Worcester Polytechnic Institute Electronic Theses and Dissertations Collection

Title page for ETD etd-050714-212720


Document Typethesis
Author NameSabonis, Cynthia Anne
Email Address casabonis at wpi.edu
URNetd-050714-212720
TitleNumerical Scheme for the Solution to Laplace's Equation using Local Conformal Mapping Techniques
DegreeMS
DepartmentMathematical Sciences
Advisors
  • Burt S. Tilley, Advisor
  • Keywords
  • Laplace's equation
  • numerical methods
  • conformal map
  • Date of Presentation/Defense2014-05-07
    Availability restricted

    Abstract

    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.

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