# Topology Optimization of Piezoelectric Energy Harvesters for Enhanced Open-Circuit Voltage Subjected to Harmonic Excitations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Analysis of Piezoelectric Structures

## 3. Optimization Model and Sensitivity Analysis

$\mathrm{find}:$ | ${\mathbf{\rho}}_{\mathrm{pzt}}$ | (11) |

$\mathrm{max}:$ | ${\eta}_{\mathrm{energy}}\mathrm{or}\text{}{\eta}_{\mathrm{voltage}}$ | |

$\mathrm{s}.\mathrm{t}.$ | ${\overline{K}}_{uu}U+{K}_{u\varphi}\Phi =F$ ${K}_{\varphi u}U-{K}_{\varphi \varphi}\Phi =Q$ $V\left({\mathbf{\rho}}_{\mathrm{pzt}}\right)/{V}_{0}\le \overline{V}$ $0\le {\mathbf{\rho}}_{\mathrm{pzt}}\le 1$ |

## 4. Numerical Examples

_{11}= 120.3 GPa, c

_{12}= 75.2 GPa, c

_{13}= 75.1 GPa, c

_{33}= 110.8 GPa, c

_{44}= 21.1 GPa, c

_{66}= 22.6 GPa, e

_{31}= −5.4 C/m

^{2}, e

_{33}= 15.8 C/m

^{2}, e

_{15}= 12.3 C/m

^{2}, κ

_{11}= 919.1, κ

_{33}= 826.6, κ

_{0}= 8.55 × 10

^{−12}F/m, and the density is 7500 kg/m

^{3}. The Young’s modulus, density and Poisson’s ratio of the aluminum are 71 GPa, 2700 kg/m

^{3}and 0.33, respectively. A harmonic pressure load with an amplitude of 10 kPa was applied on the bottom surface. For the optimization process, we used the in-house MATLAB code that considers the dynamic pressure loads. The developed FE code was then verified using commercial software COMSOL. The parameters of the applied piezoelectric material PZT-5A are given as follows:

- Elastic matrix: ${C}_{2\mathrm{D}}^{\mathrm{E}}=\left[\begin{array}{ccc}120.3& 75.1& 0\\ 75.1& 110.8& 0\\ 0& 0& 21.1\end{array}\right]$ GPa;
- Piezoelectric coupling matrix: ${e}_{2\mathrm{D}}^{\mathrm{T}}=\left[\begin{array}{c}\begin{array}{ccc}0& 0& 12.3\end{array}\\ \begin{array}{ccc}-5.4& 15.8& 0\end{array}\end{array}\right]$ C/m
^{2}; - Dielectric matrix: ${\mathsf{\kappa}}_{2\mathrm{D}}^{\mathrm{s}}=\left[\begin{array}{cc}919.1& 0\\ 0& 826.6\end{array}\right]\times 8.55\times {10}^{-12}$ F/m.

#### 4.1. Design for Enhanced Energy Conversion Efficiency

#### 4.2. Design for Enhanced Open-Circuit Voltage

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 2.**Schematic diagram and frequency response curve of PEH. (

**a**) Schematic cross-section of a harmonically pressured clamped–clamped piezoelectric plate; (

**b**) frequency responses of the open-circuit voltage and the energy conversion efficiency.

**Figure 3.**Influence of the equipotential boundary conditions on voltage contour. (

**a**) Without equipotential constraint f = 4 kHz; (

**b**) without equipotential constraint f = 16 kHz; (

**c**) with equipotential constraint f = 4 kHz; (

**d**) with equipotential constraint f = 16 kHz.

**Figure 4.**Energy conversion efficiency enhanced designs for variant excitation frequencies. (

**a**) f = 4 kHz; (

**b**) f = 8 kHz; (

**c**) f = 12 kHz; (

**d**) f = 16 kHz.

**Figure 6.**Iterative history and topological evolution for energy conversion efficiency designs. (

**a**) f = 4 kHz; (

**b**) f = 16 kHz.

**Figure 8.**Open-circuit voltage enhanced designs for variant excitation frequencies. (

**a**) f = 4 kHz; (

**b**) f = 6 kHz; (

**c**) f = 10 kHz; (

**d**) f = 16 kHz.

**Figure 10.**Iterative history and topological evolution for open-circuit voltage designs. (

**a**) f = 4 kHz; (

**b**) f = 16 kHz.

**Figure 11.**Frequency response results of the open-circuit voltage and energy conversion efficiency in different designs. (

**a**) f = 4 kHz; (

**b**) f = 16 kHz.

Boundary Conditions | Frequency | Energy Conversion Efficiency | Open-Circuit Voltage |
---|---|---|---|

Without equipotential constraint | f = 4 kHz | 4.84% | Not applicable |

f = 16 kHz | 4.33% | Not applicable | |

With equipotential constraint | f = 4 kHz | 1.66% | 0.26 V |

f = 16 kHz | 1.09% | 0.19 V |

Frequency | Energy Conversion Efficiency | Open-Circuit Voltage |
---|---|---|

f = 4 kHz | 16.85% | 2.46 V |

f = 8 kHz | 15.21% | 8.91 V |

f = 12 kHz | 12.31% | 0.51 V |

f = 16 kHz | 14.12% | 0.13 V |

Frequency | Voltage | Efficiency | 1st Eigenfrequency | 2nd Eigenfrequency |
---|---|---|---|---|

f = 4 kHz | 18.78 V | 12.02% | 8755 Hz | 17,344 Hz |

f = 6 kHz | 21.07 V | 7.42% | 7075 Hz | 18,815 Hz |

f = 10 kHz | 41.78 V | 13.53% | 9188 Hz | 18,966 Hz |

f = 16 kHz | 3.26 V | 12.05% | 6450 Hz | 16,509 Hz |

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**MDPI and ACS Style**

He, M.; He, M.; Zhang, X.; Xia, L.
Topology Optimization of Piezoelectric Energy Harvesters for Enhanced Open-Circuit Voltage Subjected to Harmonic Excitations. *Materials* **2022**, *15*, 4423.
https://doi.org/10.3390/ma15134423

**AMA Style**

He M, He M, Zhang X, Xia L.
Topology Optimization of Piezoelectric Energy Harvesters for Enhanced Open-Circuit Voltage Subjected to Harmonic Excitations. *Materials*. 2022; 15(13):4423.
https://doi.org/10.3390/ma15134423

**Chicago/Turabian Style**

He, Meng, Mu He, Xiaopeng Zhang, and Liang Xia.
2022. "Topology Optimization of Piezoelectric Energy Harvesters for Enhanced Open-Circuit Voltage Subjected to Harmonic Excitations" *Materials* 15, no. 13: 4423.
https://doi.org/10.3390/ma15134423