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Robust virtual element methods for coupled stress-assisted diffusion problems

Rekha Khot, Andres E. Rubiano, Ricardo Ruiz-Baier

TL;DR

This work develops a robust framework for the coupled stress-assisted diffusion problem, integrating a Herrmann-pressure formulation of linear elasticity with a nonlinear, stress-dependent diffusion model. The authors establish continuous well-posedness via an extended Banach fixed-point argument on perturbed saddle-point problems and then design a comprehensive virtual element discretisation (VEM) for the two subproblems, ensuring divergence-conforming discretisation for the elasticity component. A rigorous a priori error analysis demonstrates convergence on general polygonal meshes with robustness with respect to material parameters, and numerical experiments on diverse meshes validate the theoretical results and illustrate multiphysics interactions such as lithiation. The combination of parameter-robust theory, VE discretisations, and mixed formulations provides a stable, flexible framework for simulating coupled diffusion-elasticity phenomena in heterogeneous materials.

Abstract

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babuška--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.

Robust virtual element methods for coupled stress-assisted diffusion problems

TL;DR

This work develops a robust framework for the coupled stress-assisted diffusion problem, integrating a Herrmann-pressure formulation of linear elasticity with a nonlinear, stress-dependent diffusion model. The authors establish continuous well-posedness via an extended Banach fixed-point argument on perturbed saddle-point problems and then design a comprehensive virtual element discretisation (VEM) for the two subproblems, ensuring divergence-conforming discretisation for the elasticity component. A rigorous a priori error analysis demonstrates convergence on general polygonal meshes with robustness with respect to material parameters, and numerical experiments on diverse meshes validate the theoretical results and illustrate multiphysics interactions such as lithiation. The combination of parameter-robust theory, VE discretisations, and mixed formulations provides a stable, flexible framework for simulating coupled diffusion-elasticity phenomena in heterogeneous materials.

Abstract

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babuška--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.
Paper Structure (31 sections, 20 theorems, 145 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 31 sections, 20 theorems, 145 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. Recurrent notation and Sobolev spaces
  4. Governing equations
  5. Model statement
  6. Weak formulation
  7. Two uncoupled perturbed saddle-point problems
  8. Well-posedness of the continuous problem
  9. Unique solvability of the decoupled sub-problems
  10. Fixed-point strategy
  11. Virtual element discretisation
  12. Mesh assumptions and recurrent notation
  13. Discrete formulation for the elasticity problem
  14. VE spaces and DoFs
  15. Projection operators
  16. ...and 16 more sections

Key Result

Theorem 1

\newlabelth:unique-solvability0 Let $V,Q_b$ be Hilbert spaces endowed with the (possibly parameter-dependent) norms $\lVert\cdot\rVert_{V}$ and $\lVert\cdot\rVert_{Q_b}$, let $Q$ be a dense (with respect to the norm $\lVert\cdot\rVert_{Q_b}$) linear subspace of $Q_b$ and three bilinear forms $a(\c Assume that $Q$ is complete with respect to the norm $\lVert\cdot\rVert_{Q}^2:=\lVert\cdot\rVert_{Q_

Figures (7)

  • Figure 1: An illustration of six distinct coarse meshes used in the numerical tests.
  • Figure 2: Illustration of the DoFs on a square element with $k_1=2$ and $k_2=1$.
  • Figure 3: Example 1. Snapshot of the approximated solution with the displacement magnitude $\boldsymbol{u}_h$, flux magnitude $|\boldsymbol{\zeta}_h|$, Herrmann pressure $\tilde{p}_h$, and concentration $\varphi_h$, shown over an in-house kangaroo mesh for $k_1=2$ and $k_2=0,1$.
  • Figure 4: Example 2. Error history of the total error $\overline{\text{e}}_*$ for $k_1=2$ and $k_2=0,1$ is illustrated by examining the impact of varying the physical parameters $\lambda$, $\mu$, and $\theta$ (first, second, and third row) for a variety of meshes.
  • Figure 5: Example 2. Error history for the partial error with $k_1=2$ and $k_2=0,1$ for the non-convex mesh is illustrated by examining the impact of varying the physical parameters $\lambda$, $\mu$, and $\theta$ (first, second, and third row).
  • ...and 2 more figures

Theorems & Definitions (42)

  • Theorem 1
  • Remark 3.1
  • Lemma 2
  • Proof 1
  • Lemma 3
  • Proof 2
  • Remark 3.2
  • Lemma 4
  • Proof 3
  • Lemma 5
  • ...and 32 more