Wave Pressure 3D

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

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.

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

1.2. Initial conditions

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

2. Inputs

Table 1. 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

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

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. Discretization

Here are the different configurations we worked on. The parameter Incomp defines if we use the incompressibility 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.

5. Results

Documentation pending

5.1. Conclusion

Documentation pending

6. 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.