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Normal-normal continuous symmetric stress approximation in three-dimensional linear elasticity

Carsten Carstensen, Norbert Heuer

TL;DR

This work develops a conforming mixed finite element framework for 3D linear elasticity on tetrahedra with symmetric stress and normal-normal continuity across faces. It introduces a $30$-DOF H(div;S) stress element, reducible to $12$ via static condensation, and a compatible space $P^2_{nn}(\mathcal{T})$ with a commuting interpolation, enabling stable discretization independent of the Poisson ratio. The resulting scheme is locking-free and achieves quasi-optimal convergence, supported by a Poisson-ratio robust well-posedness theory and detailed element-level analysis. Numerical experiments on a unit cube verify locking-free behavior and show the expected $O(h)$-type convergence for stress and trace, and $O(h^2)$ for displacement and divergence of stress.

Abstract

We present a conforming setting for a mixed formulation of linear elasticity with symmetric stress that has normal-normal continuous components across faces of tetrahedral meshes. We provide a stress element for this formulation with 30 degrees of freedom that correspond to standard boundary conditions. The resulting scheme converges quasi-optimally and is locking free. Numerical experiments illustrate the performance.

Normal-normal continuous symmetric stress approximation in three-dimensional linear elasticity

TL;DR

This work develops a conforming mixed finite element framework for 3D linear elasticity on tetrahedra with symmetric stress and normal-normal continuity across faces. It introduces a -DOF H(div;S) stress element, reducible to via static condensation, and a compatible space with a commuting interpolation, enabling stable discretization independent of the Poisson ratio. The resulting scheme is locking-free and achieves quasi-optimal convergence, supported by a Poisson-ratio robust well-posedness theory and detailed element-level analysis. Numerical experiments on a unit cube verify locking-free behavior and show the expected -type convergence for stress and trace, and for displacement and divergence of stress.

Abstract

We present a conforming setting for a mixed formulation of linear elasticity with symmetric stress that has normal-normal continuous components across faces of tetrahedral meshes. We provide a stress element for this formulation with 30 degrees of freedom that correspond to standard boundary conditions. The resulting scheme converges quasi-optimally and is locking free. Numerical experiments illustrate the performance.

Paper Structure

This paper contains 10 sections, 12 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction and model problem
  2. Analytical framework
  3. Notation, spaces and trace operators
  4. Normal-normal continuous stress space and dual trace operator
  5. Mixed formulation
  6. Finite element approximation
  7. An $H(\operatorname{div};\mathbb{S})$ element
  8. Approximation space
  9. Finite element scheme and locking-free convergence
  10. Numerical experiments

Key Result

Proposition 1

Any function $\varphi\in H^{1/2}(\mathcal{S};\mathbb{R}^3)$ satisfies $\varphi\in \widetilde{H}^{1/2}_{0,n}(\mathcal{S};\mathbb{R}^3)$ if and only if $\pi_t\varphi|_F=0$, $\varphi\cdot n|_F\in \widetilde{H}^{1/2}(F)$ for all $F\in\mathcal{F}(\Omega)$, and $\varphi|_\Gamma=0$.

Figures (1)

  • Figure 1: Relative errors for the manufactured example with different values of $\nu$.

Theorems & Definitions (23)

  • Proposition 1: $\widetilde{H}^{1/2}_{0,n}(\mathcal{S};\mathbb{R}^3)$
  • proof
  • Proposition 2: $\widetilde{H}^{1/2}_{0,t}(\mathcal{S};\mathbb{R}^3)$
  • proof
  • Proposition 3: density
  • proof
  • Proposition 4: inclusion
  • proof
  • Proposition 5: duality
  • proof
  • ...and 13 more