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

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Finite volumes and finite elements for the numerical simulation of wave breaking F. Golay University of Toulon, France ANAM/MNC. Plan. Numerical simulation of wave breaking Finite volume and finite element code Mesh refinement. Numerical simulation of wave breaking.

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## Finite volumes and finite elements for the numerical simulation of wave breaking F. Golay

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

F. Golay

University of Toulon, France

ANAM/MNC

Plan

• Numerical simulation of wave breaking
• Finite volume and finite element code
• Mesh refinement

Numerical simulation of wave breaking

• Mathematical model
• Numerical model
• Numerical results

Numerical simulation of wave breaking: Mathematical model

(

)

=

g

j

-

re

-

g

j

p

j

p

(

)

1

(

)

(

)

1

1

1

g

+

p

(

p

)

=

j

+

j

-

(

1

)

=

c

g

j

-

g

-

g

-

(

)

1

1

1

r

w

a

g

j

p

j

g

p

g

p

(

)

(

)

w

w

a

a

=

j

+

j

-

(

1

)

g

j

-

g

-

g

-

(

)

1

1

1

w

a

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

where

Sound velocity

Equation Of State: stiffened gaz

(Abgrall-Saurel, 1996)

Ci

Cj

The system has the form of a system of conservation laws

We solve it by a standard finite volume scheme

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

Numerical simulation of wave breaking: Test case

In the air sound velocity c=20m/s, p=105 Pa

pa=-99636 Pa, ga=1.1

In the water sound velocity c=20m/s, p=105 Pa

pw=263636 Pa, gw=1.1

Numerical simulation of wave breaking: Numerical results wave propagation

Mesh: 2000x150

• Simple and efficient method: no interface tracking
• The same code can be used for (un)compressible multifluid flows
• Improvements:
• Unstructured mesh, automatic mesh refinement
• A posteriori error
• Physical interaction
• Mixed numerical method

Integration in a finite element code

Finite volume in a finite element code

• Finite element formulation
• Finite volume formulation
• Software architecture
• Validation

Finite volume/element formulation

Discontinuous finite element formulation

Baumann, Oden (2000)

Finite volume formulation

Finite element formulation

FV & FE: Finite Volume formulation

Geometrical node with no dof

Centroid node with 5 dof

N+1

5

2

1

4

3

4

2

3

1

Compute numerical flux

exact Godunov scheme

Helluy, Barberon, Rouy 2003

• Estimation of U with slope limiter
• Display the result

FV & FE: Software architecture

Object element

Identifieur

Object

model

number

zone

Id material properties

Id geometric properties

Id element properties

Id interpolation function

Id save vector

List of nodes

Mesh refinement parameter edges number

List of neighbour elements

Name

Identifieur

Template:

Character array

Real array

Integer array

Object node

Identifieur

Id kinematic condition

number

X coordinate

Y coordinate

Z coordinate

Degree of freedom

Nodal properties

Equation numbers

List of elements

Object oriented finite element code: SIC (Systeme Interactif de Conception)

Touzot, Aunay, Breitkopf 1985

• Exemple of object :
• - a node
• a element
• a kinematic condition
• a matrix
• a vector
• a command
• a model
• An object could be:
• - created
• duplicated
• listed
• modified

http://sic.univ-tln.fr

FV & FE: Validation

Test 2

Test 1

Stationnary choc

Mesh refinement / unrefinement / adaptation

• Finite element mesh refinement
• example: topologic optimization
• Unrefinement

Mesh refinement: Finite elementmesh refinement

e3

e4

e1

e3

e4

e2

e1

e2

e1

e3

e1

e1

e1

e1

e1

e1

e1

e2

e1

e1

e2

e2

e1

conformity

e2

e1

e1

e3

e1

e1

e2

e3

e4

e2

Refinement

Mesh refinement: Mesh refinement Test

2

ò

]

1

Criterion 1:

[

ò

2

2

2

h

=

+

e

å

r

R

de

h

(

u

)

D

dl

e

ce

e

u

2

Î

e

face

e

e

CriterionR. Verfürth (2000)

1

ì

ü

2

2

£

h

å

error

K

í

ý

e

î

þ

e

Error

P=0,2,4

P=4,6,8

P=8,10,12

P=12,14,16

Criterion 2: Verfürth

Initial Mesh

+1

W

+1/2

y

+1

ux=0

uy=0

+1

x

?

+1

ux=0

P=0,2,4

399 nodes

130 elements

P=4,6,8

708 nodes

257 elements

P=8,10,12

1016 nodes

389 elements

P=12,14,16

1472 nodes

589 elements

time cpu improved

• best precision
• « static » front captured
• but conformity!
• local unrefinement is difficult

• Loop on volume to set a refinement criteria
• Loop on nodes to find patch to unrefine
• 4 volumes at same hierarchical level
• 4 edge at same hierarchical level

Modification of the central node

Destruction of the other central nodes

Destruction of the central edge elements

Modification of the peripheral edges

Loop on the nodes to merge edges if necessary

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