Monte Carlo Based Isogeometric Stochastic Finite Element Method for Uncertainty Quantization in Vibration Analysis of Piezoelectric Materials

Abstract

In this study, a Monte Carlo simulation (MCs)-based isogeometric stochastic Finite Element Method (FEM) is proposed for uncertainty quantification in the vibration analysis of piezoelectric materials. In this method, deterministic solutions (natural frequencies) of the coupled eigenvalue problem are obtained via isogeometric analysis (IGA). Moreover, MCs is employed to solve various uncertainty parameters, including separate elastic and piezoelectric constants and their combined cases.

1. Introduction

Piezoelectric materials are the most widely used smart materials in engineering structures. Their application has reached a mature level, and they have been extensively used in sensors/transducers [1,2,3,4,5,6], energy harvesters [7,8,9] and resonators [10,11,12]. The application of piezoelectric materials is often related to their vibration analysis. Therefore, it is crucial to investigate the effects of electroelastic coupling on the vibration modes of piezoelectric structures. The coupling effect influences the lattice structure of the piezoelectric material and enhances the structure stiffness by a so-called “piezoelectric stiffening” effect [13]. This results in an increase in the natural frequencies of the vibration modes. Piezoelectric ceramics are among primary piezoelectric materials owing to their low price and good coupling. The green body of piezoelectric ceramics is prepared by compression; therefore, the distribution of grain size, density, and local composition is inhomogeneous, resulting in the fluctuation of material parameters [14]. Although some efforts have been made to improve piezoelectric properties [15], the effects of heterogeneity cannot be completely eliminated. Owing to these uncertain factors, the uncertainty quantification of the vibration analysis of piezoelectric materials is extremely important.

Owing to the inability of deterministic analysis to describe random fields, stochastic analysis techniques have been extensively researched to enhance the credibility of computational predictions for uncertainty problems [16]. There are three main approaches to stochastic analysis: perturbation-based techniques [17], stochastic spectral approaches [18,19], and Monte Carlo simulations [20,21,22,23]. Among them, sample-based Monte Carlo simulation (MCs) is considered the most versatile and simple approach. MCs is used to calculate responses from random sampling data to obtain statistical characteristics (expected value, variance, etc.).

Isogeometric analysis (IGA) is an important advancement in computational mechanics and an extension of the Finite Element Method (FEM) [24]. Using the basic features of computer-aided design (CAD), such as Non-Uniform Rational B-splines (NURBS), to discretize the partial differential equations [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41], one can perform numerical analysis directly from the CAD models, which reduces the preprocessing time for MCs to construct geometric models [23] while maintaining geometric accuracy.

This paper lays out a procedure for solving the uncertainty problems of vibration analysis of piezoelectric materials. This approach consists of two novel points:

• IGA-FEM is employed for the coupled eigenvalue problems of piezoelectric materials.
• We investigate the influence of different factors on coupled eigenvalue problems.

The remainder of the paper is organized as follows. Section 2 gives the fundamentals of MCs in uncertainty quantification. Section 3 presents the isogeometric finite element theory of the vibration analysis of piezoelectric materials. Several numerical examples are provided in Section 4 to obtain the statistical characteristics of the natural frequencies in the vibration analysis of piezoelectric materials, followed by conclusions in Section 5.

2. Uncertainty Quantification Based on Monte Carlo Simulation

Monte Carlo simulation (MCs) is used to directly characterize uncertainty by calculating the expectation and variance of many samples. In general uncertainty quantification, considering a given random field $r\left(X\right)$, the probability density function is $\tilde{\mathcal{P}}\left(r\right)$, and the first two probabilistic moments (expected value and covariance) are defined as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \mathcal{E}\left(r\left(X\right)\right)={\int }_{-\infty }^{+\infty }r\tilde{\mathcal{P}}\left(r\right)\mathrm{d}r,\hfill \\ \hfill & Cov(X,Y)=\mathcal{E}(X-\mathcal{E}(X\left)\right)\mathcal{E}(Y-\mathcal{E}(Y\left)\right).\hfill \end{array}\end{array}$$

(1)

According to the law of large numbers, when more sampling points are selected, the average of the results obtained from multiple sampling points should converge to the desired value, which is the theoretical basis for MCs. Assuming $\tilde{\mathcal{H}}\left(r\right)$ to be an arbitrary function of random variable r. The expected value and variance of $\tilde{\mathcal{H}}\left(r\right)$ can be approximated as:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \mathcal{E}\left[\tilde{\mathcal{H}}\left(r\right)\right]\approx \frac{1}{n_a}\sum _{i=1}^{n_a}{\tilde{\mathcal{H}}}_i\left(r\right),\hfill \\ \hfill & V\left[\tilde{\mathcal{H}}\left(r\right)\right]\approx \frac{1}{n_a-1}\sum _{i=1}^{n_a}{\left[{\tilde{\mathcal{H}}}_i\left(r\right)-\mathcal{E}(\tilde{\mathcal{H}}\left(r\right)\right]}^2.\hfill \end{array}\end{array}$$

(2)

where $n_a$ is the sample size, and the order of convergence rate is $O\left({n_a}^{-1/2}\right)$.

3. Linear Piezoelectricity Vibration Analysis Theory with IGA-FEM

3.1. Linear Piezoelectricity Formulation

For linear piezoelectricity, the electric enthalpy density, $h(S_{ij},E_i)$, is a function of the strain tensor $\mathbf{S}$ and the electric field $\mathbf{E}$, expressed as follows:

$$\begin{array}{c}\hfill h(S_{ij},E_i)=\frac{1}{2}{C}_{ijkl}{S}_{ij}{S}_{kl}-e_{kij}{E}_k{S}_{ij}-\frac{1}{2}{\kappa }_{ij}{E}_i{E}_j,\end{array}$$

(3)

in which $C_{ijkl}$ are the components of the fourth-order elastic tensor $\mathbf{C}$; $e_{kij}$ denote the components of the third-order piezoelectric tensor $\mathbf{e}$; $i,j,k$ and l vary from 1 to 3. The ${\kappa }_{ij}$ are the components of the second-order dielectric tensor $\mathbf{\kappa }$. $E_i$ represent components of electric field $\mathbf{E}$, and $S_{ij}$ are components of strain tensor $\mathbf{S}$. The Stresses/electric displacements $T_{ij}/D_i$ is then defined through the following relations [42]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & T_{ij}=\frac{\partial h}{\partial S_{ij}},\hfill \\ \hfill & D_i=-\frac{\partial h}{\partial E_i}.\hfill \end{array}\end{array}$$

(4)

Consider integration over the domain $\mathsf{\Omega }$, the total electrical enthalpy is

$$\begin{array}{c}\hfill H={\int }_{\mathsf{\Omega }}h(S_{ij},E_i)\mathrm{d}\mathsf{\Omega }={\int }_{\mathsf{\Omega }}[\frac{1}{2}{C}_{ijkl}{S}_{ij}{S}_{kl}-e_{kij}{E}_k{S}_{ij}-\frac{1}{2}{\kappa }_{ij}{E}_i{E}_j]\mathrm{d}\mathsf{\Omega }.\end{array}$$

(5)

The generalized geometric equations are given as

$$\begin{array}{cc}\hfill & S_{ij}=\frac{1}{2}[u_{i,j}+u_{j,i}],\hfill \end{array}$$

(6a)

$$\begin{array}{cc}\hfill & E_i=-{\phi }_{,i},\hfill \end{array}$$

(6b)

where $u_i$ are components of displacement $\mathbf{u}$. The ${(·)}_{,i}$ denotes the partial derivative with respect to the coordinate $x_i$. The kinetic energy is defined by

$$\begin{array}{c}\hfill \mathcal{K}\left(\mathbf{u}\right)=\frac{\rho }{2}{\int }_{\mathsf{\Omega }}\frac{\partial u_i}{\partial t}\frac{\partial u_i}{\partial t}\mathrm{d}\mathsf{\Omega },\end{array}$$

(7)

where $\rho$ denotes the mass density per unit volume. The external work is defined by

$$\begin{array}{c}\hfill W_{ext}(\mathbf{u},\phi )={\int }_{{\mathsf{\Gamma }}_t}{u}_i{\overline{f}}_i\mathrm{d}{\mathsf{\Gamma }}_t+{\int }_{{\mathsf{\Gamma }}_d}\omega \phi \mathrm{d}{\mathsf{\Gamma }}_d,\end{array}$$

(8)

where ${\overline{f}}_i$ is the components of mechanical tractions. The value $\omega$ is the surface charge density. ${\mathsf{\Gamma }}_t$ and ${\mathsf{\Gamma }}_d$ are boundaries of $\mathsf{\Omega }$ corresponding to mechanical tractions and electric displacements, respectively.

3.2. IGA-FEM Theory for Linear Piezoelectricity Vibration Analysis

3.2.1. Variational Setting

From Hamilton’s principle, the variation of linear piezoelectricity is expressed as

$$\begin{array}{c}\hfill \delta \left({\int }_{t_0}^{t_1}\mathcal{L}(\mathbf{u},\phi )\mathrm{dt}\right)=0,\end{array}$$

(9)

where $\delta (·)$ represents the variational operator and the Lagrangian is defined as

$$\begin{array}{c}\hfill \mathcal{L}(\mathbf{u},\phi )=\mathcal{K}\left(\mathbf{u}\right)-H(\mathbf{u},\phi )+W_{ext}(\mathbf{u},\phi ).\end{array}$$

(10)

Thus, Equation (9) expands as

$$\begin{array}{c}\hfill \delta {\int }_{t_0}^{t_1}\mathcal{K}\left(\mathbf{u}\right)\mathrm{d}t-\delta {\int }_{t_0}^{t_1}H(\mathbf{u},\phi )\mathrm{d}t+\delta {\int }_{t_0}^{t_1}{W}_{ext}(\mathbf{u},\phi )\mathrm{d}t=0,\end{array}$$

(11)

where the variation of kinetic energy and external work are expressed as

$$\begin{array}{c}\hfill \delta {\int }_{t_0}^{t_1}K\left(\mathbf{u}\right)\mathrm{d}t=-{\int }_{t_0}^{t_1}\rho \left[{\int }_{\mathsf{\Omega }}\delta u_i\frac{{\partial }^2{u}_i}{\partial t^2}\mathrm{d}\mathsf{\Omega }\right]\mathrm{d}t,\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \delta {\int }_{t_0}^{t_1}{W}_{ext}(\mathbf{u},\phi )\mathrm{d}t={\int }_{t_0}^{t_1}\left[{\int }_{{\mathsf{\Gamma }}_t}{\overline{f}}_i\delta u_i\mathrm{d}{\mathsf{\Gamma }}_t+{\int }_{{\mathsf{\Gamma }}_d}\omega \delta \phi \mathrm{d}{\mathsf{\Gamma }}_d\right]\mathrm{d}t.\end{array}$$

(13)

From Equations (5) and (6b), the variation of the total enthalpy is as following

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \delta {\int }_{t_0}^{t_1}H(\mathbf{u},\phi )\mathrm{d}t=& {\int }_{t_0}^{t_1}\left[{\int }_{\mathsf{\Omega }}{\widehat{C}}_{abcd}\delta S_{ab}{S}_{cd}\mathrm{d}\mathsf{\Omega }\right]\mathrm{d}t+\delta {\int }_{t_0}^{t_1}\left[{\int }_{\mathsf{\Omega }}{\widehat{e}}_{abc}\delta {\phi }_{,a}{S}_{bc}\mathrm{d}\mathsf{\Omega }\right]\mathrm{d}t\hfill \\ \hfill & +\delta {\int }_{t_0}^{t_1}\left[{\int }_{\mathsf{\Omega }}{\widehat{e}}_{abc}{\phi }_{,a}\delta S_{bc}\mathrm{d}\mathsf{\Omega }\right]\mathrm{d}t-\delta {\int }_{t_0}^{t_1}\left[{\int }_{\mathsf{\Omega }}{\widehat{\kappa }}_{ab}\delta {\phi }_{,a}{\phi }_{,b}\mathrm{d}\mathsf{\Omega }\right]\mathrm{d}t.\hfill \end{array}\end{array}$$

(14)

The modified elastic, piezoelectric and dielectric tensors are shown in the Appendix A where $a,b,c$ and d denote the surface variable components. The values for these indicators are 1 and 3. To satisfy Equation (11) for all possible $\delta \mathbf{u}$ and $\delta \phi$, the weak form of the linear piezoelectric vibration analysis controlling equation follows as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}\delta u_i\frac{{\partial }^2{u}_i}{\partial t^2}\mathrm{d}\mathsf{\Omega }+{\int }_{\mathsf{\Omega }}{\widehat{C}}_{abcd}\delta S_{ab}{S}_{cd}\mathrm{d}\mathsf{\Omega }+{\int }_{\mathsf{\Omega }}{\widehat{e}}_{abc}\delta {\phi }_{,a}{S}_{bc}\mathrm{d}\mathsf{\Omega }+{\int }_{\mathsf{\Omega }}{\widehat{e}}_{abc}{\phi }_{,a}\delta S_{bc}\mathrm{d}\mathsf{\Omega }\hfill \\ \hfill & -{\int }_{\mathsf{\Omega }}{\widehat{\kappa }}_{ab}\delta {\phi }_{,a}{\phi }_{,b}\mathrm{d}\mathsf{\Omega }-{\int }_{{\mathsf{\Gamma }}_t}\overline{f_i}\delta u_i\mathrm{d}{\mathsf{\Gamma }}_t-{\int }_{{\mathsf{\Gamma }}_d}\omega \delta \phi \mathrm{d}{\mathsf{\Gamma }}_d=0.\hfill \end{array}\end{array}$$

(15)

3.2.2. IGA-FEM Discretization

Given a set of non-decreasing coordinates $\mathsf{\Xi }=[{\xi }_0,{\xi }_1,\dots ,{\xi }_{n+p+1}]$ in the parameter space, and is called the knot vector. The B-spline basis functions are defined as

$$\begin{array}{c}\hfill N_{i,0}\left(\xi \right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{\xi }_i\le \xi <{\xi }_{i+1}\\ 0\hfill & \mathrm{otherwise}\end{array}\right.,\end{array}$$

(16)

and

$$N_{i,p}\left(\xi \right)=\frac{\xi -{\xi }_i}{{\xi }_{i+p}-{\xi }_i}{N}_{i,p-1}\left(\xi \right)+\frac{{\xi }_{i+p+1}-\xi }{{\xi }_{i+p+1}-{\xi }_{i+1}}{N}_{i+1,p-1}\left(\xi \right),$$

(17)

where $\xi$ denotes the parametric coordinate, n is the number of basis functions, and p is the polynomial order. Figure 1 gives an example of a k-refinement of a B-spline, in which different color lines represent different basis functions. From this example, we can intuitively feel the richness of B-spline functions in IGA. Considering the control points ${\mathbf{P}}_{i,j}\in \mathbb{R}{}^d$, all parametric spaces can be reduced to a unit interval (d = 1), square (d = 2) or cube (d = 3) [43]. In this paper, d is set to be 2. Considering two knot vectors ${\mathsf{\Xi }}^1=[{\xi }_0,{\xi }_1,\dots ,{\xi }_{n+p+1}]$ and ${\mathsf{\Xi }}^2=[{\eta }_0,{\eta }_1,\dots ,{\eta }_{m+q+1}]$ and control points ${\mathbf{P}}_{i,j}\in \mathbb{R}{}^d$, a B-spline surface is defined as

$$\overline{\mathbf{S}}(\xi ,\eta )=\sum _{i=1}^n\sum _{j=1}^m{N}_{i,p}\left(\xi \right)M_{j,q}\left(\eta \right){\mathbf{P}}_{i,j},$$

(18)

where $N_{i,p}$ and $M_{j,q}$ are univariate B-spline basis functions with polynomial order p and q, respectively. The NURBS surface is defined with the following:

$$\tilde{\mathbf{S}}(\xi ,\eta )=\sum _{i=1}^n\sum _{j=1}^m{R}_{i,j}^{p,q}(\xi ,\eta ){\mathbf{P}}_{i,j},$$

(19)

and

$$R_{i,j}^{p,q}(\xi ,\eta )=\frac{N_{i,p}\left(\xi \right)M_{j,q}\left(\eta \right){\mathcal{W}}_{i,j}}{{\displaystyle \sum _{i^{\prime }=1}^n}{\displaystyle \sum _{j^{\prime }=1}^m}{N}_{i^{\prime },p}\left(\xi \right)M_{i^{\prime },q}\left(\eta \right){\mathcal{W}}_{i^{\prime },j^{\prime }}},$$

(20)

where $R_{i,j}^{p,q}(\xi ,\eta )$ are the bivariate NURBS basis functions, ${\mathcal{W}}_i$ is referred to as the ith weight. The displacement and electric potential $\phi$ are discretized using the NURBS basis functions as

$$\begin{array}{c}\hfill \mathbf{u}=\sum _{B=0}^{n_b-1}{N}_B{\mathbf{U}}_B,\phi =\sum _{B=0}^{n_b-1}{N}_B{\mathsf{\Phi }}_B,\end{array}$$

(21)

where $n_b$ is the number of basis functions. ${\mathbf{U}}_B$ is the Bth nodal coefficients of displacement, and the ${\mathsf{\Phi }}_B$ represents the nodal coefficients of the electric potential.

Figure 1. An example of k-refinement of B-spline, in which different color lines represent different basis functions.

From Equation (6a), the strain components are computed as

$$S_{ab}=\sum _B^{n_b-1}\frac{1}{2}[N_{B,b}+N_{B,a}]·{\mathbf{U}}_B.$$

(22)

Following a Bubnov-Galerkin method, the NURBS basis functions are also used for the $\delta \mathbf{u}$ and $\delta \phi$, and Equation (15) can be written as [44]

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\mathrm{M}& 0\\ 0& 0\end{array}\right]\left\{\begin{array}{c}\ddot{\mathrm{u}}\hfill \\ 0\hfill \end{array}\right\}+\left[\begin{array}{cc}\mathrm{K}& {\mathrm{C}}_{u\phi }\\ {\mathrm{C}}_{\phi u}& {\mathrm{D}}_{\phi \phi }\end{array}\right]\left\{\begin{array}{c}\mathrm{u}\hfill \\ \psi \hfill \end{array}\right\}=\left\{\begin{array}{c}{\mathrm{f}}_u\hfill \\ {\mathrm{f}}_{\phi }\hfill \end{array}\right\},\end{array}$$

(23)

where $\mathrm{M}$ is the global mass matrix. The value $\ddot{\mathrm{u}}$ represents global acceleration vector. $\mathrm{K}$ is the global stiffness matrix, ${\mathrm{D}}_{\phi \phi }$ represents the global dielectric system matrix, ${\mathrm{C}}_{u\phi }$ and ${\mathrm{C}}_{\phi u}$ denote the direct and converse piezoelectric coupling matrices, respectively. Values $\mathrm{u}$ and $\psi$ are the global vectors of displacement, and electric potential coefficients, respectively. Values ${\mathrm{f}}_u$ and ${\mathrm{f}}_{\phi }$ are the global structural and electrical load vectors. To improve the computational efficiency, the Schur complement ${\mathrm{C}}_{u\phi }{\mathrm{D}}_{\phi \phi }^{-1}{\mathrm{C}}_{\phi u}$ is used to modify the system of equations. Thus, the problem for $\mathrm{u}$ can be written as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{M}\ddot{\mathrm{u}}+[\mathrm{K}-{\mathrm{C}}_{u\phi }{\mathrm{D}}_{\phi \phi }^{-1}{\mathrm{C}}_{\phi u}]\mathrm{u}={\mathrm{f}}_u-{\mathrm{C}}_{u\phi }{\mathrm{D}}_{\phi \phi }^{-1}{\mathrm{f}}_{\phi }.\end{array}\end{array}$$

(24)

Then, the new global system matrices are defined as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{A}=\mathrm{K}-{\mathrm{C}}_{u\phi }{\mathrm{D}}_{\phi \phi }^{-1}{\mathrm{C}}_{\phi u}.\end{array}\end{array}$$

(25)

The system of equations is thus defined as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{M}\ddot{\mathrm{u}}+\mathrm{A}\mathrm{u}={\mathrm{f}}_u-{\mathrm{C}}_{u\phi }{\mathrm{D}}_{\phi \phi }^{-1}{\mathrm{f}}_{\phi }.\end{array}\end{array}$$

(26)

The free vibration problem of linear piezoelectricity can be obtained by assuming harmonic motions, and the expression is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill [-{\overline{\omega }}^2\mathrm{M}+\mathrm{A}]\mathrm{u}={\mathrm{f}}_u-{\mathrm{C}}_{u\phi }{\mathrm{D}}_{\phi \phi }^{-1}{\mathrm{f}}_{\phi },\end{array}\end{array}$$

(27)

where $\overline{\omega }$ is the angular frequency. For the free vibration analysis problem, by setting the external mechanical and electrical loads to zero, the above equation can be reduced as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill [-{\overline{\omega }}^2\mathrm{M}+\mathrm{A}]=0.\end{array}\end{array}$$

(28)

4. Numerical Examples

4.1. Piezoelectric Tapered Panel

4.1.1. Piezoelectric Tapered Panel Free Vibration Analysis

In this section, the free-vibration analysis of a clamped tapered panel model is discussed, as shown in Figure 2. The boundary conditions are similar to those of cantilever structures. The PZT-4 piezoelectric material is used to create this model, and the material parameters are listed in Table 1. Nine initial control points are considered, and the coordinates and weights for these control points are listed in Table 2. The lower surface electric potential is specified as 0V. FEM employs the traditional Lagrangian basis functions, which generally require many mesh divisions to obtain high accuracy, but the computational efficiency is low. However, IGA-FEM uses the NURBS basis functions to ease the preprocessing time by increasing the order, thus improving the computational efficiency. This is the fundamental reason for choosing IGA-FEM in this paper.

Figure 2. The free-vibration analysis of a clamped tapered panel model.

Table 1. Material parameters for tapered panel.

Name	PZT-4
Mass density $\rho$	7500 kg/m${}^3$
Elastic constants
$C_{1111}$	139 GPa
$C_{1122}$	77.8 GPa
$C_{1133}$	74 GPa
$C_{3333}$	115 GPa
$C_{1313}$	25.6 GPa
Piezoelectric constants
$e_{113}$	12.7 C/m${}^2$
$e_{311}$	−5.2 C/m${}^2$
$e_{333}$	15.7 C/m${}^2$
Permittivity
${\kappa }_{11}$	6.46 × 10${}^{-9}$ C/(Vm)
${\kappa }_{33}$	5.62 × 10${}^{-9}$ C/(Vm)

Table 2. Control point coordinates and weights for the tapered panel.

i	${\mathit{P}}_{\mathit{i},1}$ (mm, mm)	${\mathit{P}}_{\mathit{i},2}$ (mm, mm)	${\mathit{P}}_{\mathit{i},3}$ (mm, mm)	${\mathcal{W}}_{\mathit{i},1}$	${\mathcal{W}}_{\mathit{i},2}$	${\mathcal{W}}_{\mathit{i},3}$
1	(0, 0)	(24, 22)	(48, 44)	1	1	1
2	(0, 22)	(24, 37)	(48, 52)	1	1	1
3	(0, 44)	(24, 52)	(48, 60)	1	1	1

Figure 3 shows the first six displacement eigenmodes of a purely elastic problem for the tapered structure. Figure 4 and Figure 5 show the first six eigenmodes of the piezoelectric tapered structure. The magnitude of the displacement and electric potential function $\phi$ distribution on the tapered panel are plotted. Compared to the purely elastic tapered panel, the modal displacement did not exhibit a notable change. Table 3 lists the natural frequencies of the elastic and electroelastic coupling problems. As can be seen from Table 3, the coupling effect causes an increase in the eigenmode frequency, known as “piezoelectric stiffening”.

Figure 3. Eigenmodes distribution of purely elastic displacement of tapered panel structure.

Figure 4. Eigenmodes distribution of coupled displacement of tapered panel structure.

Figure 5. Eigenmodes distribution of electric potential of the tapered panel structure.

Table 3. Comparison of natural frequency IGA results for elastic and electroelastic coupling of tapered panel.

Eigenvalue Number	Elastic (Hz)	Piezoelectric Coupled (Hz)
1	3.90 × 10${}^4$	4.23 × 10${}^4$
2	1.00 × 10${}^5$	1.19 × 10${}^5$
3	1.21 × 10${}^5$	1.28 × 10${}^5$
4	2.00 × 10${}^5$	2.29 × 10${}^5$
5	2.55 × 10${}^5$	2.73 × 10${}^5$
6	2.83 × 10${}^5$	3.09 × 10${}^5$

4.1.2. Uncertainty Quantification for Piezoelectric Tapered Panel Model with MCs

In this section, the tapered panel model presented in Section 4.1.1, is selected for uncertainty quantization models. We consider the elastic constant $C_{111}$ and piezoelectric constant $e_{333}$ as random input variables. These random variables satisfy the expected values $\mu$ = 1.39 × 10${}^{11}$ and $\mu =15.1$ Gaussian distribution and the standard deviation $\sigma =\mu \times \gamma$, where $\gamma$ represents the coefficient of variation. The natural frequency has a large influence on the stability of the structure. Therefore, in this paper, we investigate the factors influencing the natural frequency of piezoelectric materials.

Table 4 and Table 5 list all the input parameters with their range, where the scale of the datasets is determined according to the $3\sigma$ principle. Table 6 gives the size of samples for single random input variable denoted by “1-D” and multidimensional input variables “2-D”, respectively. The number of sampling points denoted by $n_a$ is chosen as 500 for “1-D” analysis, but ${25}^2$ for “2-D” analysis. The computational efficiency of Monte Carlo simulation depends on the size of the sample points, too few sample points cannot guarantee the computational accuracy while too many sample points reduce the computational efficiency. We found through experimental comparison that 500 sample points can guarantee the computational accuracy while ensuring the computational efficiency. In the multidimensional input variables analysis, we increased the number of sample points to verify the effect of sample point fluctuations. When the number of samples increases, the calculation accuracy can be improved, but the calculation efficiency becomes worse.

Table 4. Definitions and the statistical properties of the random input variables ($\gamma =0.02$ and 0.06).

Random Input Variables	$\mathit{\mu }$	The Limits of Input Variables: [Lower, Upper]
		$\mathit{\gamma }$ = 0.02	$\mathit{\gamma }$ = 0.06
$C_{1111}$	1.39 × 10 ${}^{11}$	[1.31 × 10${}^{11}$, 1.46 × 10${}^{11}$]	[1.14 × 10${}^{11}$, 1.66 × 10${}^{11}$]
$e_{333}$	15.1	[14.3, 16.2]	[12.2, 17.7]

Table 5. Definitions and the statistical properties of the random input variables ($\gamma =0.1$, 0.14, and 0.18).

Random Input Variables	$\mathit{\mu }$	The Limits of Input Variables: [Lower, Upper]
		$\mathit{\gamma }$ = 0.1	$\mathit{\gamma }$ = 0.14	$\mathit{\gamma }$ = 0.18
$C_{1111}$	1.39 × 10${}^{11}$	[8.96 × 10${}^{10}$, 1.85 × 10${}^{11}$]	[8.59 × 10${}^{10}$, 2.08 × 10${}^{11}$]	[7.81 × 10${}^{10}$, 2.17 × 10${}^{11}$]
$e_{333}$	15.1	[10.5, 19.5]	[9.1, 20.8]	[7.5, 22.0]

Table 6. Size of the samples.

Dimension of Input	Variable	Samples
Variables		(${\mathit{n}}_{\mathit{a}}$,${\mathit{n}}_{\mathit{a}}$)
1-D	$C_{1111}$	(500, 500)
	$e_{333}$	(500, 500)
2-D	$C_{1111}$+$e_{333}$	(${25}^2$, 625)

Figure 6 and Figure 7 show the expected values and standard deviations of the second and fourth nonzero eigenvalues (natural frequency) under different coefficients of variation. This includes a single random variable, $C_{1111}$ or $e_{333}$ and the two combined case. It can be observed from these figures that the variation in the piezoelectric constants had little effect on the expected values and standard deviations of the responses. Notably, the standard deviation was the largest when considering the uncertainty of the single random variable $C_{1111}$ and increased rapidly with an increase in the coefficient of variation. Therefore, fluctuations in elastic constants significantly affect the dynamic characteristics of piezoelectric structures.

Figure 6. Expected values for separate $C_{1111}$, $e_{333}$ and the two uncertainties combined case with different coefficient of variation (tapered panel model).

Figure 7. Standard variation for separate $C_{1111}$, $e_{333}$ and the two uncertainties combined case with different coefficient of variation (tapered panel model).

4.2. Infinite Plate with Circular Hole

4.2.1. Infinite Plate with Circular Hole Free Vibration Analysis

In this section, we examine the influence of the coupling effects on the natural frequencies and eigenmodes using an infinite plate with a circular hole. The detailed material parameters are listed in Table 7. Owing to the symmetry of the infinite plate, a quarter-plate model with a quarter-hole is employed, as shown in Figure 8. The size of the quarter-plate is much larger than that of the hole. The control points on the left edge have only one translational DoF of the $x_3$-direction and the control points on the bottom edge have only one translational DoF of the $x_1$-direction, i.e., sliding supports are applied on the symmetrical edges of the quarter-plate model. The electric potential on the left surface of the quarter-plate is prescribed as 0 V. The coarsest mesh, ${\mathsf{\Xi }}^1\times {\mathsf{\Xi }}^2$, is defined by the knot vector ${\mathsf{\Xi }}^1={\mathsf{\Xi }}^2=[0,0,0,0.5,1,1,1]$. The control point relationships are listed in Table 8. We selected this sample size in Section 4.1.2.

Table 7. Material parameters for infinite plate with circular hole.

Name	PZT-5H
Mass density $\rho$	7500 kg/m${}^3$
Elastic constants
$C_{1111}$	126 GPa
$C_{1122}$	79.1 GPa
$C_{1133}$	83.9 GPa
$C_{3333}$	117 GPa
$C_{1313}$	23 GPa
Piezoelectric constants
$e_{113}$	17.0 C/m${}^2$
$e_{311}$	−6.5 C/m${}^2$
$e_{333}$	23.3 C/m${}^2$
Permittivity
${\kappa }_{11}$	1.505 × 10${}^{-8}$ C/(Vm)
${\kappa }_{33}$	1.302 × 10${}^{-8}$ C/(Vm)

Figure 8. Infinite plate with circular hole model.

Table 8. Control point coordinates and weights for the plate with a circular hole.

i	${\mathit{P}}_{\mathit{i},1}$ (mm, mm)	${\mathit{P}}_{\mathit{i},2}$ (mm, mm)	${\mathit{P}}_{\mathit{i},3}$ (mm, mm)	${\mathit{P}}_{\mathit{i},4}$ (mm, mm)	${\mathcal{W}}_{\mathit{i},1}$	${\mathcal{W}}_{\mathit{i},2}$	${\mathcal{W}}_{\mathit{i},3}$	${\mathcal{W}}_{\mathit{i},4}$
1	(0, 1)	(0, 3.4278)	(0, 7.75)	(0, 10)	1	1	1	1
2	(0.4142, 1)	(0.5954, 3.4278)	(5.375, 7.75)	(10, 10)	0.8536	1	1	1
3	(1, 0.4142)	(3.4278, 0.5954)	(7.75, 5.375)	(10, 10)	0.8536	1	1	1
4	(1, 0)	(3.4278, 0)	(7.75, 0)	(10, 0)	1	1	1	1

The distributions for the first six eigenmodes of the purely elastic and coupled problems are shown in Figure 9 and Figure 10. The distribution of eigenmodes of the electric potential is shown in Figure 11. It can be seen from these figures that distribution of displacement eigenmodes were approximately the same for the purely elastic problem and the coupled problem. The first six eigenvalues (natural frequencies) of the purely elastic and coupled problems are listed in Table 9. From the results, we can still observe the piezoelectric “stiffening effect”.

Figure 9. Eigenmodes distribution of purely elastic displacement of a quarter plate with circular hole.

Figure 10. Eigenmodes distribution of coupled displacement of a quarter plate with circular hole.

Figure 11. Eigenmodes distribution of electric potential of a quarter plate with circular hole.

Table 9. Comparison of natural frequency IGA results for elastic and electroelastic coupling of infinite plate with circular hole.

Eigenvalue Number	Elastic (Hz)	Piezoelectric Coupled (Hz)
1	3.33 × 10${}^5$	3.46 × 10${}^5$
2	4.13 × 10${}^5$	4.28 × 10${}^5$
3	5.23 × 10${}^5$	5.34 × 10${}^5$
4	7.26 × 10${}^5$	8.10 × 10${}^5$
5	8.49 × 10${}^5$	9.26 × 10${}^5$
6	1.01 × 10${}^5$	1.05 × 10${}^5$

4.2.2. Uncertainty Quantification for Infinite Plate with Circular Hole Model with MCs

In this section, we further investigate the effect of the coefficient of variation on the eigenvalues (natural frequencies) of the coupled problem using an infinite plate with a hole model. We selected $C_{1111}$ and $e_{333}$ as random variables. The expected values of these random variables were $\mu$ = 1.26 × 10${}^{11}$ and $\mu$ = 23.3, and they satisfied the Gaussian distribution. Table 10 and Table 11 list all the input parameters with their range, where the scale of the datasets is determined according to the 3$\sigma$ principle.

Table 10. Definitions and the statistical properties of the random input variables ($\gamma =0.02$ and 0.06).

Random Input Variables	Symbol	$\mathit{\mu }$	The Limits of Input Variables: [Lower, Upper]
			$\mathit{\gamma }$ = 0.02	$\mathit{\gamma }$ = 0.06
Elastic constants	$C_{1111}$	1.26 × 10${}^{11}$	[1.18 × 10${}^{11}$, 1.33 × 10${}^{11}$]	[1.04 × 10${}^{11}$, 1.53 × 10${}^{11}$]
piezoelectric constants	$e_{333}$	23.3	[22.0, 24.8]	[19.1, 29.0]

Table 11. Definitions and the statistical properties of the random input variables ($\gamma =0.1$ and 0.14).

Random Input Variables	Symbol	$\mathit{\mu }$	The Limits of Input Variables: [Lower, Upper]
			$\mathit{\gamma }$ = 0.1	$\mathit{\gamma }$ = 0.14
Elastic constants	$C_{1111}$	1.26 × 10${}^{11}$	[8.86 × 10${}^{10}$, 1.61 × 10${}^{11}$]	[8.06 × 10${}^{10}$, 1.72 × 10${}^{11}$]
piezoelectric constants	$e_{333}$	23.3	[17.3, 31.0]	[12.9, 34.1]

In this problem, we examined the third and sixth non-zero eigenvalues. For the elastic constant $C_{1111}$, piezoelectric constant $e_{333}$, and their combinations, the expected values and standard deviations of the responses in relation to the coefficient of variation are shown in Figure 12 and Figure 13, respectively. It can be observed that the expected values and standard deviations of the responses are similar to the results obtained from the tapered panel model in Section 4.1.2. The statistical characteristics of the natural frequencies in the vibration analysis of piezoelectric materials are better reflected by different examples.

Figure 12. Expected values for separate $C_{1111}$, $e_{333}$ and the two uncertainties combined case with different coefficient of variation (one-fourth of the plate model).

Figure 13. Standard variation for separate $C_{1111}$, $e_{333}$ and the two uncertainties combined case with different coefficient of variation (one-fourth of the plate model).

5. Conclusions

In this study, a Monte Carlo simulation-based isogeometric stochastic Finite Element Method was used for uncertainty quantification in the vibration analysis of piezoelectric materials. The smoothness of the IGA basis functions was used to discretize the governing equations of the coupled problem. The IGA-FEM eliminates repetitive meshing procedure in uncertainty quantification and retains geometric accuracy. In general, the natural frequencies of piezoelectric structures are higher than that in the absence of piezoelectric effects. This “piezoelectric stiffening” effect is crucial for certain modes. By comparing the statistical characteristics of the natural frequencies, the fluctuations in the elastic constants have the greatest impact on the results of the statistical characteristics in the three cases. According to the numerical cases, the change in dielectric constants has a negligible effect on the expected values and standard deviations of the natural frequencies. Additionally, the present method will be applied to three-dimensional piezoelectric problems, two-dimensional and three-dimensional flexoelectric problems.