In this paper, we propose a time-marching multi-level Variational Multiscale-Tensor Decomposition (VMS-TD) algorithm to solve the heat equation with a moving heat source model that arises from additive manufacturing. First, we take a second-order centered difference for time semi-discretization. The temperature field is resolved by multiple levels of spatial grids. Then we adopt the VMS formulation [19] for the resulting elliptic problem to obtain a Galerkin weak form and design a VMS-TD algorithm to effectively solve it. Furthermore, to comply with the TD solution scheme, special inter-scale data transfers are made at the scale interface and in the moving fine-scale subdomains to bypass the tensor decomposition deficiency. Numerical results demonstrate that the multi-level VMS-TD algorithm is much more efficient than the fully resolved TD algorithm, let alone traditional direct numerical simulation methods such as finite difference or finite element analysis. Compared with the well-known multi-grid methods or more recent GO-MELT framework [24], the three-level VMS-TD algorithm uses much smaller degrees of freedom to obtain accurate results. A multi-time-scale extension of VMS-TD algorithm is also proposed.
