# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 3 An Algorithm to Estimate Failure Paths
From these figures, it can be inferred that the proposed method is able to predict multiple failure paths for a given initial crack configuration. Figure 6 shows example results from the graph theory approach for cases where there was only one predicted failure path with comparison to results from HOSS. Figure 6a shows the initial crack configuration. Figure 6b shows the failure path prediction using the proposed method. Figure 6c shows the results obtained from the HOSS simulations. It should be noted that it is not always possible to predict all paths of failure. For example, the HOSS results shown in Figure 6c contain complex failure paths that could be considered two distinct pathways. However, our method provided only a single failure path. There are 43 out of 190 simulations where the graph approach has predicted a single failure path and matched perfectly with atleast one failure path of HOSS simulation. By perfect match, we mean that all of the initial cracks or crack tips in the graph predicted failure path are also in the HOSS predicted path. Figure 4: Proposed method description: The above figures show a step-by-step description of proposed method for estimating failure paths based on the method outline in Section 3. (**a**) Cracks perpendicular to tensile loading (0-degree cracks); (**b**) Failure zone identification; (**c**) Node-to-Node connectivity; (**d**) Weighted graph based on 2-\(NN\); (**e**) Failure path-1 prediction; (**f**) Failure path-2 prediction; (**g**) Failure path-3 prediction; (**h**) Failure path-4 prediction. Figure 7 shows example results from the graph theory approach for cases where there was only a partial match. Quantitatively, by partial match we mean atleast 50% of the initial cracks are in the

Figure 5: Failure path prediction (best scenarios, multiple predicted failure paths): The above figures show the prediction of multiple failure paths using the proposed method. Figure 6: Failure path prediction (best scenarios, one failure path exact match): The above set of figures show the prediction of failure paths with exact match to the HOSS results. The left set of figures provide the initial crack configuration. The middle set of figures provide the prediction by the proposed method. The right set of figures show the HOSS results at failure. failure paths in comparison to the HOSS simulations. Predictions that have more than 50% of the initial cracks in the failure paths and also have a zone of failure similar to that of HOSS are a total of 75 out of 190. Note that these 75 are separate from the previously mentioned 43 simulations. Figure 8 shows example scenarios where the proposed method failed to match the HOSS results. Predictions that fall in this category are the remaining 72 out of 190. Table 1 summarizes the accuracy of the proposed method, which is the number of reasonably accurate predictions made by the Algorithm 1. Out of 190 simulation, the proposed method predicted the failure paths of 118 simulations with reasonable (50% of the failure path) or accurate prediction (100% of the failure path). The prediction accuracy of failure zones by the proposed method is 143 simulations out of 190 simulations. It should be noted that the time taken by the proposed method to predict a failure path is a couple of seconds, which is \(\approx\mathcal{O}(10^{6})\) times less than a single HOSS simulation. For Mode I loading, we can roughly determine the cases where there is a chance that the surrogate fails a priori. For example, if there are small number of 0-degree angle cracks or if the 0-degree angle cracks are far away from each other, then the Euclidean distance between a pair of 0-degree angle cracks is greater than the length of FPZ. Hence, there is no interaction between the pair of 0-degree angle cracks, and it will be more difficult for the graph-based surrogate model to reliably predict the failure pathway. This highlights a need for improved feature engineering, perhaps including some information about long-range interactions between pre-existing cracks.