Many Objective Optimization of a Magnetic Micro–Electro–Mechanical (MEMS) Micromirror with Bounded MP-NSGA Algorithm
Abstract
:1. Introduction
- (i)
- The electrostatic excitation, based on an electric field which causes the controlled displacement of a movable component or the deformation of an elastic membrane;
- (ii)
- The thermal excitation, which exploits the difference between the thermal expansion coefficients featuring two elastic materials when subject to a temperature gradient;
- (iii)
- The magnetic excitation, based on the Lorentz force acting on a loop of current placed in an external magnetic field.
2. Materials and Methods
2.1. The Magnetic Micromirror
2.2. Field Analysis
2.3. Optimization Problem
2.4. Optimization Algorithms
3. Results
4. Discussion
Author Contributions
Funding
Conflicts of Interest
References
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Design Variable | Minimum | Maximum | Constraint | Threshold |
---|---|---|---|---|
x1 | 50 µm | 150 µm | Holding torque | >10−8 Nm |
x2 | 1 mm | 2 mm | ||
x3 | 25 µm | 75 µm | Actuation torque | >10−9 Nm |
x4 | 25 µm | 75 µm | ||
x5 | 100 µm | 300 µm | Current density | <5 × 109 Am−2 |
x6 | 300 µm | 900 µm | ||
x7 | 0 A | 150 A |
MP-NSGA | NSGA-III | ||||
---|---|---|---|---|---|
f1 | f2 | f3 | f1 | f2 | f3 |
1.91 × 10−3 | 1.11 × 10−9 | 5.67 × 10−10 | 10 | 1.38 × 10−7 | 9.8 × 10−10 |
0.31 | 6.71 × 10−9 | 4.00 × 10−10 | 1.8 | 8.31 × 10−8 | 1.14 × 10−9 |
1.05 × 10−3 | 1.38 × 10−9 | 1.22 × 10−9 | 3.39 | 9.79 × 10−8 | 9.92 × 10−10 |
12.0 | 1.43 × 10−7 | 1.28 × 10−9 | 7.68 | 1.28 × 10−7 | 9.71 × 10−10 |
1.14 | 3.96 × 10−8 | 2.41 × 10−9 | 7.27 | 1.37 × 10−7 | 1.04 × 10−9 |
7.96 | 1.36 × 10−7 | 1.01 × 10−9 | 1.72 | 7.76 × 10−8 | 9.25 × 10−10 |
3.76 | 8.88 × 10−8 | 1.15 × 10−9 | 3.44 | 9.82 × 10−8 | 9.37 × 10−10 |
10.3 | 1.13 × 10−7 | 8.66 × 10−10 | 4.59 | 1.11 × 10−7 | 9.61 × 10−10 |
6.93 | 1.13 × 10−7 | 1.30 × 10−9 | 5.91 | 1.2 × 10−7 | 9.71 × 10−10 |
5.01 | 3.88 × 10−8 | 7.66 × 10−10 | 6.14 | 1.3 × 10−7 | 1.04 × 10−9 |
11.1 | 1.41 × 10−7 | 1.13 × 10−9 | 2.94 | 9.36 × 10−8 | 1.06 × 10−9 |
1.75 | 5.08 × 10−8 | 2.02 × 10−9 | 4.59 | 1.11 × 10−7 | 9.74 × 10−10 |
4.81 | 1.01 × 10−7 | 9.39 × 10−10 | 5.35 | 1.25 × 10−7 | 1.07 × 10−9 |
1.62 | 4.71 × 10−8 | 2.07 × 10−9 | 3.03 | 9.11 × 10−8 | 7.29 × 10−10 |
6.64 | 1.07 × 10−7 | 9.38 × 10−10 | 4.59 | 1.15 × 10−7 | 9.80 × 10−10 |
0.97 | 1.83 × 10−8 | 1.63 × 10−9 | 5.90 | 1.20 × 10−7 | 9.71 × 10−10 |
1.95 | 6.78 × 10−8 | 1.00 × 10−9 | 2.26 | 9.02 × 10−8 | 1.10 × 10−9 |
1.47 | 4.22 × 10−8 | 2.34 × 10−9 | 3.76 | 1.04 × 10−7 | 9.71 × 10−10 |
1.56 | 3.94 × 10−8 | 1.90 × 10−9 | 0.96 | 5.76 × 10−8 | 9.22 × 10−10 |
3.32 | 7.47 × 10−8 | 8.85 × 10−10 | 3.95 | 1.08 × 10−7 | 9.69 × 10−10 |
Design Variable | P | SPL | FPL | Objective Function | P | SPL | FPL |
---|---|---|---|---|---|---|---|
x1 [µm] | 100 | 144 | 140 | f1 power losses [W] | 1.04 | 2.36 × 10−2 | 1.05 × 10−3 |
x2 [mm] | 1.2 | 1.17 | 1.24 | ||||
x3 [µm] | 50 | 33.8 | 28.9 | f2 actuation torque [Nm] | 2.53 × 10−8 | 7.76 × 10−9 | 1.38 × 10−9 |
x4 [µm] | 50 | 34.6 | 29.2 | ||||
x5 [µm] | 200 | 178 | 175 | f3 system volume [m3] | 1.15 × 10−9 | 1.16 × 10−9 | 1.22 × 10−9 |
x6 [µm] | 600 | 532 | 485 | ||||
x7, current [A] | 10 | 1.89 | 0.35 |
Design Variable | SAT = SSV | FAT | FSV | Objective Function | SAT = SSV | FAT | FSV |
---|---|---|---|---|---|---|---|
x1 [µm] | 134 | 150 | 50 | f1 power losses [W] | 9.34 | 12 | 0.31 |
x2 [mm] | 1.02 | 1.18 | 1 | ||||
x3 [µm] | 45.9 | 25 | 25 | f2 actuation torque [Nm] | 4.8 × 10−8 | 1.43 × 10−7 | 6.71 × 10−9 |
x4 [µm] | 75 | 55 | 25 | ||||
x5 [µm] | 156 | 222 | 300 | f3 system volume [m3] | 1.07 × 10−9 | 1.28 × 10−9 | 4 × 10−10 |
x6 [µm] | 827 | 300 | 900 | ||||
x7, current [A] | 55.6 | 60.1 | 8.16 |
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Di Barba, P.; Mognaschi, M.E.; Sieni, E. Many Objective Optimization of a Magnetic Micro–Electro–Mechanical (MEMS) Micromirror with Bounded MP-NSGA Algorithm. Mathematics 2020, 8, 1509. https://doi.org/10.3390/math8091509
Di Barba P, Mognaschi ME, Sieni E. Many Objective Optimization of a Magnetic Micro–Electro–Mechanical (MEMS) Micromirror with Bounded MP-NSGA Algorithm. Mathematics. 2020; 8(9):1509. https://doi.org/10.3390/math8091509
Chicago/Turabian StyleDi Barba, Paolo, Maria Evelina Mognaschi, and Elisabetta Sieni. 2020. "Many Objective Optimization of a Magnetic Micro–Electro–Mechanical (MEMS) Micromirror with Bounded MP-NSGA Algorithm" Mathematics 8, no. 9: 1509. https://doi.org/10.3390/math8091509
APA StyleDi Barba, P., Mognaschi, M. E., & Sieni, E. (2020). Many Objective Optimization of a Magnetic Micro–Electro–Mechanical (MEMS) Micromirror with Bounded MP-NSGA Algorithm. Mathematics, 8(9), 1509. https://doi.org/10.3390/math8091509