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