Data Driven Prognosis of Fracture Dynamics Using Tensor Train and Gaussian Process Regression

Accurate prediction of the mechanical failure of structural components plays an important role in the design of engineering structures. However, the fracture process is challenging to model numerically due to the existence of an elastic singularity at the crack tip. The phase field model (PFM) is one promising approach for modeling brittle fracture using an auxiliary field variable to regularize discontinuities associated with sharp cracks. Unfortunately, it is generally computationally expensive due to the need to solve a fourth-order partial differential equation. In order to reduce the computational burden for parameterized problems, different reduced order modeling approaches such as proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM), have been proposed. However, these frameworks are model-based and intrusive approaches and need in-depth efforts in modifying existing complex simulation codes. In this work, a tensor train (TT) approach is proposed in combination with Gaussian process regression (GPR) for modeling and predicting the dynamics of fracture in composite materials. As opposed to POD and DEIM, the TT-GPR approach is a fully data-driven and non-intrusive approach. In this work, we study the fracture of a brittle elastic 2D rectangular slab with a pre-existing crack under Mode I (crack opening) loading conditions. The high dimensional training data for the TT model was generated using PFM with the finite difference method. The predictions by the TT-GPR model are compared with the results from the finite difference method. The TT-GPR is robust enough to predict the catastrophic evolution of the crack growth trend with an accuracy of up to 95% with computational load reduced by two orders of magnitude.