## 3 Results
### Microstructure  
Two subsets of the performed simulations were taken for analysis: a series of austenite samples with increasing grain diameters, and a comparable set of ferrite samples with three different grain diameters. These are the series A and B from Table 2. Grain diameters are estimated from the data, ignoring the truncated grains at the edges. Assuming spherical grains, an average grain diameter \(\hat{D}\) can be deduced from the average grain volume \(\hat{V}\) as  
\[\hat{D}=\sqrt[3]{\frac{6\hat{V}}{\pi}}. \tag{7}\]  
The grain diameters retrieved from the austenitic steel samples are 10.0, 15.0 and 20.0 \(\upmu\)m, while the ferritic samples have diameters of 5.4, 10.0 and 15.0 \(\upmu\)m, respectively. For each of those six samples, simulations have been performed at four different temperatures (27 \({}^{\circ}\)C, 300 \({}^{\circ}\)C, 550 \({}^{\circ}\)C, and 900 \({}^{\circ}\)C). These temperatures have been used in the simulations to study the same temperature domain for ferrite and austenite. Note that the phase transition temperature has not been taken into account, and that some of these (pure) steel phases do not actually exist at the chosen temperatures. Per set, the underlying microstructures only differ in grain size. The texture is isotropic in all samples. The variation in between the average elasticity matrices of the three samples within each set, is less than 0.1%, taken from samples at the same temperature. For the pure polycrystalline materials austenite, ferrite and martensite, the corresponding Voigt-Reuss-Hill averages [55] of the elasticity matrices at 27 \({}^{\circ}\)C are shown in Figure 11. [PERSON]'s anisotropy metric \(\mathcal{Z}\), as defined in Equation (1), of all three samples is 1.000, which indicates perfect isotropy. \begin{table}
\begin{tabular}{c c c c} \hline
**Microstructure** & **Average Grain** & **Material Properties** & **Subset IDs** \\
**Geometry ID** & **Diameter [\(\upmu\)m]** & **(Phases and Phase Ratios)** & **Subset IDs** \\ \hline
1 & 10.0 & 100\% Austenite & A, D \\
2 & 15.0 & 100\% Austenite & A, C, D \\
3 & 20.0 & 100\% Austenite & A, D \\
4 & 5.4 & 100\% Ferrite & B, D \\
5 & 10.0 & 100\% Ferrite & B, D \\
6 & 15.0 & 100\% Ferrite & B, D \\
7 & A: 15.0, M: 10.0 & 75\% Austenite, 25\% & C, D \\  & & Martensite & \\
8 & A: 15.0, M: 10.0 & 50\% Austenite, 50\% & C, D \\  & & Martensite & C, D \\
9 & A: 15.0, M: 10.0 & 25\% Austenite, 75\% & C, D \\  & & Martensite & C, D \\
10 & 10.0 & 100\% Martensite & C, D \\ \hline \end{tabular}
\end{table} 
Table 2: Ten simulation cases for LUS simulations. Each case has been simulated at temperatures of 27 \({}^{\circ}\)C, 300 \({}^{\circ}\)C, 550 \({}^{\circ}\)C, and 900 \({}^{\circ}\)C. The subset IDs refer to the four cases in Section 3. The wave propagation velocity can be determined as a function of the temperature by finding the time delay between two consecutive echoes. By dividing the travelled distance \(2H\) (i.e., double the sample thickness) by this time delay \(\Delta t\), the velocity \(c\) is obtained, as in Equation (3). Results are shown in Figures 12 and 13, where the observed velocities are plotted either as function of grain diameter or as a function of temperature. As can be seen, the temperature has a significant effect on the wave velocity. This is due to the strong temperature dependencies of both the elasticity matrix and the density. Note that the individual components of the elasticity matrix and the density are nonlinearly dependent of the temperature, as can be seen in Figure 4, leading to nonlinear velocity curves in Figure 13. On the other hand, the grain size variation has a much smaller impact on the wave velocity than temperature. Grain size affects the wave velocity due to small time delays induced by scattering of waves, leading to relatively small changes in the resulting wave velocity. The larger the grains, the greater the delay caused by multiple scattering and hence the lower the resulting wave velocity.