# Ultrasonic Deicing Efficiency Prediction and Validation for a Flat Deicing System
## 2 Basic Theory and Model Establishment
### FE Model and Simulation  
In this section, harmonic analysis and dynamic analysis were performed to model deicing effect and deicing process, respectively. Harmonic analysis was implemented to calculate the stress tensor at expected frequency range. The finite element model equation of the PZT actuator bonded to the host structure can be written as Equation (6) [29]:  
\[F=([K_{nm}-w^{2}[M]-jw[R])U+[K_{n\Phi}]\Phi\bigg{(}\frac{I}{jw}\bigg{)}=[K_{n \Phi}]U+[K_{\Phi\Phi}]\Phi, \tag{6}\]  
where \(F\) is applied mechanical force, \(I\) is excitation current, \(U\) is elastic displacement field, \(\Phi\) is electric potential, \([K_{nm}]\) is elastic rigidity matrix, \([K_{n\Phi}]\) is piezoelectric stiffness matrix, \([K_{\Phi\Phi}]\) is permittivity matrix, \([M]\) is mass matrix, \([R]\) is dissipation matrix, and \(w\) is the angular frequency. The equation of harmonic analysis used in FE software can be expressed as follows:  
\[\left[ \begin{array}{cc}K_{nm}-w^{2}M&K_{n\Phi}\\ K_{n\Phi}&K_{\Phi\Phi}\end{array} \right]\left[ \begin{array}{c}U\\ \Phi\end{array} \right]=\left[ \begin{array}{c}0\\ -Q\end{array} \right]. \tag{7}\]  
The work started from the evaluation of the effectiveness of guided waves technology to produce the ice detaching from the structure. The 3D model of finite element simulation is shown in Figure 2.  
The geometry size of the aluminum plate and ice layer was 200 mm \(\times\) 150 mm \(\times\) 2 mm and 140 mm \(\times\) 80 mm \(\times\) 3 mm. The material of the piezoelectric disc was selected to be PZT-4. The material properties of the aluminum and ice are shown in Table 1.  
Figure 2: The 3D model of finite element simulation. Material properties for PZT-4 are shown as follows:  
\[\rho =7500\ \mathrm{kg/m^{3}}\ \ c^{E}=\left[ \begin{array}{ccccc}13.9&7.78&7.43 &\\ 7.78&13.9&7.43&\\ 7.43&7.43&11.5&\\ &&&2.56&\\ &&&&2.56&\\ &&&&2.56&\\ &&&&2.56\end{array} \right]\times 10^{10}\mathrm{Pa}\]  
\[e =\left[ \begin{array}{ccccc}0&0&-5.2\\ 0&0&-5.2\\ 0&0&15.1\\ 0&12.7&0\\ 12.7&0&0\\ 0&0&0\end{array} \right]\ c/\mathrm{m^{2}}\ \ \varepsilon=\left[ \begin{array}{ccccc}762.5&\\ &762.5&\\ &&663.2\end{array} \right],\]  
where \(\rho\) is density, \(c^{E}\) is elasticity matrix, \(e\) is piezoelectric coupling constant, and \(\varepsilon\) is dielectric matrix. The FE simulations were performed with an Abaqus(r) during processing. The host plate and ice layer were meshed by using 20-node quadratic brick elements with reduced integration (C3D20R) [30] and PZT actuator was meshed by using 20-node quadratic piezoelectric brick elements with reduced integration (C3D20 RE). In the frequency range of research (20-240 kHz), the wavelengths of the Lamb wave and SH wave were taken into consideration. According to the theory of guided waves in layered structures in Ref. [27], the global matrix method can be used to solve the dispersion curves of the ice-plate layered structure when ice accretes on the plate. By solving dispersion curve equations for both Lamb wave and SH wave, the phase velocity of both Lamb wave and SH wave can be calculated. The results for the 2 mm aluminum plate and 2 mm aluminum plate with a 3 mm ice layer are shown in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **Density (kg/m\({}^{3}\))** & **Young’s Modulus (GPa)** & **Poisson’s Ratio** \\ \hline Ice & 920 & 9 & 0.28 \\ Aluminum & 2700 & 70 & 0.30 \\ \hline \end{tabular}
\end{table} 
Table 1: The material properties. Figure 3: Phase velocity of the aluminum plate and ice-plate layered structure: (**a**) 2 mm aluminum plate, (**b**) 2 mm aluminum plate with 3 mm ice. According to phase velocity, the shortest wavelengths should be around 8.0 mm for the 2 mm aluminum plate with a 3 mm ice layer and around 7.8 mm for the 2 mm aluminum plate at frequency 240 kHz. After several times trial calculations, the global mesh seeding size was chosen as 0.74 mm, smaller than one over ten the smallest wavelength.