# A Fast Multiobjective Optimization Strategy for Single-Axis Electromagnetic MOEMS Micromirrors

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Constitutive Equations

#### 2.2. Choice of the Optimization Objectives

#### 2.3. Choice of the Design Variables

## 3. MO Implementation and Results

## 4. FEM Validation

^{−1}) flowing in the active areas. We then applied at each mesh node of the active area a pressure equal to ${\sigma}_{c}{B}_{y}\left(x,y\right)$, where ($x,y$) were the coordinates of the node, and the expression of ${B}_{y}$ was determined through the formulas in Appendix A. To include the effect of large rotations of the plate, which produces a reduction of the effective arm of the magnetic torque with respect to the rest position, the pressure was further reduced by a factor $\mathrm{cos}\phi $, with $\phi $ being the target angle (10°). A typical deflected shape for static simulations is shown in Figure 4. For modal FEM simulations, the frequency of the lowest mode was computed, and it was verified that it corresponded to the torsional mode.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Structure of the mirror assembly with the two permanent magnets (the front magnet is cut away for clarity). The direction of the actuation magnetic field ${B}_{y}$ and the mirror thickness ${t}_{S}$ are also shown. The pictured number of loops, ${n}_{c}$, is 2½.

**Figure 2.**Top view of the mirror. Actuation coil segments are in green, areas for the calculation of the equivalent magnetic field are in red.

**Figure 3.**Interpolated Pareto frontier for the mirror problem, as determined by numerical solution of (17), with curves at constant ${l}_{Ty}$ plotted over the surface. Each black dot is a Pareto-optimal solution lying (by definition) on the Pareto frontier.

**Figure 6.**Relative error for the resonance frequency between the FEM case and the MO (Multiobjective Optimization) case.

Symbol | Unit | Definition | Value |
---|---|---|---|

${A}_{c}$ | m^{2} | equivalent area of coil (are such that ${T}_{x}={I}_{c}{B}_{y}{A}_{c}$) | - |

${a}_{k}$ | mm | distance of kth active coil segment from plate center | - |

B_{R} | T | magnetic material remanence | 1.4 |

${B}_{y}$ | T | magnetic field along $y$ at the center of active area (Figure 2) | - |

${d}_{x}$ | mm | length of the reflective surface | 2.5 |

${d}_{y}$ | mm | width of the reflective surface | 1.5 |

${f}_{\phi}$ | Hz | angular resonance frequency of main torsional mode | - |

g_{l} | μm | distance (gap) between metal lines | 10 |

${g}_{m}$ | mm | distance (gap) between the mirror and the magnets | |

${G}_{Si}$ | GPa | shear modulus of silicon along $x$ | 79.5 |

${I}_{c}$ | mA | coil current | - |

${J}_{Px}$ | kg·m^{2} | moment of inertia of the plate | - |

${J}_{S\phi}$ | m^{4} | torsional constant of the spring cross-section | - |

${k}_{\phi}$ | N·m | torsional spring constant of one spring along $\phi $ | - |

${l}_{c}$ | mm | Total length of the coil | - |

${l}_{My}$ | mm | magnet width along $y$ | - |

${l}_{M\left\{x,z\right\}}$ | mm | magnet dimensions along $\left\{x,z\right\}$ | {5, 5} |

${l}_{s}$ | m | spring length | - |

${l}_{Ty}$ | mm | total assembly width (mirror + magnets) | - |

${l}_{x}$ | m | length of mirror plate (including room for the coil) | - |

${l}_{y}$ | m | width of mirror plate (including room for the coil) | - |

${M}_{c}$ | kg | total mass of the coil | - |

${n}_{c}$ | - | number of coils (counting also half-coils) | - |

${p}_{l}$ | μm | metal line pitch | - |

${s}_{k}$ | mm | length of kth active coil segment | - |

${t}_{l}$ | μm | Metal thickness | 10 |

${t}_{s}$ | μm | thickness of plate and springs | 30 |

${T}_{x}$ | N·m | torque on the plate | - |

${w}_{l}$ | μm | metal line width | - |

${w}_{s}$ | μm | spring width | 30 |

$\phi $ | rotation along main axis (x axis) | 10 | |

${\mu}_{l}$ | kg/m^{3} | aluminum density | 2700 |

${\mu}_{Si}$ | kg/m^{3} | silicon density | 2330 |

Variable | Max | Min |
---|---|---|

${l}_{s}$ | 0.2 mm | 2 mm |

${w}_{l}$ | 20 μm | 200 μm |

${n}_{c}$ | 1.5 | 199.5 |

${l}_{My}$ | 0.3 mm | 1 mm |

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**MDPI and ACS Style**

Pieri, F.; Cilea, A.
A Fast Multiobjective Optimization Strategy for Single-Axis Electromagnetic MOEMS Micromirrors. *Micromachines* **2018**, *9*, 2.
https://doi.org/10.3390/mi9010002

**AMA Style**

Pieri F, Cilea A.
A Fast Multiobjective Optimization Strategy for Single-Axis Electromagnetic MOEMS Micromirrors. *Micromachines*. 2018; 9(1):2.
https://doi.org/10.3390/mi9010002

**Chicago/Turabian Style**

Pieri, Francesco, and Alessandro Cilea.
2018. "A Fast Multiobjective Optimization Strategy for Single-Axis Electromagnetic MOEMS Micromirrors" *Micromachines* 9, no. 1: 2.
https://doi.org/10.3390/mi9010002