A super--convergent hybridisable discontinuous Galerkin method for linear elasticity

TL;DR

This paper introduces a super-convergent hybridisable discontinuous Galerkin method for linear elasticity that enforces stress symmetry via Voigt notation, achieving high accuracy and robustness in low order approximations.

Contribution

It presents the first super-convergent HDG method for linear elasticity that uses the same degree for primal and mixed variables and enforces stress symmetry without complex constraints.

Findings

Achieves super-convergence in displacement fields for low order approximations

Demonstrates robustness and locking-free performance in nearly-incompressible cases

Validates optimality and super-convergence through extensive numerical tests

Abstract

The first super-convergent hybridisable discontinuous Galerkin (HDG) method for linear elastic problems capable of using the same degree of approximation for both the primal and mixed variables is presented. The key feature of the method is the strong imposition of the symmetry of the stress tensor by means of the well-known and extensively used Voigt notation, circumventing the use of complex mathematical concepts to enforce the symmetry of the stress tensor either weakly or strongly. A novel procedure to construct element-by-element a super-convergent post-processed displacement is proposed. Contrary to other HDG formulations, the methodology proposed here is able to produce a super-convergent displacement field for low order approximations. The resulting method is robust and locking-free in the nearly-incompressible limit. An extensive set of numerical examples is utilised to provide evidence of the optimality of the method and its super-convergent properties in two and three dimensions and for different element types.