A Higher Order Equilibrium Finite Element Method

TL;DR

This paper introduces a mixed spectral element method for 2D linear elasticity that ensures continuous tractions and enforces equilibrium conditions through a variational approach, improving accuracy and consistency in stress analysis.

Contribution

It presents a novel higher order finite element formulation with integrated traction degrees of freedom that guarantees equilibrium pointwise for forces and weakly for moments.

Findings

Continuous tractions across element boundaries.

Accurate stress fields on complex domains.

Effective handling of singularities.

Abstract

In this paper a mixed spectral element formulation is presented for planar, linear elasticity. The degrees of freedom for the stress are integrated traction components, i.e. surface force components. As a result the tractions between elements are continuous. The formulation is based on minimization of the complementary energy subject to the constraints that the stress field should satisfy equilibrium of forces and moments. The Lagrange multiplier which enforces equilibrium of forces is the displacement field and the Lagrange multiplier which enforces equilibrium of moments is the rotation. The formulation satisfies equilibrium of forces pointwise if the body forces are piecewise polynomial. Equilibrium of moments is weakly satisfied. Results of the method are given on orthogonal and curvilinear domains and an example with a point singularity is given.