Tensor Train Decomposition for Data-Driven Prognosis of Fracture Dynamics in Composite Materials

It is important to be able to accurately predict the evolution of damage in structural components to evaluate the mechanical reliability of engineering structures. This requires modeling complex mechanisms in damage including crack nucleation and propagation. These pose significant computational challenges to simulation, specifically the singular crack tip field as well as the moving boundary problem inherent in crack propagation. In order to address these problems, many different approaches in computational mechanics have been developed including the cohesive zone method, the extended finite element method and the phase-field method, although all these methods are still relatively expensive in computational effort. In order to reduce the computational burden, reduced order models based on the proper orthogonal decomposition (POD) approach can be used to exploit the spatial correlation to get a set of modes characterizing the spatial structure of the model. For the multidimensional problem, there is a need for vectorization of the solution for derivation of the POD modes. This leads to difficulty in explanation of the model. Tensor train (TT) or matrix product states is a better representation of the multidimensional solution using the product of three-dimensional tensors. In this work, the TT methodology is proposed for modeling and predicting the dynamics of fracture in composite materials. We consider a rectangular slab with a pre-existing line crack subject to Mode-I loading condition. Uniaxial strains are applied to the top and bottom edges of the slab. The phase-field method (PFM) with finite-difference (FD) is used for generating the high dimensional data for training the TT method. The predictions using the TT method are then compared with the results from the finite difference method with phase-field to verify the correctness of the TT. Our results show that the TT can predict the crack growth trends based on the finite difference method with an accuracy of 95-98% while reducing the computational load by up to 2-5 orders of magnitude.