# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 1 Introduction
Similarly, there are combined finite-discrete element methods (FDEM) that also have discrete material blocks, which can deform themselves due to further resolution of each block with finite elements [49; 50; 51; 52; 53]. In the FDEM methods, the discontinuities of interest (e.g., micro-cracks) must be on similar scale to the computational domain. Of all these approaches available in the fracture mechanics literature, we adopt the FDEM approach in this work. FDEM merges many features of other methods listed above. The high-fidelity FDEM model used in this work that accounts for fracture propagation was developed in a multi-physics software tool called Hybrid Optimization Software Suite (HOSS) [54; 55]. Although, FDEM models have been shown to accurately model brittle fracture, the use of these models (e.g., HOSS) in realistic simulations is computationally intractable as they resolve all the individual cracks with highly resolved meshes and small time-steps at large-scales. Despite the increased realism in the physics and parallel implementation, we are still unable to capture the full range of spatial or temporal scales due to the large computational requirements of our applications, such as hydraulic fracturing [7] or underground nuclear explosion monitoring [56]. Coarsening of the domain and simplification of the physics [57; 58] are commonly used workarounds, but these methods eliminate critical topological features (critical effects of crack-to-crack interactions) and as a result, predictions routinely fail to match observations [59; 60]. One then has to resort to upscaling methods where the domain is split into several grid cells and the material properties in each grid cell are obtained from QoI such as failure paths and damage statistics, which are still computationally challenging. For example, to obtain the results for each HOSS simulation presented in this work, 4 hours of computation time on 400 processors were necessary. In addition, the amount of data being generated is also quite significant. That is, each simulation took up to 11.5 GB of disk space to store all the information needed to characterize different fracture propagation stages. Even for a very coarse upscaled domain of say 1000 cells, this leads to 1.6 M CPU-hours and 11.5 TB of disk space. For this reason, we need an approach that can provide key quantities of interest (QoIs) in a matter of seconds and with reasonable accuracy. This is the aim of the present work. We develop surrogate models for estimating the damage statistics along failure paths to inform continuum models. These surrogate models are built using data from HOSS high-fidelity numerical simulations. The proposed methods presented in this work are applicable only under low-strain rates and Mode I failure or tensile loading conditions. In these conditions, a single dominant fracture perpendicular to tensile loading is observed. A limitation of the proposed methodology is that it is not applicable for other modes of fracture and high strain rates loading conditions but the overall approach to build the surrogate models can be used. In our previous works [61; 62], we have utilized machine learning approaches to efficiently emulate the high-fidelity model. The key QoIs [PERSON] et. al. [61] focused on were time to failure and fracture coalescence predictions. The previous work [61] did not estimate damage, which is a key input needed for upscaling micro-scale information to macro-scale models. Numerical codes such as FLAG [63; 64; 65] need information on how effective moduli degrade over time to simulate brittle damage at continuum-scales. Current upscaling approaches in literature use empirical/phenomenological models based on experimental data or homogenization or statistical averaging techniques to inform damage to continuum models [58]. However, detailed micro-scale information about the damage evolution of the system is lost during this homogenization process [66; 67].