# Finite Element Modellingand Simulations of Piezoelectric Actuators Responses with Uncertainty Quantification

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulations

#### 2.1. Governing Equations for Piezoelectric Materials

#### 2.2. Finite Element Discretization

#### 2.3. Error and Uncertainty Quantification

## 3. Verification and Validation

#### 3.1. Single-Layer Piezoelectric Beam

#### 3.2. Bimorph Piezoelectric Beam

#### 3.3. Three-Layer Actuator Beams

## 4. Uncertainty Quantification of the Piezoelectric Composite Plate Actuator’s Response

#### 4.1. Characterization and Comparison of Actuator Responses

#### 4.2. Uncertainty Quantification of Actuator Responses

#### 4.2.1. Sources of Uncertainties

#### 4.2.2. Quantification of Numerical Errors

#### 4.2.3. Propagation of Uncertainties through the Model

#### 4.2.4. Model Uncertainty

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of hexahedral (LSPiezo) and tetrahedral (LTRPiezo) reference element for piezo-elastic coupling implemented in OOFEM.

**Figure 2.**Single-layer beam. The poling direction of the beam is the z-axis. The length L, height H, and width D of the beam are 10 mm, 2 mm, and 5 mm, respectively.

**Figure 5.**Comparison of FEM results with analytical data for the bimorph beam case: (

**a**) deflection profile along the x-axis and (

**b**) deflection error along the length of the bimorph beam with the increasing number of elements in the simulation.

**Figure 6.**Three-layered beam with two smaller piezoelectric beams of the same width (shown as gray color) as the non-piezoelectric beam (shown as white color). Beam length $L=60$ mm; piezoelectric layer thickness ${t}_{p}=0.5$ mm, and the thickness of the non-piezoelectric beam was ${t}_{b}=1.0$ mm; and the applied voltage $v=100$ V.

**Figure 7.**Comparison of FEMwith analytical results for the three-layer beam case: (

**a**) deflection profile along the beam span and (

**b**) error of deflection for meshes with different numbers elements.

**Figure 8.**Problem setup and computational domain of the composite piezoelectric actuator. Note that the computational domain is taken as a quarter of the disc with symmetric planes.

**Figure 9.**Mesh convergence of FEM analysis for a unimorph piezoelectric actuator in comparison with the experiment and analytical results [18]. The plots show the deflection profile with normalized radius along the bottom shim plate.

**Figure 11.**Prediction of maximum deflection at the center of the plate as a function of the radius ratio between the PZT layer and shim layer. The comparison between analytical and current FEM model for different thickness ratios ${h}_{p}/{h}_{s}$ shows a consistent trend in the variation of maximum displacement with the radius ratio.

**Figure 12.**Scatter plot of maximum deflection at the center of the disc with geometrical and material property parameters using the Latin hypercube sampling technique.

**Figure 13.**Distribution of the maximum response with uncertainties and its numerical discretization error obtained from Richardson extrapolation. The displacement error is shown as error bars in the graph.

**Figure 14.**Cumulative distribution function of the maximum deflection response obtained from different UQ approaches. The green line is the CDF of the experiment constructed from two data points. The shaded area is the area validation metric.

Mechanical | Electrical | |
---|---|---|

Natural Boundary Conditions | ${\sigma}_{ij}{n}_{j}={f}_{i}^{b}$ on ${S}_{f}$ | ${D}_{i}{n}_{i}={q}^{s}$ on ${S}_{q}$ |

Essential Boundary Conditions | ${u}_{i}={u}_{i}^{s}$ on ${S}^{u}$ | $\varphi ={\varphi}^{s}$ on ${S}_{\varphi}$ |

$\mathit{\rho}=7500$ | (kg/m${}^{3}$) |
---|---|

$\left[c\right]=\left[\begin{array}{cccccc}12.6& 7.95& 8.41& 0& 0& 0\\ 7.95& 12.6& 8.41& 0& 0& 0\\ 8.41& 8.41& 11.7& 0& 0& 0\\ 0& 0& 0& 2.3& 0& 0\\ 0& 0& 0& 0& 2.3& 0\\ 0& 0& 0& 0& 0& 2.325\end{array}\right]$ | × 10${}^{10}$ (Pa) |

$\left[\epsilon \right]=\left[\begin{array}{ccc}1700{\epsilon}_{0}& 0& 0\\ 0& 1700{\epsilon}_{0}& 0\\ 0& 0& 1470{\epsilon}_{0}\end{array}\right]$ | |

$\left[e\right]=\left[\begin{array}{cccccc}0& 0& 0& 0& 17& 0\\ 0& 0& 0& 17& 0& 0\\ -6.5& -6.5& 23.3& 0& 0& 0\end{array}\right]$ | (Cb/m${}^{2}$) |

**Table 3.**Displacement prediction from the FEM solver with different types of elements (tetrahedral and hexahedral) for a single-layer PZT beam under tensile and shear load.

Tensile | Shear | |||||
---|---|---|---|---|---|---|

Δ (mm) | ${\mathit{N}}_{\mathit{e}}$ | ${\mathit{u}}_{\mathit{x}}$ | ${\mathit{u}}_{\mathit{y}}$ | ${\mathit{u}}_{\mathit{z}}$ | |$\mathit{u}$| | |

Tet | 1 | 780 | 1.37 × 10${}^{-7}$ | 6.85 × 10${}^{-8}$ | 5.93 × 10${}^{-8}$ | 3.900 × 10${}^{-7}$ |

- | 0.2 | 49,960 | 1.37 × 10${}^{-7}$ | 6.85 × 10${}^{-8}$ | 5.93 × 10${}^{-8}$ | 3.900 × 10${}^{-7}$ |

Hex | 1 | 100 | 1.37 × 10${}^{-7}$ | 6.85 × 10${}^{-8}$ | 5.93 × 10${}^{-8}$ | 3.696 × 10${}^{-7}$ |

- | 0.2 | 10,000 | 1.37 × 10${}^{-7}$ | 6.85 × 10${}^{-8}$ | 5.93 × 10${}^{-8}$ | 3.696 × 10${}^{-7}$ |

Analytical | - | - | 1.37 × 10${}^{-7}$ | 6.85 × 10${}^{-8}$ | 5.93 × 10${}^{-8}$ | 3.695 × 10${}^{-7}$ |

$\left[d\right]=\left[\begin{array}{cccccc}0& 0& 0& 0& 3.27& 0\\ 0& 0& 0& 3.27& 0& 0\\ -1.28& -1.28& 3.28& 0& 0& 0\end{array}\right]$ × 10${}^{-10}$ | (m/V) |

**Table 5.**Dimensions and material properties of the APC 850 actuator extracted from [18].

Geometric Properties | ||

Outer radius | ${R}_{out}$ (mm) | 11.7 |

Radius of piezoelectric | ${R}_{in}$ (mm) | 10.0 |

Radius of silver | ${R}_{s}$ (mm) | 9.2 |

Thickness of shim | ${h}_{s}$ (mm) | 0.221 |

Thickness of piezoelectric | ${h}_{p}$ (mm) | 0.234 |

Thickness of silver | h (mm) | 0.015 |

Material Properties | ||

Elastic modulus of shim | ${E}_{s}$ (GPa) | 90 |

Poisson’s ratio of shim | ${\lambda}_{s}$ (-) | 0.32 |

Density of shim | ${\rho}_{s}$ (kg/m${}^{3}$) | 8700 |

Elastic modulus of piezoelectric | ${E}_{p}$ (GPa) | 63 |

Poisson’s ratio of piezoelectric | ${\lambda}_{p}$ (-) | 0.31 |

Density of piezoelectric | ${\rho}_{p}$ (kg/m${}^{3}$) | 7700 |

Electrical Properties | ||

Relative dielectric constant | ${\u03f5}_{r}$ (-) | 1750 |

Piezoelectric constant | ${d}_{31}$ (m/V) | $-175\times {10}^{-12}$ |

**Table 6.**Mesh sensitivity study and convergence index for the unimorph actuator. GCI, Grid Convergence Index.

Grid | ${\mathit{N}}_{\mathit{r}1}$ | ${\mathit{N}}_{\mathit{r}2}$ | ${\mathit{N}}_{\mathit{h}}$ | ${\mathit{u}}_{\mathit{z},\mathit{max}}$ ($\mathsf{\mu}$m/V) | GCI (%) |
---|---|---|---|---|---|

G1 | 40 | 10 | 4 | 0.08809 | - |

G2 | 60 | 20 | 6 | 0.09167 | 2.89 |

G3 | 80 | 30 | 8 | 0.09653 | 6.03 |

G4 | 100 | 40 | 10 | 0.09918 | 4.44 |

Asymptotic | - | - | - | 0.10499 | - |

**Table 7.**Uncertain input parameters and correlation coefficients obtained from the Latin hypercube sampling study for maximum displacement of the APC plate. The simple and partial correlation coefficients are computed on ranked data.

Parameter | Mean | Probability Distribution | LHS 200 | Corr | LHS300 | Corr |
---|---|---|---|---|---|---|

Partial | Simple | Partial | Simple | |||

${h}_{p}$ ($\mathsf{\mu}$m) | 220.0 | normal, $\sigma =10.0$ | −0.8808 | −0.4305 | −0.8628 | −0.4855 |

${h}_{pzt}$ ($\mathsf{\mu}$m) | 230.0 | normal, $\sigma =12.5$ | −0.9043 | −0.4933 | −0.8438 | −0.4610 |

${d}_{31}^{E}$ (pC/m) | −175 | normal, $\sigma =15.0$ | −0.9522 | −0.7250 | −0.9302 | −0.7209 |

${d}_{33}^{E}$ (pC/m) | 395 | normal, $\sigma =5.0$ | −0.16793 | −0.0376 | −0.0672 | 0.0272 |

**Table 8.**Moments of maximum deflection estimated using different Uncertainty Quantification (UQ) approaches.

Method | No. of Evaluations | Mean ($\mathsf{\mu}$m/V) | Std Dev ($\mathsf{\mu}$m/V) | Skewness | Kurtosis |
---|---|---|---|---|---|

LHS200 | 200 | 1.0755 × 10${}^{-7}$ | 1.2538× 10${}^{-8}$ | 1.1756 | 6.8497 |

LHS300 | 300 | 1.0543 × 10${}^{-7}$ | 1.1204 × 10${}^{-8}$ | 2.2445 × 10${}^{-1}$ | 6.9922 × 10${}^{-1}$ |

Polynomial chaos expansion | 135 | 1.0543 × 10${}^{-6}$ | 1.0590 × 10${}^{-7}$ | 1.5774× 10${}^{-1}$ | 6.8291 × 10${}^{-2}$ |

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**MDPI and ACS Style**

Nguyen, V.-T.; Kumar, P.; Leong, J.Y.C.
Finite Element Modellingand Simulations of Piezoelectric Actuators Responses with Uncertainty Quantification. *Computation* **2018**, *6*, 60.
https://doi.org/10.3390/computation6040060

**AMA Style**

Nguyen V-T, Kumar P, Leong JYC.
Finite Element Modellingand Simulations of Piezoelectric Actuators Responses with Uncertainty Quantification. *Computation*. 2018; 6(4):60.
https://doi.org/10.3390/computation6040060

**Chicago/Turabian Style**

Nguyen, Vinh-Tan, Pankaj Kumar, and Jason Yu Chuan Leong.
2018. "Finite Element Modellingand Simulations of Piezoelectric Actuators Responses with Uncertainty Quantification" *Computation* 6, no. 4: 60.
https://doi.org/10.3390/computation6040060