Solving phase-field models in the tensor train format to generate microstructures of bicontinuous composites

Abstract

Phase-field models permit accounting for the underlying physics of the microstructure evolution process when simulating the emergent microstructure of a variety of multiscale materials. As an inherent characteristic, the individual phases (or states) of the material are not known in advance and only emerge upon evolution. To account for this matter, structured grids are typically used when solving phase-field models. In particular for large three-dimensional grids and long time scales, the used memory and computational time can be considerable, instigating research on alternative approaches.The work at hand studies the Tensor Train (TT) format as a possible remedy for phase-field simulations. More precisely, the TT format is a specific tensor format which permits – small so-called TT rank provided – to store huge arrays of data and comes with specific algorithms replacing the linear algebra operations familiar for fully stored arrays.We investigate the Cahn-Hilliard equation and use a semi-implicit discretization in time which is based on solving a single linear system per time step. We compare this strategy with a classic implicit Euler discretization using a Newton scheme as a nonlinear solver. Upon a finite-difference discretization in space, the resulting equations are recast in the Quantics Tensor Train (QTT) format and solved by dedicated linear solvers. We study the performance of the algorithms and investigate the effective properties of the resulting bicontinuous composites via computational homogenization. We show that the Cahn-Hilliard equation can be solved using the TT format and that runtimes scale well with the grid size.