From: Ed McBride Newsgroups: sci.math.num-analysis Subject: Re: Conformal mapping Date: Tue, 15 Dec 1998 08:12:46 -0700 Ed McBride wrote: > > Charles Bond wrote: > > > > I have a class of problems which represent cross sections of magnetic > > structures. These are all polygonal and so they will have conformal > > mappings onto the unit circle or, preferably, the upper half plane. > > What I was looking for was a software package which would take > > the structure boundaries and solve the mapping problem so I could > > generate field plots. > > Once upon a time (around 1967) I solved this exact problem for reservoir > analysis at Arco Oil & Gas. My solution was an iterative mapping from > the polygon to the unit circle, usually requiring 50-100 iterations for > a "normal" polygon, and this resulted in all points on the polygon lying > between R = 0.99 and R = 1.00. If this would help, let me know and I'll > try to remember how I did it. It wasn't very complicated. BTW, the > analysis took around one hour on an IBM 360/50, so it wouldn't be > CPU-intensive by today's standards. Ed McBride, P.E., Consulting > Engineer, Colo Spgs Charles e-mailed me and said it might be helpful, so here goes. I'm posting it here because I don't remember the details and somebody else may be able to finish what I start. The iteration proceeded as follows: Rotate the boundary until the boundary point nearest the origin is on the real axis, then perform the mapping (discussed later). The point on the real axis mapped to (1,0) and every other point inside the unit circle, including obviously all other boundary points, mapped to a point such that the absolute value of the mapped point was > the absolute value of the original point. Then take the mapped boundary points, check for distances from origin, rotate, etc. I continued until every point on the boundary had absolute value > 0.95 (I said 0.99 in the original post; I think that's wrong), then mapped into the upper half plane, placed images in the lower half plane, etc., but with todays computers, a higher number might be appropriate. Note that there is no inverse mapping, so I generated a rectangular mesh inside the unit square, threw away those outside the original polygon, and carried the rest along with all the iterative mappings. Then I had the mapped points, calculated the potentials, etc. Now for the tough part, the mapping. It was something like w = 1/[1-(z-a)/(z+a)] where a is a real number, the coordinate of the point on the real axis. It was thirty-some years ago, and even though my Alzheimer's is in remission, I simply can't remember the details. Note that the above function does map the point on the real axis to (1,0) so, realizing that even a blind hog finds an acorn now and then, it's possible that that this is the right answer. I am simply too busy at the moment, with a couple of jobs I promised to get done before Christmas, to cary this further; can anybody else help? Thanks, Ed McBride