# Identification of MEMS Geometric Uncertainties through Homogenization

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## Abstract

**:**

## 1. Introduction

## 2. Asymptotic Homogenization

## 3. MEMS Filter

#### 3.1. Mechanical Design

#### 3.2. Homogenization of the Auxetic Core

#### 3.3. Validation of Effective Properties

## 4. Identification of Geometric Uncertainties

#### 4.1. Experimental Results

#### 4.2. Optimization Procedure

#### 4.3. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FEA | Finite Element Analysis |

MEMS | Micro-Electro-Mechanical Systems |

PCB | Printed Circuit Board |

PLCC68 | 68-Pin Plastic-Leaded-Chip-Carrier |

SEM | Scanning Electron Microscopy |

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**Figure 1.**(

**a**) Single-phase heterogeneous medium with periodic structure. (

**b**) Close-up view of its unit cell.

**Figure 2.**SEM immage of the MEMS structure (

**left**) with a close-up view of the stators S1, S2 (

**mid**) and the auxetic unit cell (

**right**).

**Figure 3.**Contour of the displacement magnitude over the (magnified) deformed shape of the solution of cell problems (

**a**) ${\mathbf{\chi}}^{11}$, (

**b**) ${\mathbf{\chi}}^{22}$ and (

**c**) ${\mathbf{\chi}}^{12}$.

**Figure 4.**Polar plots of the homogenized Young’s modulus (

**a**) and Poisson’s ratio (

**b**) as a function of $\vartheta $ for different values of the over-etch. Negative values of ${\nu}^{0}$ are indicated with dashed lines.

**Figure 5.**Contours of the horizontal (

**a**,

**b**) and vertical (

**c**,

**d**) displacement of the MEMS auxetic core (

**a**,

**c**) and the corresponding homogenized medium (

**b**,

**d**).

**Figure 6.**(

**a**) Block diagram of the experimental setup employed for the static characterization of the auxetic MEMS device; (

**b**) image of tailored printed circuit board (PCB) containing the MEMS device housed in a PLCC68 package.

**Figure 7.**Capacitance–voltage curves, between the rotor and the different stators (

**a**) Exp. $\Delta {C}_{D1-R}$; Exp. $\Delta {C}_{S2-R}$, (

**b**) Exp. $\Delta {C}_{D2-R}$; Exp. $\Delta {C}_{S1-R}$, (

**c**) Exp. $\Delta {C}_{Tu1-R}$; Exp. $\Delta {C}_{Td2-R}$, (

**d**) Exp. $\Delta {C}_{Tu2-R}$; Exp. $\Delta {C}_{Td1-R}$, obtained experimentally (dotted), and the optimal analytical ones (black continuos).

**Figure 8.**(

**a**) Schematic representation of the analytical model; (

**b**) contour of the error as a function of the over-etch and horizontal shift.

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**MDPI and ACS Style**

Faraci, D.; Zega, V.; Nastro, A.; Comi, C.
Identification of MEMS Geometric Uncertainties through Homogenization. *Micro* **2022**, *2*, 564-574.
https://doi.org/10.3390/micro2040037

**AMA Style**

Faraci D, Zega V, Nastro A, Comi C.
Identification of MEMS Geometric Uncertainties through Homogenization. *Micro*. 2022; 2(4):564-574.
https://doi.org/10.3390/micro2040037

**Chicago/Turabian Style**

Faraci, David, Valentina Zega, Alessandro Nastro, and Claudia Comi.
2022. "Identification of MEMS Geometric Uncertainties through Homogenization" *Micro* 2, no. 4: 564-574.
https://doi.org/10.3390/micro2040037