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Higher-order multiscale method and its convergence analysis for nonlinear thermo-electric coupling problems of composite structures

Hao Dong, Zongze Yang, Yufeng Nie

Abstract

This paper proposes a higher-order multiscale computational method for nonlinear thermo-electric coupling problems of composite structures, which possess temperature-dependent material properties and nonlinear Joule heating. The innovative contributions of this work are the novel multiscale formulation with the higher-order correction terms for periodic composite structures and the global error estimation with an explicit rate for higher-order multiscale solutions. By employing the multiscale asymptotic approach and the Taylor series technique, the higher-order multiscale method is established for time-dependent nonlinear thermo-electric coupling problems, which can keep the local balance of heat flux and electric charge for high-accuracy multiscale simulation. Furthermore, an efficient numerical algorithm with off-line and on-line stages is presented in detail, and corresponding convergent analysis is also obtained. Two- and three-dimensional numerical experiments are conducted to showcase the competitive advantages of the proposed method for simulating the time-dependent nonlinear thermo-electric coupling problems in composite structures, not only exceptional numerical accuracy, but also less computational cost.

Higher-order multiscale method and its convergence analysis for nonlinear thermo-electric coupling problems of composite structures

Abstract

This paper proposes a higher-order multiscale computational method for nonlinear thermo-electric coupling problems of composite structures, which possess temperature-dependent material properties and nonlinear Joule heating. The innovative contributions of this work are the novel multiscale formulation with the higher-order correction terms for periodic composite structures and the global error estimation with an explicit rate for higher-order multiscale solutions. By employing the multiscale asymptotic approach and the Taylor series technique, the higher-order multiscale method is established for time-dependent nonlinear thermo-electric coupling problems, which can keep the local balance of heat flux and electric charge for high-accuracy multiscale simulation. Furthermore, an efficient numerical algorithm with off-line and on-line stages is presented in detail, and corresponding convergent analysis is also obtained. Two- and three-dimensional numerical experiments are conducted to showcase the competitive advantages of the proposed method for simulating the time-dependent nonlinear thermo-electric coupling problems in composite structures, not only exceptional numerical accuracy, but also less computational cost.
Paper Structure (16 sections, 8 theorems, 95 equations, 10 figures, 3 tables)

This paper contains 16 sections, 8 theorems, 95 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Higher-order multiscale computational model of nonlinear thermo-electric coupling problems
  3. The statement of multiscale nonlinear coupling system
  4. Higher-order multiscale analysis for nonlinear coupling problem
  5. The error analyses of multiscale asymptotic solutions
  6. The proof of local balance of heat flux and electric charge by local error analysis
  7. The convergence proof by global error estimation
  8. Two-stage numerical algorithm
  9. Off-line stage: computation for microscopic cell problems
  10. On-line stage: computation for macroscopic homogenized problem
  11. On-line stage: computation for higher-order multiscale solutions
  12. Error estimation for two-stage multiscale numerical algorithm
  13. Numerical examples and results
  14. Validation of computational accuracy and efficiency
  15. Application to 3D composite structure
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let ${u}^\varepsilon(\bm{x},t)$ and $\phi^\varepsilon(\bm{x},t)$ be the weak solutions of multiscale nonlinear equations (1.1), ${u}^{(0)}(\bm{x},t)$ and $\phi^{(0)}(\bm{x},t)$ be the weak solutions of corresponding homogenized equations (2.13), ${u^{(2\varepsilon)} }(\bm{x},t)$ and ${\phi^{(2\varep

Figures (10)

  • Figure 1: (a) Integral periodic domain $\Omega_0$; (b) non-integral periodic domain $\Omega$ with boundary layer $\Omega _1$ and ${\Omega_0}=\cup_{\mathbf{z}\in I_{\varepsilon}}\varepsilon(\mathbf{z}+\bar{\Theta})$, namely $\bar{\Omega}= {\bar{\Omega} _0} \cup {\bar{\Omega} _1}$.
  • Figure 1: The schematic diagram of two-stage multiscale numerical algorithm.
  • Figure 1: (a) The 2D composite structure $\Omega$; (b) PUC $\Theta$; (c) homogenized structure $\Omega$.
  • Figure 2: The temperature field at $t=1.0$: (a) $u_{0}$; (b) $u^{(1\varepsilon)}$; (c) $u^{(2\varepsilon)}$; (d) $u_{\rm{DNS}}^\varepsilon$.
  • Figure 3: The electric potential field at $t=1.0$: (a) $\phi_{0}$; (b) $\phi^{(1\varepsilon)}$; (c) $\phi^{(2\varepsilon)}$; (d) $\phi_{{\rm{DNS}}}^\varepsilon$.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Remark 5
  • Lemma 5.1
  • Lemma 5.2
  • ...and 3 more