A higher-order three-scale computational method for efficient nonlinear thermo-mechanical coupling simulation of heterogeneous structures with multiple spatial scales

TL;DR

The paper develops a higher-order three-scale (HOTS) framework to efficiently simulate time-dependent nonlinear thermo-mechanical coupling in heterogeneous structures with macro-, meso-, and micro-scale hierarchies, incorporating temperature-dependent material properties. It constructs a macro-meso-micro correlative model via recursive two-scale analyses, derives higher-order correction terms, and provides a pointwise local error/well-balanced analysis. A two-stage offline-online algorithm is proposed, enabling offline computation of microscopic/mesoscopic cell functions and online evaluation of macroscopic homogenized problems with HOTS corrections, yielding substantial gains in accuracy and computational efficiency. Numerical experiments in 2D and 3D demonstrate that HOTS accurately captures microscopic oscillations, scales robustly with problem size, and significantly reduces memory and time requirements compared to full FEM, making it practical for large-scale nonlinear multiphysics simulations. The approach holds promise for real-world heterogeneous materials and future extensions to random media and radiation/convection effects with parallelization.

Abstract

Classical multi-scale methods involving two spatial scales face significant challenges when simulating heterogeneous structures with complicated three-scale spatial configurations. This study proposes an innovative higher-order three-scale (HOTS) computational method, aimed at accurately and efficiently computing the transient nonlinear thermo-mechanical coupling problems of heterogeneous structures with multiple spatial scales. In these heterogeneous structures, temperature-dependent material properties have an important impact on the thermo-mechanical coupling responses, which is the particular interest in this work. At first, the detailed macro-meso-micro correlative model with higher-order correction terms is established by recursively two-scale analysis between macro-meso and meso-micro scales, which enables high-accuracy analysis of temperature-dependent nonlinear thermo-mechanical behaviors of heterogeneous structures with complicated three-scale configurations. The local error analysis mathematically illustrates the well-balanced property of HOTS computational model, endowing it with high computational accuracy. In addition, a two-stage numerical algorithm with off-line and on-line stages is proposed in order to efficiently simulate the nonlinear thermo-mechanical responses of heterogeneous structures with three-level spatial scales and accurately capture their highly oscillatory information at micro-scale. Finally, the high computational efficiency, high numerical accuracy and low computational cost of the presented higher-order three-scale computational approach are substantiated via representative numerical experiments. It can be summarized that this scalable and robust HOTS computational approach offers a reliably numerical tool for nonlinear multiphysics simulation of large-scale heterogeneous structures in real-world applications.

Figures (24)

• Figure 1: The illustration of inhomogeneous structures: (a) braided structure with two-scale spatial hierarchy; (b) superconducting bulk with three-scale spatial hierarchy R11.
• Figure 2: The illustration of macro-meso two-scale modeling.
• Figure 3: The illustration of meso-micro two-scale modeling.
• Figure 4: The illustration of macro-meso-micro three-scale modeling.
• Figure 5: (a) Macroscopic domain $\Omega$ ; (b) Mesoscopic unit cell $Y$ (left side) and the microscopic unit cell $Z$ (right side).

Theorems & Definitions (2)

• Remark 1
• Remark 2