# Numerical Optimization of a Nonlinear Nonideal Piezoelectric Energy Harvester Using Deep Learning

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## Abstract

**:**

## 1. Introduction

## 2. Accurate System Simulations of the PEH

#### 2.1. FEM Simulation

#### 2.2. ECS Simulation

#### 2.3. FEM-ECS Coupling

## 3. Definition of the Optimization Problem

## 4. DNN Models to Approximates the System Simulations

#### 4.1. Generation of Training Data

#### 4.2. Training of the DNNs

#### 4.3. Performance Evaluation of the DNNs

## 5. Optimization Results

## 6. Discussion of the Results

- The maximum harvested energies are obtained for configurations that result in maximum principle stresses close to the constraint value.
- The capacitances of the electric circuits have a strong influence on the optimal length and tip mass of the beam.
- For the Greinacher circuit, the optimal design does not depend on capacitance ${C}_{2}^{G}$.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. FEM Simulation

## Appendix B. PZT-5A Parameters

## Appendix C. Diode Parameters

Diode saturation current ${I}_{S}^{D}$ [pA] | 1 |

Diode ideality factor n | 1 |

Diode thermal voltage ${V}_{T}$ [mV] | 26 |

## References

- Shirvanimoghaddam, M.; Shirvanimoghaddam, K.; Abolhasani, M.M.; Farhangi, M.; Barsari, V.Z.; Liu, H.; Dohler, M.; Naebe, M. Towards a green and self-powered Internet of Things using piezoelectric energy harvesting. IEEE Access
**2019**, 7, 94533–94556. [Google Scholar] [CrossRef] - Safaei, M.; Sodano, H.A.; Anton, S.R. A review of energy harvesting using piezoelectric materials: State-of-the-art a decade later (2008–2018). Smart Mater. Struct.
**2019**, 28, 113001. [Google Scholar] [CrossRef] - Dietl, J.M.; Garcia, E. Beam shape optimization for power harvesting. J. Intell. Mater. Syst. Struct.
**2010**, 21, 633–646. [Google Scholar] [CrossRef] - Wein, F.; Kaltenbacher, M.; Stingl, M. Topology optimization of a cantilevered piezoelectric energy harvester using stress norm constraints. Struct. Multidiscip. Optim.
**2013**, 48, 173–185. [Google Scholar] [CrossRef] - Miller, L.M.; Elliott, A.D.; Mitcheson, P.D.; Halvorsen, E.; Paprotny, I.; Wright, P.K. Maximum performance of piezoelectric energy harvesters when coupled to interface circuits. IEEE Sens. J.
**2016**, 16, 4803–4815. [Google Scholar] [CrossRef] [Green Version] - Bagheri, S.; Wu, N.; Filizadeh, S. Application of artificial intelligence and evolutionary algorithms in simulation-based optimal design of a piezoelectric energy harvester. Smart Mater. Struct.
**2020**, 29, 105004. [Google Scholar] [CrossRef] - Leadenham, S.; Erturk, A. Mechanically and electrically nonlinear non-ideal piezoelectric energy harvesting framework with experimental validations. Nonlinear Dyn.
**2020**, 99, 625–641. [Google Scholar] [CrossRef] - Stanton, S.C.; Erturk, A.; Mann, B.P.; Dowell, E.H.; Inman, D.J. Nonlinear nonconservative behavior and modeling of piezoelectric energy harvesters including proof mass effects. J. Intell. Mater. Syst. Struct.
**2012**, 23, 183–199. [Google Scholar] [CrossRef] - Stanton, S.C.; Erturk, A.; Mann, B.P.; Inman, D.J. Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification. J. Appl. Phys.
**2010**, 108, 074903. [Google Scholar] [CrossRef] [Green Version] - Gammaitoni, L.; Neri, I.; Vocca, H. Nonlinear oscillators for vibration energy harvesting. Appl. Phys. Lett.
**2009**, 94, 164102. [Google Scholar] [CrossRef] - Cai, W.; Harne, R.L. Vibration energy harvesters with optimized geometry, design, and nonlinearity for robust direct current power delivery. Smart Mater. Struct.
**2019**, 28, 075040. [Google Scholar] [CrossRef] - Zhou, S.; Lallart, M.; Erturk, A. Multistable vibration energy harvesters: Principle, progress, and perspectives. J. Sound Vib.
**2022**, 116886. [Google Scholar] [CrossRef] - Hegendörfer, A.; Steinmann, P.; Mergheim, J. An implicitly coupled finite element—electronic circuit simulator method for efficient system simulations of piezoelectric energy harvesters. J. Intell. Mater. Syst. Struct.
**2023**, accepted. [Google Scholar] - Arndt, D.; Bangerth, W.; Clevenger, T.C.; Davydov, D.; Fehling, M.; Garcia-Sanchez, D.; Harper, G.; Heister, T.; Heltai, L.; Kronbichler, M.; et al. The deal.II library, Version 9.1. J. Numer. Math.
**2019**, 27, 203–213. [Google Scholar] [CrossRef] - Hegendörfer, A.; Steinmann, P.; Mergheim, J. Nonlinear finite element system simulation of piezoelectric vibration-based energy harvesters. J. Intell. Mater. Syst. Struct.
**2022**, 33, 1292–1307. [Google Scholar] [CrossRef] - Wood, W.; Bossak, M.; Zienkiewicz, O. An alpha modification of Newmark’s method. Int. J. Numer. Methods Eng.
**1980**, 15, 1562–1566. [Google Scholar] [CrossRef] - Hibbitt, H.; Karlsson, B. Analysis of Pipe Whip; Technical Report; Hibbitt and Karlsson: RI, USA, 1979; Available online: https://inis.iaea.org/search/search.aspx?orig_q=RN:11538109 (accessed on 19 December 2022).
- Erturk, A.; Inman, D.J. An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations. Smart Mater. Struct.
**2009**, 18, 025009. [Google Scholar] [CrossRef] - Berlincourt, D.; Krueger, H.; Near, C. Properties of Morgan Electro Ceramic Ceramics; Technical Publication TP-226; Morgan Electro Ceramics: Cleveland, OH, USA, 2000. [Google Scholar]
- Qiu, J.; Jiang, H.; Ji, H.; Zhu, K. Comparison between four piezoelectric energy harvesting circuits. Front. Mech. Eng. China
**2009**, 4, 153–159. [Google Scholar] [CrossRef] - Abidin, N.A.K.Z.; Nayan, N.M.; Azizan, M.M.; Ali, A. Analysis of voltage multiplier circuit simulation for rain energy harvesting using circular piezoelectric. Mech. Syst. Signal Process.
**2018**, 101, 211–218. [Google Scholar] [CrossRef] - MATLAB. Version 9.12.0 (R2022a); The MathWorks Inc.: Natick, MA, USA, 2022. [Google Scholar]
- Chimeh, H.E.; Nabavi, S.; Al Janaideh, M.; Zhang, L. Deep-Learning-Based Optimization for a Low-Frequency Piezoelectric MEMS Energy Harvester. IEEE Sens. J.
**2021**, 21, 21330–21341. [Google Scholar] [CrossRef] - Nabavi, S.; Zhang, L. Design and optimization of a low-resonant-frequency piezoelectric MEMS energy harvester based on artificial intelligence. Multidiscip. Digit. Publ. Inst. Proc.
**2018**, 2, 930. [Google Scholar]

**Figure 1.**The nonlinear bimorph electromechanical structure with the Greinacher electric circuit. The four design variables ${m}_{t}^{G}$, ${l}^{G}$, ${C}_{1}^{G}$ and ${C}_{2}^{G}$ are indicated.

**Figure 2.**The nonlinear bimorph electromechanical structure with the standard electric circuit. The three design variables ${m}_{t}^{S}$, ${l}^{S}$ and ${C}_{1}^{S}$ are indicated.

**Figure 3.**Triangular shock-like excitation $a\left(t\right)$ with a magnitude of 0.3 mm and a duration of 8 ms.

**Figure 4.**Typical wave forms of electrode voltage ${\phi}_{el}$ and electric current ${\dot{Q}}_{el}$ evolutions for the Greinacher circuit charging a storage capacitor when a harmonic base excitation is applied.

**Figure 5.**Typical wave forms of electrode voltage ${\phi}_{el}$ and electric current ${\dot{Q}}_{el}$ evolutions for the standard circuit charging a storage capacitor when a harmonic base excitation is applied.

**Figure 6.**DNN structure for the approximation of the harvested energy ${\mathcal{E}}^{G}$ for the configuration with Greinacher circuit.

**Figure 7.**Comparison between true values and predictions of the DNNs for the harvested energy $\mathcal{E}$ for the configurations with Greinacher and standard circuit.

**Figure 8.**Comparison between true values and predictions of the DNNs for the maximum mechanical stress ${\sigma}_{1,max}$ for the configurations with Greinacher and standard circuit.

**Figure 9.**Time signals of the electrode voltage for the optimal Greinacher ${\widehat{\phi}}_{el}^{G}$ and standard circuit ${\widehat{\phi}}_{el}^{S}$ configurations.

**Figure 10.**Time signals of the harvested energy for the optimal Greinacher ${\widehat{\mathcal{E}}}^{G}$ and standard circuit ${\widehat{\mathcal{E}}}^{S}$ configurations.

**Figure 11.**Harvested energy ${\mathcal{E}}^{S}$ of the standard circuit configuration for the optimal value of the capacitor ${\widehat{C}}_{1}^{S}=0.60$ µF with respect to different tip mass ${m}_{t}^{S}$ and beam length ${l}^{S}$ values. The optimum point is indicated with a cyan dot.

**Figure 12.**Harvested energy ${\mathcal{E}}^{G}$ of the Greinacher configuration for optimal values for the capacitor ${\widehat{C}}_{2}^{G}=5.79$ µF and the length ${\widehat{l}}^{G}$ = 48.46 mm and varying values for the mass ${m}_{t}^{G}$ and the capacitor ${C}_{1}^{G}$. The optimum point is indicated with a cyan dot.

**Figure 13.**Harvested energy ${\mathcal{E}}^{G}$ of the Greinacher configuration for optimal values for the tip mass ${\widehat{m}}_{t}^{G}=10.83$ g and the length ${\widehat{l}}^{G}$ = 48.46 mm and varying values for the capacitors ${C}_{1}^{G}$ and ${C}_{2}^{G}$. The optimum point is indicated with a cyan dot.

**Table 1.**Design variables ${\mathcal{X}}^{G}$ with lower and upper bounds for the configuration with Greinacher circuit.

Design Variables ${\mathcal{X}}^{\mathcal{G}}$ | ${\mathcal{X}}_{\mathit{min}}^{\mathit{G}}$ | ${\mathcal{X}}_{\mathit{max}}^{\mathit{G}}$ |
---|---|---|

length ${l}^{G}$ | 40 mm | 80 mm |

tip mass ${m}_{t}^{G}$ | 5 g | 15 g |

capacitance ${C}_{1}^{G}$ | 0 µF | 1 µF |

capacitance ${C}_{2}^{G}$ | 0 µF | 8 µF |

**Table 2.**Design variables ${\mathcal{X}}^{S}$ with lower and upper bounds for the configuration with standard circuit.

Design Variables ${\mathcal{X}}^{\mathcal{S}}$ | ${\mathcal{X}}_{\mathit{min}}^{\mathit{S}}$ | ${\mathcal{X}}_{\mathit{max}}^{\mathit{S}}$ |
---|---|---|

length ${l}^{S}$ | 40 mm | 80 mm |

tip mass ${m}_{t}^{S}$ | 5 g | 15 g |

capacitance ${C}_{1}^{S}$ | 0 µF | 1 µF |

**Table 3.**Optimal values for the design variables for the Greinacher configuration ${\widehat{\mathcal{X}}}^{G}$ and for the standard circuit configuration ${\widehat{\mathcal{X}}}^{S}$.

${\widehat{\mathcal{X}}}^{\mathit{G}}$ | ${\widehat{\mathcal{X}}}^{\mathit{S}}$ |
---|---|

${\widehat{l}}^{G}=48.46$ mm | ${\widehat{l}}^{S}=47.64$ mm |

${\widehat{m}}_{t}^{G}=10.83$ g | ${\widehat{m}}_{t}^{S}=12.91$ g |

${\widehat{C}}_{1}^{G}=0.19$ µF | ${\widehat{C}}_{1}^{S}=0.60$ µF |

${\widehat{C}}_{2}^{G}=5.79$ µF | - |

**Table 4.**Comparison between the results for the optimal design varaibles $\widehat{\mathcal{X}}$ of the DNNs against that of the FEM-ECS coupling for the harvested energy $\widehat{\mathcal{E}}$ and the maximum mechanical stress ${\widehat{\sigma}}_{1,max}$.

Quantity | DNNs ($\widehat{\mathcal{X}}$) | FEM-ECS ($\widehat{\mathcal{X}}$) | Rel. Error | Abs. Error |
---|---|---|---|---|

${\widehat{\mathcal{E}}}^{G}$ | 24.8 µJ | 24.1 µJ | 2.9% | 0.7 µJ |

${\widehat{\sigma}}_{1,max}^{G}$ | 18.4 MPa | 19.1 MPa | 3.7% | 0.7 MPa |

${\widehat{\mathcal{E}}}^{S}$ | 26.0 µJ | 24.8 µJ | 4.8% | 1.2 µJ |

${\widehat{\sigma}}_{1,max}^{S}$ | 18.4 MPa | 18.2 MPa | 1.1% | 0.2 MPa |

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**MDPI and ACS Style**

Hegendörfer, A.; Steinmann, P.; Mergheim, J.
Numerical Optimization of a Nonlinear Nonideal Piezoelectric Energy Harvester Using Deep Learning. *J. Low Power Electron. Appl.* **2023**, *13*, 8.
https://doi.org/10.3390/jlpea13010008

**AMA Style**

Hegendörfer A, Steinmann P, Mergheim J.
Numerical Optimization of a Nonlinear Nonideal Piezoelectric Energy Harvester Using Deep Learning. *Journal of Low Power Electronics and Applications*. 2023; 13(1):8.
https://doi.org/10.3390/jlpea13010008

**Chicago/Turabian Style**

Hegendörfer, Andreas, Paul Steinmann, and Julia Mergheim.
2023. "Numerical Optimization of a Nonlinear Nonideal Piezoelectric Energy Harvester Using Deep Learning" *Journal of Low Power Electronics and Applications* 13, no. 1: 8.
https://doi.org/10.3390/jlpea13010008