Forecasting phase-field variable in brittle fracture problems by autoregressive integrated moving average technique

Phase-field modeling is a powerful and versatile computational approach for modeling the evolution of cracks in solids. However, phase-field modeling requires high computational cost for accurately capturing how cracks develop under increasing loads. In brittle fracture mechanics, crack initiation and propagation can be considered as a time series forecasting problem so they can be studied by observing changes in the phase-field variable, which represents the level of material damage. In this paper, we develop a rather simple approach utilizing the autoregressive integrated moving average (ARIMA) technique to predict variations of the phase-field variable in an isothermal, linear elastic and isotropic phase-field model for brittle materials. Time series data of the phase-field variable is extracted from numerical results using coarse finite-element meshes. Two ARIMA schemes are introduced to exploit the structure of the collected data and provide a prediction for changes in phasefield variable when using a finer mesh. This finer mesh gives a better results in terms of accuracy but requires significantly higher computational cost.