## 3 Numerical Experiments on a Turbine Blade
### Simulation of Ice-Coupled Ultrasonic Testing in a Turbine Blade  
In this study, an open-source code k-wave toolbox was applied to conduct the numerical simulation of ice-coupled ultrasonic testing to record the full wavefield [30]. This toolbox solves the acoustic wave equation based on a k-space pseudospectral method. Compared with the conventional finite difference method, the memory and computational cost were reduced dramatically. The turbine blade was encased by ice as shown in Figure 2b. The material properties of the turbine blade and ice are shown in Table 1[11]. The size of the whole model was 185 mm \(\times\) 85 mm. A series of ultrasonic transducers were deployed evenly around the turbine blade. Considering that the length of the specimen in the horizontal direction was approximately half that of the vertical direction, 6 transducers were arranged at the top and bottom sides and 12 transducers at the left and right sides, shown in Figure 3. Such an arrangement of transducers can reduce the transducer number and assure the imaging quality. At four corners, the spacing of two adjacent transducers was set as \(d_{1}=d_{2}=5\) mm. The PML boundary was set around the calculation domain to absorb outgoing waves and avoid reflection. To avoid uncertain factors further from the absorbing layer, all transducers were shifted inward from all sides by 1 mm. It was an effective simplification to simulate the wave propagation in a large domain because the reflections from the boundaries beyond the area of interest could be excluded by applying appropriate time gates to the recorded signals. The spacing of the transducers was \(d_{x}=14.6\) mm in a horizontal direction and \(d_{z}=15.73\) mm in a vertical direction. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Mass Density \(\rho\) (kg/m\({}^{3}\))** & **Wave Speed \(c\) (m/s)** \\ \hline Steel & 7800 & 5875 \\ \hline Ice (at \(-5\,^{\circ}\)C) & 917 & 4000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Acoustic properties of the turbine blade and ice [11]. Figure 2: (a) Photo of a turbine blade and (b) typical cross section. The ultrasonic wave was excited with the following signal:  
\[R(t)=[1-2\pi^{2}f_{c}^{2}(t-t_{0})^{2}]e^{-\pi^{2}f_{c}^{2}(t-t_{0})^{2}} \tag{8}\]  
where \(f_{c}\) was the center frequency and \(t_{0}\) referred to the time delay. As the theoretical resolution of the FWI is a half wavelength [20; 21], an ultrasonic wave with a center frequency of 1 MHz was applied to inspect a defect as small as 3 mm in the turbine blade. When \(f_{c}=1\)MHz and \(t_{0}=1.5/f_{c}\), this excitation waveform and the corresponding amplitude spectrum are depicted in Figure 4. It could be seen that the dominant energy (above the value of a \(-6\) dB drop from the maximum amplitude) was within the effective frequency band of 0.48 MHz-1.6 MHz. A quadrilateral mesh was applied to discretize the spatial domain and the mesh size was carefully chosen to satisfy the Courant-Friedrichs-Lewy (CFL) condition. The mesh size was chosen as \(L_{element}=0.2\) mm in both directions for an accurate acoustic wave calculation up to a frequency of 2 MHz (about 10 elements per minimum wavelength) [31]. To ensure numerical stability, the time step should be chosen carefully according to the smallest element and the wave speed. Here, time step was set as \(\Delta t=0.5L_{element}/c_{max}\), where \(c_{max}\) was the wave speed in the steel. Figure 4: Excitation signal (\(f_{c}=1\) MHz and \(t_{0}=1.5/f_{c}\)). (**a**) Waveform in the time domain and (**b**) the corresponding normalized amplitude spectrum. Figure 3: Configuration of the ultrasonic transducer array. The resolution and sensitivity of tomography strongly depend upon 'ray density' [32]. For this reason, a full matrix capture (FMC) technique was adopted to perform the ultrasonic data acquisition, which collected complete datasets including all transmitter-receiver combinations. Therefore, each transducer was excited successively and all transducers recorded the ultrasonic signals simultaneously.