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

\[\boldsymbol{\sigma}_{f} \boldsymbol{n}_f = \left\{ \begin{aligned} & \left(-\frac{1.3332 \cdot 10^4}{2} \left( 1 - \cos \left( \frac{ \pi t} {1.5 \cdot 10^{-3}} \right) \right), 0 \right)^T \quad & \text{ if } t < 0.003 \\ & \boldsymbol{0} \quad & \text{ else } \end{aligned} \right.\]
\[ \frac{\partial \boldsymbol{\eta_{s}} }{\partial t} - \boldsymbol{u}_f \circ \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(1)}\]
\[ \boldsymbol{F}_{s} \boldsymbol{\Sigma}_{s} \boldsymbol{n}^*_s + J_{\mathcal{A}_{f}^t} \hat{\boldsymbol{\sigma}}_f \boldsymbol{F}_{\mathcal{A}_{f}^t}^{-T} \boldsymbol{n}^*_f = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(2)}\]
\[ \boldsymbol{\varphi}_s^t - \mathcal{A}_{f}^t = \boldsymbol{0} , \quad \text{ on } \Gamma_{fsi}^* \times \left[t_i,t_f \right] \quad \boldsymbol{(3)}\]

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


Young’s modulus

\(3 \times 10^6 \)



Poisson’s ratio




fluid tube radius




solid tube radius




tube length




A coordinates




B coordinates












proximal resistance



distal resistance

\(6.2 \times 10^3\)



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

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.



































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.