Investigation of SPH and ALE Methods for FSI Problems M.Souli, R

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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4
5
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
9
10
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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
23
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 ??
26
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
27
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
28
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
29
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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