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Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

Dmitry Ammosov, Tina Mai, Juan Galvis

Abstract

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.

Generalized multiscale finite element method for a nonlinear elastic strain-limiting Cosserat model

Abstract

For nonlinear Cosserat elasticity, we consider multiscale methods in this paper. In particular, we explore the generalized multiscale finite element method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting property (ensuring bounded linearized strains even under high stresses). Such strain-limiting Cosserat model can find potential applications in solids and biological fibers. However, Cosserat media with naturally rotational degrees of freedom, nonlinear constitutive relations, high contrast, and heterogeneities may produce challenging multiscale characteristics in the solution, and upscaling by multiscale methods is necessary. Therefore, we utilize the offline and residual-based online (adaptive or uniform) GMsFEM in this context while handling the nonlinearity by Picard iteration. Through various two-dimensional experiments (for perforated, composite, and stochastically heterogeneous media with small and big strain-limiting parameters), our numerical results show the approaches' convergence, efficiency, and robustness. In addition, these results demonstrate that such approaches provide good accuracy, the online GMsFEM gives more accurate solutions than the offline one, and the online adaptive strategy has similar accuracy to the uniform one but with fewer degrees of freedom.
Paper Structure (27 sections, 1 theorem, 93 equations, 8 figures, 18 tables)

This paper contains 27 sections, 1 theorem, 93 equations, 8 figures, 18 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A nonlinear strain-limiting Cosserat model
  4. General elastic Cosserat model
  5. Elastic Cosserat planar model
  6. Nonlinear planar strain-limiting Cosserat elasticity energy
  7. Monotonicity
  8. Field equations for plane nonlinear strain-limiting Cosserat elasticity
  9. Variational problem of plane nonlinear strain-limiting Cosserat elasticity
  10. Fine-grid discretization and Picard iteration for linearization
  11. Coarse-grid approximation using the GMsFEM
  12. Overview
  13. Offline GMsFEM
  14. Snapshot spaces
  15. Multiscale space
  16. ...and 12 more sections

Key Result

Proposition 3.1

For any $\boldsymbol{\chi} \in \mathbb{R}^2$ and $\boldsymbol{\gamma} \in \mathbb{R}^{2 \times 2}$ in the form of vectorization $\boldsymbol{\gamma} \in \mathbb{R}^{4}$ such that $Q(\boldsymbol{\chi}, \boldsymbol{\gamma}) <1\,,$ it holds that the Hessian of the strain-energy density is positive semi-definite.

Figures (8)

  • Figure 1: Coarse grid $\mathcal{T}^H$, coarse block $K_j^H$, and local domain (coarse neighborhood) $\omega_i\,.$
  • Figure 2: Computational grids for the perforated media: left -- the coarse grid (each small square is a coarse block), middle -- the fine grid, right -- a local domain.
  • Figure 3: Distributions of the microrotations $\Phi$ and displacements $u_1$ and $u_2$ (from left to right) for Case 1 (small strain-limiting parameter) of the perforated media.
  • Figure 4: Computational grids for the composite media: left -- the coarse grid (each small square is a coarse block), middle -- the fine grid ($\Omega_1$ is blue, $\Omega_2$ is red), right -- a local domain.
  • Figure 5: Distributions of the microrotations $\Phi$ and displacements $u_1$ and $u_2$ (from left to right) for Case 1 (small strain-limiting parameter) of the composite media.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Proposition 3.1
  • proof
  • Remark 6.1
  • Remark 7.1
  • Remark 7.2