An Eulerian-Lagrangian approach for fluid-structure coupled systems

ΩS. ΩS. Γ. Γ. Closed structure. Open structure t t £. ∆t. ΩF. ΩF. ΩF. 8th USNCCM, 25-27th July 2005, Austin. 2/ 26. Page 3. Introduction. Basic ideas.
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An Eulerian-Lagrangian approach for fluid-structure coupled systems using X-FEM Antoine Legay / CNAM-Paris Collaborations: A. Tralli, P. Gaudenzi, Universita` di Roma ”La Sapienza”, Italia

8th USNCCM, 25-27th July 2005, Austin

Introduction

Position of the problem







 

 

 













































































 

 









 

 

















































































 

 







 

 







 

 







 

 







 

 







 

























































































































 







 

 







 

 







 

 





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ΩF

Open structure Closed structure

t ΩF

ΩF

ΩS ΩS

∆t Γ t Γ

Transient dynamics Fluid-structure interaction

Introduction

Basic ideas Fluid: Eulerian description, fixed mesh Structure: Lagragian description Incompatible meshes ¨ Belytschko 99] Enrichment of the fluid fields using X-FEM [Moes,

Previous work: Compressible fluid, thin structure Fluid on one side of the structure No enrichment [Legay, Chessa and Belytschko 2005]

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Introduction

Other approaches Classical ALE-approach: need to update the fluid mesh mesh distortions: can not deal with large relative structure displacement Similar methods: Immersed boundaries [Peskin 89, 02] Fictitious domain [Glowinski 94] [Bertrand 97] [Baaijens 01] Immersed FEM [Wang 04] [Zhang 04] Do not enrich the fluid field around the structure

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Description of the problem

Strong form Fluid: incompressible, viscous, Eulerian description 

ρF v˙i ρF vi v j τi j vi i 0 in ΩF τi j pδi j 2µei j

gi in ΩF



j



 











Structure: thin, Lagrangian description 

ρS u¨i σi j j gi in ΩS ui ugi on ∂u ΩS σi j nSj Fig on ∂F ΩS 







Fluid-structure interface 2

u˙i



Γ

vi



β δ penalty term: 2



0 on Γ





u˙i

0 on Γ









vi

σi j n j



τi j



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Description of the problem

Weak form Fluid: Find vi such that δvi  



u˙i dΓ 

0





Γ

δvi vi



β

δvi j τi j dΩF

ΩF



δvi τi j nFj dSF



∂ΩF

dΩF

j









ΩF

δvi gi dΩF

vi v j



ΩF

δvi ρF v˙i



 

penalty term Structure: Find ui such that δui 

ΩS

δui gi dΩS 

u˙i dΓ

0



Γ

δui vi



β



∂u

ΩS

δui σi j nSj dSS







∂F

ΩS

ΩS

δui j σi j dΩS 

g

δui Fi dSS



ΩS

δui ρS u¨i dΩS



 

penalty term This leads to the strong form [Legay, Chessa and Belytschko 2005] 8th USNCCM, 25-27th July 2005, Austin

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Discretization

Space discretization: finite element method Fluid: mixed formulation 9/4-node element:

Pressure: Ni4 x





Velocity: Ni9 x

Structure: 1D beam Incompatible meshes: Enrichment of the fluid (X-FEM)

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Discretization

Time discretization: fractional time step method Semi-implicit scheme, 3-steps process: tn

tn

Un 1

2

3

Pn



Pn

t 

Un

1

1 1

Collaboration with A. Tralli and P. Gaudenzi, Universit a` di Roma, Italia [Tralli and Gaudenzi, 3rd MIT conf., 2005] [Tralli and Gaudenzi, submitted to IJNME, 2005]

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Discretization

Localization of the interface The position of the structure is described by a level-set function



0 on one side of the



φxt

structure 



φxt

0



Discretization of φ x t :

0





0 on the other side



φxt

x



φxt







φxt

0 on the interface Γ



φxt

0



Γt :φxt



φxt

n





 

Normal on Γ: n

8th USNCCM, 25-27th July 2005, Austin

Grad φ



NI9 x φI t



φxt

φ: signed distance to the interface

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Enrichment

Partition of unity 

Set of functions f i x defined on ΩPU such that

x

1





∑ fi

ΩPU

x

i

Obviously, 



ψx



xψx 

∑ fi i



New approximation of g x in ΩPU :

j









g f x ψ x A ∑i i



∑Nj x Gj



gx

i







 



regular part

gx



1



ψx





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0 et Agi 



note that : G j

enriched part

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Enrichment

Choice of enrichment Velocity

Pressure

continuous discontinuous derivative



 



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sign φ x



φx

v

Enrichment: sign φ x



Enrichment: φ x

discontinuous

p

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Enrichment

Velocity enrichment φ x Avj 



∑ j N 4j x







v

∑i Ni9 x Vi



v



Enrichment: Regular part

Partition of unity

continuous, discontinuous derivative

Note: This choice of P.u. avoids problems in blending elements [Legay, Wang and Belytschko, 2005] 8th USNCCM, 25-27th July 2005, Austin

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Enrichment

Pressure enrichment 

sign φ x





∑ j N 4j x





A pj



p

∑i Ni4 x Pi



p



Partition of unity Regular part

Enrichment: discontinuous

Note: This enrichment does not introduce extra-terms in blending elements

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Applications

Driven Cavity The fluid domain is divided into 2 parts: the left part is a driven cavity the right part should not move The structure is a rigid wall Reynolds number: 10

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Applications

Driven Cavity, streamlines

Compatible mesh, no enrichment.

Non-compatible mesh, velocity and pressure enrichment.

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Applications

Driven Cavity, velocity and pressure Comparison of velocities

Comparison of pressure

— compatible mesh

— compatible mesh

— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment

— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment

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Applications

Driven Cavity, zoom around structure Comparison of velocities



0 04m.s

Comparison of pressure

1

— compatible mesh

— compatible mesh

— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment

— non-comp. mesh, with no enrichment — non-comp. mesh, with enrichment

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Applications

Fixed structure in a driven cavity 1 4m 

1m.s

1

The structure is fixed Fluid mesh: 24

12

Reynolds number: 10

31deg 0 5m

2m

2 6m

4m

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Applications

Fixed structure in a driven cavity, streamlines

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Applications

Translating straight structure 4m

Fluid mesh: 36

18

Reynolds number: 10

time=0s 

Structure velocity: 1m.s 1

2m

time=2s

0 88m 1m

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2m

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Applications

Translating structure

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0.03 s

0.7 s

1.4 s

2.0 s 21/ 26

Applications

Rotating structure 0 7m

Fluid mesh: 12

12

The structure is a rotating straight

2m

line

1rad.s





Angular velocity: Ω

1

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Applications

Rotating structure

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Applications

Two Rotating structures 0 7m Fluid mesh: 24

12

Two rotating straight lines Angular velocities:

1rad.s



Ω2



0 5rad.s



Ω1

1

2m

0 8m

1

Ω2

Ω1



4m

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Applications

Two Rotating structures

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Conclusions and future work

Conclusions Eulerian-Lagrangian fluid-structure approach take advantage of the existing formulations for both fluid and structure Incompatible meshes no fluid mesh updating no mesh distortion Enrichment of fluid field around the interface using X-FEM improve accuracy around the interface

Future work Further validation Improve the enrichment at the structure tip Rigid structures with inertia terms, flexible structures 8th USNCCM, 25-27th July 2005, Austin

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References A. Legay, J. Chessa and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, in press, 2005. A. Tralli and P. Gaudenzi. Simulation of unsteady incompressible flows by a fractional-step FEM. International Journal for Numerical Methods in Engineering, submitted, 2005.

Contacts [email protected]

8th USNCCM, 25-27th July 2005, Austin

[email protected]