## 2 Modeling a Fluid-Conveying Pipe with PVDF Piezoelectric Film Sensor
### Electromechanical Coupling  
The internal fluid pressure can be transferred through the fluid-structure interface to load the pipe in the form of stress and strain [7]. Moreover, the contact region between pipe and PVDF piezoelectric thin films is set to bonded contact region, which means that there is no normal separation and relative tangential sliding on the contact surface. Thus, the continuity of displacement, stress, and strain is imposed at the contact region.  
The stress and strain of the pipe causes mechanical deformation of the polarized piezoelectric film, and electrical charges are generated on its polarized surface simultaneously. The mechanical deformation can be attributed to mechanical behavior, and charge generation can be attributed to electrical behavior. In piezoelectricity theory, the electromechanical coupling between mechanical quantities (stress, strain) and electrical quantities (electric displacement, electric field intensity) of the piezoelectric film is described by piezoelectric equation.  
The piezoelectric equation of the PVDF piezoelectric thin film is  
\[\{D\}=[d]\{T\}+[\varepsilon]^{\mathsf{T}}\{E\}, \tag{6}\]  
where \(\{D\}\) denotes the electric displacement vector, \([d]\) is the piezoelectric constant matrix, \(\{T\}\) denotes the stress vector, \([\varepsilon]\) is the dielectric permittivity matrix, and \(\{E\}\) denotes the electric field intensity vector. There is no external electric field when the piezoelectric film is used as a receiving sensor, i.e., \(E\) is set to zero. Equation (6) can be simplified to  
\[\{D\}=[d]\{T\}, \tag{7}\]  
Applying element discretization and using the Hamilton's principle, we derive the finite element dynamic equation for the piezoelectric film as [20]  
\[\left[\begin{array}{cc}M_{\text{d}}&0\\ 0&0\end{array}\right]\left\{\begin{array}{c}\ddot{X}\\ \ddot{U}\end{array}\right\}+\left[\begin{array}{cc}C_{\text{d}}&0\\ 0&0\end{array}\right]\left\{\begin{array}{c}\dot{X}\\ \dot{U}\end{array}\right\}+\left[\begin{array}{cc}K_{\text{d}}&K_{\text{dv} }\\ K_{\text{dv}}^{\mathsf{T}}&K_{\text{v}}\end{array}\right]\left\{\begin{array}[ ]{c}X\\ U\end{array}\right\}=\left\{\begin{array}{c}F\\ Q\end{array}\right\}, \tag{8}\]  
where \([M_{\text{d}}]\) is the mass matrix of piezoelectric film, \([C_{\text{d}}]\) is the damping matrix of piezoelectric film, \([K_{\text{d}}]\) is the stiffness matrix of piezoelectric film, \([K_{\text{v}}]\) is the dielectric permittivity matrix, \([K_{\text{dv}}]\) is the piezoelectric coupling matrices, \(\{X\}\) denotes the displacement vector, \(\{U\}\) denotes the electric potential vector, \(\left\{\ddot{U}\right\}\) denotes the second derivative of electric potential vector with respect to time, \(\left\{\dot{U}\right\}\) denotes the first derivative of electric potential vector with respect to time, \(\{F\}\) denotes the force vector, and \(\{Q\}\) denotes the charge vector.  
Measuring the pressure fluctuation of the internal fluid using piezoelectric sensors is a complex problem. The key to solving this problem is finding the correlation between fluid pressure and voltage of PVDF film, and thus we first studied the correlation between fluid pressure and stress and strain of the pipe on the basis of the simulation results of FSI. Then, we investigated the correlation between voltage of PVDF film and stress and strain of the pipe on the basis of the simulation results of electromechanical coupling. Finally, we analyzed the feasibility of using PVDF piezoelectric thin film sensors to measure the pressure fluctuation of the internal fluid on the basis of the above studies.