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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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