Development and Application of bundle-valued forms in hybrid mimetic spectral element method

Abstract

One of the novel methodologies in computational physics research is to use mimetic discretisation techniques. Among these, the mimetic spectral element method holds special promise as it not only has the benefits of mimetic methods but also the additional benefit of higher-order discretisations using higher polynomial degrees. These methods are aided by the development of algebraic dual polynomials, resulting in a sparser system for better computational efficiency. This combination was used to develop a formulation that would result in topological relations for equilibrium of forces as well as the symmetry of the stress tensor for linear elasticity as well as the first steps for Stokes flows in an orthogonal domain. As a result, this study was extended to look at how a modified formulation would behave for an unsteady linear elastic solid, with the intention to extend this method to Fluid-Structure Interaction cases. However, the choice of both primal and nodal basis functions makes it impossible to undertake this challenge, demanding a rethink in strategy towards looking at linear elastic solids when the physical domain is not orthogonal. With the use of bundle-valued forms to represent physical quantities, a new hybrid formulation is developed where the equivalent of the physical problem is computed on a reference domain, which is orthogonal and thus can utilise the spectral bases defined before. The physical problem is defined on a skewed domain, where partial transformation of components results in a formulation that can conserve linear momentum point-wise, but not conservation of angular momentum, although angular momentum does converge on refinement of polynomial degree and mesh parameters. A change in bases with partial transformation aiming to make angular momentum conservation topological is not fruitful, although the value of the error decreases in the process. The final attempt is through full transformation, which results in a formulation with an inherent error in the formulation, owing to erroneous assumptions.