POCHODNA TOPOLOGICZNA – TEORIA I ZASTOSOWANIA

The paper is devoted to present some mathematical aspects of the topological derivative and its applications in different fields of sciences such as shape optimization and inverse problems. First the definition of the topological derivative is given and the shape optimization problem is formulated. Next the form of the topological derivative is evaluated for a mixed boundary value problem defined in a geometrical domain. Finally, an example of an application of the topological derivative in the electric impedance tomography is presented.W pracy przedstawiono matematyczne aspekty dotyczące pochodnej topologicznej oraz jej zastosowań w różnych dziedzinach nauki, takich jak optymalizacja kształtu czy problemy odwrotne. W pierwszej części podano nieformalna˛ definicje˛ pochodnej topologicznej oraz sformułowano problem optymalizacji kształtu. Następnie wyprowadzono postać pochodnej topologicznej dla mieszanego problemu brzegowego. W ostatniej części przedstawiono przykład zastosowania pochodnej topologicznej dla problemu elektrycznej tomografii impedancyjnej

Topology Optimization of Irregular Shaped Pressure Vessels Using a Level-Set Method

Advances in manufacturing capabilities, such as additive manufacturing, have expanded the design freedom given to engineers enabling more efficient designs through the use of complex geometries. However, determining the optimal geometric structure for a given set of performance criteria can be quite challenging when given such design freedom. One technique to do so is with the use of topology optimization methods, in which optimal material distribution within a given design space is determined. Many established topology optimization methods are developed such that a set of boundary conditions are prescribed to the design domain and remain fixed throughout the optimization process of determining the material distribution. This eliminates the ability to implement design dependent loading conditions, such as pressure loading, which requires tracking (following) the pressure surface as the geometry evolves during the optimization process. In this thesis, a level-set topology optimization method is implemented based on voxel elements on design domains in R^3 subjected to internal pressure loading, such as in the case of a non-spherical or cylindrical pressure vessel. Following a thorough literature review, a level-set function was chosen to define a crisp material/void boundary for identifying loading conditions caused by the applied pressure. This pressure loading is calculated as an applied traction across all material elements, excluding exterior surface nodes. This results in an equal and opposite cancelation throughout the material domain and leaving forces only at desired nodes along the material/void boundary. This implementation only requires material elements to be meshed, allowing for remeshing throughout the process to increase accuracy while saving computational cost by excluding void regions. Additionally, to improve convergence, the Lagrangian formulation of a penalty is replaced by a method analogous to PID-control systems as the algorithm hones in on convergence. To test the effectiveness of the method and the practicality of designing an irregular pressure vessel, the gas storage tanks of the MK16 rebreather for the US NAVY were redesigned within the current system’s geometric constraints in an effort to increase gas storage capacity. To do this, an outside domain geometry of the irregular shaped pressure vessel was defined, and not subject to change, while the optimization code was executed on the interior structure to minimize compliance subjected to an overall volume fraction constraint. This was done at various target volume fractions, and then stresses and compliance values were analyzed and compared to the existing pressure vessel of the MK16. The findings of this research concluded that designing an irregular shaped pressure vessel is a viable means of increasing storage capacity although future work would need to be executed to manufacture and experimentally validate these findings

Physically consistent modelling of surface tension forces in the Volume-of-Fluid method for three or more phases

\ua9 2024 The Authors. Multiphase numerical simulations have become a widely sought methodology for modelling capillary flows due to their scientific relevance and multiple industrial applications. Much progress has been achieved using different approaches, and the volume of fluid is one of the most popular methods widely used for modelling two or more phases due to its simplicity, accuracy and robustness. However, when prescribing the forces emerging from three or more fluid-fluid interfaces, the force balance is not guaranteed and can lead to spurious self-propulsion. Here, a new approach to account for the surface tension forces for multiphase flows with a correct force balance is proposed. The newly proposed method is successfully validated for a wide range of tests, including contact angles for the fluid-fluid and fluid-solid triple line. Additionally, complete spreading phenomena of fluid on fluid and fluid on solid have been found to emerge naturally from the newly proposed surface tension force model. Finally, simulation results are compared against experiments of lubricant-impregnated surfaces to demonstrate the practical applicability of the newly proposed method

Haydys-Witten Instantons and Symplectic Khovanov Homology

An influential conjecture by Witten states that there is a Floer theory based on Haydys-Witten instantons that provides a gauge theoretic approach to Khovanov homology. This thesis explores a novel approach towards a potential proof of this claim. One of the key insights is the existence of a Hermitian Yang-Mills structure for a ‘decoupled’ version of the Haydys-Witten and Kapustin-Witten equations. It is shown that, in favourable circumstances, any Haydys-Witten solution is already a solution of the decoupled equations. This utilizes a dichotomy that is proved to be satisfied by θ-Kapustin-Witten solutions on any ALE or ALF space, generalizing a corresponding result on ℝ^4. The Hermitian Yang-Mills structure gives rise to a Kobayashi-Hitchin-like correspondence. It is proposed that solutions are classified by intersections of Lagrangian submanifolds in the moduli space of solutions of the extended Bogomolny equations. In that interpretation, Haydys-Witten instantons are in correspondence with pseudo-holomorphic discs, leading to a conjectural equivalence with a Lagrangian intersection Floer homology. A physically motivated argument suggests that the latter is fully determined in a finite-dimensional model space, given by a Grothendieck-Springer resolution of the nilpotent cone inside the underlying Lie algebra. This provides a relation to symplectic Khovanov homology, which is known to be isomorphic to a grading-reduced version of Khovanov homology

A phase-field model for active contractile surfaces

The morphogenesis of cells and tissues involves an interplay between chemical signals and active forces on their surrounding surface layers. The complex interaction of hydrodynamics and material flows on such active surfaces leads to pattern formation and shape dynamics which can involve topological transitions, for example during cell division. To better understand such processes requires novel numerical tools. Here, we present a phase-field model for an active deformable surface interacting with the surrounding fluids. The model couples hydrodynamics in the bulk to viscous flow along the diffuse surface, driven by active contraction of a surface species. As a new feature in phase-field modeling, we include the viscosity of a diffuse interface and stabilize the interface profile in the Stokes-Cahn-Hilliard equation by an auxiliary advection velocity, which is constant normal to the interface. The method is numerically validated with previous results based on linear stability analysis. Further, we highlight some distinct features of the new method, like the avoidance of re-meshing and the inclusion of contact mechanics, as we simulate the self-organized polarization and migration of a cell through a narrow channel. Finally, we study the formation of a contractile ring on the surface and illustrate the capability of the method to resolve topological transitions by a first simulation of a full cell division

Topological effects in particle physics phenomenology

This thesis is devoted to the study of topological effects in quantum field theories, with a particular focus on phenomenological applications. We begin by deriving a general classification of topological terms appearing in a non-linear sigma model based on maps from an arbitrary worldvolume manifold to a homogeneous space $G/H$ (where $G$ is an arbitrary Lie group and $H \subset G$). Such models are ubiquitous in phenomenology; in three or more dimensions they cover all cases in which only some subgroup $H$ of a dynamical symmetry group $G$ is linearly realized in vacuo. The classification is based on the observation that, for topological terms, the maps from the worldvolume to $G/H$ may be replaced by singular homology cycles on $G/H$. We find that such terms come in one of two types, which we refer to as `Aharonov-Bohm' (AB) and `Wess-Zumino' (WZ) terms. We derive a new condition for their $G$-invariance, which we call the `Manton condition', which is necessary and sufficient when the Lie group $G$ is connected. Armed with this classification of topological terms, we then apply it to Composite Higgs models based on a variety of coset spaces $G/H$ and discuss their phenomenology. For example, we point out the existence of an AB term in the minimal Composite Higgs model based on $SO(5)/SO(4)$, whose phenomenological effects arise only at the non-perturbative level, and lead to $P$ and $CP$ violation in the Higgs sector. Consideration of the Manton condition leads us to discover a rather subtle anomaly in a non-minimal model based on $SO(5)\times U(1)/SO(4)$ (a model which does, however, feature an AB term not previously noticed in the literature). A particularly rich topological structure, with six distinct terms of various types, is uncovered for the model based on $SO(6)/SO(4)$, which features two Higgs doublets and one singlet. Perhaps most importantly for phenomenology, measuring the coefficients of WZ terms that appear in any of these Composite Higgs models can allow one to probe the gauge group of the underlying microscopic theory. As a further application of our results, we analyse quantum mechanics models featuring such topological terms. In this context, a topological term couples the particle to a background magnetic field. The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. We show how both locality and symmetry can be made manifest, by passing to an otherwise redundant description on a principal bundle over the original configuration space, and by promoting the original symmetry group to a central extension thereof. We then demonstrate how harmonic analysis on the extended symmetry group can be used to solve the Schr{\"o}dinger equation. To conclude our study of topological terms in sigma models, we show that the classification we have proposed may be rigorously justified (and generalised) using differential cohomology theory. In doing so, we introduce the notion of the `$G$-invariant differential characters' of a manifold $M$. Within this language, the Manton condition follows from the homotopy formula for differential characters, and we show that it remains necessary and sufficient under weaker conditions than connectedness of $G$. We prove that the abelian group of $G$-invariant differential characters sits inside various exact sequences and commutative diagrams, which thus provide us with some powerful algebraic tools for classifying topological terms in quantum field theories. In the remainder of the thesis we depart from the topic of sigma models and turn to gauge theories. We analyse anomalies (which may be understood as arising from topological effects) in both the Standard Model (SM) and theories Beyond the Standard Model (BSM). This analysis consists of two parts, in which we consider `local' and `global' anomalies in a gauge symmetry $G$; the former depend only on the Lie algebra of $G$, while the latter are sensitive also to its global structure, {\em i.e.} its topology. We first chart the space of anomaly-free extensions of the SM by a flavour-dependent $U(1)$ gauge symmetry, using arithmetic techniques from Diophantine analysis to cancel all possible local anomalies. We then develop some of these anomaly-free theories into phenomenological models featuring a heavy $Z^\prime$ gauge boson, that can account for a collection of recent measurements involving $b\rightarrow s\mu\mu$ transitions which are discrepant with SM predictions. We discuss how these models might also explain coarse features of the fermion mass problem, such as the heaviness of the third family. We then turn to global anomalies, which we analyse using the Dai-Freed theorem. Our principal tool here is to compute the bordism groups of the classifying spaces of various Lie groups, preserving particular spin structures, using the Atiyah-Hirzebruch spectral sequence. We show that there are no global anomalies (beyond the Witten anomaly associated with the electroweak factor) in four different `versions' of the SM, in which the gauge group is taken to be $G_{\text{SM}}/\Gamma$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $\Gamma \in \{0,\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_6\}$. We also show that there are no new global anomalies in $U(1)^m$ extensions of the SM, which feature multiple $Z^\prime$ bosons, or in the Pati-Salam model.Vice-Chancellor's Award (Cambridge Trust

Invariance and Invertibility in Deep Neural Networks

Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input is altered. Furthermore, we want the network to learn the appropriate invariance from training data, instead of being explicitly constructed to achieve invariance to a pre-defined transformation type. The second part of our work is centered on two recently proposed types of deep networks: neural ordinary differential equations and invertible residual networks. These networks are invertible, that is, we can reconstruct the input from the output. However, there are some classes of functions that these networks cannot approximate. We show how to modify these two architectures to provably equip them with the capacity to approximate any smooth invertible function

Single and two-cells shape analysis from energy functionals for three-dimensional vertex models

© 2023 The Authors. International Journal for Numerical Methods in Biomedical Engineering published by John Wiley & Sons Ltd.Vertex models have been extensively used for simulating the evolution ofmulticellular systems, and have given rise to important global properties con-cerning their macroscopic rheology or jamming transitions. These models arebased on the definition of an energy functional, which fully determines the cellu-lar response and conclusions. While two-dimensional vertex models have beenwidely employed, three-dimensional models are far more scarce, mainly due tothe large amount of configurations that they may adopt and the complex geomet-rical transitions they undergo. We hereinvestigate the shape of single and two-cells configurations as a function of the energy terms, and we study the depen-dence of the final shape on the model parameters: namely the exponent of theterm penalising cell-cell adhesion and surface contractility. In single cell analysis,we deduce analytically the radius and limit values of the contractility for linearand quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetri-cal and asymmetrical, we deduce the evolution of the aspect ratio and the relativeradius. While in functionals with linear surface terms yield the same aspect ratioin 2D and 3D, the configurations when usingquadratic surface terms are distinct.We relate our results with well-known solutions from capillarity theory, and verifyour analytical findings with a three-dimensional vertex model.Generalitat de Catalunya; Ministerio de Ciencia e Innovaci on; Ministry of Federal Education and Professional Training; Higher Education Commission (under the Ministry of Federal Education and Professional Training) of Pakistan; Spanish Ministry of Science and Innovation, Grant/Award Numbers: CEX2018-000797-S, PID2020-116141GB-I00; Catalan government Generalitat de Catalunya, Grant/Award Number: 2021 SGR 01049Peer ReviewedPostprint (published version

A state of the art review of the Particle Finite Element Method (PFEM)

The particle finite element method (PFEM) is a powerful and robust numerical tool for the simulation of multi-physics problems in evolving domains. The PFEM exploits the Lagrangian framework to automatically identify and follow interfaces between different materials (e.g. fluid–fluid, fluid–solid or free surfaces). The method solves the governing equations with the standard finite element method and overcomes mesh distortion issues using a fast and efficient remeshing procedure. The flexibility and robustness of the method together with its capability for dealing with large topological variations of the computational domains, explain its success for solving a wide range of industrial and engineering problems. This paper provides an extended overview of the theory and applications of the method, giving the tools required to understand the PFEM from its basic ideas to the more advanced applications. Moreover, this work aims to confirm the flexibility and robustness of the PFEM for a broad range of engineering applications. Furthermore, presenting the advantages and disadvantages of the method, this overview can be the starting point for improvements of PFEM technology and for widening its application fields.Technology Innovation Program funded by the Ministry of Trade, Industry & Energy (MI, Korea), Grant/Award Number: 10053121; Universiti Teknologi PETRONAS (UTP) Internal Grant, Grant/Award Number: URIF 0153AAG24Peer ReviewedPostprint (published version