Fluide Structure Interaction

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A FLUIDSTRUCTURE INTERACTION

METHOD FOR HIGHLY DEFORMABLE

SOLIDS


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Introduction IEFGM (Immersed Element-Free

Galerkin Method ) Formulation of the IEFGM

A lication exam les

PRESENTATION OVERVIEW

Comparison with analytical solution

Conclusion

References


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INTRODUCTION

IEFGM Necessary ???

ALE FEM formulation

Limitations : Fluid flow or large strain continuum deformation

Mesh free particle methods (2003)

SPH method(1994)


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IMMERSED ELEMENT-FREE GALERKIN METHOD

(IEFGM)

Numerical method to model FSI problems involving highly

flexible solids and slender solid body. Solid domain - updated-Lagrangian IEFG formulation

The fluid-solid interaction force is represented as a volumetric

A Lagrangian solid domain moves on top of an Eulerian fluid

domain which spans over the entire computational region

Coupling (continuity) between the solid and fluid domains

Moving Least Squares local approximationMoving Least Squares local approximation


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FORMULATION OF THE IEFGM

-------------(1)

Fig. Definition of the solid (Lagrangian) domain and the fluid (Eulerian) domain.

Eulerian configuration - characterized by the time invariant position vector x,

Lagrangian configuration - characterized by the current position vector xs


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ASSUMPTIONS :

The materials in both the solid and fluid domains are

incompressible No-slip condition between solid and fluid regions

The union of the two domains can be treated as one continuum

, ,

and temperature fields

The fluid occupies the entire computational domain and that

the solid domain is placed on top of the fluid domain.


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DOMAIN MODELING

The fluid domainThe fluid domain - Modelled using the finite element method

with an Eulerian formulation.positionx

actual time tThe independent variables

velocity v

pressure p

temperature T

The dependent variables


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DOMAIN MODELING

The solid domainThe solid domain - modelled using the element-free Galerkinmethod with an updated-Lagrangian formulation

pos onx

actual time t,

particles displacement us

us defined as the difference between the current and previous

position.

The independent variables

The dependent variables


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GOVERNING EQUATIONSSolid domain

The solid domain is assumed to be fully submerged into the

fluid domain at all times, it is considered to be incompressible,and it moves at the same speed as the fluid domain (non-slip

condition between fluid and solid)

Where, fiFSI,s Fluidstructure interaction force

The fluidstructure interaction force is treated as an additional

body force acting on the solid


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Computational domain

Momentum equation for the entire computational domainNavierStokes equation

------- (3)

GOVERNING EQUATIONS

Computational domain is incompressibleIncompressibility constrain ----- (4)

Equation 3,4---- Strong forms of the governing equation for the

following domain


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COUPLING BETWEEN THE SOLID AND FLUIDDOMAINS

Coupling = Development of a numerical code capable of

simulating fluidsolid interaction problems with the IEFGformulation

Two critical variables relevant to this coupling are

1. Solid domain Velocit vs x t and

2. The interaction force acting upon the fluid in the overlapping domain fFSI

--- - -- - -- -- - -- - (5).

- - - - - - (6) .


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COUPLING BETWEEN THE SOLID AND FLUID

DOMAINSLocal Approximation of x component of fluid model

a) local approximation of the x-component of the fluid nodal velocity (v,x).

b) Influence domains associated with each node


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--------------- -- - -- -- - -- - (5) .

COUPLING BETWEEN THE SOLID AND FLUID

DOMAINS

- - - - - - (6) .

-----------(7) .


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DISTRIBUTION OF THE INTERACTION FORCE, FFSI,

IN THE FLUID DOMAIN

-------(8) .

--------(9) .


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NUMERICAL EXAMPLES

Case 1:

Infinitely rigid solid disk falling by gravity while submerged in aviscous Newtonian fluid

Case 2:

-


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The properties of the solid disk

Disk dia = 2mm density = 12,000 kg/m3

Poisons ratio = 0.3

CASE 1 : INFINITELY RIGID SOLID DISK FALLING

The properties of the fluid

Density = 8800 kg/m3

l = 2.0e3 N s/m2.

The Eulerian grid consisted of 3321 rectangular bilinear

elements (10 mm wide X 20 mm high)


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COMPARISON WITH ANALYTICAL SOLUTION

The average solid velocity

The analytical velocity


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Comparison between the velocity histories of a rigid solid disk

falling in a viscous fluid

COMPARISON WITH ANALYTICAL SOLUTION


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CASE 2 : SOFT DISK FALLING IN A VISCOUS FLUID

a) Position and shape of the solid body at nine different times are shown,

b)


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CASE 2 : SOFT DISK FALLING IN A VISCOUS FLUID


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CONCLUSIONS

Very robust for modelling the fluid dynamics of flexible slender

bodies.

Large rotations, translations and deformations of the solid body

can be captured

represents an important factor affecting the movement of thefluid. Details of the velocity field and the shape and position of

the solid domain as a function of time, which for the highly thin

and flexible film shown in this work would be difficult to model

with finite element-based formulations, were effectively

captured by the IEFG method


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REFERENCES1. APPLICATIONS OF THE IMMERSED ELEMENT-FREE GALERKIN

METHOD, Claudio M. Pita, Sergio D. Felicelli, Department of

Mechanical Engineering, Mississippi State University, USA

2. Comparison of various fluidstructure interaction methods for

deformable bodies , Computers & Structures,Volume 85, Issues

11-14, June-July 2007, Pages 833-843

3.

Coupling of the improved element-free Galerkin and boundaryelement methods for two-dimensional elasticity problems

Engineering Analysis with Boundary Elements, Volume 32, Issue 2,

February 2008, Pages 100-107

4.

On the use of element-free Galerkin Method for problems involvingincompressibility, Engineering Analysis with BoundaryElements,

Volume 31, Issue 2, February 2007, Pages 103-115


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LAGRANGIAN BACKGROUND MESH

Detail of the Lagrangian background mesh for the solid disk on top of the Eulerian

fluid mesh

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Fluide Structure Interaction