# Multi-Objective Topology Optimization of a Broadband Piezoelectric Energy Harvester

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## Abstract

**:**

## 1. Introduction

## 2. Definition and Modeling of Multi-Modal Piezoelectric Energy Harvester

#### 2.1. Finite Element Modeling of Piezoelectric Energy Harvester

#### 2.2. Folded-Beam Resonator

## 3. Topology Optimization Procedure

#### 3.1. Objective Function and Sensitivity Analysis

#### 3.2. Filters

#### 3.2.1. Density Filter

#### 3.2.2. Threshold Projection

## 4. Optimization Results

## 5. Discussion

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PEH | Piezoelectric Energy Harvester |

TO | Topology Optimization |

## Appendix A

## Appendix B

## References

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**Figure 1.**Folded-beam design introduced in [13]. The geometry is shown on the left. It is composed of a steel structure (gray), two tip masses (black), and three piezoelectric patches (mustard). The first two mode shapes of the structure are shown on the right.

**Figure 2.**An ideal case for a frequency range of efficient operation, formed by two overlapping resonant modes (log scale).

**Figure 3.**Filter techniques in TO.

**Left**: A TO-obtained structure without any filter.

**Center**: Same optimization with a density filter.

**Right**: Same optimization with both filters active.

**Figure 5.**Design space of the problem. The areas beneath the piezo (-electric) patches and the masses M are excluded from the design space.

**Figure 7.**Mode shapes of the first two resonant modes of the optimal structure after the conversion to a manufacturable design value at ${f}_{1}=74.573\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$ and ${f}_{2}=79.054\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$.

**Figure 8.**Open- circuit voltage output of the structure obtained through harmonic analysis. The structure is excited with 0.2 g at its fixed end. The damping is set to 0.008.

**Figure 9.**Maximum power output of the structure at optimal load value computed from values presented in Figure 8. Both outer patches are connected in parallel. The total power output is computed as the sum of both patches.

**Table 1.**Mechanical properties of commercial MFC piezoelectric patches [43].

Density (kg/m${}^{3}$) | Piezoelectric Constants (C/m${}^{2}$) | ||
---|---|---|---|

$\rho $ | 4700 | ${e}_{31}$ | −2.227 |

Young’s Modulus (GPa) | ${e}_{32}$ | −0.671 | |

${E}_{1}$ | 45.21 | ${e}_{33}$ | 16.665 |

${E}_{2}$ | 12.39 | ${e}_{24}$ | 0.0258 |

${E}_{3}$ | 40.44 | ${e}_{15}$ | 13.668 |

Shear Modulus (GPa) | Dielectric Relative Constants | ||

${G}_{12}$ | 6.03 | ${\epsilon}_{11}^{T}/{\epsilon}_{0}$ | 1574.8 |

${G}_{23}$ | 6.68 | ${\epsilon}_{22}^{T}/{\epsilon}_{0}$ | 24.7 |

${G}_{31}$ | 17.01 | ${\epsilon}_{33}^{T}/{\epsilon}_{0}$ | 1528.7 |

Poisson’s Ratio | |||

${\nu}_{12}$ | 0.39 | ||

${\nu}_{23}$ | 0.17 | ||

${\nu}_{13}$ | 0.44 |

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**MDPI and ACS Style**

Hu, S.; Fitzer, U.; Nguyen, K.C.; Hohlfeld, D.; Korvink, J.G.; Bechtold, T.
Multi-Objective Topology Optimization of a Broadband Piezoelectric Energy Harvester. *Micromachines* **2023**, *14*, 332.
https://doi.org/10.3390/mi14020332

**AMA Style**

Hu S, Fitzer U, Nguyen KC, Hohlfeld D, Korvink JG, Bechtold T.
Multi-Objective Topology Optimization of a Broadband Piezoelectric Energy Harvester. *Micromachines*. 2023; 14(2):332.
https://doi.org/10.3390/mi14020332

**Chicago/Turabian Style**

Hu, Siyang, Ulrike Fitzer, Khai Chau Nguyen, Dennis Hohlfeld, Jan G. Korvink, and Tamara Bechtold.
2023. "Multi-Objective Topology Optimization of a Broadband Piezoelectric Energy Harvester" *Micromachines* 14, no. 2: 332.
https://doi.org/10.3390/mi14020332