MultiLevel Variational MultiScale (ML-VMS) framework for large-scale simulation

TL;DR

This work introduces ML-VMS, a multilevel variational multiscale framework that couples a global coarse mesh with localized fine patches using the C-HiDeNN interpolation, enabling high-order accuracy on linear meshes without increasing degrees of freedom. It provides rigorous elliptic and space-time error analyses, and extends to space-time reduced-order modeling via C-HiDeNN-TD, delivering substantial speedups on large-scale LPBF-like problems. Numerical experiments demonstrate that ML-VMS with C-HiDeNN and TD outperforms conventional FEM baselines in accuracy and efficiency, including 3D LPBF-like heat-transfer simulations with orders-of-magnitude reductions in DoFs. A space-time formulation with TD enables large time steps and simultaneous resolution of all time steps, while a data-driven error model and potential LLM-assisted tooling offer practical pathways for hyperparameter optimization and automation. The framework thus provides a scalable, high-fidelity surrogate for multiscale simulations in additive manufacturing and related engineering domains, with clear avenues for multiphysics, parametric, and digital-twin extensions.

Abstract

In this paper, we propose the MultiLevel Variational MultiScale (ML-VMS) method, a novel approach that seamlessly integrates a multilevel mesh strategy into the Variational Multiscale (VMS) framework. A key feature of the ML-VMS method is the use of the Convolutional Hierarchical Deep Neural Network (C-HiDeNN) as the approximation basis. The framework employs a coarse mesh throughout the domain, with localized fine meshes placed only in subdomains of high interest, such as those surrounding a source. Solutions at different resolutions are robustly coupled through the variational weak form and interface conditions. Compared to existing multilevel methods, ML-VMS (1) can couple an arbitrary number of mesh levels across different scales using variational multiscale framework; (2) allows approximating functions with arbitrary orders with linear finite element mesh due to the C-HiDeNN basis; (3) is supported by a rigorous theoretical error analysis; (4) features several tunable hyperparameters (e.g., order $p$, patch size $s$) with a systematic guide for their selection. We first show the theoretical error estimates of ML-VMS. Then through numerical examples, we demonstrate that ML-VMS with the C-HiDeNN takes less computational time than the FEM basis given comparable accuracy. Furthermore, we incorporate a space-time reduced-order model (ROM) based on C-HiDeNN-Tensor Decomposition (TD) into the ML-VMS framework. For a large-scale single-track laser powder bed fusion (LPBF) transient heat transfer problem that is equivalent to a full-order finite element model with $10^{10}$ spatial degrees of freedom (DoFs), our 3-level ML-VMS C-HiDeNN-TD achieves an approximately 5,000x speedup on a single CPU over a single-level linear FEM-TD ROM.

Figures (25)

• Figure 1: MultiLevel Variational MultiScale (ML-VMS) method: (a) capturing disparate length scales in LPBF poses a significant challenge to current available methods; (b) ML-VMS provides an integrated framework for modeling a general $m$ level system; (c) ML-VMS with C-HiDeNN-TD achieves significant computational acceleration as a reduced-order model; (d) The proposed ML-VMS C-HiDeNN-TD achieves superior performance in all 5 performance metrics.
• Figure 2: Neural network structure of Convolution Hierarchical Deep-learning Neural Network (C-HiDeNN) for solving one-dimensional (1D) Poisson’s equation. This figure is borrowed from zhang2025multi.
• Figure 3: The schematic of the two-level VMS method.
• Figure 4: The schematic of the $m$-level VMS method.
• Figure 5: The schematic of the two-level mesh and the exact solution.

Theorems & Definitions (14)

• Theorem 1
• Corollary 1
• Corollary 2
• Corollary 3
• Theorem 2
• Remark 1
• Theorem 3
• Theorem 4
• proof
• Corollary 4