  Real-world physical systems, like composite materials and porous media,
exhibit complex heterogeneities and multiscale nature, posing significant
computational challenges. Computational homogenization is useful for predicting
macroscopic properties from the microscopic material constitution. It involves
defining a representative volume element (RVE), solving governing equations,
and evaluating its properties such as conductivity and elasticity. Despite its
effectiveness, the approach can be computationally expensive. This study
proposes a tensor-train (TT)-based asymptotic homogenization method to address
these challenges. By deriving boundary value problems at the microscale and
expressing them in the TT format, the proposed method estimates material
properties efficiently. We demonstrate its validity and effectiveness through
numerical experiments applying the proposed method for homogenization of
thermal conductivity and elasticity in two- and three-dimensional materials,
offering a promising solution for handling the multiscale nature of
heterogeneous systems.
