DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE MULTIPHASE FLOW WITH PHASE CHANGE

Phase change problems arise in many practical applications such as air-conditioning and refrigeration, thermal energy storage systems and thermal management of electronic devices. The physical phenomenon in such applications are complex and are often difficult to be studied in detail with the help of only experimental techniques. The efforts to improve computational techniques for analyzing two-phase flow problems with phase change are therefore gaining momentum. The development of numerical methods for multiphase flow has been motivated generally by the need to account more accurately for (a) large topological changes such as phase breakup and merging, (b) sharp representation of the interface and its discontinuous properties and (c) accurate and mass conserving motion of the interface. In addition to these considerations, numerical simulation of multiphase flow with phase change introduces additional challenges related to discontinuities in the velocity and the temperature fields. Moreover, the velocity field is no longer divergence free. For phase change problems, the focus of developmental efforts has thus been on numerically attaining a proper conservation of energy across the interface in addition to the accurate treatment of fluxes of mass and momentum conservation as well as the associated interface advection. Among the initial efforts related to the simulation of bubble growth in film boiling applications the work in \cite{Welch1995} was based on the interface tracking method using a moving unstructured mesh. That study considered moderate interfacial deformations. A similar problem was subsequently studied using moving, boundary fitted grids \cite{Son1997}, again for regimes of relatively small topological changes. A hybrid interface tracking method with a moving interface grid overlapping a static Eulerian grid was developed \cite{Juric1998} for the computation of a range of phase change problems including, three-dimensional film boiling \cite{esmaeeli2004computations}, multimode two-dimensional pool boiling \cite{Esmaeeli2004} and film boiling on horizontal cylinders \cite{Esmaeeli2004a}. The handling of interface merging and pinch off however remains a challenge with methods that explicitly track the interface. As large topological changes are crucial for phase change problems, attention has turned in recent years to front capturing methods utilizing implicit interfaces that are more effective in treating complex interface deformations. The VOF (Volume of Fluid) method was adopted in \cite{Welch2000} to simulate the one-dimensional Stefan problem and the two-dimensional film boiling problem. The approach employed a specific model for mass transfer across the interface involving a mass source term within cells containing the interface. This VOF based approach was further coupled with the level set method in \cite{Son1998}, employing a smeared-out Heaviside function to avoid the numerical instability related to the source term. The coupled level set, volume of fluid method and the diffused interface approach was used for film boiling with water and R134a at the near critical pressure condition \cite{Tomar2005}. The effect of superheat and saturation pressure on the frequency of bubble formation were analyzed with this approach. The work in \cite{Gibou2007} used the ghost fluid and the level set methods for phase change simulations. A similar approach was adopted in \cite{Son2008} to study various boiling problems including three-dimensional film boiling on a horizontal cylinder, nucleate boiling in microcavity \cite{lee2010numerical} and flow boiling in a finned microchannel \cite{lee2012direct}. The work in \cite{tanguy2007level} also used the ghost fluid method and proposed an improved algorithm based on enforcing continuity and divergence-free condition for the extended velocity field. The work in \cite{sato2013sharp} employed a multiphase model based on volume fraction with interface sharpening scheme and derived a phase change model based on local interface area and mass flux. Among the front capturing methods, sharp interface methods have been found to be particularly effective both for implementing sharp jumps and for resolving the interfacial velocity field. However, sharp velocity jumps render the solution susceptible to erroneous oscillations in pressure and also lead to spurious interface velocities. To implement phase change, the work in \cite{Hardt2008} employed point mass source terms derived from a physical basis for the evaporating mass flux. To avoid numerical instability, the authors smeared the mass source by solving a pseudo time-step diffusion equation. This measure however led to mass conservation issues due to non-symmetric integration over the distributed mass source region. The problem of spurious pressure oscillations related to point mass sources was also investigated by \cite{Schlottke2008}. Although their method is based on the VOF, the large pressure peaks associated with sharp mass source was observed to be similar to that for the interface tracking method. Such spurious fluctuation in pressure are essentially undesirable because the effect is globally transmitted in incompressible flow. Hence, the pressure field formation due to phase change need to be implemented with greater accuracy than is reported in current literature. The accuracy of interface advection in the presence of interfacial mass flux (mass flux conservation) has been discussed in \cite{tanguy2007level,tanguy2014benchmarks}. The authors found that the method of extending one phase velocity to entire domain suggested by Nguyen et al. in \cite{nguyen2001boundary} suffers from a lack of mass flux conservation when the density difference is high. To improve the solution, the authors impose a divergence-free condition for the extended velocity field by solving a constant coefficient Poisson equation. The approach has shown good results with enclosed bubble or droplet but is not general for more complex flow and requires additional solution of the linear system of equations. In current thesis, an improved approach that addresses both the numerical oscillation of pressure and the spurious interface velocity field is presented by featuring (i) continuous velocity and density fields within a thin interfacial region and (ii) temporal velocity correction steps to avoid unphysical pressure source term. Also I propose a general (iii) mass flux projection correction for improved mass flux conservation. The pressure and the temperature gradient jump condition are treated sharply. A series of one-dimensional and two-dimensional problems are solved to verify the performance of the new algorithm. Two-dimensional and cylindrical film boiling problems are also demonstrated and show good qualitative agreement with the experimental observations and heat transfer correlations. Finally, a study on Taylor bubble flow with heat transfer and phase change in a small vertical tube in axisymmetric coordinates is carried out using the new multiphase, phase change method.

Self-Organizing Map Neural Architectures Based on Limit Cycle Attractors

Recent efforts to develop large-scale neural architectures have paid relatively little attention to the use of self-organizing maps (SOMs). Part of the reason is that most conventional SOMs use a static encoding representation: Each input is typically represented by the fixed activation of a single node in the map layer. This not only carries information in an inefficient and unreliable way that impedes building robust multi-SOM neural architectures, but it is also inconsistent with rhythmic oscillations in biological neural networks. Here I develop and study an alternative encoding scheme that instead uses limit cycle attractors of multi-focal activity patterns to represent input patterns/sequences. Such a fundamental change in representation raises several questions: Can this be done effectively and reliably? If so, will map formation still occur? What properties would limit cycle SOMs exhibit? Could multiple such SOMs interact effectively? Could robust architectures based on such SOMs be built for practical applications? The principal results of examining these questions are as follows. First, conditions are established for limit cycle attractors to emerge in a SOM through self-organization when encoding both static and temporal sequence inputs. It is found that under appropriate conditions a set of learned limit cycles are stable, unique, and preserve input relationships. In spite of the continually changing activity in a limit cycle SOM, map formation continues to occur reliably. Next, associations between limit cycles in different SOMs are learned. It is shown that limit cycles in one SOM can be successfully retrieved by another SOM’s limit cycle activity. Control timings can be set quite arbitrarily during both training and activation. Importantly, the learned associations generalize to new inputs that have never been seen during training. Finally, a complete neural architecture based on multiple limit cycle SOMs is presented for robotic arm control. This architecture combines open-loop and closed-loop methods to achieve high accuracy and fast movements through smooth trajectories. The architecture is robust in that disrupting or damaging the system in a variety of ways does not completely destroy the system. I conclude that limit cycle SOMs have great potentials for use in constructing robust neural architectures.

Language-based Techniques for Practical and Trustworthy Secure Multi-party Computations

Secure Multi-party Computation (MPC) enables a set of parties to collaboratively compute, using cryptographic protocols, a function over their private data in a way that the participants do not see each other's data, they only see the final output. Typical MPC examples include statistical computations over joint private data, private set intersection, and auctions. While these applications are examples of monolithic MPC, richer MPC applications move between "normal" (i.e., per-party local) and "secure" (i.e., joint, multi-party secure) modes repeatedly, resulting overall in mixed-mode computations. For example, we might use MPC to implement the role of the dealer in a game of mental poker -- the game will be divided into rounds of local decision-making (e.g. bidding) and joint interaction (e.g. dealing). Mixed-mode computations are also used to improve performance over monolithic secure computations. Starting with the Fairplay project, several MPC frameworks have been proposed in the last decade to help programmers write MPC applications in a high-level language, while the toolchain manages the low-level details. However, these frameworks are either not expressive enough to allow writing mixed-mode applications or lack formal specification, and reasoning capabilities, thereby diminishing the parties' trust in such tools, and the programs written using them. Furthermore, none of the frameworks provides a verified toolchain to run the MPC programs, leaving the potential of security holes that can compromise the privacy of parties' data. This dissertation presents language-based techniques to make MPC more practical and trustworthy. First, it presents the design and implementation of a new MPC Domain Specific Language, called Wysteria, for writing rich mixed-mode MPC applications. Wysteria provides several benefits over previous languages, including a conceptual single thread of control, generic support for more than two parties, high-level abstractions for secret shares, and a fully formalized type system and operational semantics. Using Wysteria, we have implemented several MPC applications, including, for the first time, a card dealing application. The dissertation next presents Wys*, an embedding of Wysteria in F*, a full-featured verification oriented programming language. Wys* improves on Wysteria along three lines: (a) It enables programmers to formally verify the correctness and security properties of their programs. As far as we know, Wys* is the first language to provide verification capabilities for MPC programs. (b) It provides a partially verified toolchain to run MPC programs, and finally (c) It enables the MPC programs to use, with no extra effort, standard language constructs from the host language F*, thereby making it more usable and scalable. Finally, the dissertation develops static analyses that help optimize monolithic MPC programs into mixed-mode MPC programs, while providing similar privacy guarantees as the monolithic versions.

Nested Sampling and its Applications in Stable Compressive Covariance Estimation and Phase Retrieval with Near-Minimal Measurements

Compressed covariance sensing using quadratic samplers is gaining increasing interest in recent literature. Covariance matrix often plays the role of a sufficient statistic in many signal and information processing tasks. However, owing to the large dimension of the data, it may become necessary to obtain a compressed sketch of the high dimensional covariance matrix to reduce the associated storage and communication costs. Nested sampling has been proposed in the past as an efficient sub-Nyquist sampling strategy that enables perfect reconstruction of the autocorrelation sequence of Wide-Sense Stationary (WSS) signals, as though it was sampled at the Nyquist rate. The key idea behind nested sampling is to exploit properties of the difference set that naturally arises in quadratic measurement model associated with covariance compression. In this thesis, we will focus on developing novel versions of nested sampling for low rank Toeplitz covariance estimation, and phase retrieval, where the latter problem finds many applications in high resolution optical imaging, X-ray crystallography and molecular imaging. The problem of low rank compressive Toeplitz covariance estimation is first shown to be fundamentally related to that of line spectrum recovery. In absence if noise, this connection can be exploited to develop a particular kind of sampler called the Generalized Nested Sampler (GNS), that can achieve optimal compression rates. In presence of bounded noise, we develop a regularization-free algorithm that provably leads to stable recovery of the high dimensional Toeplitz matrix from its order-wise minimal sketch acquired using a GNS. Contrary to existing TV-norm and nuclear norm based reconstruction algorithms, our technique does not use any tuning parameters, which can be of great practical value. The idea of nested sampling idea also finds a surprising use in the problem of phase retrieval, which has been of great interest in recent times for its convex formulation via PhaseLift, By using another modified version of nested sampling, namely the Partial Nested Fourier Sampler (PNFS), we show that with probability one, it is possible to achieve a certain conjectured lower bound on the necessary measurement size. Moreover, for sparse data, an l1 minimization based algorithm is proposed that can lead to stable phase retrieval using order-wise minimal number of measurements.