# On the Optimal Control of Stationary Fluid–Structure Interaction Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### Optimality System

- On ${\mathcal{A}}_{ad}^{\prime}$ we have $\chi <E<\omega $ and$${\mathcal{J}}^{\prime}\left(\overline{\mathbf{\eta}}\left(\overline{E}\right)\right)\xb7\tilde{E}=0\phantom{\rule{2.em}{0ex}}\forall \tilde{E}\in {L}^{2}\left({\mathsf{\Omega}}^{s}\right)\phantom{\rule{0.166667em}{0ex}},$$
- On ${\mathcal{A}}_{ad}-{\mathcal{A}}_{ad}^{\prime}$ we have $s=0$ which implies $E=\chi $ or $E=\omega $.

## 3. Numerical Implementation and Results

Algorithm 1 Description of the Steepest Descent algorithm. | |

1. Set a state $({\mathit{v}}^{0},{p}^{0},{\mathbf{\eta}}^{0})$ satisfying (16) and (17) | ▹ Setup of the state - Reference case |

2. Compute the functional ${\mathcal{J}}^{0}$ in (20) | |

3. Set ${r}^{0}=1$ | |

for$i=1\to {i}_{max}$do | |

4. Solve the system (46) and (47) to obtain the adjoint state $({\mathit{v}}_{a}^{i},{p}_{a}^{i})$ | |

5. Compute the control update $\delta {g}^{i}$ with (42)–(44) | |

6. Set ${r}^{i}={r}^{0}$ | |

while ${\mathcal{J}}^{i}({g}^{i-1}+{r}^{i}\delta {g}^{i})>{\mathcal{J}}^{i-1}\left({g}^{i-1}\right)$ do | ▹ Line search |

7. Set ${r}^{i}=\rho \phantom{\rule{0.166667em}{0ex}}{r}^{i}$ | |

8. Solve (16) and (17) for the state $({\mathit{v}}^{i},{p}^{i},{\mathbf{\eta}}^{i})$ with ${g}^{i}$ | |

if ${r}^{i}<toll$ then | |

Line search not successful | ▹ End of the algorithm |

end if | |

end while | |

end for |

#### 3.1. Test Case Configuration

#### 3.2. Pressure Boundary Control

#### 3.3. Distributed Control

#### 3.4. Parameter Estimation Problem

#### Control with Gradient Regularization

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Geometry and controlled region defined by dotted square on the right ${\mathsf{\Omega}}_{d}$ (

**left**). Reference case with velocity profiles and streamlines in the liquid (

**center**). Solid displacement field ${\eta}_{x}$ for the same reference configuration (

**right**).

**Figure 2.**Control pressure profile with $\alpha ={10}^{-8}$ (A), $\alpha ={10}^{-9}$ (B) and $\alpha ={10}^{-10}$ (C).

**Figure 3.**Controlled case with velocity profiles $\mathit{v}$ and streamlines in the liquid (

**left**) and displacement $\mathbf{\eta}$ in the solid (

**center**). Adjoint velocity ${\mathit{v}}_{a}$ field and streamlines for the reference case configuration (

**right**).

**Figure 4.**Controlled case with velocity profiles $\mathit{v}$ and streamlines in the liquid (

**left**) and displacement $\mathbf{\eta}$ in the solid (

**center**). Force vectors $\mathbf{f}$ in black and magnitude in colors (

**right**).

**Figure 5.**Young modulus E fields, discretization $l=2$ with different ${E}_{min}=10$ Pa (

**left**), 50 Pa (

**center**), 100 Pa (

**right**). E and ${E}_{min}$ are scaled by $2.4\xb7{10}^{3}$.

**Figure 6.**Young modulus E fields, discretization $l=5$ with different ${E}_{min}=10$ Pa (

**left**), 50 Pa (

**center**), 100 Pa (

**right**). E and ${E}_{min}$ are scaled by $2.4\xb7{10}^{3}$.

**Figure 7.**Young modulus profile on the solid vertical mid-line (

**left**) and on the interface ${\mathsf{\Gamma}}_{i}$ (

**right**), with different ${E}_{min}=10$ Pa (A), 50 Pa (B), 100 Pa (C). E and ${E}_{min}$ are scaled by $2.4\xb7{10}^{3}$.

**Figure 8.**Young modulus E fields, discretization $l=5$ with ${E}_{min}=10$ Pa and $\lambda ={10}^{-1}$ (

**left**), $\lambda ={10}^{-3}$ (

**middle**). Young modulus profile on the solid vertical mid-line (

**right**), with ${E}_{min}=10$ Pa and $\lambda ={10}^{-1}$ (A), $\lambda ={10}^{-3}$ (B). E and ${E}_{min}$ are scaled by $2.4\xb7{10}^{3}$.

**Table 1.**Objective functional. The reference case with no control is labeled with $\alpha =\infty $.

$\mathit{\alpha}$ | $\mathcal{J}(\mathit{\eta},\mathit{p})$ | ${\overline{\mathit{\eta}}}_{\mathit{x}}\phantom{\rule{4pt}{0ex}}\left[\mathit{m}\right]$ |
---|---|---|

∞ | $1.292\xb7{10}^{-5}$ | 0.0180 |

${10}^{-8}$ | $9.192\xb7{10}^{-6}$ | 0.0202 |

${10}^{-9}$ | $1.515\xb7{10}^{-6}$ | 0.0469 |

${10}^{-10}$ | $6.454\xb7{10}^{-7}$ | 0.0497 |

**Table 2.**Objective functional $\mathcal{J}$ and average x-displacement over the controlled region ${\mathsf{\Omega}}_{d}$ computed with no control ($\beta =\infty $) and different $\beta $ values.

$\mathit{\beta}$ | ∞ | ${10}^{-11}$ | ${10}^{-12}$ | ${10}^{-13}$ |
---|---|---|---|---|

$\mathcal{J}(\mathbf{\eta},\mathbf{f})\xb7{10}^{8}$ | 1292.4 | 25.854 | 5.8864 | 2.3875 |

${\overline{\eta}}_{x}\phantom{\rule{0.166667em}{0ex}}\left[m\right]$ | 0.0180 | 0.0494 | 0.0498 | 0.0499 |

**Table 3.**Objective functional. ${E}_{min}$ is scaled by $2.4\xb7{10}^{3}$. The reference case with no control is labeled with NC.

${\mathit{E}}_{\mathit{min}}\left[\mathit{Pa}\right]$ | |||||
---|---|---|---|---|---|

Level | 10 | 50 | 100 | 200 | NC |

2 | 1.63$\xb7{10}^{-7}$ | 2.08$\xb7{10}^{-7}$ | 1.82$\xb7{10}^{-7}$ | 2.73$\xb7{10}^{-6}$ | 1.23$\xb7{10}^{-5}$ |

3 | 1.41$\xb7{10}^{-7}$ | 1.12$\xb7{10}^{-7}$ | 1.53$\xb7{10}^{-7}$ | 2.58$\xb7{10}^{-6}$ | 1.23$\xb7{10}^{-5}$ |

4 | 1.23$\xb7{10}^{-7}$ | 1.62$\xb7{10}^{-7}$ | 1.18$\xb7{10}^{-7}$ | 2.58$\xb7{10}^{-6}$ | 1.23$\xb7{10}^{-5}$ |

5 | 1.19$\xb7{10}^{-7}$ | 9.17$\xb7{10}^{-8}$ | 1.16$\xb7{10}^{-7}$ | 2.52$\xb7{10}^{-6}$ | 1.23$\xb7{10}^{-5}$ |

**Table 4.**Objective functionals. ${E}_{min}$ is scaled by $2.4\xb7{10}^{3}$. The reference case with no control is labeled with NC.

${\mathit{E}}_{\mathit{min}}\left[\mathit{Pa}\right]$ | |||||
---|---|---|---|---|---|

$\mathbf{\lambda}$ | 10 | 50 | 100 | 200 | NC |

${10}^{-1}$ | 3.29$\xb7{10}^{-7}$ | 3.29$\xb7{10}^{-7}$ | 3.51$\xb7{10}^{-7}$ | 2.68$\xb7{10}^{-6}$ | 1.23$\xb7{10}^{-5}$ |

${10}^{-3}$ | 9.79$\xb7{10}^{-8}$ | 9.20$\xb7{10}^{-8}$ | 3.22$\xb7{10}^{-7}$ | 2.68$\xb7{10}^{-6}$ | 1.23$\xb7{10}^{-5}$ |

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Chirco, L.; Manservisi, S.
On the Optimal Control of Stationary Fluid–Structure Interaction Systems. *Fluids* **2020**, *5*, 144.
https://doi.org/10.3390/fluids5030144

**AMA Style**

Chirco L, Manservisi S.
On the Optimal Control of Stationary Fluid–Structure Interaction Systems. *Fluids*. 2020; 5(3):144.
https://doi.org/10.3390/fluids5030144

**Chicago/Turabian Style**

Chirco, Leonardo, and Sandro Manservisi.
2020. "On the Optimal Control of Stationary Fluid–Structure Interaction Systems" *Fluids* 5, no. 3: 144.
https://doi.org/10.3390/fluids5030144