Discretisation of Hodge Laplacians in the elasticity complex

Thesis

The elasticity differential complex associated with a 2- or 3-dimensional domain is a sequence of function spaces connected by differential operators, which together encode topological properties of the domain. Associated with any complex is a sequence of partial differential equations, known as the Hodge Laplace equations, which include and generalise many important elliptic equations arising in continuum mechanics. This thesis addresses the discretisation of the Sobolev spaces and Hodge Laplacian problems associated with the elasticity complex using finite elements. We demonstrate the broad utility of such efforts via applications to linear elasticity, linear irreversible thermodynamics, and defect elasticity.

First, we address the classical problem of enforcing the symmetry and div-conformity of the elastic stress tensor. The exactly symmetric Arnold–Winther elements were one of the key early breakthroughs of the finite element exterior calculus, but have never been systematically implemented, as their dual bases are not preserved by the Piola pullback; we develop abstract transformation theory which enables the first robust and composable implementations of these exotic elements. We then apply these tensor-valued elements to discretise the viscous stress in the compressible Stokes equations, a crucial coupling variable for the incorporation of convection into modelling the molecular diffusion of multicomponent single-phase fluids. We derive a novel variational formulation, called the Stokes–Onsager–Stefan–Maxwell system, with appropriate finite element discretisations which represent the first ever rigorous numerics for the coupling of non-ideal multicomponent diffusion with compressible convective flow. Finally, we turn our attention to the discretisation of the strain space in the elasticity complex, and analyse the incompatibility operator acting on strain tensor fields; the Hodge Laplacian boundary value problem we study comprises initial steps towards a canonical well-posed model of linearised defect elasticity.

Authors: Aznaran, FRA

• DOI: 10.5287/ora-qzrddxrvb
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: finite element exterior calculus; finite element method; elasticity complex
• Subjects: Finite element method