# Surrogate Models for Estimating Failure in Brittle and Quasi-Brittle Materials
## 3 An Algorithm to Estimate Failure Paths
This condition can be relaxed by considering other pre-existing cracks that may be advantageously orientated for growth with respect to the loading conditions, and include such cracks in estimating the failure paths. ## 4 An Algorithm to Estimate Damage

For upscaling to continuum-scale constitutive models, statistical information that describes damage accumulated over time is needed. In this section, we provide an algorithm to estimate damage accumulation along failure paths from the HOSS simulation data. As mentioned in Section 2, within our FDEM framework, cracks are formed along the edges of the mesh elements. Opening and/or shearing between two elements is determined with multiple cohesive points, which are modeled as non-linear springs. For each pair of finite elements, we have a user specified number of non-linear springs. For our HOSS simulations, we have assumed four normal and four shear springs. The number four has been chosen because it has been used previously [PERSON] _et al._ (2018) and produced good results in comparison to experimental results with reasonable computational expense. Figure 1 shows a schematic of the stress-displacement curve, which is representative of each cohesive point's normal response. This response represents reversible elastic damage and softening due to damage growth and coalescence. As the crack grows, the interfacial springs go through a non-linear elastic regime, where the maximum elastic opening (or shearing) of the spring is \(\delta_{n}^{e}\). The total maximum opening (or shearing) of the spring is \(\delta_{n}^{max}\). Between the maximum elastic opening and total max displacement is the strain softening regime. The area under this portion of the curve corresponds to the strain energy density (i.e, energy per length) dissipated during damage evolution. The maximum stress that the spring can withstand in either tension (opening) or shearing is represented by \(\sigma_{n}^{max}\). Damage accumulated per unit length between two finite elements is evaluated based on the strain of the springs that exceed \(\delta_{n}^{e}\). This value ranges from zero to one. Zero damage value corresponds to undamaged springs (\(\delta_{n}\leq\delta_{n}^{e}\)) and damage value of one corresponds to completely broken springs (\(\delta_{n}>\delta_{n}^{max}\)). We use a non-parametric approach for determining the evolution of damage. This is achieved by constructing a confidence interval over time. 150 HOSS simulations are used for training and 40 simulations are used for testing the damage evolution model. Let \(t\) be the time index, our 95% confidence interval for damage at \(t\) is given by \(\left(D_{L}^{(t)},D_{U}^{(t)}\right)\). The damage \(D^{(t)}\) is the accumulated damage along the failure path as a function of time in our damage evolution surrogate model. Accumulated damage along the failure path is calculated by summing the growth of all propagating cracks at every time-step within the HOSS simulation time. \(D_{L}^{(t)}\) and \(D_{U}^{(t)}\) correspond to lower and upper estimate on damage at a given instance of time. We use bootstrapping [PERSON] and [PERSON] (2018); [PERSON] _et al._ (2018) to estimate \(D_{L}^{(t)}\), \(D_{U}^{(t)}\), and mean damage \(\overline{D_{M}^{(t)}}\) from our training data. If a parametric distribution on the damage is desired, then one can prescribe a Gaussian distribution using the quantities, \(\overline{D_{M}^{(t)}}\), \(D_{L}^{(t)}\) and \(D_{U}^{(t)}\) as follows, \(D_{M}^{(t)}\sim\mathcal{N}\left(\overline{D_{M}^{(t)}},\sigma^{(t)}\right)\). Mean and variance of the distribution being \(\overline{D_{M}^{(t)}}\) and \(\sigma^{(t)}\), where \(\sigma^{(t)}=\max\left(0.5*\left(\overline{D_{M}^{(t)}}-D_{L}^{(t)}\right),0.5* \left(D_{U}^{(t)}-\overline{D_{M}^{(t)}}\right)\right)\). Although we have not made it explicit, most damage occurs within our predicted path to failure. Of course other cracks will have some propagation, but in most cases failure is driven by a dominant fracture pathway.