Stochastic perturbation of a 2D brittle fracture model

We use a phase field method to study fracture problems. Within this framework, the linear momentum equations are coupled with a diffusion type equation, which describes the evolution of the fracture variable. The coupled nonlinear system of partial differential equations are solved in a ‘staggered’ approach. Following previous studies, we use the open-source finite element software FEniCS to implement the phase field method for brittle fracture model. The model depends on a constant k related to the volume fraction of the material. This constant is now a random variable and we require to establish how the uncertainty in k is propagated into the brittle properties of the material. A direct simulation-solve approach is unfeasible given the high computational burden of the numeric solver. Instead, we evaluate the model in some values for k and in turn build a surrogate model for the subsequent Monte Carlo evaluations. Common out-of-the-box surrogates (eg. Gaussian processes, neural networks etc.) have the problem of not following the physical characteristics of the problem and deliver clearly unrealistic interpolators. We experiment with a purpose made surrogate model, that tries to maintain some features of the underlying physical model, specifically the fracture fase change. We present some preliminary results.