The scalar wave equation is investigated for a scalar field propagating in a spacetime background ds²=e^{2a}(-dt²+dr²)+R(e^{-2ψ}dφ²+e^{2ψ}dz²). The metric is compactified in the radial direction. The spacetime slices of constant φ and z are foliated into outgoing null hypersurfaces by the null coordinate transformation u=t-r. The scalar field imitates the amplitude behavior of a light ray, or a gravitational wave, traveling along a null hypersurface when the area function R is a constant or is a function of u. These choices for R restrict the gravitational wave factor ψ to being an arbitrary function of u.

Since their conception by Collins in 1975, Cylindrical Algebraic Decompositions (CADs) have been used to analyse the real algebraic geometry of systems of polynomials. Applications for CAD technology range from quantifier elimination to robot motion planning. Although of great use in practice, the CAD algorithm was shown to have doubly exponential complexity with respect to the number of variables for the problem, which limits its use for large examples. Due to the high complexity of CAD, much work has been done to improve its performance. In this thesis new advances will be discussed that improve the practical efficiency of CAD for a variety of problems, with a new complexity result for one set of algorithms. A new invariance condition, truth table invariance (TTICAD), and two algorithms to construct TTICADs are given and shown to be highly efficient. The idea of restricting the output of CADs, allowing for greater efficiency, is formalised as sub-decompositions and two particular ideas are investigated in depth. Efficient selection of various formulation choices for a CAD problem are discussed, with a collection of heuristics investigated and machine learning applied to assist in choosing an optimal heuristic. The mathematical expression of a problem is shown to be of great importance, with preconditioning and reformulation investigated. Finally, these advances are collected together in a general framework for applying CAD in an efficient manner to a given problem. It is shown that their combination is not cumulative and care must be taken. To this end, a prototype software CADassistant is described to help users take advantage of the advances without knowledge of the underlying theory. The effects of the various advances are demonstrated through a guiding example originally considered by Solotareff, which describes the approximation of a cubic polynomial by a linear function. Naïvely applying CAD to the problem takes 916.1 seconds of construction (from which a solution can easily be derived), which is reduced to 20.1 seconds by combining various advances from this thesis.

This paper presents the layout optimization of passive constrained layer damping (PCLD) treatment for vibration control of cylindrical shells under a broadband force excitation. The equations governing the vibration responses are derived using the energy approach and assumed-mode method. These equations provided relationship between the integrated displacement response over the whole structural volume, i.e. the structural volume displacement (SVD), of a cylindrical shell to structural parameters of base structure and multiple PCLD patches, Genetic algorithms (GAs) based penalty function method is employed to find the optimal layout of rectangular PCLD patches with minimize the maximum displacement response of PCLD-treated cylindrical shells. Optimization solutions of PCLD patches’ locations and shape are obtained under the constraint of total amount of PCLD in terms of percentage added weight to the base structure. Examination of the optimal layouts reveals that the patches tend to increase their coverage in the axial direction and distribute over the whole surface of the cylindrical shell for optimal control of the structural volume displacement. / Singapore-MIT Alliance (SMA)

The interaction of nanoparticles with biological systems, especially interactions with cell membranes, has been a subject of active research due to its numerous applications in many areas of soft-matter and biological systems. Within only a few relevant physical parameters profound structural properties have been discovered in the context of simple coarse-grained theoretical models. In this Thesis we study the structure of a tubular membrane adhering to two rigid cylindrical particles on a basis of a free-energy model that uses Helfrich energy for the description of the membrane. A numerical procedure is developed to solve the shape equations that determine the state of lowest energy. Several phase transitions exist in the system, arising from the competition between the bending energy of the membrane and the adhesion energy between the membrane and the particles. A continuous adhesion transition between the free and bound states, as well as several discontinuous shape transitions are identified, depending on the physical parameters of the system. The results are then generalized into a single phase diagram separating free, symmetric- and asymmetric-wrapping states in the phase space of the size of the particles and the adhesion energy. We show that for a relatively small size of the membrane tube the interaction between the cylinders becomes attractive in the strong curvature regime, leading to aggregation of the particles in the highly curved area of the tube that is characteristically different from the aggregation in a related three-dimensional system. For a relatively large membrane tube size the cylinders prefer to have a non-zero separation, even in the completely engulfed state. This indicates that, i) the spontaneous curvature of the membrane may play a role in the sign of the interaction of two colloidal particles adhered to a membrane and ii) cylindrical particles can aggregate on membrane tubes and vesicles if the curvature of the membrane around the aggregation region is sufficiently large.

In this study, the energy conservation equation in a cylindrical coordinate system and the moving heat source from the tool are used to establish a steady-state three-dimensional heat transfer model for the friction stir welding (FSW). Then, the simplified momentum conservation equation is employed to predict the material flow model for the FSW. Combining the effects of heat transfer and material flow, this numerical model successfully predicts the weld temperature field and the material flow for the FSW. Numerical results show that increasing the welding or translational speed of the tool has the effect of decreasing the magnitude of the temperature within the workpiece, while increasing the rotating speed has the opposite effect. During the feeding process, the material located on the back of the tool pin has higher temperature than that on the front. Moreover, the temperature profile are asymmetrical between the advancing and retreating sides due to the material flow stirred by the tool, and this temperature difference depends on the speed of material flow under the tool shoulder.

This thesis is aimed to apply cylindrical polishing system to a large flat workpiece (ex:wafer) to obtain high degree of planarization, low surface roughness and no crack layer. First, a mathematical model is presented which describes the axially symmetric form error compensation by cylindrical polishing process. The dwelling time-distribution of tool and the machining depth distribution are solved using the methods of simultaneous equations and least square error with non-negativity constraints. Then, using the simulation analysis and experimental method to examine the machining rate distribution effects on machining precision and the dwelling time-distribution of the tool when the workpiece is machining by cylindrical tool. The examined effects will include machining length effect, boundary effect and inclined effect. Under the range of the machining precision required, the approach to smooth the dwelling time-distribution of tool that will benefit layer-by-layer removing strategy. After these analyses, an adequacy dwelling time-distribution of the tool can be designed according to the workpiece form error. In addition, the relative between the machining probability due to the an-isotropic polishing property and surface roughness will be discussed. The contents of this thesis include three parts. First, a mathematical model is presented which describes the axially symmetric form error compensation by cylindrical polishing process. And, the polishing angles and polishing probability of the an-isotropic polishing property will be identified according to the mathematical model. Second, the design of dwelling time-distribution of tool and the analysis of the geometric effects will be discussed by computer simulation. Third, the experimental results will show the suitable of the machining strategy and compare the machining effects with the simulated results. And, the planarization of the workpiece that has axially symmetric form error will be done.

This work was motivated by questions raised from multifrequency microwave interferometer measurements taken in a cylindrical plasma on the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) project. Standard data analysis based on a thin beam model neglecting refraction yields inconsistent electron densities and density profiles for different frequencies. This work focuses on the development of a model for the wave propagation through cylindrical plasmas when the plasma radius is on the order of the beam waist. For the calculations presented a Gaussian beam profile and plasma spatial profile were assumed. Both refraction by density gradients and finite beam sizes are found to play important roles and explain polychromatic differences in the electron densities and profiles. Calculations for the new model are compared to a thin beam model not accounting for refraction and experimental data from VASIMR.

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No description available.

Polyferrocenylsilane (PFS) diblock copolymers self-assemble in selective solvents to form one-dimensional micelles for a broad range of polymer compositions and experimental conditions, driven by the crystallization of the PFS block that forms the micelle core. The most striking feature of these micelles is that they remain active for further growth. They can be extended in length when additional polymer, dissolved in a good solvent, is added to a solution of the pre-existing micelles. This thesis describes several studies investigating the factors that affect the growth and fragmentation of PFS diblock copolymer micelles in solution, with a particular emphasis on polyisoprene-PFS (PI-PFS) diblock copolymers. The goal of my research was trying to provide deeper understanding of this crystallization-driven self-assembly (CDSA) process. In an attempt to understand the growth kinetics of the PI-PFS cylindrical micelles, I added tiny amount of short micelle seeds into supersaturated solution of the same polymer, and followed the micelle growth by light scattering. The data analysis showed that the increase of micelle length could be described by an expression with two exponential decay terms. In another attempt to examine the factors that may affect the growth behavior of the PI-PFS micelles, I found that PI-PFS long micelles underwent fragmentation when they were subjected to external stimuli, e.g. addition of polar solvent, or heating. During the course of studying the effect of heating on the micelles, I developed a new approach to generate cylindrical micelles with controllable and uniform length, a one-dimensional analogue of self-seeding of crystalline polymers. I carried out a systematic study to investigate the self-seeding behavior of PFS block copolymers.