Mathematician Solves a Big One After 140 Years

TaeKwonDood notes that ScientificBlogging.com has just written about a development in applied math that was published last year. "The Schwarz-Christoffel transformation is an elegant application of conformal mapping to make complex problems faster to solve. But it didn't do well with irregular geometries or holes, so it simplified too much for a lot of modern-day mechanical engineering applications. 140 years after Schwarz and Christoffel's work, a professor at Imperial College London has generalized the equation. MatLab users rejoice!"

Re:Math Forfront

[bjorniac]

Really? Leibniz invented physics?

OK, I know what you're saying, but really, Newton takes too much credit here. In his early work he even credited Leibniz then in a later edition of his work removed the statement.

Article text

[melikamp]

The article [ic.ac.uk] is available at the author's website [ic.ac.uk].

As far as I can tell, the original result provided a conformal map [wikipedia.org] from a disk onto a polygon. Prof. Crowdy extended this result to provide a map from a disk with circular holes poked in it onto a domain with polygonal holes. Why is it useful? I am sure someone in the applied camp would know.

Re:Article text

[tqft]

Only for purchase linked except this one which has more detail (under News)
http://sinews.siam.org/old-issues/2008/januaryfebruary-2008/breakthrough-in-conformal-mapping [siam.org]

Re:Math Forfront

[Anonymous Coward]

Your math history is actually completely wrong. Hamilton actually was looking for a way of extending an algebra of vectors to 3 dimensional space to do stuff with classical mechanics. In fact, for awhile during the 19th century that was the way to do it, and there was also a bit of dispute about using the vector calculus methods vs. quaternions as well. So no, they actually did come about for a reason.

Re:Design

[Zed is not Zee]

I am a designer for a large gas turbine engine manufacturer, and I have to agree that there is still a lot that we just don't understand well enough or can't model adequately. Combustion noise, liquid atomization, fatigue/creep interaction, etc. We do all kinds of FEA and CFD analysis, but still spend tens of millions of dollars on testing to back up those simulations.

Re:Not quite a breakthrough

[gardyloo]

Indeed. See this 1956 paper: http://links.jstor.org/sici?sici=0002-9947(195605)82%3A1%3C128%3AOTCMOM%3E2.0.CO%3B2-P [jstor.org] (warning: links to only an abstract on JSTOR).

      Conformal mapping is pretty easy to explain to a lay audience (no, not necessarily hookers); the original article did a horrible job.

Re:Design

[ceoyoyo]

Well, the 757 was designed in 1983. Certain versions of it have a reputation for being very fuel efficient. The U2 and SR-71 were designed and built in the 40s and 50s, and the SR-71 is still the fastest aircraft to take off under its own power. The H-4 Hercules was designed and built in the 40s and has the largest wingspan and height of any aircraft in history. The 747, one of the most successful commercial aircraft, was designed during the 60s.

So it depends what you mean by "math." The Wright brothers undoubtedly needed to add and subtract measurements to build their plane. That's math. Those designers in the 50s and 60s used pencils, slide rules and tables to work out some equations to help guide them (there was some talk of using the new electronic computers, but aircraft designers weren't overly enamored of them). The big aircraft manufacturers started developing 2D computational fluid dynamics software in the 70s, and two major packages were developed in the 80s.

So what about today? Well, you won't find a test pilot who's willing to fly a new design that hasn't been tested in a wind tunnel. There's no way I would fly on an aircraft that hadn't been tested in real flight, unless I was being paid (and trained) as a test pilot. Aircraft companies spend billions on wind tunnels. It seems even today the math is awfully useful but it's no substitute for putting an aircraft in an airstream and seeing what happens.

Sources:
http://en.wikipedia.org/wiki/Computational_fluid_dynamics [wikipedia.org]
Cosner, RR and Roetman, EL, "Application of Computational Fluid Dynamics to Air Vehicle Design and Analysis", IEEE Aerospace Proceedings, 2: 129-42 (2000).