# Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems

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## Abstract

**:**

## 1. Introduction

- The fully Eulerian formulation is derived in Section 1.
- In Section 2 the equations are discretized implicitly in time as in [18,19]; then the displacements are eliminated and a non-linear system for the velocities and pressures remains. Spatial discretization is obtained by using stable finite element spaces like linear pressures and quadratic velocities on a tetrahedral mesh.
- In Section 3 it is shown that the energy decays at each time step, which is an indication that the method is robust.
- Finally, in Section 4, the method is implemented in 2D for axisymmetric systems and in 3D for general systems. Energy decrease, mass conservation and convergence are analyzed on an axisymmetric case: an elastic torus in a cylindrical canister filled with a fluid at a Reynolds number of a few hundred. Then, the method is carefully evaluated on the test case proposed in [20] and comparisons with previous publications are made.

## 2. Derivation of the Formulation from the General Laws of Continuum Mechanics

#### 2.1. Notations

- ${\Sigma}^{t}={\overline{\Omega}}_{f}^{t}\cap {\overline{\Omega}}_{s}^{t}$ be the fluid–structure interface, and $\partial {\Omega}^{t}$ be the boundary of ${\Omega}^{t}$,
- $\Gamma $ be the part of $\partial {\Omega}^{t}$ where the structure is clamped or the fluid does not slip. It is assumed to be independent of t.

- $\mathbf{X}\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}{\Omega}^{0}\times \left(0,\phantom{\rule{3.33333pt}{0ex}}T\right),\mapsto {\Omega}^{t}\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}\mathbf{X}\left({x}^{0},\phantom{\rule{3.33333pt}{0ex}}t\right)$ is the Lagrangian position at t of ${x}^{0}$,
- $\mathbf{d}=\mathbf{X}\left({x}^{0},\phantom{\rule{3.33333pt}{0ex}}t\right)-{x}^{0}$ is the displacement,
- $\mathbf{u}(X({x}^{0},\phantom{\rule{3.33333pt}{0ex}}t),\phantom{\rule{3.33333pt}{0ex}}t)={\partial}_{t}\mathbf{X}({x}^{0},\phantom{\rule{3.33333pt}{0ex}}t)$ is the Eulerian velocity of the deformation at t and $x=X({x}^{0},t)$,
- ${\mathbf{F}}_{ji}={\partial}_{{x}_{i}^{0}}{\mathbf{X}}_{j}$ is the transposed gradient of the deformation,
- $J={\mathrm{det}}_{\mathbf{F}}$ is the Jacobian of the deformation.

- $\rho \left(x,\phantom{\rule{3.33333pt}{0ex}}t\right)={\mathbf{1}}_{{\Omega}_{f}^{t}}(x,\phantom{\rule{3.33333pt}{0ex}}t){\rho}_{f}\left(x,t\right)+{\mathbf{1}}_{{\Omega}_{s}^{t}}(x,\phantom{\rule{3.33333pt}{0ex}}t){\rho}_{s}\left(x,t\right)$, the density,
- $\mathbf{\sigma}\left(x,t\right)={\mathbf{1}}_{{\Omega}_{f}^{t}}(x,\phantom{\rule{3.33333pt}{0ex}}t){\mathbf{\sigma}}_{f}\left(x,\phantom{\rule{3.33333pt}{0ex}}t\right)+{\mathbf{1}}_{{\Omega}_{s}^{t}}(x,\phantom{\rule{3.33333pt}{0ex}}t){\mathbf{\sigma}}_{s}\left(x,\phantom{\rule{3.33333pt}{0ex}}t\right)$, the stress tensor.

#### 2.2. Conservation Laws

#### 2.3. Constitutive Equations

- For a Newtonian incompressible fluid, ${\mathbf{\sigma}}_{f}=-{p}_{f}\mathbf{I}+\mu \mathbf{Du}$
- For an hyperelastic incompressible material, ${\mathbf{\sigma}}_{s}=-{p}_{s}\mathbf{I}+{\partial}_{\mathbf{F}}\phantom{\rule{0.166667em}{0ex}}\Psi \phantom{\rule{0.166667em}{0ex}}{\mathbf{F}}^{T}$

#### 2.4. The Mooney–Rivlin 3D Stress Tensor

#### 2.5. From 3D to 2D

#### 2.6. Variational Formulation

## 3. Numerical Schemes

#### 3.1. Characteristic-Galerkin Derivatives

#### 3.2. A Monolithic Time–Discrete Variational Formulation

**Proposition**

**1.**

**Some comments**

- One may wonder why the scheme is applied to $\mathbf{u}$ and not to $\rho \mathbf{u}$. Note that $\rho ={\rho}_{f}{\mathbf{1}}_{{\Omega}_{f}^{t}}+{\rho}_{s}{\mathbf{1}}_{{\Omega}_{s}^{t}}$ is convected by the velocity $\mathbf{u}$. Hence ${\rho}^{n+1}\left(x\right)={\rho}^{n}\circ {\mathbb{Y}}^{n+1}\left(x\right)$. This shows that discretizing the total derivative of $\mathbf{u}$ or the total derivative of $\rho \mathbf{u}$ gives the same scheme:$$\frac{1}{\delta t}\left({\mathbf{w}}^{n+1}\left(x\right)-\mathbf{w}\left({\mathbb{Y}}^{n+1}\left(x\right)\right)\right)=\left({\partial}_{t}\mathbf{w}+\mathbf{u}\xb7\nabla \mathbf{w}\right){|}_{x,{t}^{n+1}}+O\left(\delta t\right)\mathrm{with}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathbf{w}=\mathbf{u}\phantom{\rule{0.277778em}{0ex}}\mathrm{or}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathbf{w}=\rho \mathbf{u}.$$

**Proof.**

#### 3.3. Spacial Discretization with Finite Elements

#### 3.4. Solution Algorithm

- Set $\rho ={\rho}_{n}$, ${c}_{3}={c}_{3}^{n}$, ${\Omega}^{r}={\Omega}_{r}^{n},\phantom{\rule{3.33333pt}{0ex}}r=f,s$, $\mathbf{u}={\mathbf{u}}_{h}^{n}$, $\mathbb{Y}\left(x\right)=x-\mathbf{u}\delta t$,
- Solve system (21). In this study a direct solver is used for the linear system.
- Set $\mathbf{u}={\mathbf{u}}_{h}^{n+1}$, $\mathbb{Y}\left(x\right)=x-\mathbf{u}\delta t$, ${\Omega}^{r}={\mathbb{Y}}^{-1}\left({\Omega}_{r}^{n}\right),\phantom{\rule{0.277778em}{0ex}}r=s,\phantom{\rule{0.277778em}{0ex}}f$; update ${c}_{3}$ and $\rho $.
- If not converged return to step 2.

**Remark**

**1.**

**Remark**

**2.**

## 4. Stability of the Scheme

#### 4.1. Conservation of Energy

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

#### 4.2. Stability of the Scheme Discretized in Time

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 4.3. Energy Inequality for the Fully Discrete Scheme

## 5. Numerical Tests

#### 5.1. An Axisymmetric Torus

#### 5.2. Clamped Structure in a Fluid

#### 5.2.1. Free-Fall of a Clamped Structure in Vacuum

#### 5.2.2. Free-Fall of a Clamped Beam in a Fluid

#### 5.3. The Benchmark

#### 5.3.1. Phase I

#### 5.3.2. Phase II

#### 5.3.3. Variation of Coefficient ${c}_{3}$

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Deformation of a rubber-like torus in a fluid due to initial centripetal velocity of the torus; snapshots at t = 0 (

**a**), 0.12 (

**b**), 0.28 (

**c**), 1.45 (

**d**) show also the four meshes, from finest to coarsest. The torus moves towards its axis (left boundary) and then bounces to the right to a maximum position and return to its initial position after a few oscillations. Notice the change of area to preserve mass.

**Figure 2.**(

**a**): Energy vs. time computed on mesh${}_{1}$ with 1, 2, 3 and 4 iterations in the solver of Section 3.4. On the same plot energy vs. time is also plotted for mesh${}_{3}$ and mesh${}_{5}$. (

**b**): Mass conservation vs. time when mesh${}_{1}$, (lower curve), mesh${}_{3}$ and mesh${}_{5}$ are used. Curves 3 and 5 are overlapping.

**Figure 3.**(

**a**): Evolution of the farthest point on the right of the torus versus time computed with mesh${}_{i}$, i = 1 (oscillations), 3 and 5. (

**b**): Mesh${}_{7}$ is used as a reference computations. The errors on the farthest point on the right in the torus versus time are shown for mesh${}_{1}$ (highest curve, its mean absolute value is $0.013$), mesh${}_{3}$ (lowest curve, its mean absolute value is $0.0023$), and mesh${}_{5}$ (flat curve, its mean absolute value is $4.1\times {10}^{-5}$).

**Figure 4.**Free-falling beam in vacuum; snapshots at time (

**a**) t = 10, (

**b**) t = 20, (

**c**) t = 40, (

**d**) t = 70, (

**e**) t = 80, (

**f**) t = 90.

**Figure 5.**Free-falling beam in fluid; snapshots at time (

**a**) t = 25, (

**b**) t = 50, (

**c**) t = 75, (

**d**) t = 100.

**Figure 7.**Computational results for phase I (

**a**) position of center line of the beam along z direction (results and experiments overlap), (

**b**) velocity norms about the symmetry plane.

**Figure 8.**Experimental curves of the peak inflow of inflow velocities used for the boundary condition in the computations.

**Figure 9.**Deflection of the silicone filament in phase $\mathrm{II}$ (

**a**) t = 0.721, (

**b**) t = 1.153, (

**c**) t = 1.585, (

**d**) t = 2.017, (

**e**) t = 2.449, (

**f**) t = 2.881.

**Figure 10.**Phase $\mathrm{II}$ results (

**a**) maximum and minimum magnitude of $2{c}_{3}$ and relative volume error multiplied by 1000, (

**b**) maximum magnitude of displacement $\mathbf{d}$.

**Figure 11.**Deflection $\mathbf{d}=\left({d}_{1},\phantom{\rule{3.33333pt}{0ex}}{d}_{2},\phantom{\rule{3.33333pt}{0ex}}{d}_{3}\right)$ and ${a}_{2}=2{c}_{3}$ of the silicone filament in phase $\mathrm{II}$ at (

**a**) t = 0.721, (

**b**) t = 1.153, (

**c**) t = 1.585, (

**d**) t = 2.017, (

**e**) t = 2.449, (

**f**) t = 2.881.

**Table 1.**For phase $\mathrm{II}$, curve–fitting coefficients of inlet peak velocity for ${\widehat{v}}_{k}\left(t\right)={\mathbf{\sigma}}_{i=1}^{3}{n}_{i}{t}^{i}/{\mathbf{\sigma}}_{j=0}^{4}{b}_{j}{t}^{j}$ with ${\widehat{v}}_{k}=0$ for $t\in \mathrm{I}\setminus {\mathrm{I}}_{k}$. Note that the flow in y direction is applied only in the upper inlet.

$\widehat{\mathit{v}}$ | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | ${\mathit{n}}_{3}$ | ${\mathit{d}}_{0}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{b}}_{3}$ | ${\mathit{b}}_{4}$ | ${\widehat{\mathbf{I}}}_{\mathit{k}}$ |
---|---|---|---|---|---|---|---|---|---|

${\widehat{v}}_{x}$ | $-11.37$ | $-28.99$ | 7.73 | 1.38 | 0.24 | 3.59 | $-3.14$ | 1 | $\left[0,\phantom{\rule{3.33333pt}{0ex}}4.07\right]$ |

${\widehat{v}}_{y}$ | 14.95 | 11.88 | $-2.17$ | 2.06 | $-2.0$ | 4.95 | $-3.50$ | 1 | $\left[0,\phantom{\rule{3.33333pt}{0ex}}5.51\right]$ |

${\widehat{v}}_{z}$ | 367.10 | 363.40 | $-62.24$ | 1.21 | $-0.38$ | 3.76 | $-3.19$ | 1 | $\left[0,\phantom{\rule{3.33333pt}{0ex}}5.27\right]$ |

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**MDPI and ACS Style**

Chiang, C.-Y.; Pironneau, O.; Sheu, T.W.H.; Thiriet, M.
Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems. *Fluids* **2017**, *2*, 34.
https://doi.org/10.3390/fluids2020034

**AMA Style**

Chiang C-Y, Pironneau O, Sheu TWH, Thiriet M.
Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems. *Fluids*. 2017; 2(2):34.
https://doi.org/10.3390/fluids2020034

**Chicago/Turabian Style**

Chiang, Chen-Yu, Olivier Pironneau, Tony W. H. Sheu, and Marc Thiriet.
2017. "Numerical Study of a 3D Eulerian Monolithic Formulation for Incompressible Fluid-Structures Systems" *Fluids* 2, no. 2: 34.
https://doi.org/10.3390/fluids2020034