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
TitleNumerical Scheme for the Solution to Laplace's Equation using Local Conformal Mapping Techniques
DepartmentMathematical Sciences
  • Burt S. Tilley, Advisor
  • Keywords
  • Laplace's equation
  • numerical methods
  • conformal map
  • Date of Presentation/Defense2014-05-07
    Availability restricted


    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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