This thesis investigates a new approach to lattice basis reduction suggested by M. Seysen. Seysen's algorithm attempts to globally reduce a lattice basis, whereas the Lenstra, Lenstra, Lovasz (LLL) family of reduction algorithms concentrates on local reductions. We show that Seysen's algorithm is well suited for reducing certain classes of lattice bases, and often requires much less time in practice than the LLL algorithm. We also demonstrate how Seysen's algorithm for basis reduction may be applied to subset sum problems. Seysen's technique, used in combination with the LLL algorithm, and other heuristics, enables us to solve a much larger class of subset sum problems than was previously possible.

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We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.

Knowledge of energy dissipation and relaxation in electron, spin, and lattice degrees of freedom is of fundamental importance from both a technological and scientific point of view. In this dissertation, the electronic and magnetization dynamics of photoexcited colloidal cobalt substituted iron oxide nanocrystals, CoxFe3-xO4, were investigated through transient absorption and pump-probe Faraday rotation measurements. In this dissertation, linearly polarized femtosecond optical pulses at 780 nm were used to excite the weak absorption originating from the intervalence charge transfer transition (IVCT) between Fe2+ and Fe3+ ions of Fe3O4 nanocrystals. The timescale and corresponding relaxation processes of electronic relaxation dynamics of the excited IVCT state were first discussed. Size effect on electronic relaxation dynamics in Fe3O4 nanocrystals is not distinct on the basis of result from this study. One interesting feature of electronic dynamics data of photoexcited Fe3O4 nanocrystals is the creation of coherent acoustic phonons. Information on lattice temperature was obtained by measuring the period of coherent acoustic phonon as a function of excitation fluence and fit into a simple model based on Lamb’s theory. Since optical control of the magnetization can be either through optical or heating mechanisms, quantitative estimation of degree of demagnetization caused by lattice temperature is made by using Langevin function. The result from such estimation indicates the effect of lattice temperature rise on magnetization is too small to significantly affect the magnetization of Fe3O4 nanocrystals. Magnetization dynamics were studied via pump-probe Faraday rotation measurements. Optical excitation with near-infrared pulse resulted in an ultrafast demagnetization in 100fs. The energy of the excited state then relaxed through spin-lattice relaxation (SLR). Effects of surface spin and chemical tuning on the SLR were investigated by comparing the magnetization recovery timescales of nanocrystal with different particle sizes and cobalt concentration respectively. The experimental result is explained by a simple model where interior and surface spins contributed to the spin-lattice relaxation process differently. The observations suggest that spin-orbit coupling of the surface is stronger and less sensitive to stoichiometric variation than the interior spins of the nanocrystals.

This thesis deals with the lattice-Boltzmann method (LBM) in combination with other methods to solve thermal flow problems. The three primary, current approaches for thermal lattice-Boltzmann method (TLBM) will be introduced. The three approaches are the multispeed approach by McNamara and Alder , the passive scalar approach by Shan, and the thermal distribution model proposed by He et al. Shi et al. simplified the thermal distribution model for incompressible thermal flows. The model proposed by Shi et al. was simulated and compared to a hybrid LBM and energy equation model proposed by Khiabani et al. The thermal lattice-Boltzmann method will be compared to the temperature fields generated by the energy equation of the hybrid method. To determine which method is better suited from computer simulations the two will be compared for computational demands, and the speed of both convergence and computation.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references and index.

Particle suspensions are common both in nature and in various technological applications. The complex nature of hydrodynamic interactions between particles and the solvent makes such analysis difficult that often requires numerical modeling to understand the behavior of particle suspensions. In this dissertation, we employ a hybrid computational model that integrates a lattice spring model for solid mechanics and a lattice Boltzmann model for fluid dynamics. We use this model to study several practical problems in which the dynamics of spherical and spheroidal particles and deformable capsules in dilute suspensions plays an important role. The results of our studies yield new information regarding the dynamics of solid particle in pressure-driven channel flows and disclose the nonlinear effects associated with fluid inertia leading to particle cross-stream migration. This information not only give us a fundamental insight into the dynamics of dilute suspensions, but also yield engineering guidelines for designing high throughput microfluidic devices for sorting and separation of synthetic particles and biological cells. We first demonstrate that spherical particles can be size-separated in ridged microchannels. Specifically, particles with different sizes follow distinct trajectories as a result of the nonlinear inertial effects and secondary flows created by diagonal ridges in the channel. Then, separation of biological cells by their differential stiffness is studied and compared with experimental results. Cells with different stiffness squeeze through narrow gaps between solid diagonal ridges and channel wall, and migrate across the microchannel with different rates depending on their stiffness. This deformability-based microfluidic platform may be valuable for separating diseased cells from healthy cells, as a variety of cell pathologies manifest through the change in mechanical cell stiffness. Finally, the dynamics of spheroid particles in simple shear and Poiseuille flows are studied. Stable rotational motion, cross-stream migration, and equilibrium trajectories of non-spherical particles in flow are investigated. Effects of particle and fluid inertia on dynamics of particles are disclosed. The dependence of equilibrium trajectory on particle shape reveals a potential application for shape based particle separation.

Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms. We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings. Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing. Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.

The advent of rapid manufacturing has enabled the realization of countless products that have heretofore been infeasible. From customized clear braces to jet fighter ducts and one-off dental implants, rapid manufacturing allows for increased design complexity and decreased manufacturing costs. The manufacturing capabilities of this process have evolved to the point that they have surpassed current design capabilities. Meso-scale lattice structures can now be built that contain more lattice struts than it is reasonable to efficiently define. This work has attempted to create a method for designing such lattice structures that is efficient enough to allow for the design of large or complex problems. The main hindrance to the design of complex meso-scale lattice problems is essentially the need to define the strut diameters. While it is obvious that a large design would contain more struts than can be specified by hand, designs also quickly surpass the current capabilities of computational optimization routines. To overcome this problem, a design method has been developed that uses a unit-cell library correlated to finite element analysis of the bounding geometry to tailor the structure to the anticipated loading conditions. The unit-cell library is a collection of base lattice primitives, or unit-cells, that have been specialized for certain applications. In this case, primitives have been created that perform best under the types of stress analyzed by finite element analysis. The effectiveness of this process has been demonstrated through several example problems. In all cases, the unit-cell library approach was able to create structures in less time than current methods. The resulting structures had structural performance slightly lower than similar models created through optimization methods, although the extent of this degradation was slight. The method developed in this work performs extremely well, and is able to create designs for even the most complex lattice structures. There is room for future development, however, in the streamlining of the design process and consideration of higher-order affects within unit-cells.

Advances in additive manufacturing technologies have brought a new paradigm shift to both design and manufacturing. There is a much bigger design space in which designers can achieve a level of complexity and customizability, which are infeasible using traditional manufacturing processes. One application of this technology is for fabrication of meso-scale lattice structures (MSLS). These types of structures are designed to have material where it is needed for specific applications. They are suitable for any weight-critical applications, particularly in industries where both low weight and high strength are desired. MSLS can easily have hundreds to thousands of individual strut, where the diameter of each strut can be treated as a design variable. As a result, the design process poses a computational challenge. Since the computational complexity of the design problem often scales exponentially with the number of design variables, topological optimization that requires multi-variable optimization algorithm is infeasible for large-scale problems. In previous research, a new method was presented for efficiently optimizing MSLS by utilizing a heuristic that reduces the multivariable optimization problem to a problem of only two variables. The method is called the Size Matching and Scaling (SMS) method, which combines solid-body analysis and predefined unit-cell library to generate the topology of the structure. However, the method lacks a systematic methodology to generate the initial ground geometry for the design process, which limits the previous implementations of the SMS method to only simple, axis-aligned structures. In this research, an augmented SMS method is presented. The augmented method includes the integration of free-mesh approach in generating the initial ground geometry. The software that embodies that ground geometry generation process is integrated to commercial CAD system that allows designer to set lattice size parameters through graphical user interface. In this thesis, the augmented method and the unit-cell library are applied to various design examples. The augmented SMS method can be applied effectively in the design of conformal lattice structure with highly optimized stiffness and volume for complex surface. Conformal lattice structures are those conformed to the shape of a part's surface and that can used to stiffen or strengthen a complex and curved surface. This design approach removes the need for a rigorous topology optimization, which is a main bottleneck in designing MSLS.