A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

Abstract

Summary The first superconvergent hybridisable discontinuous Galerkin 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 superconvergent postprocessed displacement is proposed. Contrary to other hybridisable discontinuous Galerkin formulations, the methodology proposed here is able to produce a superconvergent 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 superconvergent properties in two and three dimensions and for different element types.

A hybridizable discontinuous Galerkin phase‐field model for brittle fracture with adaptive refinement

“…For the equilibrium equation, we choose the HDG formulation for linear elasticity proposed by Soon et al, adding the damage degradation function in the stress‐strain constitutive equation. However, the proposed strategy could also be applied with alternative HDG formulations in the literature . For the damage equation, we add the reaction term to the HDG formulation for diffusion by Cockburn et al In this section, we briefly recall both formulations and present the algorithm used to solve the coupled system in Equation .…”

“…For this formulation, u converges with order p +1 in L 2 norm and J with order p +1/2, if an approximation of degree p is used . In a more recent formulation, Sevilla et al report convergence of order p +1 for J based on numerical experiments.…”

“…However, the proposed strategy could also be applied with alternative HDG formulations in the literature. 25,26 For the damage equation, we add the reaction term to the HDG formulation for diffusion by Cockburn et al 27 In this section, we briefly recall both formulations and present the algorithm used to solve the coupled system in Equation (3).…”

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“…where we have used the second inequality in (29) together with the definition (28) of the triple-bar strain norm to conclude. Recalling the expression (2) of the strain-stress law, we get, for any T ∈ T h ,…”

A three‐dimensional coupled Euler‐PIC method for penetration problems

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