# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 3 An Algorithm to Estimate Failure Paths
Figure 7: Failure path prediction (one failure path, greater than 50% match): The above set of figures show results where the initial cracks in the predicted failure path have a greater than 50% match with those in the HOSS predicted failure path. Figure 9 shows the failure path prediction for an initial configuration with 50 pre-existing cracks. This figure describes the application of the proposed method for estimating failure paths for larger number of cracks in the domain. The process to predict failure paths is the same as the 20 crack scenario. Figure 9b shows a failure path prediction, which agrees with HOSS results (see Figure 9c). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario Description** & **Number of Simulations** & **\% of Failure Path Match** \\ \hline Accurate prediction of failure path & 43 & 100\% match \\ Reasonable prediction of failure path & 75 & \textgreater{} 50\% match \\ Non-matching failure paths & 72 & \textless{} 50\% match \\ \hline Total & 190 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the proposed method to estimate failure paths. Among various failure paths provided by the proposed method, if a failure path exactly matches the HOSS results, then we consider that as an accurate prediction. Meaning that, all the initial cracks or their crack tips identified by the proposed method fall within the dominant HOSS fracture path. If the proposed method predicts 50% of the initial cracks or their crack tips to be within the HOSS failure path, then we consider it as a reasonable prediction. A non-matching failure path is a scenario where the prediction of the proposed method is less than 50% match. Figure 8: Failure path prediction (less than a 50% match): The above set of figures show the scenarios where the proposed method failed to predict the HOSS results. ### Estimating Damage Along the Failure Paths

Figure 9(a) shows the estimated damage evolution vs time, confidence interval, and prediction on unseen data. In this figure, the mean estimated damage \(\overline{D_{M}^{(t)}}\) is shown with a blue line. The magenta region shows the damage confidence interval \(\left(D_{L}^{(t)},D_{U}^{(t)}\right)\), which is constructed based on the method discussed in Section 4. The test data from HOSS simulations are shown in gray lines. Figure 9(a) shows that our confidence interval captures the damage from the test set. Figure 9: 50 crack problem: The above figures describe the application of the proposed method for estimating failure paths for 50 crack problem. Figure 10: Damage statistics: The above figure provide the estimated damage and empirical coverage on data from 40 unseen HOSS simulations. From these two figures, it is clear that for most of the times we have more than 90% prediction accuracy. Figure 10b provides the corresponding empirical coverage of the proposed surrogate model for damage. The blue line corresponds to the empirical coverage on the test dataset from HOSS. The red line corresponds to 95% confidence interval. As discussed in Section 4, empirical coverage is a measure of how well our predicted confidence interval captures unseen data. From this figure, we see that we are able to capture more than 90% of the test data over time except for a small region around time 1.75 ms where we under cover (this is because the cracks along the failure path start to grow around this time). Macro-scale models can take advantage of the proposed methodology as follows: The proposed surrogate models provide statistics on the micro-scale damage evolution for each representative volume element (RVE). This statistical information obtained from the surrogate models is used in calculating macro-scale effective moduli. Specifically, the elastic moduli are degraded using probability distribution functions that describe the evolving lengths and orientations of all the micro-cracks within a RVE. The prediction accuracy (>90%) of the damage evolution surrogate model implies that the proposed method can be used to develop probability density functions.