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 decomposed according to multiple space resolution levels, each represented by the TD method. Then we adopt the VMS formulation [T.J.R. Hughes, G.R. Feijoo, L. Mazzei, J.B. Quincy. Comput. Methods Appl. Mech. Engrg. 166:3-24 (1998)] for the resulting elliptic problem to obtain a Galerkin weak form, and design VMS-TD algorithm to solve it. Furthermore, to comply with the TD solution scheme, special inter-scale data transfers are made at the scale interface and moving fine-scale subdomains. 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 [J.P. Leonor, G.J. Wagner. Comput. Methods Appl. Mech. Engrg, 426:116977 (2024)], the three-level VMS-TD uses much smaller degrees of freedom to reach accurate results. A multi-time-scale extension of VMS-TD algorithm is also proposed.

翻译：本文提出了一种时间推进多层级变分多尺度-张量分解算法，用于求解增材制造中出现的移动热源模型热传导方程。首先，对时间变量采用二阶中心差分进行半离散化处理。温度场依据多个空间分辨率层级进行分解，每个层级均采用张量分解方法表示。随后，针对所得椭圆型问题，采用变分多尺度构造方法 [T.J.R. Hughes, G.R. Feijoo, L. Mazzei, J.B. Quincy. Comput. Methods Appl. Mech. Engrg. 166:3-24 (1998)] 获得其 Galerkin 弱形式，并设计变分多尺度-张量分解算法进行求解。此外，为匹配张量分解求解格式，在尺度界面及移动的细尺度子域上实施了特殊的尺度间数据传递。数值结果表明，多层级变分多尺度-张量分解算法相较于全分辨率张量分解算法效率显著提升，更优于有限差分或有限元分析等传统直接数值模拟方法。与经典多重网格方法或近期提出的 GO-MELT 框架 [J.P. Leonor, G.J. Wagner. Comput. Methods Appl. Mech. Engrg, 426:116977 (2024)] 相比，三层级变分多尺度-张量分解算法能以更少的自由度获得精确结果。本文还进一步提出了该算法的多时间尺度扩展形式。