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A low-order locking-free multiscale finite element method for isotropic elasticity

Antônio Tadeu Azevedo Gomes, Weslley da Silva Pereira, Frédéric Valentin

TL;DR

This work develops a low-order, locking-free multiscale hybrid-mixed (MHM) finite element framework for isotropic elasticity with heterogeneous coefficients. By embedding face-based global degrees of freedom and local Neumann problem-driven multiscale bases into a Galerkin Least Squares (GaLS) stabilized scheme, the method achieves well-posedness and optimal convergence while remaining robust as the Poisson ratio approaches the nearly incompressible limit $\nu\to\tfrac12$. Theoretical results include stability, a priori error estimates, and locking-free behavior, complemented by two-dimensional numerical tests that verify improved stress accuracy and overall robustness relative to standard low-order Galerkin and non-stabilized MHM variants. The approach enables efficient, parallelizable multiscale elasticity computations on polytopal meshes with heterogeneous materials, offering a practical tool for nearly incompressible and high-contrast settings. Overall, the paper advances stabilized, low-order MHM techniques as effective, locking-free solvers for complex elasticity problems.

Abstract

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a family of low-order finite elements for the linear elasticity problem which are free from Poisson locking. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient refinement levels on the fine-scale mesh such that the MHM method is well-posed, optimally convergent under local regularity conditions, and locking-free. Two-dimensional numerical tests assess theoretical results.

A low-order locking-free multiscale finite element method for isotropic elasticity

TL;DR

This work develops a low-order, locking-free multiscale hybrid-mixed (MHM) finite element framework for isotropic elasticity with heterogeneous coefficients. By embedding face-based global degrees of freedom and local Neumann problem-driven multiscale bases into a Galerkin Least Squares (GaLS) stabilized scheme, the method achieves well-posedness and optimal convergence while remaining robust as the Poisson ratio approaches the nearly incompressible limit . Theoretical results include stability, a priori error estimates, and locking-free behavior, complemented by two-dimensional numerical tests that verify improved stress accuracy and overall robustness relative to standard low-order Galerkin and non-stabilized MHM variants. The approach enables efficient, parallelizable multiscale elasticity computations on polytopal meshes with heterogeneous materials, offering a practical tool for nearly incompressible and high-contrast settings. Overall, the paper advances stabilized, low-order MHM techniques as effective, locking-free solvers for complex elasticity problems.

Abstract

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a family of low-order finite elements for the linear elasticity problem which are free from Poisson locking. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient refinement levels on the fine-scale mesh such that the MHM method is well-posed, optimally convergent under local regularity conditions, and locking-free. Two-dimensional numerical tests assess theoretical results.
Paper Structure (17 sections, 9 theorems, 81 equations, 3 figures, 6 tables)

This paper contains 17 sections, 9 theorems, 81 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The elasticity problem
  3. The MHM method
  4. Global-local formulation
  5. MHM method's discrete formulation
  6. Locking-free finite elements for the MHM method
  7. Preliminaries
  8. The Galerkin Least Squares formulation
  9. Summary
  10. Well-posedness of the local formulation
  11. Well-Posedness of the MHM method
  12. A priori error estimates for the MHM method
  13. Numerical Results
  14. The Poisson locking phenomenon
  15. $H$-convergence tests
  16. ...and 2 more sections

Key Result

Lemma 4.3

\newlabelboundnessLemma0 There is a positive constant $C$ such that, for all $({\boldsymbol u}_h,p_h), ({\boldsymbol v}_h,q_h)\in \widetilde{\mathbf V}_h(K) \times Q_h(K)$, it holds

Figures (3)

  • Figure 1: A two-dimensional polygon partitioned by the meshes $\mathscr{P}$ and $\mathcal{T}_h^{}$. The fine-scale meshes, ${\mathcal{T}_{h}}^{K_1}$ and ${\mathcal{T}_{h}}^{K_2}$, are defined over $K_1, K_2 \in \mathscr{P}$, respectively. Elements $K_1$ and $K_2$ belong to $\mathscr{P}$; faces $E$ and $F$ are in $\mathcal{E}$ and $\mathcal{E}_H^{}$, respectively; the simplexes $\tau$ belong to the affine mesh $\mathcal{T}_h^{}$.
  • Figure 1: $\upepsilon$-convergence in the four different methods. Orders $k = 1,2,3$ and mesh sizes $h = 2^{k-4.5}$, from top to bottom. The global partition's size in the MHM methods is $H = 2^{-1.5}$.
  • Figure 2: $h$-convergence in the MHM-GaLS and GaLS methods using $\nu = 0.4999$, and orders $k = 1$ (left) and $2$ (right). The size of the global partition in the MHM-GaLS is $\mathcal{H} = 2^{3-k}\,h$.

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.3: Boundness of $B_K$
  • Proof 1
  • Lemma 4.4
  • Proof 2
  • ...and 18 more