Time-marching multi-level variational multiscale tensor decomposition algorithm for heat conduction with moving heat source

TL;DR

The paper addresses accurate and efficient simulation of heat conduction with a moving heat source in additive manufacturing. It develops a time-marching, multi-level variational multiscale tensor decomposition (VMS-TD) framework that resolves the near-field fine-scale region around the moving source while coarsening the far field, and imposes seamless inter-scale coupling via specialized interface and moving-subdomain treatments. The approach includes two-level and three-level extensions with TD representations for both coarse and fine scales, and demonstrates substantial reductions in degrees of freedom and computational effort while preserving accuracy relative to a full fine-mesh TD reference in 2D and 3D tests. The results indicate strong potential for large-scale, real-time or near-real-time thermal simulations in LPBF and related additive manufacturing processes, with avenues for further enhancement through space-time TD formulations and multiple-time-step schemes.

Abstract

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.

Figures (17)

• Figure 1: Schematical plot of a VMS in 2D.
• Figure 2: Interface treatment. upper-left: coarse grid on the whole domain; upper-right: fine grid; upper-middle: the interface; lower-middle: three types of grid points at the interface.
• Figure 3: Moving fine-scale subdomain. blue: $U^{n-1}$; dark-blue: reserved data; light-blue: discarded data; yellow: re-built data in $\Theta^n$; red- and black-circled: occupy rectangular regions allowing variable-separated form.
• Figure 4: Schematical plot of a three-level mesh in 2D.
• Figure 5: Two dimensional results $u(x,y,t)$ by two-level VMS-TD algorithm with $N_1 = N_2 = 64$, $Q=2$ at $t = 0.0625, 0.375, 1$.