[Paper Review] The simplest mixed finite element method for linear elasticity in the symmetric formulation on $n$-rectangular grids

Jun Hu, Hongying Man+1 more|arXiv (Cornell University)|Apr 19, 2013

Advanced Numerical Methods in Computational Mathematics16 references11 citations

TL;DR

This paper proposes a family of minimal, symmetric, nonconforming mixed finite elements for linear elasticity in any space dimension on rectangular grids, using minimal polynomial spaces: span{1, x_i} for normal stresses σ_ii, span{1, x_i, x_j} for shear stresses σ_ij, and span{1} for displacements. The method achieves optimal convergence rates and exhibits unexpected superconvergence in numerical tests, outperforming prior elements with significantly fewer degrees of freedom.

ABSTRACT

A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress components and the displacement. The discrete spaces for the normal stress $σ_{ii}$, the shear stress $σ_{ij}$ and the displacement $u_i$ are $\operatorname{span}\{1,x_i\}$, $\operatorname{span}\{1,x_i,x_j\}$ and $\operatorname{span}\{1\}$, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, this element is nothing else but the 1D Raviart-Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element is 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. The previous record of the least degrees of freedom is, 13 plus 4 in 2D, and 54 plus 12 in 3D, on the rectangular grid. These elements are the simplest element for any space dimension. The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved for both pure displacement and traction problems. Numerical tests in 2D and 3D are presented to show a superiority of the new element over others, as a superconvergence is surprisingly exhibited.

Motivation & Objective

• To develop a stable, conforming mixed finite element method for linear elasticity in the symmetric stress formulation on n-dimensional rectangular grids.
• To minimize degrees of freedom while preserving optimal convergence and stability for both pure displacement and traction boundary conditions.
• To overcome the challenges of nonconformity and discrete inf-sup condition in symmetric mixed methods for high-dimensional problems.
• To provide a unified, dimension-independent formulation that works identically across 1D, 2D, and 3D.
• To demonstrate superior performance through numerical experiments, including unexpected superconvergence.

Proposed method

• The method uses minimal polynomial finite element spaces: span{1, x_i} for normal stress components σ_ii, span{1, x_i, x_j} for shear stresses σ_ij, and span{1} for displacement u_i on each rectangular element.
• The discrete spaces are constructed to match the directional derivatives of stress components, ensuring consistency with the physical structure of the problem.
• A nonconforming mixed finite element formulation is employed, avoiding vertex degrees of freedom and composite elements common in conforming methods.
• The stability is proven via verification of the discrete inf-sup condition, relying on a discrete exact sequence and canonical interpolation operators.
• The method is formulated in the first-order system form of linear elasticity, using the Hellinger-Reissner variational principle.
• Numerical experiments use uniform refinement of rectangular grids and nodal interpolation operators to compute errors and convergence rates.

Experimental results

Research questions

• RQ1 Can a minimal, symmetric, nonconforming mixed finite element method be constructed for linear elasticity that is stable and convergent in any space dimension?
• RQ2 Does the proposed method achieve optimal convergence rates despite its minimal degrees of freedom and nonconformity?
• RQ3 Can the method exhibit superconvergence in numerical tests, even when theoretical analysis only guarantees first-order convergence?
• RQ4 How does the method compare in efficiency and accuracy to existing elements with more degrees of freedom?
• RQ5 Is the discrete inf-sup condition satisfiable with such minimal polynomial spaces on rectangular grids?

Key findings

• The method achieves optimal convergence rates of order 2 in the L2 norm for both displacement and stress, despite theoretical analysis only proving first-order convergence.
• Numerical tests in 2D and 3D show superconvergence, with convergence orders of 2.0 for displacement, stress, and divergence of stress error, exceeding theoretical predictions.
• The new element has only 7+2 degrees of freedom per element in 2D and 15+3 in 3D, significantly fewer than the previous record of 13+4 (2D) and 54+12 (3D).
• The method is stable and convergent for both pure displacement and pure traction problems, as confirmed by numerical results on the unit square and cube.
• In 2D, the new element outperforms Yi's element [31] in convergence rate and degrees of freedom, achieving one order higher convergence with fewer degrees of freedom.
• The 3D implementation confirms the same superconvergence behavior, with convergence orders of 2.0 for all error norms, even though the theory only guarantees first-order convergence.