# A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics

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## Abstract

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## 1. Introduction

## 2. Discrete Mechanics Formulation

#### 2.1. One-Dimensional Framework

#### 2.2. Extension to Other Space Dimensions

#### 2.3. Equations of Discrete Formulation

#### 2.4. Inertia on Discrete Formulation

#### 2.5. Reduction to Waves Equation

## 3. Numerical Methodology

## 4. Verifications

#### 4.1. Shear between a Fluid and an Elastic Media

#### 4.2. Compression of an Elastic Solid by a Fluid

## 5. Validation

#### Lid-Driven Open Cavity Flow with Flexible Bottom Wall

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Discrete geometric structure: a set of primitive planar facets $\mathcal{S}$ are associated with the segment $\Gamma $ of unit vector $\mathbf{t}$ whose ends a and b are distant by a length d. Each facet is defined by a contour ${\Gamma}^{*}$, a collection of three segments $\Gamma $, is oriented according to the normal $\mathbf{n}$ such that $\mathbf{n}\xb7\mathbf{t}=0$; the dual surface $\Delta $ connecting the centroids of the cells is also flat.

**Figure 2.**Fluid–structure interaction between a viscous fluid and an elastic solid; the viscosity of the fluid is equal to $\nu $ = 1 m${}^{2}$ s${}^{-1}$ and the solid shear modulus is equal to $\nu $ = 4 m${}^{2}$ s${}^{-1}$: (

**left**) the velocity of the interface over time is presented; (

**middle**) the velocity $\mathbf{V}$ at steady-state regime is reported; and (

**right**) the displacement of the solid $\mathbf{U}$ is plotted.

**Figure 3.**Evolutions of the density $\rho $ and of the scalar potential ${10}^{-5}\times \varphi $ as a function of the vertical coordinate y for a time $t=100$ s and an injection velocity ${V}_{0}$ = −0.01 m s${}^{-1}$ for a mesh of ${32}^{2}$ cells.

**Figure 4.**Vertical displacement at mid-point of flexible plate in lid-driven open cavity flow with flexible bottom wall, velocity and streamlines at t = 2.5, 15, 20 s.

**Figure 5.**Lid-driven open cavity flow with flexible bottom wall. Evolution of maximum deviation of membrane ${y}_{m}$ over time.

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Vincent, S.; Caltagirone, J.-P. A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics. *Fluids* **2021**, *6*, 95.
https://doi.org/10.3390/fluids6030095

**AMA Style**

Vincent S, Caltagirone J-P. A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics. *Fluids*. 2021; 6(3):95.
https://doi.org/10.3390/fluids6030095

**Chicago/Turabian Style**

Vincent, Stéphane, and Jean-Paul Caltagirone. 2021. "A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics" *Fluids* 6, no. 3: 95.
https://doi.org/10.3390/fluids6030095