A Meshless Solution For Potential Equations
Using A Continuous-valued Circular Line Source
Author(s)
P. Mitic & Y. F. Rashed
Abstract
We find solutions to 2 0 U ∇ = in a simply-connected 2-D domain D, using a
continuous line source associated with a concentration function comprising n
undetermined parameters. This choice reduces ill-conditioning effects by
reducing the number of parameters involved. The choice of a continuous circular
line source C around D follows from previous results indicating that, when
solving the same problem with discrete point sources, the result is independent of
precise placement of sources. The circle is associated with a concentration
function that is constrained to satisfy the problem’s boundary conditions.
Accuracy is achievable using a number of parameters which, had discrete
sources been used, would be insufficient to represent the geometry of D, thus
giving inaccurate results. Empirical investigations with various forms of
concentration function show that with some domains, the error in calculated
values of U can be less than 0.1%: an order of magnitude improvement over
discrete methods. More complex domains yield less accuracy, and, after testing
on a range of domains, we formulate an empirical rule for an appropriate form
for the concentration function for a generic domain. Code requiring highprecision
arithmetic was developed in Mathematica, which also simplifies
routine tasks of solving linear systems and integrations.
1 Introduction
Previous research has shown that when using meshless discrete sources in the
MFS, the configuration of sources relative to the domain D is extremely flexible.
In [1] we showed that, within certain limits, the source distribution can be
random, and in [2] we showed that sources \“at infinity” (i.e. a large distance
Keywords
Related Book
Other papers in this volume
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