Discrete approaches to mechanics and physics of solids

Abstract

In this PhD thesis I analyse discretisations of continuum formulations and fully discrete formulations of problems in solid mechanics, specifically diffusion-like scalar transport phenomena and phenomena involving conservation of momenta like elasticity. In the beginning I discuss different types of continuum formulations (strong, integral, weak) and the various approaches to their discretisations. The discussion is completed when I consider various fully discrete formulations, their similarities and differences with discretisations of continuum formulations. The dissertation continues by considering representations of transport phenomena in solids, including heat transfer through conduction, mass diffusion, charge transport, volume transport. These phenomena share a common formalism and I consider heat transport as a model problem. The discussion continues with methods for discretisation. As a starting point I consider the finite difference approximation for the strong elliptic formulation. Then I consider two weak formulations (elliptic and saddle point) and lay the common ground of two numerical methods (finite and virtual element methods). I discuss their characteristics, and the procedure of implementation and analysis of both finite and virtual element methods. After that I review a novel discrete modelling approach (Berbatov et al. Diffusion in multi-dimensional solids using Forman’s combinatorial differential forms. Applied Mathematical Modelling. 2022;110:172-192.), based on Forman's combinatorial differential forms, for which I developed the mathematical background. This approach allows for further interactions between the cells of a polytopal mesh with the notion of combinatorial differential forms. The space of discrete forms is supplied with a discrete derivative which corresponds to the coboundary operator on an extended mesh, called the Forman subdivision. I first summarise the intrinsic topological operations that I generalised and developed, specifically the cup product of cochains (which pulls back to a wedge product of discrete forms). Then I consider the notion of an (intrinsic) discrete vector field and prove that a discrete analogue of the interior product satisfying the graded Leibniz rule does not exist. As a consequence, a discrete Lie derivative satisfying Cartan's magic formula does not exist as well. I then summarise the geometric operators I developed: a discrete inner product based on a discrete metric tensor, from which an adjoint coboundary operator, discrete Laplacian, and discrete Hodge star are derived. Then I propose a discrete inner product calculated without the need of a discrete metric tensor, which I discovered after the publication of the mentioned article. It leads to discrete operators that mimic better some properties of their smooth counterparts, especially a discrete Laplacian that works optimally on meshes of rectangular cells with different sizes. The purely geometric Laplacian is modified with material properties in order to specify a hybrid model for transport phenomena with continuous time but discrete space. The chapter is completed with an exact copy of the previously mentioned published article. The last topic of the dissertation is representations of phenomena involving conservation of momenta, e.g., linear and angular momenta. The model problem that I consider is (linear) elasticity. Firstly, I formulate the continuum mathematical model and comment on possible weak formulations, especially primal and mixed weak formulations. Then I provide a copy of an article that I coauthored (Berbatov et al. A guide to the finite and virtual element methods for elasticity. Applied Numerical Mathematics. 2021;169:351-395.), concerned with: (a) the description and implementation of finite and virtual element methods for these formulations; (b) a comparison between those methods for 3 different problems. Finally, I state a speculative formulation of discrete elasticity but without cons

Date of Award	1 Aug 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Andrey Jivkov (Supervisor) & Andrew Hazel (Supervisor)