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Higher-order multi-scale computational method and its convergence analysis for hygro-thermo-mechanical coupling problems of quasi-periodic composite structures

Hao Dong, Yifei Ding, Jiale Linghu, Yufeng Nie, Yaochuang Han

TL;DR

The paper addresses HTM coupling in quasi-periodic composites by developing a higher-order multi-scale (HOMS) framework that includes second-order corrections to capture micro-scale oscillations. It builds a two-stage FEM-based algorithm (offline cell problems and online macro solves) together with rigorous pointwise and integral error analyses, showing O(ε) convergence in key norms. Numerical experiments in 2D and 3D demonstrate that HOMS outperforms lower-order and classical homogenization methods in accuracy and can drastically reduce computational cost, especially for large-scale structures. The results establish a solid theoretical and computational foundation for applying HOMS to complex multi-scale HTM problems and highlight its potential for engineering design and reliability assessment.

Abstract

This paper proposes a novel higher-order multi-scale (HOMS) computational method, which is highly targeted for efficient, high-accuracy and low-computational-cost simulation of hygro-thermo-mechanical (H-T-M) coupling problems in quasi-periodic composite structures. The first innovation of this work is that the establishment of the high-accuracy multi-scale model incorporating the higher-order correction terms for H-T-M coupling problems of quasi-periodic composite structures. The second innovation of this work is that the error analyses in the point-wise and integral senses are rigorously derived for multi-scale asymptotic solutions. Especially from the point-wise error analysis, the primary impetus for current study to develop the HOMS approach for quasi-periodic composite structures is illustrated. Furthermore, an high-accuracy multi-scale numerical algorithm is developed based on finite element method, while corresponding convergent analysis is also obtained. Finally, extensive numerical experiments are conducted to validate the computational performance of the proposed HOMS computational approach, demonstrating not only exceptional numerical accuracy, but also reduced computational cost.

Higher-order multi-scale computational method and its convergence analysis for hygro-thermo-mechanical coupling problems of quasi-periodic composite structures

TL;DR

The paper addresses HTM coupling in quasi-periodic composites by developing a higher-order multi-scale (HOMS) framework that includes second-order corrections to capture micro-scale oscillations. It builds a two-stage FEM-based algorithm (offline cell problems and online macro solves) together with rigorous pointwise and integral error analyses, showing O(ε) convergence in key norms. Numerical experiments in 2D and 3D demonstrate that HOMS outperforms lower-order and classical homogenization methods in accuracy and can drastically reduce computational cost, especially for large-scale structures. The results establish a solid theoretical and computational foundation for applying HOMS to complex multi-scale HTM problems and highlight its potential for engineering design and reliability assessment.

Abstract

This paper proposes a novel higher-order multi-scale (HOMS) computational method, which is highly targeted for efficient, high-accuracy and low-computational-cost simulation of hygro-thermo-mechanical (H-T-M) coupling problems in quasi-periodic composite structures. The first innovation of this work is that the establishment of the high-accuracy multi-scale model incorporating the higher-order correction terms for H-T-M coupling problems of quasi-periodic composite structures. The second innovation of this work is that the error analyses in the point-wise and integral senses are rigorously derived for multi-scale asymptotic solutions. Especially from the point-wise error analysis, the primary impetus for current study to develop the HOMS approach for quasi-periodic composite structures is illustrated. Furthermore, an high-accuracy multi-scale numerical algorithm is developed based on finite element method, while corresponding convergent analysis is also obtained. Finally, extensive numerical experiments are conducted to validate the computational performance of the proposed HOMS computational approach, demonstrating not only exceptional numerical accuracy, but also reduced computational cost.

Paper Structure

This paper contains 16 sections, 6 theorems, 123 equations, 24 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Novel higher-order multi-scale computational model
  3. Problem setting and governing equations
  4. HOMS modeling for quasi-periodic composite structures
  5. The error analyses of multi-scale asymptotic solutions
  6. The error analysis in the point-wise sense
  7. The error analysis in the integral sense
  8. Multi-scale finite element algorithm
  9. Off-line microscale computation
  10. On-line macroscale and multi-scale computation
  11. Error estimation of multi-scale finite element algorithm
  12. Numerical examples and results
  13. Example 1: 2D quasi-periodic composite structure
  14. Example 2: 3D quasi-periodic composite structure
  15. Example 3: 3D quasi-periodic composite plate
  16. ...and 1 more sections

Key Result

Lemma 3.1

Defining three differential operators $\sigma_{TY}(\chi)=n_i k_{ij}(\mathbf{x},\mathbf{y})\frac{\partial \chi}{\partial y_j}$, $\sigma_{cY}(\chi)=n_i g_{ij}(\mathbf{x},\mathbf{y})\frac{\partial \chi}{\partial y_j}$ and $\sigma_{iY}(\bm{\phi})=n_jD_{ijkl}(\mathbf{x},\mathbf{y})\frac{\partial \phi_{k}

Figures (24)

  • Figure 1: The schematic of composite structure: (a) composite structure $\Omega$; (b) PUC $Y$.
  • Figure 2: Temperature increment field with scale separation: (a) $T^{(0)}$; (b) $T^{(1,\epsilon)}$; (c) $T^{(2,\epsilon)}$; (d) $T^{\epsilon}$.
  • Figure 3: Moisture field with scale separation: (a) $c^{(0)}$; (b) $c^{(1,\epsilon)}$; (c) $c^{(2,\epsilon)}$; (d) $c^{\epsilon}$.
  • Figure 4: First displacement field component with scale separation: (a) $u_1^{(0)}$; (b) $u_1^{(1,\epsilon)}$; (c) $u_1^{(2,\epsilon)}$; (d) $u_1^{\epsilon}$.
  • Figure 5: Temperature increment field under scale coupling: (a) $T^{(0)}$; (b) $T^{(1,\epsilon)}$; (c) $T^{(2,\epsilon)}$; (d) $T^{\epsilon}$.
  • ...and 19 more figures

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 3.1
  • Theorem 1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 2