# Ultrasonic Deicing Efficiency Prediction and Validation for a Flat Deicing System
## 2 Basic Theory and Model Establishment
### FE Model and Simulation
Meanwhile, the mesh type adapted in simulations was 20-node quadratic brick elements with reduced integration, rather 8-node linear brick elements with reduced integration to ensure accuracy. Thus, the amount of calculation would be several times larger than using mesh type 8-node linear brick elements with reduced integration. After meshing the model, the number of elements was more than 310,000. The FE mesh and reference points (A and B) for FE study is shown in Figure 4. In the model, the voltage applied to the positive and negative electrode of the actuator was 25 and 0 V, respectively. The left and right edge of the aluminum plate were fixed. To make out the effect of different actuators on deicing and stress distributions, the volume of the PZT actuator was kept as 100 \(\pi\) mm\({}^{3}\), while their radius (\(r\)) and thickness (\(t\)) were changing as seven different configurations. The radius for seven configurations were 4, 5, 6, 7, 8, 9, and 10 mm, respectively. Then, the deicing efficiency factor (DEF) was defined. The deicing efficiency for these seven configurations were compared. Based on simulation results in Section 4, the most efficient configuration was confirmed to be configuration 2. To verify the rationality of DEF, four different icing cases as shown in Figure 5 were studied. For these four cases, the radius of the PZT actuator was 5 mm. Theicing case (a) in Figure 5 was the same as configuration 2 mentioned above. The size of the icing area and the centroid of ice layer for these four cases were the same. For the cases (b-d) in Figure 5, the distance between each ice block was 4 mm. Meanwhile, the applied voltage and frequency were the same for these four cases. Based on these, shear stresses and DEF were computed and verified. Figure 4: FE mesh and reference points (A and B) of the deicing system for the FE parametric study. Figure 5: Different icing cases for configuration 2: (**a**) One ice block; (**b**) Two ice blocks separated along X axis; (**c**) Two ice blocks separated along Y axis; (**d**) Four ice blocks. Finally, Abaqus standard-explicit co-simulation was conducted to model wave propagation and the deicing process for configuration 2. The aluminum plate and ice layer were defined as the parent model to conduct transient dynamic analysis in Abaqus explicit, while the PZT actuator was defined as the child model to conduct Abaqus implicit analysis. An interaction interface between the PZT actuator and aluminum plate was used to manage the data exchange between the two solvers. A bilinear cohesive zone model (CZM) [23] was considered to simulate delamination behavior at the interface of the ice layer and aluminum plate. Cohesive elements with zero thickness were inserted between the aluminum plate and ice layer. The constitutive behavior of the cohesive layer was defined as a traction-separation law to model the detachment at the interface of the ice layer and aluminum plate. It is assumed that there is a linear elastic traction separation law before failure, and the material stiffness will gradually degrade after failure. To describe cohesive material behavior, damage initiation and damage evolution should be defined. Initiation of interface damage is based on quadratic traction interaction failure criterion, written as Equation (8):  
\[ \begin{array}{l}\sqrt{\left(\frac{\sigma_{z}}{\tau}\right)^{2}+\left(\frac{ \tau_{yz}}{S}\right)^{2}+\left(\frac{\tau_{yz}}{S}\right)^{2}}=1\ (\sigma_{z}>0),\\ \sqrt{\left(\frac{\tau_{yz}}{S}\right)^{2}+\left(\frac{\tau_{yz}}{S}\right)^{2 }}=1\ (\sigma_{z}\leq 0)\end{array}  \tag{8}\]  
where \(\sigma_{z}\), \(\tau_{xz}\), \(\tau_{yz}\) represent the normal stress, shear stress in the x and y direction, separately. \(T\) and \(S\) are the tensile strength and shear strength, respectively. The damage evolution satisfies the mixed-mode fracture energy approach [31; 32]. The fracture energy for the aluminum/ice interface was reported to be 1.0 N/m [33].