## 2 Computational Methods  
Figure 1 shows the computational model, including a high-temperature flat specimen at the bottom and a rectangular waveguide rod at the top. The simulation process was based on the ultrasonic pulse reflection principle to measure the thickness of the specimen. The high-temperature specimen was fixed to the waveguide rod. The ultrasonic probe located at the top of the waveguide rod emitted a pulse signal, here using an SH-guided wave. The ultrasonic pulse signal propagated along the waveguide rod and the specimen to be measured, reflecting when it reached the bottom surface and generating a pulse echo. The pulse echo was also received by the same ultrasonic probe, and the specific waveform can be displayed by an oscilloscope or a curve recorder [29]. The thickness of the tested specimen can be determined by measuring the time of ultrasonic wave propagation in the material, which can be expressed as:  
\[\delta=\frac{1}{2}c\Delta t-L \tag{1}\]  
where \(c\) is the propagation speed of the ultrasonic wave in the specimen; \(\Delta t\) is the time difference between two adjacent wave peaks of the echo pulse signal; \(L\) is the length of the waveguide rod. The waveguide rod exists to avoid permanent failure of the ultrasonic probe due to direct contact with the high-temperature specimen and was in a state of steady heat conduction. The governing equation can be expressed as:  
\[\
abla^{2}t=0 \tag{2}\]  
Figure 1: Computational model of the waveguide structure. In the model, the temperature of the specimen was determined. The bottom end of the waveguide rod was in contact with the high-temperature specimen, and heat transfer occurred with the air on the remaining surfaces. The ambient temperature was \(T_{amb}\), the length of the waveguide rod was \(L\), the width was \(w\), and the thickness was \(d\). The flat specimen was stainless steel 316 L with a size of 250 mm \(\times\) 180 mm \(\times\) 50 mm, while the material, length and thickness of the waveguide rod needed to be designed by using COMSOL Multiphysics. Three materials were chosen as candidates, which are shown in Table 1.  
Figure 2 shows the schematic diagram of the mesh division. The waveguide rod used a structured mesh, and the specimen used a free tetrahedral mesh structure. The whole model contained 13,058 domain cells, 15,390 boundary elements and 920 edge cells. Grid sensitivity tests were validated. The ambient temperature was 20 deg C and the temperature at the bottom surface of the specimen was 500 deg C. Define all the other surfaces of the specimen and the waveguide rod as the heat flow boundary conditions. At high temperatures, radiation also plays an important role, and it can be combined with the convection and take the empirical value of heat transfer coefficient as \(h_{com}=5\) W/ \(\left(\text{m}^{2}\cdot\text{K}\right)\). Therefore, the thermal boundary conditions can be expressed as:  
\[q=h_{com}(T-T_{amb}) \tag{3}\]  
When the waveguide structure was determined, the same model was used to analyze the propagation of ultrasonic waves in the waveguide rod and the specimen. As the waveguide structure was fixed on the specimen by a clamping device with a strong pressing force, so the contact surface was coupled for the two parts in the model. The ultrasonic wave propagation was simulated using a time-domain display solver based on the interrupted Galliakin method (dG-FEM). The model used the velocity-strain formulation to solve the governing equations for a general linear elastic material [23; 24] as:  
\[\rho\frac{\partial\text{v}}{\partial t}-\
abla\cdot\mathbf{S}=\mathbf{F}_{ \text{v}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **SS316L** & **Aluminum** & **Copper** \\ \hline Density (kg/m\({}^{3}\)) & 7800 & 2700 & 8960 \\ Young’s modulus (GPa) & 211 & 70 & 110 \\ Poisson’s ratio & 0.286 & 0.33 & 0.35 \\ Constant pressure heat capacity (J/(kg\(\cdot\)K)) & 475 & 900 & 385 \\ Thermal conductivity (W/(m\(\cdot\)K)) & 44.5 & 238 & 400 \\ \hline \hline \end{tabular}
\end{table}