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A $C^0$-continuous nonconforming virtual element method for linear strain gradient elasticity

Jianguo Huang, Yue Yu

TL;DR

This work develops a $C^0$-continuous nonconforming virtual element method for a two-dimensional strain gradient elasticity problem, enabling stable computation on general polygonal meshes under a broad geometric assumption. The method uses Green's formulas to define elliptic projections, constructs a nonconforming local space with a lifting space for computability, and assembles a stabilized, $k$-consistent bilinear form to discretize the SG elasticity variational problem. A thorough error analysis, including a Strang-type lemma and Korn-type stability results, yields robustness with respect to the microscopic parameter $\iota$ and the Lamé constant $\lambda$, and, under regularity assumptions, a uniform error bound in the lowest-order case. Numerical experiments corroborate the theoretical findings, showing linear convergence for large $\iota$, quadratic convergence as $\iota\to0$, and locking-free behavior with respect to $\lambda$, confirming the practical effectiveness of the proposed VEM on polygonal meshes for gradient-elastic problems.

Abstract

A robust $C^0$-continuous nonconforming virtual element method (VEM) is developed for a boundary value problem arising from strain gradient elasticity in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). The stability condition of the VEMs is derived by establishing Korn-type inequalities and inverse inequalities. Some crucial commutative relations for locking-free analysis as in elastic problems are derived. The sharp and uniform error estimates with respect to both the microscopic parameter and the Lamé coefficient are achieved in the lowest-order case, which is also verified by numerical results.

A $C^0$-continuous nonconforming virtual element method for linear strain gradient elasticity

TL;DR

This work develops a -continuous nonconforming virtual element method for a two-dimensional strain gradient elasticity problem, enabling stable computation on general polygonal meshes under a broad geometric assumption. The method uses Green's formulas to define elliptic projections, constructs a nonconforming local space with a lifting space for computability, and assembles a stabilized, -consistent bilinear form to discretize the SG elasticity variational problem. A thorough error analysis, including a Strang-type lemma and Korn-type stability results, yields robustness with respect to the microscopic parameter and the Lamé constant , and, under regularity assumptions, a uniform error bound in the lowest-order case. Numerical experiments corroborate the theoretical findings, showing linear convergence for large , quadratic convergence as , and locking-free behavior with respect to , confirming the practical effectiveness of the proposed VEM on polygonal meshes for gradient-elastic problems.

Abstract

A robust -continuous nonconforming virtual element method (VEM) is developed for a boundary value problem arising from strain gradient elasticity in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). The stability condition of the VEMs is derived by establishing Korn-type inequalities and inverse inequalities. Some crucial commutative relations for locking-free analysis as in elastic problems are derived. The sharp and uniform error estimates with respect to both the microscopic parameter and the Lamé coefficient are achieved in the lowest-order case, which is also verified by numerical results.

Paper Structure

This paper contains 15 sections, 23 theorems, 135 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Strain gradient elastic model
  3. A $C^0$-continuous virtual element method
  4. Green's formulas and the induced projectors
  5. The nonconforming virtual element space
  6. Construction of the discrete problem
  7. Some estimates of a $C^0$-continuous and $H^2$-nonconforming virtual element
  8. Korn's inequality and norm equivalence
  9. Korn's inequality
  10. Norm equivalence
  11. Error analysis
  12. An abstract Strang-type lemma
  13. Error estimates
  14. Uniform error estimate in the lowest order case
  15. Numerical examples

Key Result

Lemma 3.1

For all $\boldsymbol u \in \boldsymbol H^2(K)$ and for all $\boldsymbol p \in (\mathbb{P}_k(K))^2$, there holds the following Green's formula where $[ M_{\boldsymbol{tn}}^k(\boldsymbol p) ](z_i) = M_{\boldsymbol{tn}}^k(\boldsymbol p) |_{z_i^ -}^{z_i^ +}$ is the jump at the vertex $z_i$ along the boundary of $K$, and

Figures (1)

  • Figure 1: The error orders for Example \ref{['divfree']}. Left: Polygonal meshes; Right: Triangular meshes.

Theorems & Definitions (44)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.1
  • ...and 34 more