Fluid-Structure Interaction

Turek's benchmark FSI3

Fluid (Navier-Stokes)
viscosity \mu^F=1 Pa s
density \rho^F=1000 Kg/(m^3)

Structure (linear elasticity)
length l=0.35 m,
thickness h=0.02 m,
Young modulus E=5.6*10^6 Pa,
Poisson ratio \nu^S=0.4,
density \rho^S=1000 Kg/(m^3)

time step dt=0.002 s
the no of time steps N=4000

fluid-structure velocities (format mpg),

fluid pression (format mpg),

fluid-structure meshes zoom (format mpg)

Nonlinear elasticity ( compressive Neo Hookean )<-> Navier-Stokes

Blood
viscosity \mu=0.035 g/(cm s)
density \rho^F=1 g/(cm^3)

Vessel
thickness h^S=0.1 cm,
the Young modulus E=3000000 g/(cm\cdot s^2),
the Poisson ratio \nu =0.3,
the density \rho^S=1.1 g/(cm^3).

time step dt=0.001 s
the no of time steps N=100

inflow pressure Pin=3*10^3*(1-\cos(2\pi t/0.025)), for 0 < t < 0.025

Straight vessel

fluid-structure velocities (format mpg),

fluid pression (format mpg),

fluid-structure meshes (format mpg)

Nonlinear elasticity ( St Venant-Kirchhoff )<-> Navier-Stokes

inflow pressure Pin=3*10^3*(1-\cos(2\pi t/0.025)), for 0 < t < 0.025

Semi-implicit strategy for time advancing schema. At each time step the fluid-structure coupled problem is solved by the BFGS method.

Straight vessel

fluid-structure meshes (format mpg),

fluid velocities (format mpg) ,

fluid pression (format mpg),

structure velocities (format mpg)

Cerebral aneurysm

fluid-structure meshes (format mpg),

fluid velocities (format mpg),

fluid pression (format mpg),

structure velocities (format mpg)

Linear elasticity <-> Navier-Stokes

1. Implicit strategy for time advancing schema. At each time step the fluid-structure coupled problem is solved by the fixed point method with relaxation.

relaxation parameter omega=0.03

fluid-structure meshes (format mpg) ,

fluid pression (format mpg).

2. Semi-implicit strategy for time advancing schema. At each time step the fluid-structure coupled problem is solved by a monolithic method.

fluid-structure velocities (format mpg) ,

fluid pression (format mpg).

3. Semi-implicit strategy for time advancing schema. At each time step the fluid-structure coupled problem is solved by the BFGS method.

The case when the left and the right sides of the structure are fixed.

fluid-structure meshes (format mpg),

fluid velocities (format mpg) ,

fluid pression (format mpg),

structure velocities (format mpg)

The case when the right side of the structure is free.

fluid-structure meshes (format mpg),

fluid velocities (format mpg) ,

fluid pression (format mpg),

structure velocities (format mpg)

4. Semi-implicit strategy for time advancing schema. At each time step the fluid-structure coupled problem is solved by the Augmented Lagrangian method.

Blood
viscosity \mu=0.035 g/(cm s)
density \rho^F=1 g/(cm^3)

Vessel
thickness h^S=0.1 cm,
the Young modulus E=750000 g/(cm\cdot s^2),
the Poisson ratio \nu =0.3,
the density \rho^S=1.1 g/(cm^3).

time step dt=0.00005 s
the no of time steps N=1000

inflow pressure Pin=2*10^3*(1-\cos(2\pi t/0.025)), for 0 < t < 0.025

fluid velocities (format mpg) ,

fluid pression (format mpg),

structure velocities (format mpg)

Plate <-> Navier-Stokes

Implicit strategy for time advancing schema. At each time step the fluid-structure coupled problem is solved by the BFGS method.

Blood
viscosity \mu=0.035 g/(cm s)
density \rho^F=1 g/(cm^3)

Vessel
thickness h^S=0.1 cm,
the Young modulus E=750000 g/(cm\cdot s^2),
the Poisson ratio \nu =0.5,
the density \rho^S=1.1 g/(cm^3).

time step dt=0.0005 s
the no of time steps N=500

inflow pressure Pin=10^3 (1-\cos(2\pi t/0.005)), for 0 < t < 0.005

deplacements of the structure (format mpg),
fluid pressions (format mpg),
mesh evolution (format mpg),
fluid velocities (format mpg),

the Young modulus of the vessel E=3 000 000 g/(cm\cdot s^2)
inflow pressure Pin=10^3 (1-\cos(2\pi t/0.025)), for 0 < t < 0.025
The other parameters are the same as in the previous test.

fluid pression dt=0.0005 s (format mpg),
fluid pression dt=0.0010 s (format mpg),
fluid pression dt=0.0025 s (format mpg),
mesh evolution dt=0.0005 s (format mpg),
fluid velocities dt=0.0005 s (format mpg)

Benchmark for Navier-Stokes equations in a moving domain using the ALE method

Our algorithm is based on the implicit Euler approximation of the time derivative and the convective term is treated in a semi-implicit way.

We have tested on the benchmark proposed in F. Nobile, PhD, 2001, pp. 92-94.

The reference domain is (X1,X2) in [0,1] x [0,6].

The ALE maping is defined by:

x1=X1 and

x2=( 1-0.4*sin(2*pi*t/10) )*(X2-0.5)+0.5

The exact solution of Navier-Stokes equations for mass density rho=1, viscosity mu=1, external forces f=(0,0) is:

v1=-2*V*(x-6 )/(1+2*V*t)

v2= 2*V*(y-0.5)/(1+2*V*t)

p= - ( 2*V*(x-6 )/(1+2*V*t) )^2, where V=0.2 is a parameter.

On the top, left and bottom boundary, we have imposed the velocity profile, while

on the right boundary, we have imposed external forces:

lambda1=-4*V/(1+2*V*t) the horizontal component

lambda2=0 the vertical component.

We have performed the simulation for dt=1/2, ¼, 1/8, 1/16, 1/32.

For dt=1/32, at the time instant t=2, the errors in the L2 norm between the computed and the exact solution are:

5.97431e-06 for the velocity

0.00978646 for the pressure

0.000142107 for the forces exerced on the left boundary

For dt=1/32, at the time instant t=7, the errors in the L2 norm between the computed and the exact solution are:

4.38349e-06 for the velocity

0.00157697 for the pressure

0.000180705 for the forces exerced on the left boundary

mesh evolution (format mpg), fluid velocities (format mpg), fluid pression (format mpg)