In this thesis, I analyze the weight functions used in moving least squares (MLS) methods to construct implicit surfaces that interpolate or approximate polygon soup. I found that one previous method that presented an analytic solution to the integrated moving least squares method has issues with degeneracies because they changed the weight functions to decrease too slowly. Inspired by their method, I derived a bound for the choice of weight function for implicit moving least squares (IMLS) methods to avoid these degeneracies in two-dimensions and in three-dimensions. Based on this bound, I give a theoretical proof of the correctness of the moving least squares interpolation and approximation scheme with weight function used in Shen et al. when used on closed polyhedrons. Further, previous IMLS implicit surface reconstruction algorithms that ll holes and gaps create surfaces with obvious bulges due to an intrinsic property of MLS. I propose a generalized IMLS method using a Gaussian distribution function to re-weight each polygon, making nearer polygons dominate and reducing the bulges on holes and gaps.

Carbon Capture and Storage (CCS) is recognised as having a significant role to play in tackling climate change and reducing carbon dioxide (CO2) emissions. In CCS schemes, CO2 is captured from anthropogenic sources, and transported to suitable sites either for EOR (Enhanced Oil Recovery) or storage. The transport of such huge amount of CO2 causes new challenges. The main concern is in the difference between natural gas and CO2 transportation pipelines. CO2 phase behaviour during decompression, existence of different impurities and very high operating pressure are some of the new challenges for pipeline designer and operators. This PhD study has taken a systematic approach to understand the mechanics involved in the fracture of pipes containing high pressure flue-gas CO2. The work involved the development of a novel weight function stress intensity factor solution that can be used with complex stress fields induced by residual and/or thermal stresses in addition to applied pressure. In addition, the thesis reports a substantial experimented test programme which involved low temperature fracture toughness tests linked to a detailed finite element based stress analysis. Overall, the thesis presents an integrated engineering criticality means to assess the suitability or otherwise of a pipeline system to transport high pressure flue-gas CO2.

Electromagnetic flowmeters (known as magnetic flow meters) are a widely used type of flowmeter. The accuracy of magnetic flow meters are a function of several factors, not the least of which is the flow condition inside the pipe. It has been shown that disturbances in the velocity profile affects the accuracy of a magnetic flow meter (Luntta, 1998). Accordingly, manufacturers of magnetic flow meters give installation guidelines. These guidelines help prevent the user from installing the meter in a pipe configuration that is likely to cause the meter to produce inaccurate results. Although most manufacturers provide recommendations about the amount of straight pipe that is necessary upstream of the meter, little is said about the orientation of the meter in relation to upstream disturbances. This study examines the performance of magnetic flow meters when positioned at two different orientations: EIP (electrodes in plane with an upstream 90-degree short radius elbow) and EOP (electrodes out of plane). Four different meters were included in the study in which a baseline straight pipe test was first performed using over fifty diameters of straight pipe upstream of each meter. The straight pipe test was used to determine the baseline accuracy of each of the meters over a velocity range that is typical for the size and function of the meters. Meters were then installed at five different locations downstream from a 90-degree short-radius elbow. At each location the meters were tested in two orientations at five different flow rates. The intent of the research is to show that the orientation of a magnetic flow meter affects the meter’s ability to produce accurate flow readings when it is installed downstream of a flow disturbance. The results from this research showed a significant shift in measurement accuracy when the meter was in EIP and EOP orientations. All of the meters in the study produced accuracy readings at one point of another that were outside the specified accuracy from the meter manufacturer. Interestingly, the meters that had a larger manufacturer specified accuracy produced smaller shifts in accuracy when comparing the test results under EIP and EOP conditions. The results of the research are given in the section entitled “Results and Discussion” as well as in the Appendix A.

In this thesis we study Wiener’s lemma. The classical version of the lemma, whose realm is a Banach algebra, asserts that the pointwise inverse of a nonzero function with absolutely convergent Fourier expansion, also possesses an absolutely convergent Fourier expansion. The main purpose of this thesis is to investigate the validity inalgebras endowed with a quasi-norm or a p-norm.As a warmup, we prove the classical version of Wiener’s lemma using elemen-tary analysis. Furthermore, we establish results in Banach algebras concerning spectral theory, maximal ideals and multiplicative linear functionals and present a proof Wiener’s lemma using Banach algebra techniques. Let ν be a submultiplicative weight function satisfying the Gelfand-Raikov-Shilov condition. We show that if a nonzero function f has a ν-weighted absolutely convergent Fourier series in a p-normed algebra A. Then 1/f also has a ν-weightedabsolutely convergent Fourier series in A.

There has been considerable recent interest in geometric function theory, nonlinear partial differential equations, harmonic mappings, and the connection of these to minimal energy phenomena. This work explores Nitsche's 1962 conjecture concerning the nonexistence of harmonic mappings between planar annuli, cast in terms of distortion functionals. The connection between the Nitsche problem and the famous Grötzsch problem is established by means of a weight function. Traditionally, these kinds of problems are investigated in the class of quasiconformal mappings, and the assumption is usually made a priori that solutions preserve various symmetries. Here the conjecture is solved in the much wider class of mappings of finite distortion, symmetry-preservation is proved, and ellipticity of the variational equations concerning these sorts of general problems is established. Furthermore, various alternative interpretations of the weight function introduced herein lead to an interesting analysis of a much wider variety of critical phenomena -- when the weight function is interpreted as a thickness, density or metric, the results lead to a possible model for tearing or breaking phenomena in material science. These physically relevant critical phenomena arise, surprisingly, out of purely theoretical considerations.

The objective of this Master’s thesis is to find shape optimal design based on min- imizing friction force of thrust bearing by using genetic algorithm(GA) which is one of an optimization toolbox in Matlab. Reducing the friction force of thrust bearing is one way of making shaft to decreasing friction losses. With four parameters of thrust bearing geometry number of segments(m), angle of running surface(), segment inner radius(R0), and segment outer radius(R1) substitute in Reynolds’ equation. In order to know friction force, it is necessary to generate a connecting variable, oil film thickness(h0) from loading capacity(W ) and revolution per minute(rpm). Friction power loss, as well as weight func- tion conclude the final shape optimization of thrust bearing: m = 7, = 0.1, R0 = 15 mm, and R1 = 20 mm.

A weight function based modified strip-yield model is adapted to include the mode I effects of roughness induced crack closure. The weight function methodology is verified for arbitrary mixed mode loading conditions under load levels experienced in fatigue crack growth. A rough crack geometry is simulated as a sine wave fitted to a sawtooth defined by an asperity angle and period. Additional mode I crack closure due to mode II sliding of crack faces is calculated and incorporated into the strip-yield model by lengthening and shortening the crack face elements. Simulations of fatigue crack growth using the newly developed model are examined via a design and analysis of computer experiments. The effects of model parameters are identified. Combined roughness and plasticity induced crack closure for long cracks is studied.

We give a short introduction to various concepts related to the theory of p-harmonic functions on Rn, and some modern generalizations of these concepts to general metric spaces. The article Björn-Björn-Lehrbäck [6] serves as the starting point of our discussion. In [6], among other things, estimates of the variational capacity for thin annuli in metric spaces are given under the assumptions of a Poincaré inequality and an annular decay property. Most of the parameters in the various results of the article are proven to be sharp by counterexamples at the end of the article. The main result of this thesis is the verification of the sharpness of a parameter. At the center of our discussion will be a concrete metric subspace of weighted Rn, namely the so-called weighted bow-tie, where the weight function is assumed to be radial. A similar space was used in [6] to verify the sharpness of several parameters. We show that under the assumption that the variational p-capacity is nonzero for any ball centered at the origin, the p-Poincaré inequality holds in Rn if and only if it holds on the corresponding bow-tie Finally, we consider a concrete weight function, show that it is a Muckenhoupt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].

Textilbeton ist ein Verbundwerkstoff aus einer Feinbetonmatrix und einer textilen Bewehrung, die aus sogenannten Rovings besteht. Nach Reißen der spröden Matrix sind diese Rovings in der Lage, die Risse in der Matrix zu überbrücken. In diesem Beitrag wird ein analytisches Modell vorgestellt, welches den Einfluss der Rovings auf das Rissverhalten erfassen kann. Die Wirkung der Fasern wird durch rissüberbrückende Spannungen approximiert. Dabei kommt unter Annahme linear elastischen Materialverhaltens das bruchmechanische Konzept der Methode der Gewichtsfunktionen zum Einsatz. Als ein spezielles Anwendungsgebiet des vorgestellten bruchmechanischen Konzeptes werden Risse im Bereich von Übergreifungen der textilen Bewehrung untersucht. Hier kann es, abhängig von der vorhandenen Übergreifungslänge, zum Auszug der Rovings kommen. Um diesen Versagensmechanismus zu verhindern, ist eine Mindestübergreifungslänge erforderlich.

Two problems related to orthogonal polynomials and special functions are considered. For q greater than 1 it is known that continuous q-Jacobi polynomials are orthogonal on the imaginary axis. The first problem is to find proper normalization to form a system of polynomials that are orthogonal on the real line. By introducing a degree reducing operator and a scalar product one can show that the normalized continuous q-Jacobi polynomials satisfies an eigenvalue equation. This implies orthogonality of the normalized continuous q-Jacobi polynomials. As a byproduct, different results related to the normalized system of polynomials, such as its closed form,three-term recurrence relation, eigenvalue equation, Rodrigues formula and generating function will be computed. A discriminant related to the normalized system is also obtained. The second problem is related to recent results of Dilcher and Stolarky on resultants of Chebyshev polynomials. They used algebraic methods to evaluate the resultant of two combinations of Chebyshev polynomials of the second kind. This work provides an alternative method of computing the same resultant and also enables one to compute resultants of more general combinations of Chebyshev polynomials of the second kind. Resultants related to combinations of Chebyshev polynomials of the first kind are also considered.