Entangled graphs on surfaces in space

In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations. This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one. Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy. It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net. Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Thurston Polytopes

We explore a 3-manifold and link invariant called the Thurston norm which provides a deep understanding of the submanifold structure of a 3-manifold. Recently, methods to compute the Thurston norm ball (a symmetric rational polytope) have been developed, providing a doorway through which we can hope to understand more about this invariant. In particular we use these techniques in order to find patterns in these Thurston norm balls which give rise to new conjectures. We also showcase some existing literature in the field to highlight relationships between ∥ · ∥T and other properties/invariants of manifolds and links. This work embarks on a journey through low dimensional topology making stops in fields as diverse as combinatorial group theory, Floer homology, and hyperbolic geometry. In this way, we hope to convince the reader that this invariant is well worth study

Thurston Polytopes

We explore a 3-manifold and link invariant called the Thurston norm which provides a deep understanding of the submanifold structure of a 3-manifold. Recently, methods to compute the Thurston norm ball (a symmetric rational polytope) have been developed, providing a doorway through which we can hope to understand more about this invariant. In particular we use these techniques in order to find patterns in these Thurston norm balls which give rise to new conjectures. We also showcase some existing literature in the field to highlight relationships between ∥ · ∥T and other properties/invariants of manifolds and links. This work embarks on a journey through low dimensional topology making stops in fields as diverse as combinatorial group theory, Floer homology, and hyperbolic geometry. In this way, we hope to convince the reader that this invariant is well worth study

Aspects of random graphs

The present report aims at giving a survey of my work since the end of my PhD thesis "Spectral Methods for Reconstruction Problems". Since then I focussed on the analysis of properties of different models of random graphs as well as their connection to real-world networks. This report's goal is to capture these problems in a common framework. The very last chapter of this thesis about results in bootstrap percolation is different in the sense that the given graph is deterministic and only the decision of being active for each vertex is probabilistic; since the proof techniques resemble very much results on random graphs, we decided to include them as well. We start with an overview of the five random graph models, and with the description of bootstrap percolation corresponding to the last chapter. Some properties of these models are then analyzed in the different parts of this thesis

Constitutive Modelling of Non-Linear Isotropic Elasticity Using Deep Regression Neural Networks

Deep neural networks (DNNs) have emerged as a promising approach for constitutive modelling of advanced materials in computational mechanics. However, achieving physically realistic and stable numerical simulations with DNNs can be challenging, especially when dealing with large deformations, that can lead to non-convergence effects in the presence of local stretch/stress peaks. This PhD dissertation introduces a novel approach for data-driven modelling of non-linear compressible isotropic materials, focusing on predicting the large deformation response of 3D specimens. The proposed methodology formulates the underlying hyperelastic deformation problem in the principal space using principal stretches and principal stresses, in which the corresponding constitutive relation is captured by a deep neural network surrogate model. To ensure constitutive requirements of the model while preserving the robustness of underlying numerical solution schemes, several physics-motivated constraints are imposed on the architecture of the DNN, such as objectivity, growth condition, normalized condition, and Hill’s inequalities. Furthermore, the prediction phase utilizes a constitutive blending approach to overcome divergence in the Newton-Raphson process, which can occur when solving boundary value problems using the Finite Element Method. The work also presents a machine learning finite element pipeline for modelling non-linear compressible isotropic materials, involving determining automatically the hyperparameters, training, and integrating the ANN operator into the finite element solver using symbolic representation. The proposed formalism has been tested through numerical benchmarks, demonstrating its ability to describe non-trivial load-deformation trajectories of 3D test specimens accurately. Overall, the thesis presents a complete and general formalism for data-driven modelling of non-linear compressible isotropic materials that overcomes the limitations of existing approaches.9. Industry, innovation and infrastructur

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Dynamics of a class of vortex rings

The contour dynamics method is extended to vortex rings with vorticity varying linearly from the symmetry axis. An elliptic core model is also developed to explain some of the basic physics. Passage and collisions of two identical rings are studied focusing on core deformation, sound generation and stirring of fluid elements. With respect to core deformation, not only the strain rate but how rapidly it varies is important and accounts for greater susceptibility to vortex tearing than in two dimensions. For slow strain, as a passage interaction is completed and the strain relaxes, the cores return to their original shape while permanent deformations remain for rapidly varying strain. For collisions, if the strain changes slowly the core shapes migrate through a known family of two-dimensional steady vortex pairs up to the limiting member of the family. Thereafter energy conservation does not allow the cores to maintain a constant shape. For rapidly varying strain, core deformation is severe and a head-tail structure in good agreement with experiments is formed. With respect to sound generation, good agreement with the measured acoustic signal for colliding rings is obtained and a feature previously thought to be due to viscous effects is shown to be an effect of inviscid core deformation alone. For passage interactions, a component of high frequency is present. Evidence for the importance of this noise source in jet noise spectra is provided. Finally, processes of fluid engulfment and rejection for an unsteady vortex ring are studied using the stable and unstable manifolds. The unstable manifold shows excellent agreement with flow visualization experiments for leapfrogging rings suggesting that it may be a good tool for numerical flow visualization in other time periodic flows