investigation of sph and ale methods for fsi problems m.souli,...

Investigation of SPH and ALE Methods for FSI Problems

M.Souli, R. MessahelUniversite de Lille LML

A. BuffaloNEXTER Systems

Conference Lille21-22 Janvier 2015

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FEM MethodMesh and Nodes

SPH MethodParticles

Introducing SPH Method

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3D Contacts in SPH

SPH Bird SPH Bird

Shell StructureShell Structure

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SPHSPHHigh Deformation LagHigh Deformation Lag

ALEALEHigh DeformationHigh DeformationMultiMulti--materialmaterial

Flow SimulationFlow Simulation

LAGRANGIANLAGRANGIANFast and AccurateFast and Accurate

SPH ALE Lagrangian

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PlatePlate

Fluid Fluid

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Fluid Fluid SPHSPH Shell StructureShell Structure

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1) Unlike LBL , SPH method is a deterministic method and not a statistical method.

2) Statistical methods and solve for velocity Distribution, or the probability of having a specific velocity.

3) SPH Method solves for velocity pressure internal energy and state variables

Introducing SPH Method

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1) Like FEM Method, SPH method uses conservation equations for continuum Mechanics to solve for velocity, pressure and energy.

Introducing SPH Method

vdtd r..∇−= ρρ

vdtde r...1 ∇−= σρ

extfdtvd +∇−= σρ ..1r

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1) In FEM Method a weak formulation is used to solve Conservative equations

2) In SPH method we use a collocation method . to solve Conservative equations

Introducing SPH Method

vdtd r..∇−= ρρ

vdtde r...1 ∇−= σρ

extfdtvd +∇−= σρ ..1r

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Cylindrical mesh and nodes

Lagrangian and SPH Formulations

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Why do we need the mesh ?

Unlike FEM Method, because of the missing mesh the SPH method suffers Unlike FEM Method, because of the missing mesh the SPH method suffers from:from:

1) Function interpolation 1) Function interpolation

2) Support domain different from Influence Domain2) Support domain different from Influence Domain

3) Lack of Consistency3) Lack of Consistency

4) Tensile Instability4) Tensile Instability

5) Boundary Conditions5) Boundary Conditions

Question:

How to remedy to these problems in SPH ?

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In FEM we need the mesh for:

1) Function Interpolation at any location.

Function interpolationFunction interpolation

)(.)( xNuxu jj

j∑=

2) Derivative of Function at any location.

)(.)( xNuxu jj

j∇=∇ ∑

)(xN j Shape function at node j

x

x

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In SPH method, we need to define:

1) Interpolation Function

2) Derivation of function, to solve conservative equations

vdtd r..∇−= ρρ

σρ ..1 ∇−=dtvdr

vdtde r...1 ∇−= σρ

Function interpolationFunction interpolation

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Integral interpolation

δ

x

: : DIRAC function satisfiesDIRAC function satisfies::δ

∫ −=Ω

dyyxyuxu ).().()( δ

x )(xuAt any location the integral interpolation of the function is defined:

yxyx ==− 1)(δyxyx ≠=− 0)(δ

1).( =−∫Ω

dyyxδ

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Integral interpolation

∫ =Ω

1),( drhrW

The Dirac Function is approached by the Kernel Function The Dirac Function is approached by the Kernel Function ),( hrW

r

),( hrW

h2

oh→ rhrW δ→),(⇒

δ

x

0→h

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)(.1),( hd

hhdW θα=

The Kernel Function W is defined by:The Kernel Function W is defined by:

Integral interpolation

d/hd/h

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Integral interpolation

Kernel function for 2D problemKernel function for 2D problem

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xhxxWx jjj

j ≠−∑ ),(.. ,ω

1)( =∑ xN jj

Unlike FEM, SPH method cannot reproduce:

1) Constant function

u(x)=1

2)Linear function

u(x)= x xxNx jj

j =∑ )(

1),(. , ≠−∑ hxxW jj

jω

A central issue in SPH method is how to perform function interpolation with consistencywith no mesh

Interpolation Consistency

Why do we need SPH to reproduce constant and linear function ??Why do we need SPH to reproduce constant and linear function ??

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Consistency of constant function

1),(. , ≠−∑ hxxW jj

jω1)( =∑ xN jj

FEM

SPH

u constant function: u(x)=1

PhD Thesis Ecole Centrale Nantes

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Consistency of constant function

1),(. , ≠−∑ hxxW jj

jω1)( =∑ xN jj

u constant function: u(x)=1

For constant function:

FEM Interpolation is exact

SPH Interpolation is not exact.

SPH Interpolation does not reproduce constant functions

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)(),(.. , xuhxxWu jjj

j ≠−∑ ω)()( xuxNu jj

j =∑

FEM

Consistency of linear function

u linear function xxu =)(

SPHSPH

SPHSPH

ErrorError

PhD Thesis Ecole Centrale Nantes

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Lagrangian and SPH meshes in 3D

How the SPH mesh should be compared to the How the SPH mesh should be compared to the LagrangianLagrangian mesh mesh Same mesh or finer mesh ?Same mesh or finer mesh ?

waterwater

plateplate

Water impacting a plateWater impacting a plate

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Lagrangian meshes in 3D

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Lagrangian Results

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SPH meshes in 3D

Same mesh for SPH and LagrangianSame mesh for SPH and Lagrangian

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SPH meshes in 3D

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Lagrangian and SPH meshes in 3D

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Resultant force: A: SPH B: LagrangianResultant force: A: SPH B: Lagrangian

Momentum: A: SPH B: Lagrangian :Momentum: A: SPH B: Lagrangian :

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Vertical disp A: SPH B: LagrangianVertical disp A: SPH B: Lagrangian

Momentum: A: SPH B: LagrangianMomentum: A: SPH B: Lagrangian

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Finer SPH meshes in 3D

SPH 3D mesh finer than LagrangianSPH 3D mesh finer than Lagrangian

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Finer SPH meshes in 3D

SPH 3D mesh finer than LagrangianSPH 3D mesh finer than Lagrangian

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Resultant force: A: SPH B: LagrangianResultant force: A: SPH B: Lagrangian

Momentum: A: SPH B: LagrangianMomentum: A: SPH B: Lagrangian

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Vertical disp A: SPH B: LagrangianVertical disp A: SPH B: Lagrangian

Vertical vel: A: SPH B: LagrangianVertical vel: A: SPH B: Lagrangian

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Thank youThank you

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*CONTACT_AUTOMATIC_NODES_TO_SURFACE

SPH FoamSPH Foam

ContactContact

Rigid WallRigid Wall

3D Contacts in SPH

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