Microstructure modeling and finite element analysis of particulate reinforced metal matrix composites

The study of microscopic and macroscopic response of a particulate reinforced metal matrix composites (PRMMC) using finite element (FE) analysis is the aim of the current investigation. In this regard, three types of microstructure models are subjected to FE analysis. In the first part of the work, a technique is presented for the generation of artificial microstructures containing spherical and ellipsoid shaped inclusions. The problem of detection of ellipsoidal intersection is tackled using newly available algorithms. To account for higher-scale effects such as clustering, information from the micro-scale model is input into a meso-scale model.;The FE analysis of the artificial microstructure and a summary of the results form the second part of the study. It is seen that the newly developed models agree very well with the published results and that the microstructure generation technique can be reused in many computational micromechanics problems with minimum modifications. Finally, the results obtained from the new models are used to study a problem of practical significance, namely bolted joints made of MMC material

Model Uncertainty based Active Learning on Tabular Data using Boosted Trees

Supervised machine learning relies on the availability of good labelled data for model training. Labelled data is acquired by human annotation, which is a cumbersome and costly process, often requiring subject matter experts. Active learning is a sub-field of machine learning which helps in obtaining the labelled data efficiently by selecting the most valuable data instances for model training and querying the labels only for those instances from the human annotator. Recently, a lot of research has been done in the field of active learning, especially for deep neural network based models. Although deep learning shines when dealing with image\textual\multimodal data, gradient boosting methods still tend to achieve much better results on tabular data. In this work, we explore active learning for tabular data using boosted trees. Uncertainty based sampling in active learning is the most commonly used querying strategy, wherein the labels of those instances are sequentially queried for which the current model prediction is maximally uncertain. Entropy is often the choice for measuring uncertainty. However, entropy is not exactly a measure of model uncertainty. Although there has been a lot of work in deep learning for measuring model uncertainty and employing it in active learning, it is yet to be explored for non-neural network models. To this end, we explore the effectiveness of boosted trees based model uncertainty methods in active learning. Leveraging this model uncertainty, we propose an uncertainty based sampling in active learning for regression tasks on tabular data. Additionally, we also propose a novel cost-effective active learning method for regression tasks along with an improved cost-effective active learning method for classification tasks

Constrained Monotonic Neural Networks

Wider adoption of neural networks in many critical domains such as finance and healthcare is being hindered by the need to explain their predictions and to impose additional constraints on them. Monotonicity constraint is one of the most requested properties in real-world scenarios and is the focus of this paper. One of the oldest ways to construct a monotonic fully connected neural network is to constrain signs on its weights. Unfortunately, this construction does not work with popular non-saturated activation functions as it can only approximate convex functions. We show this shortcoming can be fixed by constructing two additional activation functions from a typical unsaturated monotonic activation function and employing each of them on the part of neurons. Our experiments show this approach of building monotonic neural networks has better accuracy when compared to other state-of-the-art methods, while being the simplest one in the sense of having the least number of parameters, and not requiring any modifications to the learning procedure or post-learning steps. Finally, we prove it can approximate any continuous monotone function on a compact subset of $\mathbb{R}^n$

Topological phases, non-equilibrium dynamics and parallels of black hole phenomena in condensed matter

This dissertation deals with two broad topics - Majorana modes in Kitaev chain and parallels of black hole phenomena in the quantum Hall effect. Majorana modes in topological superconductors are of fundamental importance as realizations of real solutions to the Dirac equation and for their anyonic exchange statistics. They are realised as zero energy edge modes in one-dimensional topological superconductors, modeled by the Kitaev chain Hamiltonian. Here an extensive study is made on the wavefunction features of these Majorana modes. It is shown that the Majorana wavefunction has two distinct features- a decaying envelope and underlying oscillations. The latter becomes important when one considers the coupling between the Majorana modes in a finite-sized chain. The coupled Majorana modes form a non-local Dirac fermionic state which determines the ground state fermion parity. The dependance of the fermion parity on the parameters of the system is purely determined by the oscillatory part of the Majorana wavefunctions. Using transfer matrix method, one can uncover a new boundary in the phase diagram, termed as `circle of oscillations', across which the oscillations in the wavefunction and the ground-state fermionic parity cease to exist. This is closely related to the circle that appears in the context of transverse field XY spin chain, within which the spin-spin correlations have oscillations. For a finite sized system, the circle is further split into mutliple ellipses called `parity sectors'. The parity oscillations have a scaling behaviour i.e oscillations for different superconducting gaps can be scaled to collapse to a single plot. Making use of results from random matrix theory for class D systems, one can also predict the robustness of certain features of fermion parity switches in the presence of disorder and comment on the critical properties of the MBS wavefunctions and level crossings near zero energy. These results could provide directions for making measurements on zero-bias conductance oscillations and the parameter range of operations for robust parity switches in realistic disordered system. On the front of non-equilibrium dynamics, the effect of Majorana modes on the dynamical evolution of the ground state under time variation of a Hamiltonian parameter is studied. The key result is the failure of the ground state to evolve into opposite parity sectors under the dynamical tuning of the system within the topological phase. This dramatic lack of adiabaticity is termed as ‘parity blocking’. A real-space time-dependent formalism is also developed using Pfaffian correlations, where simple momentum space methods fail. This formalism can be used for calculating the non-equilibrium quantities, such as adiabatic fidelity and the residual energy in a system with open boundaries. The consideration of Majorana modes in non-equilibrium dynamics lead to deviation from Kibble-Zurek physics and non-analyticities in adiabatic fidelity even within the topological phase. The second part of the thesis deals with uncovering structural parallels of black hole phenomena such as the Hawking-Unruh effect and quasinormal modes in quantum Hall systems. The Hawking-Unruh effect is the emergence of a thermal state when a vacuum of a quantum field theory on a given spacetime is restricted to a submanifold bounded by an event horizon. The thermal state manifests as Hawking radiation in the context of a black hole spacetime with an event horizon. The Unruh effect is a simpler example where a family of accelerating observers in Minkowski spacetime are confined by the lightcone structure and the Minkowski vacuum looks as a thermal state to them. The key element in understanding the Hawking-Unruh effect is the Rindler Hamiltonian or the boost. The boost acts as the generator of time translation for the quantum states in the Rindler wedge giving rise to thermality. In this thesis it is shown that due to an exact isomorphism between the Lorentz algebra in Minkowksi spacetime and the algebra of area preserving transformations in the lowest Landau level of quantum hall effect, an applied saddle potential acts as an equivalent to the Rindler Hamiltonian giving rise to a parallel of Hawking-Unurh effect. In the lowest Landau level, the saddle potential is reduced to the problem of scattering off an inverted harmonic oscillator(IHO) and the tunneling probability assumes the form of a thermal distribution. The IHO also has scattering resonances which are poles of the scattering matrix in the complex energy plane. The scattering resonances are states with time-decaying behavior and have purely incoming/outgoing probability current. These states are identified as quasinormal modes analogous to those occurring the scattering off an effective potential in black hole spacetimes. The quasinormal decay is an unexplored effect in quantum Hall systems and provides a new class of time-dependent probe of quantum Hall physics. The parallels between the relativistic symmetry generators and the potentials applied in the lowest Landau level also open up an avenue for studying Lorentz Kinematics and symplectic phase space dynamics in the lowest Landau level. These parallels open up new avenues of exploration in the quantum Hall effect