This paper introduces a method to determine the pressure in a fixed thickness, smooth, periodic domain; namely a lead-over-pleat cartridge filter. Finding the pressure within the domain requires the numerical solution of Laplace's equation, the first step of which is approximating, by interpolation, the curved portions of the filter to a circle in the xy plane.A conformal map is then applied to the filter, transforming the region into a rectangle in the uv plane. A finite difference method is introduced to numerically solve Laplace's equation in the rectangular domain. There are currently methods in existence to solve partial differential equations on non- regular domains. In a method employed by Monchmeyer and Muller, a scheme is used to transform from cartesian to spherical polar coordinates. Monchmeyer and Muller stress that for non-linear domains, extrapolation of existing cartesian difference schemes may produce incorrect solutions, and therefore, a volume centered discretization is used. A difference scheme is then derived that relies on mean values. This method has second order accuracy.(Rosenfeld,Moshe, Kwak, Dochan, 1989) The method introduced in this paper is based on a 7-point stencil which takes into account the unequal spacing of the points. From all neighboring pairs, a linear system of equations is constructed, which takes into account the periodic domain.This method is solved by standard iterative methods. The solution is then mapped back to the original domain, with second order accuracy. The method is then tested to obtain a solution to a domain which satisfies $y=sin(x)$ at the center, a shape similar to that of a lead-over-pleat cartridge filter. As a result, a model for the pressure distribution within the filter is obtained.

In mathematical physics, Laplace's equation plays an especially significant role. It is fundamental to the solution of problems in electrostatics, thermodynamics, potential theory and other branches of mathematical physics. It is for this reason that this investigation concerns the development of some general properties of continuous solutions of this equation.

Cracked beam problem is an elliptic boundary value problem with singularity. It is often used as a testing model for numerical methods. We use numerical and symbolic boundary approximation methods and boundary collocation method to compute its extremely high accurate solution with global error $O(10^{-100})$. This solution then can be regarded as the exact solution. On the other hand, we vary the boundary conditions of this problem to obtain several related models. Their numerical solutions are compared to those of cracked beam and Motz problems, the prototypes of singularity problems. From the comparison we can conclude the advantage of each model and decide the best testing model for numerical methods.

No description available.

We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.

For the method of fundamental solutions(MFS), many reports deal with 2D problems. Since the MFS is more advantageous for 3D problems, this thesis is devoted to Laplace's equation in 3D problems. Since the fundamental solutions(FS) £X(x,y)=1/(4£k||x-y||), x,y∈R^3 are known, the location of source points is important in real computation. In this thesis, we choose a cylinder as the solution domain, and the source points on larger cylinders and spheres. Numerical results are reported, to draw some useful conclusions. The theoretical analysis will be explored in the future.

abstract: In blindness research, the corpus callosum (CC) is the most frequently studied sub-cortical structure, due to its important involvement in visual processing. While most callosal analyses from brain structural magnetic resonance images (MRI) are limited to the 2D mid-sagittal slice, we propose a novel framework to capture a complete set of 3D morphological differences in the corpus callosum between two groups of subjects. The CCs are segmented from whole brain T1-weighted MRI and modeled as 3D tetrahedral meshes. The callosal surface is divided into superior and inferior patches on which we compute a volumetric harmonic field by solving the Laplace's equation with Dirichlet boundary conditions. We adopt a refined tetrahedral mesh to compute the Laplacian operator, so our computation can achieve sub-voxel accuracy. Thickness is estimated by tracing the streamlines in the harmonic field. We combine areal changes found using surface tensor-based morphometry and thickness information into a vector at each vertex to be used as a metric for the statistical analysis. Group differences are assessed on this combined measure through Hotelling's T2 test. The method is applied to statistically compare three groups consisting of: congenitally blind (CB), late blind (LB; onset > 8 years old) and sighted (SC) subjects. Our results reveal significant differences in several regions of the CC between both blind groups and the sighted groups; and to a lesser extent between the LB and CB groups. These results demonstrate the crucial role of visual deprivation during the developmental period in reshaping the structural architecture of the CC. / Dissertation/Thesis / M.S. Computer Science 2013

In this thesis the impact of inverted cones on a liquid surface is studied. It is known that with the right combination of velocity, geometry, and surface treatment, a cavity of air can be formed behind an impacting body and extended for a considerable distance. Other investigators have shown that the time and depth of the cone when this cavity collapses and seals follows a different power law for flat objects such as disks, then it does for slender objects such as cylinders. Intuitively it can be expected that a more slender body will have less drag and that the streamlined shape will not push the fluid out of it's way at impact to the same extent as a more blunt body, therefore forming a smaller cavity behind it. With a smaller initial cavity, the time and depth of it's eventual collapse can be expected to be less than that of a much more blunt object, such as a flat disk. To study this, a numerical model has been developed to simulate cones with the same base radius but different angles impacting on a liquid surface over a range of velocities, showing how the seal depth, time at cavity seal, and drag forces change. In order to ensure the numerical model is accurate, it is compared with experimental data including high speed video and measurements made of the force with time. It is expected that the results will fall inside the power law exponents reported by other authors for very blunt objects such as disks on one end of the spectrum, and long slender cylinders on the other. Furthermore, we expect that the drag force exerted on the cones will become lower as the L/D of the cone is increased. / Master of Science / In this thesis the impact of inverted cones on a liquid surface is studied. It is known that with the right combination of velocity, geometry, and surface treatment, a cavity of air can be formed behind an impacting body and extended for a considerable distance. Other investigators have shown that the time and depth of the cone when this cavity collapses and seals follows a different power law for flat objects such as disks, then it does for slender objects such as cylinders. Intuitively it can be expected that a more slender body will have less drag and that the streamlined shape will not push the fluid out of it’s way at impact to the same extent as a more blunt body, therefore forming a smaller cavity behind it. With a smaller initial cavity, the time and depth of it’s eventual collapse can be expected to be less than that of a much more blunt object, such as a flat disk. To study this, a numerical model has been developed to simulate cones with the same base radius but different angles impacting on a liquid surface over a range of velocities, showing how the seal depth, time at cavity seal, and drag forces change. In order to ensure the numerical model is accurate, it is compared with experimental data including high speed video and measurements made of the force with time. It is expected that the results will fall inside the power law exponents reported by other authors for very blunt objects such as disks on one end of the spectrum, and long slender cylinders on the other. Furthermore, we expect that the drag force exerted on the cones will become lower as the <i>L/D</i> of the cone is increased.

As part of the natural evolution and continued optimisation of their designs, current and future magnetic recording heads, used and proposed in technologies such as perpendicular recording, shingled magnetic recording and two-dimensional magnetic recording, often exhibit asymmetry in their structure. They consist of two semi-infinite poles separated by a gap (where the recording field is produced), with an inner gap faces inclined at an angle. Modelling of the fields from asymmetrical structures is complex, and no explicit solutions are currently available (only implicit conformal mapping solutions are available for rational inclination angles). Moreover, there is limited understanding on the correlation between the gap corner angle and the magnitude, distribution and wavelength response of these head structures. This research was therefore set out to investigate approximate analytical and semi-analytical methods for modelling the magnetic potentials and fields of two-dimensional symmetrical and asymmetrical magnetic recording heads, and deliver a quantitative understanding of the behaviour of the potentials and fields as functions of gap corner angles. The accuracy of the derived expressions (written in terms of the normalised root-mean-square deviation) was assessed by comparison to exact available solutions for limited cases, and to finite-element calculations on Comsol Multiphysics. Two analytical methods were derived to approximately model the fields from two-dimensional heads with tilted gap corners in the presence and absence of a soft magnetic underlayer (SUL): in the first method, the potential near a single, two-dimensional corner held at a constant potential is derived exactly through solution of Laplace's equation for the scalar potential in polar coordinates. Then through appropriate choice of enclosing boundary conditions, the potentials and fields of two corners at equal and opposite potentials and displaced from each other by a distance equal to the gap length were superposed to map the potential and field for asymmetrical and symmetrical heads. For asymmetrical heads, the superposition approximation provided good agreement to finite-element calculations for the limited range of exterior corner angles 0 from 0 (right-angled corner) to 45, due to the mismatch of surface charge densities on both poles for this geometry. For symmetrical head structures, the superposition approximation was found to yield remarkable agreement to exact solutions for all gap corner orientations from 0 (right-angled head) to 90 ("thin" gap head). In the second method derived in this research for modelling asymmetrical heads involved using a rational function approximation with free parameters to model the surface potential of asymmetrical heads. The free parameters and their functional dependence on corner angle were determined through fitting to finite-element calculations, enabling the derivation of analytical expressions for the magnetic fields that are in good agreement with exact solutions for all corner angels (0 to 90). To complement the two approximate methods for modelling the fields from asymmetrical and symmetrical heads, a new general approach based on the sine integral transform was derived to model the reaction of soft underlayers on the surface potential or field of any two-dimensional head structure, for sufficiently close head-to-underlayer separations. This method produces an infinite series of correction terms whose coefficients are functions of the head-to-underlayer separation and gap corner angle, that are added to the surface potential or field in the absence of an underlayer. This new approach demonstrated good agreement with finite-element calculations for sufficiently close head-to-underlayer separations, and with the classical Green's functions solutions for increasing separations. Using the derived analytical method and explicit expressions in this work, an understanding of the nature of the magnetic fields and their spectra as functions of the gap corner angles is gained. This understanding and analytical theory will benefit the modelling, design and optimisation of high performance magnetic recording heads.

This diploma thesis deals with computing of edit influences on assigned stiff body from the flow of inviscid liquid. There are derived equations for computation of the influences during translational or torsional wobble and follow-up calculation of the units of their tensors.