerer, IMFS, Barcelona 2006

OUTLINE

1. Introduction

2. Solving advective dominant transport 2.1.          Eulerian methods: Finite Volumes, Finite Elements

3. Galerkin Discontinuous Finite Elements 3.1.          1D discretization3.2.          General formulation3.3.          Numerical integration3.4. Slope limiter

4. Numerical experiments 4.1.          2D – 3D benchmarks4.2.          Comparisons with finite volumes

5. On going works

Page 3: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

3P. Ackerer, IMFS, Barcelona 2006

xjxj-1xj-2 xj+1

xj-1/2 xj+1/2Finite volumes

xjxj-1xj-2 xj+1

Finite elements

xjxj-1xj-2 xj+1

Discontinuous finite elements

x

n+1

n

n-1

t

j j+1j-1j-2

Space/time discretization

Introduction

Page 4: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

4P. Ackerer, IMFS, Barcelona 2006

C Cu 0

t x

Finite differences method (FD):

2 2

2

2 2

2

f x ff (x x) f (x) x ...

x 2 x

f x ff (x x) f (x) x ...

x 2 x

f f (x x) f (x)

x xf f (x x) f (x x)

x 2 x

n* n*n 1 nj j 1j j

C CC Cu 0

t x

n* n*n 1 nj 1 j 1j j

C CC Cu 0

t 2 x

Basic ideas:

1. Use Taylor’s (1685-1731) series 2. Replace the derivatives

Richardson (1922) was first to apply FD to weather forecasting. It required 3 months' worth of calculations to predict weather for next 24 hours.

Introduction

Page 5: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

5P. Ackerer, IMFS, Barcelona 2006

n 1 nj j n* n*

j 1/ 2 j 1/ 2

C Cx u C C

t

j 1/ 2 j j 1

1C C C

2 j 1/ 2 jC C

xjxj-1xj-2 xj+1

xj-1/2 xj+1/2

uFV have a very strong physical meaning

n* n*n 1 nj j 1j j

C CC Cu 0

t x

n* n*n 1 nj 1 j 1j j

C CC Cu 0

t 2 x

Finite Volumes methods

Introduction

Page 6: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

6P. Ackerer, IMFS, Barcelona 2006

n 1 nj j n n

j 1 j

C Cx u C C

t

Some key numbers (1D)

n 1 n nj j j 1

u tC C (1 ) C

x

2 2

2

2

2

C x CC(x x) C(x) x ...

x 2 x

C C(x x) C(x) x C

x x 2 x

C Cu 0

t x

2

2

C C(x x) C(x) x Cu u 0

t x 2 x

u xD or

2u x

Grid Peclet number 2D

To reduce numerical diffusion

u tCFL 1

x

To avoid oscillation for this scheme

(R. Courant, K. Friedrichs & H. Lewy ,1924)

Introduction

Page 7: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

7P. Ackerer, IMFS, Barcelona 2006

Galerkin Finite Elements method

Basic ideas:

1. Approximate the unknown function by a sum of ‘simple’ functionsne

j jj 1

C(x, t) (x)C (t)

i j

j ii j

1 if x x(x )

0 if x x

j jC(x , t) C (t)with so that

CL(C) uC D C 0

t

xjxj-1xj-2 xj+1

FE

2. The numerical solution should be as close as possible to the exact solution over the domain

L(C(x, t)) (x) 0

d(x)for any

iL(C(x, t)) (x) 0

dwith i=1 to ne,which leads to ne equations with ne unknowns

xjxj-1xj-2 xj+1

FV

u

Introduction

Page 8: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

8P. Ackerer, IMFS, Barcelona 2006

Basic ideas:

3. Choose i i(x) (x) which leads to

j j

jj j j j i

j j

C

u C D C d 0t

n 1 nj j j j

j j n 1 n 1j j j j i

j j

C C

u C D C d 0t

4. Standard Euler/implicit scheme for time discretization, for example

written for i=1 to ne.

The next steps are more or less easy mathematics ...

Introduction

Page 9: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

9P. Ackerer, IMFS, Barcelona 2006

Galerkin Discontinuous Finite elements method

Basic ideas:

1. Approximate the unknown function by a sum of ‘simple’ functions INSIDE an element E

xjxj-1xj-2 xj+1

FE

xjxj-1xj-2 xj+1

DFE

u

Discontinuous Finite Elements

2. Defining on node/edge/face A inside of E and on edge/face A outside of E

inAC

outAC

inj 1C

outj 1C

j jj 1

C(x, t) (x)Y (t)

Yj(t) : degree of freedom (nodal conc., ….)

Page 10: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

10P. Ackerer, IMFS, Barcelona 2006

Basic ideas:

3. Second order explicit Runge-Kutta scheme

2

tt,tt

n 1 2 nE An in nE E

E AA EE E A

QC Cw dE UC w dE C wds

t 2 A

/

, ,./

E,AQ

A

: the flux through A, positive if pointed outside

: norm of A (length, surface).

Step 1:

*,n 1 nE,An 1/ 2 in or out,n+1/2E E

E AA EE E A

QC Cw dE UC . w dE wC ds

t A

in,n+1/2A,Ein or out,n+1/2

A out,n+1/2A,E

C for outflowC

C for inflow

Step 2:

Discontinuous Finite Elements

Page 11: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

11P. Ackerer, IMFS, Barcelona 2006

Basic ideas:

4. Oscillations avoided by slope limitation

xjxj-1xj-2 xj+1

XCo

nc

55 60 65 70 75

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2DFG No Limit.

XCo

nc

55 60 65 70 75

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2DFG with Limit.

Discontinuous Finite Elements

Page 12: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

12P. Ackerer, IMFS, Barcelona 2006

C(uC)

t

C

tw dx w dx C w x C w x

E i iE

i i i i i i uC u( 1 1( ) ( ))

Hyperbolic 1D

C

tw dx uC w dx i i

EE

.( )

Variational form

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

i 1

i

x x(x)

x

i

i 1

x x(x)

x

i i i 1 i 1C(x, t) (x)C (t) (x)C (t)

Linear approximation

DGFE : 1D discretization

Page 13: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

13P. Ackerer, IMFS, Barcelona 2006

i iw (x) (x)

Galerkin formulation

*

i i iEE

*

i 1 i 1 i 1EE

Cw dx uC w dx + uC

tC

w dx uC w dx uCt

Discretization

i i 1

i i 1

n 1 n 1 n n *,n

i i 1 i

n 1 n 1 n n *,n

i i 1 i 1

u t u t 6 t2C C C (2 3 ) C (1 3 ) u C

x x xu t u t 6 t

C 2C C (1 3 ) C (2 3 ) u Cx x x

Explicit formulation leads to a local system:

xi+1xixi-1 xi+2

E

DGFE : 1D discretization

Page 14: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

14P. Ackerer, IMFS, Barcelona 2006

xi+1xixi-1 xi+2

E

DGFE : 1D discretization

t t, t t / 2 Step 1:

i i 1

i i 1

n 1/ 2 n 1/ 2 n n n

i i 1 i

n 1/ 2 n 1/ 2 n n n

i i 1 i 1

u t / 2 u t / 2 6 t / 22C C C (2 3 ) C (1 3 ) u C

x x xu t / 2 u t / 2 6 t / 2

C 2C C (1 3 ) C (2 3 ) u Cx x x

Step 2:

i i 1

i i 1

n 1 n 1 n 1/ 2 n 1/ 2 n 1/ 2,in or out

i i 1 i

n 1 n 1 n 1/ 2 n 1/ 2 n 1/ 2,in or out

i i 1 i 1

u t u t 6 t2C C C (2 3 ) C (1 3 ) u C

x x xu t u t 6 t

C 2C C (1 3 ) C (2 3 ) u Cx x x

t t, t t

Page 15: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

15P. Ackerer, IMFS, Barcelona 2006

XCo

nc

55 60 65 70 75

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2DFG No Limit.

XCo

nc

55 60 65 70 75

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2DFG with Limit.

Slope limitation

xi+1xixi-1 xi+2

En 1 n 1 n 1 n 1

i i 1 i i 1

n 1

E 1 E i E 1 E

n 1

E E 1 i 1 E E 1

C C C Cx x

2 2min(C ,C ) C max(C ,C )

min(C ,C ) C max(C ,C )

DGFE : 1D discretization

Page 16: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

16P. Ackerer, IMFS, Barcelona 2006

C

tuC .( )

General formulation

Variational formn 1 n *

E E A

E E,AA EE E A

C C C ww dE UC . w dE Q ds

t A

A : norm of A (length, surface).

A,EQ : the flux through A, positive if pointed outside

Polynomial approximation

E 1 2 3C (X, t) Y (t) xY (t) yY (t) Linear (2D):

E 1 2 3 4C (X, t) Y (t) xY (t) yY (t) xyY (t) Bi-Linear (2D):

DGFE : General formulation

Page 17: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

17P. Ackerer, IMFS, Barcelona 2006

Standard interpolation functions

(1,1)

(0,0)

1

4 3

2

Bilinear interpolation

1

2

3

4

(x,y)=(1-x)(1-y),

(x,y)=x(1-y),

(x,y)=xy,

(x,y)=y(1-x).

Linear interpolation

(1,1)

(0,0)

1

2

3

(x, y) 1,

(x, y) x x,

(x, y) y y.

E x yC (X, t) C(t) x x C (t) y y C (t)

DGFE : General formulation

Page 18: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

18P. Ackerer, IMFS, Barcelona 2006

Step 1 : t t, t t / 2

n 1/ 2 n in ,n

nE E A

E E,AA EE E A

C C CdE UC . dE Q ds

t / 2 A

A,EQ

A

: the flux through A, positive if pointed outside

: norm of A (length, surface).

Step 2 :

n 1 n in or out,n+1/2

n 1/ 2E E A

E E,AA EE K A

n

C C CdE UC . dE Q ds

t A

t t, t t

outA

inA C ,C A,EQ: depending on the sign of

DGFE : General formulation

Page 19: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

19P. Ackerer, IMFS, Barcelona 2006

x y

n

E 2E

2

C C C

1 E 0 0C w dE

(x x) 0 (x x) (x x)(y y)

(y y) 0 (x x)(y y) (y y)

n 1 n

E E

EE E

C Cw dE and UC . w dE

t

Numerical integration (1)

Exact integration in reference element for E

DGFE : Numerical integration

Page 20: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

20P. Ackerer, IMFS, Barcelona 2006

Exact numerical integration with Simpson’s rule (pol. Ordre 2)

EI f ( ) 4f ( ) f ( )

6

i jxx x j k

EI f ( ) f ( ) f ( )

3

ix xx

i

kjj

EI f ( ) 4f ( ) 16f ( )

36 i

x xx

Numerical integration (2)*

A

A

C wds

A

DGFE : Numerical integration

Page 21: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

21P. Ackerer, IMFS, Barcelona 2006

0 i 0,i 0 imin(C ,C ) C max(C ,C )

*

*

, , ,

, , .

W Ex x

N Sy y

C M C C C C C

C M C C C C C

sign( ) min( , , ), if sign( ) sign( ) sign( ),(a,b,c)=

0 otherwise.

a a b c a b cM

DGFE : Slope limiting

Page 22: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

22P. Ackerer, IMFS, Barcelona 2006

Step 3 : Multidimensional slope limiter (Bilinear function)

Ei

min(i)/max (i) : min/max ofover each element containing i

*,n 1

EC min(E)/max (E) : min/max value of

over each element which has a common node with E.

*,n 1

EC

E

nn 2n 1 n 1 n 1 *,n 1

E,1 E,nn E,i E ,ii 1

J(C ,...,C ) C C

Optimization :

Constraints :n 1 *,n 1

E EC C

n 1

E,imin(i) C max(i)

*,n 1 *,n 1

E E C max(E) or C min(E)

thenn 1 *,n 1

E,i EC C

Extrema :

DGFE : Slope limiting

Page 23: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

23P. Ackerer, IMFS, Barcelona 2006

1rd order Upwind

Centered 3rd order Upwind

ImplicitCFL=1

CFL=5

CFL=1

CFL=5

CFL=1

CFL=5

Crank-Nicholson

CFL=1

CFL=5

CFL=1

CFL=5

CFL=1

CFL=5

1rd order BDFCFL=1

CFL=5

CFL=1

CFL=5

CFL=1

CFL=5

Flux discretisation

Tim

e di

scre

tiza

tion

DGFE, CFL=1

FE, CFL=1

FE, CFL=5

DGFE : Numerical experiments

1D Benchmarks

Page 24: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

24P. Ackerer, IMFS, Barcelona 2006

Bilinear, CFL=0,6 Linear, CFL=0,6

Linear, CFL=0,6Bilinear, CFL=0,1

DGFE : Numerical experiments

2D Benchmarks

Page 25: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

25P. Ackerer, IMFS, Barcelona 2006

1 Ty x vVelocity field

DGFE : Numerical experiments

3D Benchmarks

Page 26: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

26P. Ackerer, IMFS, Barcelona 2006

Finite volume Bilin. DGFE

DGFE : Numerical experiments

Page 27: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

27P. Ackerer, IMFS, Barcelona 2006

Finite volume (CFL = 0.50) D-GFE (CFL = 0.50) CFL=0,1 CFL=0,5 CFL=1 CFL=2 CFL=5 CFL=10

E.F.D 0.363 0.684 0.887 1.109 1.383 1.549

V.F 1.276 1.343 1.406 1.491 1.622 1.713

V.F.2 0.987 1.054 1.123 1.225 1.398 1.554

EFD : 10000 cells, 30 000 unk.

VF : 10000 cells, 10 000 unk., VF 2: 40000 cells, 40 000 unk.

DGFE : Numerical experiments

Comparisons with Finite Volumes

Page 28: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

28P. Ackerer, IMFS, Barcelona 2006

Discontinuous Galerkin: well known algorithms

DGFE : Summary

Efficient in tracking fronts

Well adapted to change interpolation order from one element to the other

BUT

Explicit scheme ……

Summary

Page 29: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

29P. Ackerer, IMFS, Barcelona 2006

DGFE : On going work

Implicit upwind formulation n 1 n in or out ,*

*E E A

E E,AA EE E A

C C CdE UC . dE Q ds

t A

A,EQ

A

: the flux through A, positive if pointed outside

: norm of A (length, surface).

Time domain decomposition

Page 30: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

30P. Ackerer, IMFS, Barcelona 2006

X20 40 60 80 100

Xt

t+t

t+3t/4

t+t/2

t+t/4

Time domain decomposition

DGFE : On going work

DGFE, CFL=1

x 0.2;4.0

Page 31: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

31P. Ackerer, IMFS, Barcelona 2006

n 1 n in or out ,*

*E E A

E E,AA EE E A

C C CdE UC . dE Q ds

t A

Implicit upwind formulation

DGFE : On going work

* n n 1

E E EC (1 )C C

Page 32: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

32P. Ackerer, IMFS, Barcelona 2006

DGFE : On going work

Page 33: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

33P. Ackerer, IMFS, Barcelona 2006

DGFE : On going work

Page 34: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

34P. Ackerer, IMFS, Barcelona 2006

Next to come ….

DGFE : On going work

Page 35: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

35P. Ackerer, IMFS, Barcelona 2006

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36P. Ackerer, IMFS, Barcelona 2006

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37P. Ackerer, IMFS, Barcelona 2006


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