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Solving the Laplace Equation using Boundary-Element Methods: Electrostatics



This software solves the famous laplace equation in a homogeneous region using an integral equation method, where is the scalar electric potential. The integral equation that we seek to solve is

where the Green's function G is

.

For a more complete description of the mathematical methods, please visit my page on scattering and the Helmholtz equation.

The downloadable software has a very similar data entry method to the Helmholtz software. For convenience, I included a simple input file here.

2 ............................Number of line segments

0. 0. .......................Incident external electric field

-1. 1. -1. -1. ..........First segment endpoints

6 ...........................Number of boundary elements on the segment

-1. .........................Applied potential on the segment

1. -1. 1. 1. .............Next segment endpoints

6 ............................Number of boundary elements

1. ...........................Applied potential on this segment

-2. -2. ....................Bottom right corner of plotting area

2. 2. .......................Upper left area of plotting area

41 41 ....................Number of points in <X> and <Y> directions

10 ..........................Number of equipotential contours to plot (appearing in Equipotn.dat file, plottable with Gnuplot <plot> command)


Download the software here. Unpack using tar and compile using make. (Note, Makefile is for Gnu make. GCC must be present. To compile on Windows, you must have a C or C++ compiler and you'll need to build your own Makefile.) There are several input data files included in the tarfile for your amusement (and also to show you how to enter the geometric data).


The code is complete, but there are some improvements that can be made:

  1. Better treatment of singularities at sharp corners.

  2. Improved matrix element generation.

  3. Proper generation of stream functions.

  4. Nice user-friendy GUI. (Well, whaddya want in a free piece of software?!)