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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
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1P. Ackerer, IMFS, Barcelona 2006

P. Ackerer, A. Younès

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

[email protected]

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

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

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

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

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

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

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

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., ….)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

25P. Ackerer, IMFS, Barcelona 2006

1 Ty x vVelocity field

DGFE : Numerical experiments

3D Benchmarks

26P. Ackerer, IMFS, Barcelona 2006

Finite volume Bilin. DGFE

DGFE : Numerical experiments

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

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

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

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

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

32P. Ackerer, IMFS, Barcelona 2006

DGFE : On going work

33P. Ackerer, IMFS, Barcelona 2006

DGFE : On going work

34P. Ackerer, IMFS, Barcelona 2006

Next to come ….

DGFE : On going work

35P. Ackerer, IMFS, Barcelona 2006

36P. Ackerer, IMFS, Barcelona 2006

37P. Ackerer, IMFS, Barcelona 2006

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