SPH simulation of fluid-structure interaction...
Transcript of SPH simulation of fluid-structure interaction...
SPH simulation of fluid-structure
interaction problems
Università degli Studi di Pavia
C. Antoci, M. Gallati, S. Sibilla
Dipartimento di ingegneria idraulica e ambientale
Research project
• Problem: deformation of a plate due to the action of a fluid
(large displacement of the structure, fluid free surface)
• Numerical technique:
SPH Lagrangian: it automatically follows moving
interfaces
Relatively easy to include the simulation of different materials
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Scheme of the model
Structure Fluid
(SPH) (SPH)
Coupling conditions on the interface:
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Kinematic condition:
Dynamic condition:
sf vv&& =
nvnv sf ˆˆ ⋅=⋅ &&(perfect fluid)
ffss nn ˆˆ σσ −=
fssT
s pnn =ˆˆ σ (perfect fluid)
Continuum equations
0=+i
i
x
v
Dt
D
∂∂ρρ
j
iji
i
xg
Dt
Dv
∂∂σ
ρρ +=
(i=1,3)
(i,j=1,3)
continuity equation
equation of motion
ρ density
t time
xi position (i-component of the vector)
vi velocity (i-component of the vector)
σij stress (ij-component of the tensor)
gi gravity acceleration (i-component of the vector)
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(isothermal conditions)
fluid00 ρε=c
deviatoric stresspressure
State equation and constitutive equations
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solid
kjikjkikijijij
ijSS
dt
dSΩ+Ω+
−= εδεµ
3
12
∂∂
+∂∂
=i
j
j
iij
x
v
x
v
2
1ε
∂
∂−
∂∂
=Ωi
j
j
iij
x
v
x
v
2
1
( )02
0 ρρ −= cp
00 ρkc =
(µ shear modulus)
ijijij Sp +−= δσ (i,j=1,3)
•
• Sij
(ε compressibility modulus)
(k bulk modulus)
Sij = 0 (perfect fluid)fluid
solid
2D SPH equations (1)
(“a” and “b”: particle labels)
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•Equation of state: ( )aaa cp 02
0 ρρ −=
•Continuity equation: ∑ ∇⋅−=b
abababa Wvvm
Dt
D)(
&&ρ
2D SPH equations (2)
∑ +∂∂
+∏++=
bi
aj
abn
abijijab
b
bij
a
aij
bai g
x
WfRm
Dt
Dvδ
ρ
σ
ρ
σ22
Equation of motion:
artificial stress (Gray, Monaghan and Swift 2001)
aijijaaij Sp +−= δσ Sij calculated by an implicit scheme
from the incremental hypoelastic relation
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artificial viscosity (Monaghan and Gingold 1983)
∑ +∂∂
+−=
bi
j
abij
b
b
a
ab
ai gx
Wppm
Dt
Dv
a
δρρ 22
fluid
solid
2D SPH equations (3)
Velocity gradient:
( )a
abj
abii
b b
b
aj
i
x
Wvv
m
x
v
∂∂
−=∂∂
∑ρ
spin
rate of deformation
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[or: velocity gradient normalized to account for non uniformity
in particle distribution]
Fluid-structure interaction
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- Two sets of particles, fluid and solid
- A simple approach: to consider particles in the equations regardless
of the fact they are fluid or solid
- Interpenetration of fluid and solid particles can be prevented using
XSPH but the interaction is not well reproduced (excessive adhesion)
- Definition of the fluid-solid interface (and normal)
- Dynamic condition (action-reaction principle)
- Kinematic condition
Definition of the fluid-solid interface
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Since solid particles maintain the same regular spatial distribution (no fragmentation):
( )
−−
−−
==−+
−+
−+
−+
11
11
11
11 ,,ˆaa
aa
aa
aaayaxa
xx
yy
xx
xxttt &&&&
( )axaya ttn ,ˆ −=
aaan
dxx ˆ
2int += &&
(for every solid particle near the interface)
Dynamic condition
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( )
( )∑
∑
Ω∈
Ω∈
−
−
=
fa
fa
a
bb
b
b
bbb
b
b
hxxWm
hxxWpm
p
,
,
int
int
int&&
&&
ρ
ρ
0.5 (constant, in order to take in account
possible separation of the two media)
( ) ','intint Γ−= ∫Γ
→ dhxxWpFaasf
&&
(Ff→s/ρa added term in
the momentum equation)
Fs→f,a* = - Ff→s,a(linear interpolation)
surface term:
Kinematic condition
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( )
( )∑
∑
Ω∈
Ω∈
−
−
=
sb
sb
b
bb
b
b
bbbi
b
b
i
hxxWm
hxxWvm
v
,
,
*
*
*
int
int
int&&
&&
ρ
ρ
( )annnbn vvdvv
bb−+=
** int*
int d* = max (db/d
a,2)
atbtvv =
- a velocity distribution is assigned to solid particles in order to obtain
by SPH interpolation (on the interface) the interface velocity (normal component):
- velocity of the interface:
- “a” fluid particle “b*” solid particle (the nearest)
Other features of the code
•Correction of velocity:
-XSPH (solid)(Monaghan 1989)
-dissipative correction (fluid) (Gallati and Braschi 2000)
(on the interface particles from both media have to be included in the correction)
•Boundary conditions:
- fluid
- imaginary particles which reflect velocity
- layer of fixed particles (2h)
- solid (clamp)
- layer of fixed particles which are calculated just like
others but with velocity equal to zero
•Time integration scheme: Euler explicit (staggered)
•Kernel: cubic spline (Monaghan 1992)
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Example: Elastic gate
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≅107 PaE (Young modulus)
1100 kg/m3ρs
Sluice-gate (rubber)
0.005 mS
0.079 mL
0.14 mH
0.1 mA
Dimensions
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Simulation
ε = 2×106 N/m2 (compressible)
ρf= 1000 kg/m3
ν = 0.4 (Poisson coefficient)
E=1.2×107 N/m2
h/d=1.5
nP=6012
dt=8.34 ×10-6 st=0 s
K = 2×107 N/m2
µ = 4.27×106 N/m2
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t=0 s
Comparison between simulation and experiment
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t=0.04 s
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t=0.08 s
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t=0.12 s
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t=0.16 s
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t=0.2 s
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t=0.24 s
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t=0.28 s
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t=0.32 s
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t=0.36 s
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t=0.4 s
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Free end of the plate: displacements (1)
horizontal displacement
0
0,01
0,02
0,03
0,04
0,05
0,06
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
time (s)
h.d. (m)experiment
simulation
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Free end of the plate: displacements (2)
vertical displacement
0
0,005
0,01
0,015
0,02
0,025
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
time (s)
v.d. (m)experiment
simulation
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Free surface (1)
water level behind the gate
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
t (s)
y (m)experiment
simulation
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Free surface (2)
water level 5 cm far from the gate
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0,16
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4
time (s)
y (m)experiment
simulation
Future work
• To complete the normalization of the elastic equations with
explicit treatment of boundary conditions
• To improve the model in order to manage larger deformations
• To write the code in cylindrical coordinates in order to
simulate axially symmetric problems (“flex-flow” valves,
sloshing in cylindrical tanks…)
• To use two different spatial discretizations for the fluid and
the solid (membranes, hemodynamics)
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This research was financed by Dresser Italia