Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 1/24...

Numerical Simulation of Complex and Multiphase Flows 18th – 22nd April 20005 Porquerolles 1/24

Finite volumes and finite elements for the numerical simulation of wave breaking

F. Golay

University of Toulon, FranceANAM/MNC


• Numerical simulation of wave breaking

• Finite volume and finite element code

• Mesh refinement

Plan


• Mathematical model

• Numerical model

• Numerical results

Numerical simulation of wave breaking


Numerical simulation of wave breaking: Mathematical model

0ut

ugu)pE(divt

E

g)Ipuu(divt

u

0)u(divt

2

(x,y,t) is the density

u(x,y,t) is the velocity

g is the gravity

1E u is the energy

2is the internal energy

0 1 is the fraction of fluid

p( , , ) is the pressure

where

Equation Of State: stiffened gaz

(Abgrall-Saurel, 1996)

1)1(

11)(

)()(

1

1)1(

1

1

1)(

1

)()(1)(p

a

aa

w

ww

aw

)p(

c

Sound velocity

P. Helluy, F. Golay:”Mathematical and Numerical aspects of Low Mach Number Flows”, Porquerolles 2004


The system has the form of a system of conservation laws

11 2

1 2

( ).

( )

: approximation of in the volume at time ,

: numerical flux (exact Godunov),

( ), ( ) : length of , volume of .

i

n n ii i

i C

ni i n

i i i i

L Cw w t F G

V C

w w C t

F G

L C V C C C

1 2

21 1 1 2 1 1

22 1 2 2 2 2

2

( ) ( ) ( ),

( , , , , ),

( ) ( , , , , ( ) , ),

( ) ( , , , , ( ) , ),

( ) (0,0, , ,0).

t x yw F w G w H w

w u u E

F w u u p u u E p u u

G w u u u u p E p u u

H w g gu

We solve it by a standard finite volume scheme

Ci

Cj

•Second order extension:MUSCL•No pressure oscillation thanks to a special non-conservative discretisation of the fraction evolution.

Numerical simulation of wave breaking: Numerical model


Numerical simulation of wave breaking: Test case

In the air sound velocity c=20m/s, p=105 Paa=-99636 Pa, a=1.1

In the water sound velocity c=20m/s, p=105 Paw=263636 Pa, w=1.1


Mesh: 2000x150

Numerical simulation of wave breaking: Numerical results wave propagation


Numerical simulation of wave breaking: Numerical results wave breaking


• Simple and efficient method: no interface tracking• The same code can be used for compressible multifluid flows

Improvements:• Unstructured mesh, automatic mesh refinement• A posteriori error• Physical interaction• Mixed numerical method

Numerical simulation of wave breaking: Partial conclusion

Integration in a finite element code


• Finite element formulation

• Finite volume formulation

• Software architecture

• Validation

Finite volume in a finite element code


Finite volume/element formulation

e

e

e e

e

e

i i

e e

dUdivF S dans

dtdU

dx divFdx Sdx ...dt

dUdx Fdx div( F)dx Sdx

dt

dUdx

..dx ..dx U(x)=N

Fdx Fndx Sdxdt

Udx Fdx Sdx

..

(

.

)U

t

x

e e e

i ij j j

j

e e e

N UN dx N Fdx N Sdx ...

t

=N (x)

Finite element formulation

Finite volume formulation

e

e

ee

e

is an indicatrice function over each ele

f (U

men

,

t

dUdx Sdx U

t,n x

d)d

Discontinuous finite element formulation

e e e

e

e e e e

e

e

e e

e e e e

dUdx divFdx Sdx

dt

dUdx Fdx Fndx ... Sdx

d

..dx .

t

.dx

Baumann, Oden (2000)


FV & FE: Finite Volume formulation

0ut

ugu)pE(divt

E

g)Ipuu(divt

u

0)u(divt

x

ye

u

uU

E

e

ee

1U(x) U U(x)dx

e

e e

l rlree e

ef (U ,U

tU (t t) ,n )dU (t) x x Sd

Geometrical node with no dof

Centroid node with 5 dof

1

2

4 3

Compute numerical flux exact Godunov schemeHelluy, Barberon, Rouy 2003

1

24

3

5N+1

Compute nodal load vector Estimation of U with slope limiter Display the result


FV & FE: Software architecture

Object oriented finite element code: SIC (Systeme Interactif de Conception)Touzot, Aunay, Breitkopf 1985

ObjectNameIdentifieur

Template:Character arrayReal arrayInteger array……

An object could be:- created- duplicated- listed- modified- …

Exemple of object :- a node- a element- a kinematic condition- a matrix- a vector- a command- a model- …

Object element

Identifieur

modelnumberzoneId material propertiesId geometric propertiesId element propertiesId interpolation functionId save vectorList of nodesList of load caseMesh refinement parameter edges numberList of neighbour elements

Object node

Identifieur

Id kinematic condition Id load casenumber X coordinateY coordinateZ coordinateDegree of freedomNodal propertiesEquation numbersList of elements

http://sic.univ-tln.fr


FV & FE: Validation

x y3 2 4

, p , u , u 04 3 3

x y1 , p 1 , u 1 , u 0

Stationnary choc

Test 1

x y2 , p 2 , u 0 , u 0 x y1 , p 1 , u 0 , u 0

Test 2


1,00

1,20

1,40

1,60

1,80

2,00

-0,50 -0,25 0,00 0,25 0,50

espace °1 time °1

espace °2 time °1

espace °2 time °2

exact

1,50

1,60

1,70

1,80

1,90

2,00

- 0,30 - 0,25 - 0,20 - 0,15 - 0,10

espace °1 time °1

espace °2 time °1

espace °2 time °2

exact

1,20

1,30

1,40

1,50

1,60

0,00 0,02 0,04 0,06 0,08 0,10

espace °1 time °1

espace °2 time °1

espace °2 time °2

exact


Mesh refinement / unrefinement / adaptation

• Finite element mesh refinement

• example: topologic optimization

• Quadtree mesh refinement

• Unrefinement


Mesh refinement: Finite element mesh refinement

e1e1 e2

e3e4e1

Refinement

e1e1

e2e3

e4

e1e1

e2

e1 e1e2

e3

e1 e1

e2

e3

e1

e2e3

e4

e1

conformity

e1 e1 e2

e1e1

e2


Mesh refinement: Mesh refinement Test

P=0,2,4 P=4,6,8 P=8,10,12 P=12,14,16

Criterion 1:

Criterion 2: Verfürth

Initial Mesh

Error

eface e

2

u dlD)u(h2

1e

2ece deR 22

e r

2

1

e

2eKerror

Criterion R. Verfürth (2000)


+1

+1

+1

?

ux=0uy=0 x

y

ux=0

+1/2

+1

P=0,2,4399 nodes130 elements




Mesh refinement: Mesh refinement & topologic optimisation


• time cpu improved

• best precision

• « static » front captured

• but conformity!

• local unrefinement is difficult


Mesh refinement: Quadtree mesh refinement

Hierarchical approach on quadrilateral


1) Loop on volume to set a refinement criteria

2) Loop on nodes to find patch to unrefine- 4 volumes at same hierarchical

level- 4 edge at same hierarchical

level

Modification of the central nodeDestruction of the other central nodesDestruction of the central edge elementsModification of the peripheral edges

Loop on the nodes to merge edges if necessary

Mesh refinement: Quadtree mesh unrefinement


Mesh refinement: Wave breaking

To be continued …..New posteriori error criteria

Interface captured by the entropy jump


Conclusion

• Compressible bi-fluid model• Finite volume formulation with exact Rieman solver (integration in FE code)• Validation: simulation of wave breaking (confrontation with others models)• Integration in a finite software architecture• Quadtree mesh (un)refinement•…•…• 3D• Parallel implementation• A posteriori error• Multiphysic simulation

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Numerical Simulation of Complex and Multiphase Flows 18 th – 22 nd April 20005 Porquerolles 1/24...

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