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The MAC scheme for linear elasticity in displacement-stress formulation on non-uniform staggered grids

Hongxing Rui, Weijie Wang

TL;DR

The paper develops a MAC-E staggered-grid finite-difference method for 2D/3D linear elasticity in displacement-stress form on nonuniform grids. By placing displacements on edge midpoints, normal stresses at cell centers, and shear stresses at grid points, the scheme achieves local conservation and avoids spurious stress oscillations, with a discrete LBB-stable formulation and locking-free performance as $\lambda$ varies. It proves second-order $L^2$-error super-convergence for displacements and stresses via carefully constructed interpolants, and demonstrates robustness on both uniform and nonuniform grids through extensive 2D and 3D numerical tests. The method is computationally efficient (fewer DOFs per element) and readily extensible to 3D Stokes-like systems, offering a practical alternative to traditional finite-element approaches for compressible and nearly incompressible elasticity.

Abstract

A marker-and-cell finite difference method is developed for solving the two dimensional and three dimensional linear elasticity in the displacement-stress formulation on staggered grids. The method employs a staggered grid arrangement, where the displacement components are approximated on the midpoints of cell edges, the normal stresses are defined at the cell centers, and the shear stresses are defined at the grid points. This structure ensures local conservation properties and avoids spurious oscillations in stress approximation. A rigorous mathematical analysis is presented, establishing the stability of the scheme and proving the second-order L2-error super-convergence for both displacement and stress. The proposed method is locking-free with respect to the Lame constant, making it suitable for both compressible and nearly incompressible elastic materials. Numerical experiments demonstrate the efficiency and robustness of the finite difference scheme, and the computed results show excellent agreement with the theoretical predictions.

The MAC scheme for linear elasticity in displacement-stress formulation on non-uniform staggered grids

TL;DR

The paper develops a MAC-E staggered-grid finite-difference method for 2D/3D linear elasticity in displacement-stress form on nonuniform grids. By placing displacements on edge midpoints, normal stresses at cell centers, and shear stresses at grid points, the scheme achieves local conservation and avoids spurious stress oscillations, with a discrete LBB-stable formulation and locking-free performance as varies. It proves second-order -error super-convergence for displacements and stresses via carefully constructed interpolants, and demonstrates robustness on both uniform and nonuniform grids through extensive 2D and 3D numerical tests. The method is computationally efficient (fewer DOFs per element) and readily extensible to 3D Stokes-like systems, offering a practical alternative to traditional finite-element approaches for compressible and nearly incompressible elasticity.

Abstract

A marker-and-cell finite difference method is developed for solving the two dimensional and three dimensional linear elasticity in the displacement-stress formulation on staggered grids. The method employs a staggered grid arrangement, where the displacement components are approximated on the midpoints of cell edges, the normal stresses are defined at the cell centers, and the shear stresses are defined at the grid points. This structure ensures local conservation properties and avoids spurious oscillations in stress approximation. A rigorous mathematical analysis is presented, establishing the stability of the scheme and proving the second-order L2-error super-convergence for both displacement and stress. The proposed method is locking-free with respect to the Lame constant, making it suitable for both compressible and nearly incompressible elastic materials. Numerical experiments demonstrate the efficiency and robustness of the finite difference scheme, and the computed results show excellent agreement with the theoretical predictions.
Paper Structure (9 sections, 9 theorems, 87 equations, 1 figure, 28 tables)

This paper contains 9 sections, 9 theorems, 87 equations, 1 figure, 28 tables.

Table of Contents

  1. Introduction
  2. The Problem and Some Preliminaries
  3. The MAC-E scheme and stability
  4. Superconvergence analysis.
  5. Extension to three dimensional problems
  6. Numerical examples.
  7. Example 1
  8. Example 2
  9. Example 3

Key Result

Lemma 3.1

There is a constant $\beta>0$ independent of $h$ and $l$ such that

Figures (1)

  • Figure 1: Three different partitions

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.1
  • Theorem 3.1
  • Lemma 4.1
  • ...and 6 more