## 5 Results
### Building the Lattice-Based Surrogate Models
#### 5.1.2 Multisize Obstacles Case  
Obstacle radii: Continuous model simulations were also performed for the multisize obstacles model (Figure 5) on an RVE partitioned into four subdomains as described in Section 3.3.1. The particle radius was fixed at \(R_{\text{part}}=1.0\), with one obstacle radius fixed at \(R_{\text{obs}_{1}}=1.0\) and the other obstacle radius (\(R_{\text{obs}_{2}}\)) varied from 0.5 to 3.0 in increments of 0.5. Stationarity and linearity of \(\langle r^{2}\rangle\) versus time is illustrated, for a representative case (\(R_{\text{part}}=1.0\), \(R_{\text{obs}_{1}}=1.0\) and \(R_{\text{obs}_{2}}=1.0\)), in Figure 11. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(R_{\text{obs}}\)} & \multirow{2}{*}{\(M_{c}\)} & \multirow{2}{*}{\begin{tabular}{c} **Slope** \\ **(Continuous)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Slope** \\ **(Surrogate)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Percent** \\ **Difference** \\ \end{tabular} } \\ \hline
0.5 & 7 & 2.9984 & 2.8770 & 4.049\% \\
1.0 & 6 & 2.9044 & 2.8468 & 1.983\% \\
1.5 & 5 & 2.7507 & 2.7506 & 0.004\% \\
2.0 & 5 & 2.4731 & 2.5094 & 1.468\% \\
2.5 & 4 & 2.3376 & 2.4310 & 3.996\% \\
3.0 & 4 & 2.0765 & 2.1016 & 1.209\% \\ \hline \end{tabular}
\end{table} 
Table 2: Slope values for \(\langle r^{2}\rangle\) vs. \(t\) in the equally spaced obstacles (continuous) model (\(R_{\text{part}}=1.0\)) and the lattice-based surrogate model. The best values of \(M_{C}\) are also shown. Figure 10: Slope of \(\langle r^{2}\rangle\) versus time for equally spaced obstacles (continuous) model and the lattice-based surrogate model with best \(M_{c}\) values. Results are shown for particle radius \(R_{\text{part}}=1.0\) and varying obstacle radius \(R_{\text{obs}}\). Figure 9: Mean squared displacement and its slope for the equally spaced obstacles model. Results are shown for particle radius \(R_{\text{part}}=1.0\) and obstacle radius \(R_{\text{obs}}=2.0\): (**left**) \(\langle r^{2}\rangle\) versus time for continuous model and resulting surrogate model as the commitment index \(M_{c}\) is varied; (**right**) slope as a function of obstacle radius for an increasing number of realizations. Determining transition probabilities: Based on the approach illustrated in Figure 5, continuous model results were used to determine transition probabilities for the multisize obstacles model using (17). Commitment index calibration: Lattice-based surrogate models were built using the transition probabilities obtained from the continuous model (Section 3.3.2) by first calibrating the commitment index \(M_{C}\) (Section 4.1). For each case considered, \(M_{C}\) was varied from 3 to 8 and calibrated by comparing the continuous and surrogate plots of \(\langle r^{2}\rangle\) versus time. A sample plot illustrates increasing stationarity of the slope of \(\langle r^{2}\rangle\) versus obstacle radius as the number of realizations increases (Figure 12--left). We observe the manner in which the slope decreases with increasing obstacle radius \(R_{\text{obs}}\), which differs from that observed in the equally spaced obstacles case (Figure 10). A summary of the transition probabilities and values of \(M_{C}\) needed to build the lattice-based surrogate model for all cases considered is provided in Table 3. Note that the sum of the probabilities in each row is 1 when the second probability is weighted by a factor of 2 due to symmetries in the RVE. Comparisons of slopes for the relation between \(\langle r^{2}\rangle\), including the best values for \(M_{C}\), are shown for the continuous and surrogate models in Table 4 and Figure 12--right. These results exhibit a difference in slopes ranging from 0.6% to roughly 7%. Figure 11: Mean squared displacement versus time for the multisize obstacles model. Effects of an increasing number of realizations are shown for the case \(R_{\text{part}}=1.0\), \(R_{\text{obs}_{1}}=1.0\) and \(R_{\text{obs}_{2}}=1.5\): (**a**) curves; (**b**) linear fits to the curves.