Normal-normal continuous symmetric stresses in mixed finite element elasticity

Abstract: The classical continuous mixed formulation of linear elasticity with pointwise symmetric stresses allows for a conforming finite element discretization with piecewise polynomials of degree at least three. Symmetric stress approximations of lower polynomial order are only possible when their div-conformity is weakened to the continuity of normal-normal components. In two dimensions, this condition is meant pointwise along edges for piecewise polynomials, but a corresponding characterization for general piecewise H(div) tensors has been elusive. We introduce such a space and establish a continuous mixed formulation of linear planar elasticity with pointwise symmetric stresses that have, in a distributional sense, continuous normal-normal components across the edges of a shape-regular triangulation. The displacement is split into an $L_2$ field and a tangential trace on the skeleton of the mesh. The well-posedness of the new mixed formulation follows with a duality lemma relating the normal-normal continuous stresses with the tangential traces of displacements. For this new formulation we present a lowest-order conforming discretization. Stresses are approximated by piecewise quadratic symmetric tensors, whereas displacements are discretized by piecewise linear polynomials. The tangential displacement trace acts as a Lagrange multiplier and guarantees global div-conformity in the limit as the mesh-size tends to zero. We prove locking-free, quasi-optimal convergence of our scheme and illustrate this with numerical examples.

• The paper presents a novel mixed finite element formulation for linear elasticity that employs normal-normal continuous symmetric stresses.
• It utilizes lower-order polynomial approximations and a split displacement strategy to enhance computational efficiency and maintain accuracy.
• Computational experiments demonstrate quasi-optimal convergence and locking-free performance in both smooth and singular domains.

Normal-normal continuous symmetric stresses in mixed finite element elasticity

This paper presents a novel approach to solving linear elasticity problems using mixed finite element methods that incorporate normal-normal continuous symmetric stresses. The authors focus on deriving a continuous mixed formulation with pointwise symmetric stresses, which have continuous normal-normal components distributed across the edges of a triangulated mesh. This formulation allows for lower-order polynomial stress approximations while maintaining computational efficiency and accuracy.

Mixed Finite Element Methodology

The classical approach to mixed formulations in linear elasticity typically involves conforming finite element methods that require high polynomial degrees to ensure pointwise symmetry and conformity of the stress approximation. The paper introduces a mixed formulation that weakens the $div$-conformity by focusing on continuity of normal-normal components rather than global $div$-conformity. This is accomplished by splitting the displacement into an $L_2$ field and a tangential trace on the mesh skeleton, ensuring continuity across element interfaces.

Implementing the New Formulation

This formulation is practically deployed using a lowest-order conforming discretization scheme. The stress field is approximated by piecewise quadratic symmetric tensors, whereas the displacement is discretized using piecewise linear polynomials. The tangential displacement trace is treated as a Lagrange multiplier, enforcing global $div$-conformity in the mesh-size limit.

An interpolation operator is proposed for the stress tensor approximation, which ensures the computed stresses commute with $L_2$ projection onto piecewise linear polynomials, a crucial property for maintaining accuracy and consistency in numerical simulations.

Computational Results and Analysis

The paper demonstrates through computational tests that the proposed scheme is locking-free in the incompressible limit and converges quasi-optimally, even in cases of singular stress distribution often encountered in practical engineering problems. Numerical experiments performed in smooth and singular domains confirm the theoretical convergence rates, highlighting the robustness and efficiency of this method in solving complex elasticity problems.

Implications and Future Work

This research provides a substantial improvement in finite element elasticity modeling, specifically by reducing computational requirements and enhancing the flexibility of stress approximations. It opens the possibility for extending such formulations to three-dimensional problems, potentially demanding advances in mesh processing and degree of freedom management. Future work might explore the application of these methods beyond elasticity, into other realms of computational physics and engineering simulations where pointwise stress symmetry plays a critical role.

Conclusion

The contributions of this paper lie in providing a deeper understanding of piecewise $H(div)$ stress approximation, presenting realistic options for maintaining symmetry and continuity across elements. The developed formulation accommodates reduced polynomial degrees and offers a computationally feasible approach for accurately modeling elasticity in both regular and singular domains. This represents a significant step toward more efficient numerical simulations in mechanical engineering and material science.