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Robust virtual element methods for 3D stress-assisted diffusion problems

Andres E. Rubiano

TL;DR

Addresses the numerical analysis of stress-assisted diffusion in 3D by formulating a fully coupled nonlinear system for displacement, pressure, flux, and concentration within a 3D virtual element method framework. It develops a robust well-posedness theory using parameter-weighted norms and perturbed saddle-point analysis together with fixed-point arguments, and introduces 3D VE spaces with projection and stabilization operators. The work proves discrete solvability and convergence under mild data-smallness conditions and validates the approach with numerical tests including lithiation of a perforated particle. The results enable reliable simulations of stress-affected diffusion on general polyhedral meshes with potential impact in battery materials and soft solids.

Abstract

This paper presents an initial exploration of stress-assisted diffusion problems in three dimensions within the framework of the virtual element method (VEM). Hilbert spaces enriched with parameter-weighted norms, the extended Babuška-Brezzi-Braess theory for perturbed saddle-point problems, and Banach fixed-point theory play a crucial role in performing a robust analysis of the fully coupled non-linear system. The proposed virtual element formulations are provided with appropriate projection, interpolation, and stabilisation operators that ensures the well-posedness of the discrete problem. Numerical simulations are conducted to show the accuracy, performance, and applicability of the method.

Robust virtual element methods for 3D stress-assisted diffusion problems

TL;DR

Addresses the numerical analysis of stress-assisted diffusion in 3D by formulating a fully coupled nonlinear system for displacement, pressure, flux, and concentration within a 3D virtual element method framework. It develops a robust well-posedness theory using parameter-weighted norms and perturbed saddle-point analysis together with fixed-point arguments, and introduces 3D VE spaces with projection and stabilization operators. The work proves discrete solvability and convergence under mild data-smallness conditions and validates the approach with numerical tests including lithiation of a perforated particle. The results enable reliable simulations of stress-affected diffusion on general polyhedral meshes with potential impact in battery materials and soft solids.

Abstract

This paper presents an initial exploration of stress-assisted diffusion problems in three dimensions within the framework of the virtual element method (VEM). Hilbert spaces enriched with parameter-weighted norms, the extended Babuška-Brezzi-Braess theory for perturbed saddle-point problems, and Banach fixed-point theory play a crucial role in performing a robust analysis of the fully coupled non-linear system. The proposed virtual element formulations are provided with appropriate projection, interpolation, and stabilisation operators that ensures the well-posedness of the discrete problem. Numerical simulations are conducted to show the accuracy, performance, and applicability of the method.

Paper Structure

This paper contains 20 sections, 4 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scope.
  3. Outline.
  4. Model statement.
  5. The nonlinear terms.
  6. Weak formulation
  7. Unique solvability with parameter-weighted norms.
  8. Virtual element discretisation
  9. Mesh assumptions.
  10. Polynomial spaces.
  11. Discrete formulation for the elasticity problem.
  12. Discrete formulation for the reaction-diffusion problem.
  13. Polynomial projections.
  14. Interpolation operators.
  15. Approximation and interpolation estimates.
  16. ...and 5 more sections

Key Result

Theorem 1

Let $W =\{ w \in Q_2 \colon \lVert w\rVert_{Q_2} \leq C_2 \sqrt{M} \lVert\varphi_D\rVert_{1/2,\Gamma_D} + \lVert g\rVert_{0,\Omega}) \}$. Under the assumptions over the non-linear terms, suppose further that $1\leq \lambda$, $1\leq\mu$, $\theta \leq M^{-1}$, and $C_1 L_\ell \sqrt{2\mu} M^{2}C_2^2 L where the constants $C_1$ and $C_2$ do not depend on the physical parameters.

Figures (2)

  • Figure 4.1: Example 1. An illustration of the distinct meshes used (left), and the converge plot under uniform refinement (right).
  • Figure 4.2: Example 2. Snapshots of the variables of interest in the reference configuration (left column), and deformed configuration (remaining columns) for the unclamped (first row), and clamped cases (second row).

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2