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

AbstractThis 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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