1. No category

Investigation of SPH and ALE Methods
for FSI Problems
M.Souli, R. Messahel
Universite de Lille LML
A. Buffalo
NEXTER Systems
Conference Lille
21-22 Janvier 2015
Introducing SPH Method
FEM Method
Mesh and Nodes
SPH Method
Particles
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3D Contacts in SPH
SPH Bird
Shell Structure
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SPH ALE Lagrangian
SPH
High Deformation Lag
ALE
High Deformation
Multi--material
Multi
Flow Simulation
LAGRANGIAN
Fast and Accurate
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Plate
Fluid
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8
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Fluid SPH
Shell Structure
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Introducing SPH Method
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
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Introducing SPH Method
1) Like FEM Method, SPH method uses conservation equations for
continuum Mechanics to solve for velocity, pressure and energy.
r
dr
=-r.Ñ.v
dt
r
dv =- 1 .Ñ.s + fext
dt r
de =- 1 .s.Ñ.vr
dt r
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Introducing SPH Method
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
r
dr
=-r.Ñ.v
rdt
dv =- 1 .Ñ.s + fext
dt r
de =- 1 .s.Ñ.vr
dt r
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Lagrangian and SPH Formulations
Cylindrical mesh and nodes
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Why do we need the mesh ?
Unlike FEM Method, because of the missing mesh the SPH method suffers
from:
1) Function interpolation
2) Support domain different from Influence Domain
3) Lack of Consistency
4) Tensile Instability
5) Boundary Conditions
Question:
How to remedy to these problems in SPH ?
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Function interpolation
In FEM we need the mesh for:
1) Function Interpolation at any location.
x
u ( x ) = å u j. N j ( x )
j
x
2) Derivative of Function at any location.
Ñ u (x ) = å u jÑ .N j (x )
j
N j (x )
Shape function at node j
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Function interpolation
In SPH method, we need to define:
1) Interpolation Function
2) Derivation of function, to solve conservative equations
r
dr
=-r.Ñ.v
dt
r
dv =- 1 .Ñ.s
dt r
de =- 1 .s.Ñ.vr
dt r
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Integral interpolation
At any location
x
the integral interpolation of the function u(x) is defined:
u(x)=òu(y).d(x- y).dy
W
d:
DIRAC function satisfies:
d
d(x- y)=1 x= y
d(x- y)=0 x¹ y
d
(
x
y
).
dy
=
1
ò
W
x
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Integral interpolation
The Dirac Function is approached by the Kernel Function W(r,h)
ò W(r,h)dr=1
W
h®o
Þ W(r,h)®d r
W(r,h)
d
h®0
2h
r
x
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Integral interpolation
The Kernel Function W is defined by:
W(d,h)= 1a .q(d )
h
h
d/h
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Integral interpolation
Kernel function for 2D problem
0
2
d/h
Interpolation Consistency
A central issue in SPH method is how to perform function interpolation with consistency
with no mesh
Unlike FEM, SPH method cannot reproduce:
1) Constant function
å
u(x)=1
N j ( x ) =1
å
w j . W ( x - x ,j , h ) ¹ 1
å
w j . x j . W ( x - x ,j , h ) ¹ x
j
j
2) Linear function
u(x)= x
åx
j
j
N j (x )= x
j
Why do we need SPH to reproduce constant and linear function ??
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Consistency of constant function
u constant function: u(x)=1
å
N j ( x ) =1
j
å
w j . W ( x - x ,j , h ) ¹ 1
j
FEM
SPH
PhD Thesis Ecole Centrale Nantes
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Consistency of constant function
u constant function: u(x)=1
å
N j ( x ) =1
j
å
w j . W ( x - x ,j , h ) ¹ 1
j
For constant function:
FEM Interpolation is exact
SPH Interpolation is not exact.
SPH Interpolation does not reproduce constant functions
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Consistency of linear function
u linear function
åu
j
u (x )= x
N j (x )= u (x )
j
,
w
.
u
.
W
(
x
x
j
j
j, h ) ¹ u ( x )
å
j
SPH
Error
FEM
SPH
PhD Thesis Ecole Centrale Nantes
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Lagrangian and SPH meshes in 3D
How the SPH mesh should be compared to the Lagrangian mesh
Same mesh or finer mesh ?
Water impacting a plate
water
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 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: Lagrangian
Momentum: A: SPH B: Lagrangian :
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Vertical disp A: SPH B: Lagrangian
Momentum: A: SPH B: Lagrangian
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Finer SPH meshes in 3D
SPH 3D mesh finer than Lagrangian
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Finer SPH meshes in 3D
SPH 3D mesh finer than Lagrangian
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Resultant force: A: SPH B: Lagrangian
Momentum: A: SPH B: Lagrangian
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Vertical disp A: SPH B: Lagrangian
Vertical vel: A: SPH B: Lagrangian
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Thank you
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3D Contacts in SPH
*CONTACT_AUTOMATIC_NODES_TO_SURFACE
SPH Foam
Contact
Rigid Wall
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