# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 3 An Algorithm to Estimate Failure Paths
For testing, to show that our confidence interval captures the damage from the test set, we calculate empirical coverage over time. Empirical coverage represents the fraction of test cases that fall within our estimated confidence interval at anytime. Mathematically, it is defined as,

\[\frac{1}{N_{T}^{(t)}}\sum_{j=1}^{N_{T}^{(t)}}\mathbb{I}_{D_{j}\in\left(D_{L}^{ (t)},D_{U}^{(t)}\right)}\]

where \(N_{T}^{(t)}\) is the test set data at time \(t\) and \(D_{j}\) is one such data point of damage in our test set. \(\mathbb{I}_{D_{j}\in\left(D_{L}^{(t)},D_{U}^{(t)}\right)}\) indicates if \(D_{j}\) is in the confidence interval. The empirical coverage is one measure of how well our predicted confidence interval captures unseen data. In the next section, we provide results and compare them with HOSS simulation data. ## 5 Results

In this section, we present results of the proposed methods to estimate failure. Following are the inputs given to Algorithm 1 to calculate failure paths: width of the domain is equal to 2 m, height of the domain is equal to 3 m, NumZones = 3, initial crack tip coordinates, and length of FPZ = 0.45 m, which is quasi-brittle. We use 190 simulations to test the proposed method to estimate failure path for the 20 crack problem. In addition, we present a result for a configuration with 50 pre-existing cracks (whose crack tip coordinates are chosen randomly) to demonstrate its predictive capability. ### Estimating Failure Paths

Figure 4 provides a step-by-step description of the proposed method for estimating failure paths. First, we identify cracks that are perpendicular to the tensile loading. These are 0-degree cracks and are highlighted in green in Figure 4a. Second, we identify zone of failure, which is shown in light blue in Figure 4b. This corresponds to the region where failure occurs. As described in Section 3, we identify larger cracks that are formed after connecting pre-existing cracks that fall within the FPZ to determine failure zone. Third, we introduce boundary nodes in the failure zone (see Figure 4c). Fourth, we search for two nearest neighbors for each crack tip in the failure zone. Then, we form new edges with edge weights being the Euclidean distance. Fifth, we connect all the edges to form a weighted graph and look for failure paths connecting the boundary nodes (see Figure 4d). Finally, we compute all possible weighted shortest paths (see Figure 4e-h). The proposed method identified four possible shortest paths. Among the four paths two paths are of exact match (see Figure 5d) and other two are the next best match. Failure paths are computed with and without enforcing the constraint that the shortest path has to traverse through the maximal connected component. Herein, this constraint is enforced with some flexibility. Meaning that, we find shortest paths that contain at least one node to be present in the maximal connected component. This'soft' enforcement of the constraint is done with the intention to capture multiple failure paths (if they exist). Figures 5-8 show examples for failure path prediction and comparison with HOSS simulations. When a possibility that multiple failure paths may exist, pinpointing an exact one is very hard. Moreover, constructing a metric or criterion for an exact match can be challenging. Given an initial crack configuration, a plausible measure to assess the accuracy of the proposed method is as follows: If the predicted failure path has at least 50% of the initial cracks or their crack tips to be within the HOSS failure path, then we say that the proposed method is a reasonable prediction of the simulated results. Figure 5 presents example results in which the graph theory approach predicted two equally likely failure paths for one initial crack configuration. Two different initial crack configurations are shown in Figure 5a. The cracks perpendicular to tensile loading are highlighted in green. Figure 5b,c show the multiple failure paths predicted by the graphs for each case. The HOSS predicted failure paths for both cases are provided in Figure 5d.