1. Fluid Structure Interaction Models

The Fluid Structure models are formed from the combination of a Solid model and a Fluid model.

1.2. Fluid structure coupling conditions

In order to have a correct fluid-structure model, we need to add to the solid model and the fluid model equations some coupling conditions :

\[ \frac{\partial \boldsymbol{\eta_{s}} }{\partial t} - \boldsymbol{u}_f \circ \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(1)}\]
\[ \boldsymbol{F}_{s} \boldsymbol{\Sigma}_{s} \boldsymbol{n}^*_s + J_{\mathcal{A}_{f}^t} \hat{\boldsymbol{\sigma}}_f \boldsymbol{F}_{\mathcal{A}_{f}^t}^{-T} \boldsymbol{n}^*_f = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(2)}\]
\[ \boldsymbol{\varphi}_s^t - \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(3)}\]

\(\boldsymbol{(1)}, \boldsymbol{(2)}, \boldsymbol{(3)}\) are the fluid-struture coupling conditions, respectively velocities continuity, constraint continuity and geometric continuity.

1.2.1. Fluid structure coupling conditions with 1D reduced model

For the coupling conditions, between the 2D fluid and 1D structure, we need to modify the original ones \(\boldsymbol{(1)},\boldsymbol{(2)}, \boldsymbol{(3)}\) by

\[\dot{\eta}_s \boldsymbol{e}_r - \boldsymbol{u}_f = \boldsymbol{0} \quad \boldsymbol{(1.2)}\]
\[f_s + \left(J_{\mathcal{A}_f^t} \boldsymbol{F}_{\mathcal{A}_f^t}^{-T} \hat{\boldsymbol{\sigma}}_f \boldsymbol{n}^*_f\right) \cdot \boldsymbol{e}_r = 0 \quad \boldsymbol{(2.2)}\]
\[\boldsymbol{\varphi}_s^t - \mathcal{A}_f^t = \boldsymbol{0} \quad \boldsymbol{(3.2)}\]

1.2.2. Variables, symbols and units

Notation

Quantity

Unit

\(\boldsymbol{u}_f\)

fluid velocity

\(m.s^{-1}\)

\(\boldsymbol{\sigma}_f\)

fluid stress tensor

\(N.m^{-2}\)

\(\boldsymbol{\eta}_s\)

displacement

\(m\)

\(\boldsymbol{F}_s\)

deformation gradient

dimensionless

\(\boldsymbol{\Sigma}_s\)

second Piola-Kirchhoff tensor

\(N.m^{-2}\)

\(\mathcal{A}_f^t\)

Arbitrary Lagrangian Eulerian ( ALE ) map

dimensionless

and

\[\boldsymbol{F}_{\mathcal{A}_f^t} = \boldsymbol{\mathrm{x}}^* + \nabla \mathcal{A}_f^t\]
\[J_{\mathcal{A}_f^t} = det(\boldsymbol{F}_{\mathcal{A}_f^t})\]

Examples

2. Fluid Structure Interaction

We will interest now to the different interactions a fluid and a structure can have together with specific conditions.

2.1. Fluid structure model

To describe and solve our fluid-structure interaction problem, we need to define a model, which regroup structure model and fluid model parts.

We have then in one hand the fluid equations, and in the other hand the structure equations.

FSI
Figure 1 : FSI case example

The solution of this model are \((\mathcal{A}^t, \boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s)\).

2.2. ALE

Generally, the solid mechanic equations are expressed in a Lagrangian frame, and the fluid part in Eulerian frame. To define and take in account the fluid domain displacement, we use a technique name ALE ( Arbitrary Lagrangian Eulerian ). This allow the flow to follow the fluid-structure interface movements and also permit us to have a different deformation velocity than the fluid one.

Let denote \(\Omega^{t_0}\) the calculation domain, and \(\Omega^t\) the deformed domain at time \(t\). As explain before, we want to conserve the Lagrangian and Eulerian characteristics of each part, and to do this, we introduce \(\mathcal{A}^t\) the ALE map.

This map give us the position of \(x\), a point in the deformed domain at time \(t\) from the position of \(x^*\) in the initial configuration \(\Omega^*\).

ALE
Figure 2 : ALE map

\(\mathcal{A}^t\) is a homeomorphism, i.e. a continuous and bijective application we can define as

\[\begin{eqnarray*} \mathcal{A}^t : \Omega^* &\longrightarrow & \Omega^{t} \\ \mathbf{x}^* &\longmapsto & \mathbf{x} \left(\mathbf{x}^*,t \right) = \mathcal{A}^t \left(\mathbf{x}^*\right) \end{eqnarray*}\]

We denote also \(\forall \mathbf{x}^* \in \Omega^*\), the application :

\[\begin{eqnarray*} \mathbf{x} : \left[t_0,t_f \right] &\longrightarrow & \Omega^t \\ t &\longmapsto & \mathbf{x} \left(\mathbf{x}^*,t \right) \end{eqnarray*}\]

This ALE map can then be retrieve into the fluid-structure model.

3. 2D FSI elastic tube

This test case has originally been realised by [Pena], [Nobile] and [GerbeauVidrascu] only with the free outlet condition.

Computer codes, used for the acquisition of results, are from Vincent [Chabannes]

3.1. Problem Description

We interest here to the case of bidimensional blood flow modelisation. We want to reproduce and observe pressure wave spread into a canal with a fluid-structure interaction model.

Elastic Tube Geometry
Figure 1 : Geometry of two-dimension elastic tube.

The figure above shows us the initial geometry we will work on. The canal is represent by a rectangle with width and height, respectively equal to 6 and 1 cm. The upper and lower walls are mobile and so, can be moved by flow action.

By using the 1D reduced model, named generalized string and explained by [Chabannes], we didn’t need to define the elastic domain ( for the vascular wall ) here. So the structure domain is \(\Omega_s^*=\Gamma_{fsi}^*\)

During this benchmark, we will compare two different cases : the free outlet condition and the Windkessel model. The first one, as its name said, impose a free condition on the fluid at the end of the domain. The second one is used to model more realistically an flow outlet into our case. The chosen time step is \(\Delta t=0.0001\)

3.1.1. Boundary conditions

We set :

  • on \(\Gamma_f^{i,*}\) the pressure wave pulse \[ \boldsymbol{\sigma}_{f} \boldsymbol{n}_f = \left\{ \begin{aligned} & \left(-\frac{2 \cdot 10^4}{2} \left( 1 - \cos \left( \frac{ \pi t} {2.5 \cdot 10^{-3}} \right) \right), 0\right)^T \quad & \text{ if } t < 0.005 \\ & \boldsymbol{0} \quad & \text{ else } \end{aligned} \right. \]

  • on \(\Gamma_f^{o,*}\)

    • Case 1 : free outlet : \(\boldsymbol{\sigma}_{f} \boldsymbol{n}_f =0\)

    • Case 2 : Windkessel model ( \(P_0\) proximal pressure, see [Chabannes] ) : \(\boldsymbol{\sigma}_{f} \boldsymbol{n}_f = -P_0\boldsymbol{n}_f\)

  • on \(\Gamma_f^{i,*} \cup \Gamma_f^{o,*}\) a null displacement : \(\boldsymbol{\eta}_f=0\)

  • on \(\Gamma^*_{fsi}\) : \(\eta_s=0\)

  • We add also the specific coupling conditions, obtained from the axisymmetric reduced model, on \(\Gamma^*_{fsi}\)

\[\dot{\eta}_s \boldsymbol{e}_r - \boldsymbol{u}_f = \boldsymbol{0}\]
\[f_s + \left(J_{\mathcal{A}_f^t} \boldsymbol{F}_{\mathcal{A}_f^t}^{-T} \hat{\boldsymbol{\sigma}}_f \boldsymbol{n}^*_f\right) \cdot \boldsymbol{e}_r = 0\]
\[\boldsymbol{\varphi}_s^t - \mathcal{A}_f^t = \boldsymbol{0}\]

3.2. Inputs

Table 1. Fixed and Variable Input Parameters

Name

Description

Nominal Value

Units

\(E_s\)

Young’s modulus

\(0.75\)

\(dynes.cm^{-2}\)

\(\nu_s\)

Poisson’s ratio

\(0.5\)

dimensionless

\(h\)

walls thickness

0.1

\(cm\)

\(\rho_s\)

structure density

\(1.1\)

\(g.cm^{-3}\)

\(R_0\)

tube radius

\(0.5\)

\(cm\)

\(G_s\)

shear modulus

\(10^5\)

\(Pa\)

\(k\)

Timoshenko’s correction factor

\(2.5\)

dimensionless

\(\gamma_v\)

viscoelasticity parameter

\(0.01\)

dimensionless

\(\mu_f\)

viscosity

\(0.003\)

\(poise\)

\(\rho_f\)

density

\(1\)

\(g.cm^{-3}\)

\(R_p\)

proximal resistance

\(400\)

\(R_d\)

distal resistance

\(6.2 \times 10^3\)

\(C_d\)

capacitance

\(2.72 \times 10^{-4}\)

3.3. Outputs

After solving the fluid struture model, we obtain \((\mathcal{A}^t, \boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s)\)

with \(\mathcal{A}^t\) the ALE map, \(\boldsymbol{u}_f\) the fluid velocity, \(p_f\) the fluid pressure and \(\boldsymbol\eta_s\) the structure displacement

3.4. Discretization

\(\mathcal{F}\) is the set of all mesh faces, we denote \(\mathcal{F}_{stab}\) the face we stabilize

\[\mathcal{F}_{stab} = \mathcal{F} \bigcap \left( \left\{ (x,y) \in \mathrm{R}^2: (x - 0.3) \leqslant 0 \right\} \cup \left\{ (x,y) \in \mathrm{^2: (x - 5.7) } \geqslant 0 \right\} \right)\]

In fact, after a first attempt, numerical instabilities can be observed at the fluid inlet. These instabilities, caused by pressure wave, and especially by the Neumann condition, make our fluid-structure solver diverge.

To correct them, we choose to add a stabilization term, obtain from the stabilized CIP formulation ( see [Chabannes], Chapter 6 ).

As this stabilization bring an important cost with it, by increasing the number of non-null term into the problem matrix, we only apply it at the fluid entrance, where the instabilities are located.

Now we present the different situations we worked on.

Config

Fluid

Structure

\(N_{elt}\)

\(N_{geo}\)

\(N_{dof}\)

\(N_{elt}\)

\(N_{geo}\)

\(N_{dof}\)

\((1)\)

\(342\)

\(3~(P4P3)\)

\(7377\)

\(58\)

\(1\)

\(176~(P3)\)

\((2)\)

\(342\)

\(4~(P5P4)\)

\(11751\)

\(58\)

\(1\)

\(234~(P4)\)

For the fluid time discretization, BDF, at order \(2\), is the method we use.

And Newmark-beta method is the one we choose for the structure time discretization, with parameters \(\gamma=0.5\) and \(\beta=0.25\).

These methods can be retrieved in [Chabannes] papers.

3.4.1. Solvers

Here are the different solvers ( linear and non-linear ) used during results acquisition.

KSP

case

fluid

solid

type

gmres

relative tolerance

\(1e-13\)

max iteration

\(30\)

\(10\)

reuse preconditioner

true

false

SNES

case

fluid

solid

relative tolerance

\(1e-8\)

steps tolerance

\(1e-8\)

max iteration

\(50\)

max iteration with reuse

\(50\)

reuse jacobian

false

reuse jacobian rebuild at first Newton step

false

true

KSP in SNES

case

fluid

solid

relative tolerance

\(1e-5\)

max iteration

\(1000\)

max iteration with reuse

\(1000\)

reuse preconditioner

true

false

reuse preconditioner rebuild at first Newton step

false

PC

case

fluid

solid

type

LU

package

mumps

FSI

solver method

fix point

tolerance

\(1e-6\)

max iterations

\(1\)

3.5. Implementation

To realize the acquisition of the benchmark results, code files contained and using the Feel++ library will be used. Here is a quick look to the different location of them.

Let’s start with the main code, that can be retrieve in

    feelpp/applications/models/fsi

The configuration file associated to this test is named wavepressure2d.cfg and is located at

    feelpp/applications/models/fsi/wavepressure2d

The result files are then stored by default in

    applications/models/fsi/wavepressure2d/P2P1G1-P1G1/np_1

3.6. Results

The two following pictures have their pressure and velocity magnitude amplify by 5.

Elastic Tube Free outlet
Figure 2 : Results with free outlet conditon
Elastic Tube Windkessel
Figure 3 : Results with the Windkessel model
Inflow Outflow
Figure 4 : Evolution of the inflow and the outflow
Displacement Magnitude
Figure 5 : Maximum displacement magnitude

To draw the next two figures, we define 60 sections \(\{x_i\}_{i=0}^{60}\) with \(x_i=0.1i\).

Average Pressure
Figure 5 : Average pressure with the free outlet and the Windkessel model
Flow Rate
Figure 7 : Flow rate with the free outlet and the Windkessel model
FSI 1
Figure 8 : Implicit and semi-implicit FSI coupling comparison
FSI 2
Figure 9 : Implicit and semi-implicit FSI coupling comparison

All the files used for this case can be found in this rep [geo file, config file, fluid json file, solid json file].

3.6.1. Conclusion

Let’s begin with results with the free outlet condition ( see figure 2 ). These pictures show us how the pressure wave progresses into the tube. We can denote an increase of the fluid velocity at the end of the tube. Also, the wave eases at the same place. For the simulation with the Windkessel model, we observe a similar comportment at the beginning ( see figure 3 ). However, the outlet is more realistic than before. In fact, the pressure seems to propagate more naturally with this model. + In the two cases, the velocity field is disturbed at the fluid-structure interface. A mesh refinement around this region increases the quality. However, this is not crucial for the blood flow simulation.

Now we can interest us to the quantitative results.

The inflow and outflow evolution figure ( see figure 4 ) shows us similarities for the two tests at the inlet. At the outlet, in contrast, the flow increases for the free outlet condition. In fact, when the pressure wave arrived at the outlet of the tube, it is reflected to the other way. In the same way, when the reflected wave arrived at the inlet, it is reflected again. The Windkessel model reduce significantly this phenomenon. Some residues stay due to 0D coupling model and structure fixation.

We also have calculate the maximum displacement magnitude for the two model ( see figure 5 ). The same phenomenons explained ahead are retrieve here. We denote that, for the free outlet, the structure undergoes movements during the test time, caused by the wave reflection. The Windkessel model reduces these perturbations thanks to the 0D model.

The average pressure and the fluid flow ( see figure 6 and 7 ) show us the same non-physiological phenomenons as before. The results we obtain are in accordance with the ones proposed by [Nobile].

To end this benchmark, we will compare the two resolution algorithms used with the fluid-structure model : the implicit and the semi-implicit ones. The second configuration with Windkessel model is used for the measures.

We have then the fluid flow and the displacement magnitude ( figure 8 ) curves, which superimposed on each other. So the accuracy obtained by the semi-implicit method seems good here.

The performances of the two algorithms ( figure 9 ) are expressed from number of iterations and CPU time at each step time. The semi-implicit method is a bit ahead of the implicit one on number of iterations. However, the CPU time is smaller for 2 or 3 time, due to optimization in this method. First an unique ALE map estimation is need. Furthermore, linear terms of the Jacobian matrix, residuals terms and dependent part of the ALE map can be stored and reused at each iteration.

3.7. Bibliography

References for this benchmark
  • [Pena] G. Pena, Spectral element approximation of the incompressible Navier-Stokes equations evolving in a moving domain and applications, École Polytechnique Fédérale de Lausanne, November 2009.

  • [Nobile] F. Nobile, Numerical approximation of fluid-structure interaction problems with application to haemodynamics, École Polytechnique Fédérale de Lausanne, Switzerland, 2001.

  • [GerbeauVidrascu] J.F. Gerbeau, M. Vidrascu, A quasi-newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows, 2003.

  • [Chabannes] Vincent Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles [math.AP], Université de Grenoble, 2013.

4. 3D FSI elastic tube

Computer codes, used for the acquisition of results, are from Vincent [Chabannes].

4.1. Problem Description

As in the 2D case, the blood flow modelisation, by observing a pressure wave progression into a vessel, is the subjet of this benchmark. But this time, instead of a two-dimensional model, we use a three-dimensional model, with a cylinder

Elastic Tube Geometry
Figure 1 : Geometry of three-dimensional elastic tube.

This represents the domains into the initial condition, with \(\Omega_f\) and \(\Omega_s\) respectively the fluid and the solid domain. The cylinder radius equals to \(r+\epsilon\), where \(r\) is the radius of the fluid domain and \(\epsilon\) the part of the solid domain.

Elastic Tube Geometry
Figure 2 : Sections of the three-dimensional elastic tube.

\(\Gamma^*_{fsi}\) is the interface between the fluid and solid domains, whereas \(\Gamma^{e,*}_s\) is the interface between the solid domain and the exterior. \(\Gamma_f^{i,*}\) and \(\Gamma_f^{o,*}\) are respectively the inflow and the outflow of the fluid domain. Likewise, \(\Gamma_s^{i,*}\) and \(\Gamma_s^{o,*}\) are the extremities of the solid domain.

During this benchmark, we will study two different cases, named BC-1 and BC-2, that differ from boundary conditions. BC-2 are conditions imposed to be more physiological than the ones from BC-1. So we waiting for more realistics based results from BC-2.

4.1.1. Boundary conditions

  • on \(\Gamma_f^{i,*}\) the pressure wave pulse \[ \boldsymbol{\sigma}_{f} \boldsymbol{n}_f = \left\{ \begin{aligned} & \left(-\frac{1.3332 \cdot 10^4}{2} \left( 1 - \cos \left( \frac{ \pi t} {1.5 \cdot 10^{-3}} \right) \right), 0 \right)^T \quad & \text{ if } t < 0.003 \\ & \boldsymbol{0} \quad & \text{ else } \end{aligned} \right. \]

  • We add the coupling conditions on \(\Gamma^*_{fsi}\)

\[ \frac{\partial \boldsymbol{\eta_{s}} }{\partial t} - \boldsymbol{u}_f \circ \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(1)}\]
\[ \boldsymbol{F}_{s} \boldsymbol{\Sigma}_{s} \boldsymbol{n}^*_s + J_{\mathcal{A}_{f}^t} \hat{\boldsymbol{\sigma}}_f \boldsymbol{F}_{\mathcal{A}_{f}^t}^{-T} \boldsymbol{n}^*_f = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(2)}\]
\[ \boldsymbol{\varphi}_s^t - \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(3)}\]

Then we have two different cases :

  • Case BC-1

    • on \(\Gamma_f^{o,*}\) : \(\boldsymbol{\sigma}_{f} \boldsymbol{n}_f =0\)

    • on \(\Gamma_s^{i,*} \cup \Gamma_s^{o,*}\) a null displacement : \(\boldsymbol{\eta}_s=0\)

    • on \(\Gamma^{e,*}_{s}\) : \(\boldsymbol{F}_s\boldsymbol{\Sigma}_s\boldsymbol{n}_s^*=0\)

    • on \(\Gamma_f^{i,*}\(\)U \(\)\Gamma_f^{o,*}\) : \(\mathcal{A}^t_f=\boldsymbol{\mathrm{x}}^*\)

  • Case BC-2

    • on \(\Gamma_f^{o,*}\) : \(\boldsymbol{\sigma}_{f} \boldsymbol{n}_f = -P_0\boldsymbol{n}_f\)

    • on \(\Gamma_s^{i,*}\) a null displacement \(\boldsymbol{\eta}_s=0\)

    • on \(\Gamma^{e,*}_{s}\) : \(\boldsymbol{F}_s\boldsymbol{\Sigma}_s\boldsymbol{n}_s^* + \alpha \boldsymbol{\eta}_s=0\)

    • on \(\Gamma^{o,*}_{s}\) : \(\boldsymbol{F}_s\boldsymbol{\Sigma}_s\boldsymbol{n}_s^* =0\)

    • on \(\Gamma_f^{i,*}\) : \(\mathcal{A}^t_f=\boldsymbol{\mathrm{x}}^*\)

    • on \(\Gamma_f^{o,*}\) : \(\nabla \mathcal{A}^t_f \boldsymbol{n}_f^*=\boldsymbol{n}_f^*\)

4.1.2. Initial conditions

The chosen time step is \(\Delta t=0.0001\)

4.2. Inputs

Table 2. Fixed and Variable Input Parameters
Name Description Nominal Value Units

\(E_s\)

Young’s modulus

\(3 \times 10^6 \)

\(dynes.cm^{-2}\)

\(\nu_s\)

Poisson’s ratio

\(0.3\)

dimensionless

\(r\)

fluid tube radius

0.5

\(cm\)

\(\epsilon\)

solid tube radius

0.1

\(cm\)

\(L\)

tube length

5

\(cm\)

\(A\)

A coordinates

(0,0,0)

\(cm\)

\(B\)

B coordinates

(5,0,0)

\(cm\)

\(\mu_f\)

viscosity

\(0.03\)

\(poise\)

\(\rho_f\)

density

\(1\)

\(g.cm^{-3}\)

\(R_p\)

proximal resistance

\(400\)

\(R_d\)

distal resistance

\(6.2 \times 10^3\)

\(C_d\)

capacitance

\(2.72 \times 10^{-4}\)

4.3. Outputs

After solving the fluid struture model, we obtain \((\mathcal{A}^t, \boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s)\)

with \(\mathcal{A}^t\) the ALE map, \(\boldsymbol{u}_f\) the fluid velocity, \(p_f\) the fluid pressure and \(\boldsymbol\eta_s\) the structure displacement.

4.4. Discretization

Here are the different configurations we worked on. The parameter Incomp define if we use the incompressible constraint or not.

Config

Fluid

Structure

\(N_{elt}\)

\(N_{geo}\)

\(N_{dof}\)

\(N_{elt}\)

\(N_{geo}\)

\(N_{dof}\)

Incomp

\((1)\)

\(13625\)

\(1~(P2P1)\)

\(69836\)

\(12961\)

\(1\)

\(12876~(P1)\)

No

\((2)\)

\(13625\)

\(1~(P2P1)\)

\(69836\)

\(12961\)

\(1\)

\(81536~(P1)\)

Yes

\((3)\)

\(1609\)

\(2~(P3P2)\)

\(30744\)

\(3361\)

\(2\)

\(19878~(P2)\)

No

For the structure time discretization, we will use Newmark-beta method, with parameters \(\gamma=0.5\) and \(\beta=0.25\).

And for the fluid time discretization, BDF, at order \(2\), is the method we choose.

These two methods can be found in [Chabannes] papers.

4.5. Implementation

To realize the acquisition of the benchmark results, code files contained and using the Feel++ library will be used. Here is a quick look to the different location of them.

Let’s start with the main code, that can be retrieve in

    feelpp/applications/models/fsi

The configuration file associated to this test is named wavepressure3d.cfg and is located at

    feelpp/applications/models/fsi/wavepressure3d

The result files are then stored by default in

    applications/models/fsi/wavepressure3d/P2P1G1-P1G1/np_1

All the files used for this case can be found in this rep [geo file, config file, fluid json file, solid json file].

4.7. Bibliography

References for this benchmark
  • [Chabannes] Vincent Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles [math.AP], Université de Grenoble, 2013.

include:04-learning/FSI/Turek-Hron/README.adoc[]

5. Lid-driven Cavity Flow benchmark

This test case has originally been proposed by [MokWall].

Computer codes, used for the acquisition of results, are from Vincent [Chabannes].

This benchmark has aso been realized by [Gerbeau], [Vàzquez], [Kuttler] and [Kassiotis].

5.1. Problem Description

We study here an incompressible fluid flowing into a cavity, where its walls are elastic. We use the following geometry to represent it.

LidDriven Geometry
Figure 1 : Geometry of lid-driven cavity flow.

The domain \(\Omega_f^*\) is define by a square \( [0,1]^2 \), \(\Gamma^{i,*}_f\) and \(\Gamma^{o,*}_f\) are respectively the flow entrance and the flow outlet. A constant flow velocity, following the \(x\) axis, will be imposed on \(\Gamma_f^{h,*}\) border, while a null flow velocity will be imposed on \(\Gamma_f^{f,*}\). This last point represent also a non-slip condition for the fluid.

Furthermore, we add a structure domain, at the bottom of the fluid one, named \(\Omega_s^*\). It is fixed by his two vertical sides \(\Gamma_s^{f,*}\), and we denote by \(\Gamma_f^{w,*}\) the border which will interact with \(\Omega_f^*\).

During this test, we will observe the displacement of a point \(A\), positioned at \((0;0.5)\), into the \(y\) direction, and compare our results to ones found in other references.

5.1.1. Boundary conditions

Before enunciate the boundary conditions, we need to describe a oscillatory velocity, following the \(\)x\(\) axis and dependent of time.

\[ v_{in} = 1-cos\left( \frac{2\pi t}{5} \right)\]

Then we can set

  • on \(\Gamma^{h,*}_f\), an inflow Dirichlet condition : \(\boldsymbol{u}_{f} = ( v_{in}, 0 )\)

  • on \(\Gamma^{i,*}_f\), an inflow Dirichlet condition : \(\boldsymbol{u}_{f} = ( v_{in}(8 y -7) ,0)\)

  • on \(\Gamma^{f,*}_f\), a homogeneous Dirichlet condition : \(\boldsymbol{u}_{f} = \boldsymbol{0}\)

  • on \(\Gamma^{o,*}_f\), a Neumann condition : \(\boldsymbol{\sigma}_{f} \boldsymbol{n}_f = \boldsymbol{0}\)

  • on \(\Gamma^{f,*}_s\), a condition that imposes this boundary to be fixed : \(\boldsymbol{\eta}_{s} = \boldsymbol{0}\)

  • on \(\Gamma^{e,*}_s\), a condition that lets these boundaries be free from constraints : \(\boldsymbol{F}_{s} \boldsymbol{\Sigma}_s \boldsymbol{n}_s = \boldsymbol{0}\)

To them we also add the fluid-structure coupling conditions on \(\Gamma_{fsi}^*\) :

\[\frac{\partial \boldsymbol{\eta_{s}} }{\partial t} - \boldsymbol{u}_f \circ \mathcal{A}^t_f = \boldsymbol{0} \quad \left( \text{Velocities continuity}\right)\]
\[ \boldsymbol{F}_{s} \boldsymbol{\Sigma}_{s} \boldsymbol{n}^*_s + J_{\mathcal{A}^t_f} \boldsymbol{F}_{\mathcal{A}^t_f}^{-T} \hat{\boldsymbol{\sigma}}_f \boldsymbol{n}^*_f = \boldsymbol{0} \quad \left( \text{ Constraint continuity}\right)\]
\[ \boldsymbol{\varphi}_s^t - \mathcal{A}^t_f = \boldsymbol{0} \quad \left( \text{Geometric continuity}\right)\]

5.1.2. Initial conditions

To realize the simulations, we used a time step \(\Delta t\) equals to \(0.01\) s.

5.2. Inputs

The following table displays the various fixed and variables parameters of this test-case.

Table 3. Fixed and Variable Input Parameters

Name

Description

Nominal Value

Units

\(E_s\)

Young’s modulus

\(250\)

\(N.m^{-2}\)

\(\nu_s\)

Poisson’s ratio

\(0\)

dimensionless

\(\rho_s\)

structure density

\(500\)

\(kg.m^{-3}\)

\(\mu_f\)

viscosity

\(1\times 10^{-3}\)

\(m^2.s^{-1}\)

\(\rho_f\)

density

\(1\)

\(kg.m^{-3}\)

5.3. Outputs

\((\boldsymbol{u}_f, p_f, \boldsymbol{\eta}_s, \mathcal{A}_f^t)\), checking the fluid-structure system, are the output of this problem.

5.4. Discretization

To discretize space, we used \(P_N~-~P_{N-1}\) Taylor-Hood finite elements.

For the structure time discretization, Newmark-beta method is the one we used. And for the fluid time discretization, we used BDF, at order \(q\).

5.5. Implementation

All the codes files are into FSI

5.6. Results

We begin with a \(P_2~-~P_1\) approximation for the fluid with a geometry order equals at \(1\), and a fluid-structure stable interface.

Then we retry with a \(P_3~-~P_2\) approximation for the fluid with a geometry order equals at \(2\), and a fluid-structure stable interface.

Finally we launch it with the same conditions as before, but with a deformed interface.

5.6.1. Conclusion

First at all, we can see that the first two tests offer us similar results, despite different orders uses. At contrary, the third result set are better than the others.

The elastic wall thinness, in the stable case, should give an important refinement on the fluid domain, and so a better fluid-structure coupling control. However, the deformed case result are closer to the stable case made measure.

5.7. Bibliography

References for this benchmark
  • [MokWall] DP Mok and WA Wall, Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures, Trends in computational structural mechanics, Barcelona, 2001.

  • [Chabannes] Vincent Chabannes, Vers la simulation numérique des écoulements sanguins, Équations aux dérivées partielles [math.AP], Université de Grenoble, 2013.

  • [Gerbeau] J.F. Gerbeau, M. Vidrascu, et al, A quasi-newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows, 2003.

  • [Vazquez] J.G. Valdés Vazquèz et al, Nonlinear analysis of orthotropic membrane and shell structures including fluid-structure interaction, 2007.

  • [KuttlerWall] U. Kuttler and W.A. Wall, Fixed-point fluid–structure interaction solvers with dynamic relaxation, Computational Mechanics, 2008.

  • [Kassiotis] C. Kassiotis, A. Ibrahimbegovic, R. Niekamp, and H.G. Matthies, Nonlinear fluid–structure interaction problem ,part i : implicit partitioned algorithm, nonlinear stability proof and validation examples, Computational Mechanics, 2011.