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Multipoint stress mixed finite element methods for elasticity on cuboid grids

Ibrahim Yazici, Ivan Yotov

TL;DR

The paper addresses stable, efficient discretizations for 3D elasticity with weak stress symmetry on cuboid grids by developing two multipoint stress mixed finite element methods (MSMFE). MSMFE-0 employs the lowest-order enhanced Raviart-Thomas space $\mathcal{ERT}_0$ with vertex-based stress coupling to produce a symmetric, positive definite cell-centered system for displacement and rotation; MSMFE-1 enhances this with continuous trilinear rotations and vertex quadrature to allow elimination of rotation, yielding a cell-centered system for displacement only. A key theoretical contribution is the construction of a curl-conforming auxiliary space $\Theta_h$ forming an exact sequence with $\mathcal{ERT}_0$ and establishing a curl-based inf-sup condition that ensures stability for MSMFE-1. The analysis proves first-order convergence for all variables in their natural norms and second-order superconvergence for the displacement at cell centers, all supported by numerical experiments demonstrating robustness for discontinuous coefficients and nearly incompressible materials. The methods offer computational efficiency by reducing large saddle-point systems to SPD cell-centered systems, with practical implications for locking-free elasticity simulations and potential extensions to higher-order or hexahedral frameworks.

Abstract

We develop multipoint stress mixed finite element methods for linear elasticity with weak stress symmetry on cuboid grids, which can be reduced to a symmetric and positive definite cell-centered system. The methods employ the lowest-order enhanced Raviart-Thomas finite element space for the stress and piecewise constant displacement. The vertex quadrature rule is employed to localize the interaction of stress degrees of freedom, enabling local stress elimination around each vertex. We introduce two methods. The first method uses a piecewise constant rotation, resulting in a cell-centered system for the displacement and rotation. The second method employs a continuous piecewise trilinear rotation and the vertex quadrature rule for the asymmetry bilinear forms, allowing for further elimination of the rotation and resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both methods. For the stability analysis of the second method, a new auxiliary H-curl conforming matrix-valued space is constructed, which forms an exact sequence with the stress space. A matrix-matrix inf-sup condition is shown for the curl of this auxiliary space and the trilinear rotation space. First-order convergence is established for all variables in their natural norms, as well as second-order superconvergence of the displacement at the cell centers. Numerical results are presented to verify the theory.

Multipoint stress mixed finite element methods for elasticity on cuboid grids

TL;DR

The paper addresses stable, efficient discretizations for 3D elasticity with weak stress symmetry on cuboid grids by developing two multipoint stress mixed finite element methods (MSMFE). MSMFE-0 employs the lowest-order enhanced Raviart-Thomas space with vertex-based stress coupling to produce a symmetric, positive definite cell-centered system for displacement and rotation; MSMFE-1 enhances this with continuous trilinear rotations and vertex quadrature to allow elimination of rotation, yielding a cell-centered system for displacement only. A key theoretical contribution is the construction of a curl-conforming auxiliary space forming an exact sequence with and establishing a curl-based inf-sup condition that ensures stability for MSMFE-1. The analysis proves first-order convergence for all variables in their natural norms and second-order superconvergence for the displacement at cell centers, all supported by numerical experiments demonstrating robustness for discontinuous coefficients and nearly incompressible materials. The methods offer computational efficiency by reducing large saddle-point systems to SPD cell-centered systems, with practical implications for locking-free elasticity simulations and potential extensions to higher-order or hexahedral frameworks.

Abstract

We develop multipoint stress mixed finite element methods for linear elasticity with weak stress symmetry on cuboid grids, which can be reduced to a symmetric and positive definite cell-centered system. The methods employ the lowest-order enhanced Raviart-Thomas finite element space for the stress and piecewise constant displacement. The vertex quadrature rule is employed to localize the interaction of stress degrees of freedom, enabling local stress elimination around each vertex. We introduce two methods. The first method uses a piecewise constant rotation, resulting in a cell-centered system for the displacement and rotation. The second method employs a continuous piecewise trilinear rotation and the vertex quadrature rule for the asymmetry bilinear forms, allowing for further elimination of the rotation and resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both methods. For the stability analysis of the second method, a new auxiliary H-curl conforming matrix-valued space is constructed, which forms an exact sequence with the stress space. A matrix-matrix inf-sup condition is shown for the curl of this auxiliary space and the trilinear rotation space. First-order convergence is established for all variables in their natural norms, as well as second-order superconvergence of the displacement at the cell centers. Numerical results are presented to verify the theory.

Paper Structure

This paper contains 12 sections, 14 theorems, 85 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The model problem and its MFE approximation
  3. Mixed finite element spaces
  4. A quadrature rule
  5. The multipoint stress mixed finite element method with constant rotations (MSMFE-0)
  6. Reduction to a cell-centered displacement-rotation system
  7. The multipoint stress mixed finite element method with trilinear rotations (MSMFE-1)
  8. Well-posedness of the MSMFE-1 method
  9. Reduction to a cell-centered displacement system of the MSMFE-1 method
  10. Error Estimates
  11. Numerical Results
  12. Conclusion

Key Result

Lemma 2.1

There exist constants $0 < \alpha_0 \le \alpha_1$ independent of $h$ such that Furthermore, $(w,w)^{1/2}_Q$ is a norm in $\mathbb{W}^1_h$ equivalent to $\|\cdot\|$, and $\forall \, \tau \in \mathbb{X}_h$, $w \in \mathbb{W}^1_h$, $(\tau,w)_Q \leq C \|\tau\|\|w\|$.

Figures (5)

  • Figure 1: Degrees of freedom of $\mathbb{X}_h\times V_h\times \mathbb{W}^1_h$.
  • Figure 2: Interactions of the stress degrees of freedom in the MSMFE methods.
  • Figure 3: Degrees of freedom for $\Theta(\hat{E})$.
  • Figure 4: Computed solution for Example 1 via the MSMFE-1 method, $h = 1/16$.
  • Figure 5: Computed solution for Example 2 via the MSMFE-1 method, $h = 1/32$.

Theorems & Definitions (31)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Remark 4.1
  • Lemma 4.1
  • ...and 21 more