Application of Mixed and Hybrid Finite Element Formulations

Abstract

Partial Differential Equations (PDEs) are a fundamental tool in modelling various physical phenomena. The lack of closed form solutions in most cases continues to provide a strong incentive to seek stable and consistent numerical methods. This report deals specifically with the mixed and hybrid nite element formulations; ideas that were originally developed in the 1960's for problems in linear elasticity. Hybrid formulations have very attractive properties but are plagued with non-physical singular modes known as Spurious Kinematic Modes (SKMs). This report is devoted to the development of a hybrid scheme in linear elasticity that is free of SKMs. In the scheme, both the stress fields and kinematic fields are treated as primary variables and can hence be obtained together while also satisfying the equilibrium conditions throughout the physical domain. Subsequently, it is shown that optimal convergence is obtained for the stress and displacement fields. A versatile scheme is setup such that it can be solved efficiently by parallelisation of the system. Finally, an idea is mentioned regarding the extension to fluid dynamics.