# Feel++

 Benchmarks are available at FSI Benchmarks.

# Fluid-Structure Interaction

Feel++ Book Contributors https://github.com/feelpp/book.feelpp.org/graphs/contributors

## 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

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