Slashdot Mirror
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!"
That guy must be pretty old
It always amazes me how applicable math becomes hundreds of years after it's written. Think if Maxwell's equations, Newton's equations, Einstein's equations. Fluid Dynamics equations were probably pioneered well before they were applied to human machines. Modern-day aircraft would not operate without their understanding. Where the math goes, human technology will probably soon follow.
I give credit to all the bran I've been eating lately.
The article is available at the author's website.
As far as I can tell, the original result provided a conformal map 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.
Read the paper. This is not the first S-C formula for multiply connected regions. The claimed "key result" is a formula for a case where a formula is already known. More work will be needed to a adapt the MATLAB technology from singly- and doubly-connected regions to multiply connected regions.
This paper seems to be part of ongoing work by a small community and is probably useful, but it's not a major mathematical breakthrough -- more of an incremental step. Small technical improvements in one field of mathematics shouldn't make up a slashdot story. Just because someone put "140 year old" in the press release doesn't mean it's really important. A math story belongs on /. when a big result is announced -- on the level of Poincare's Conjecture, or the Modularity Theorem.
Designers designed planes long before they could work out the math. They experimented a lot. The math lets you make things faster, cheaper and gives you ideas for new designs. I wouldn't fly in anything based solely on the math though.
That's why I emphasized modern-day aircraft. Designing a 777 or the new 7E7 off pure experimentation would take insanely more amounts of time and money. Math makes it a LOT easier, and its probable all turbine-driven commercial craft wouldn't exist at their current efficiencies without math being in the design process. Laugh all you want about their gas-guzzling reputations, but it would be interesting to see someone design such a sophisticated aircraft without advanced math.
It makes it cheaper, but you can certainly have sophisticated turbine aircraft without the math. We've only had the computers to make a respectable stab at simulating airflow over a reasonably complex wing recently. It's great as a design aid, and invaluable as a tool for understanding, in the abstract, but the real world is often too complex for our computational capabilities. Surprises crop up all the time. The A380 wing for example. Its probably the modernest and advancedest turbine-driven commercial aircraft wing (at the moment). The wing in practice isn't as efficient as it was supposed to be. It also failed its strength certification the first time around.
In most engineering applications the math is a nice tool to let designers do a bunch of experimenting inside the computer before they have to move on to real world testing. We're not at the point yet where math is more important than experience and experiment. Not just aircraft design. I work in medical imaging and there are no shortage of ideas where the (idealized) math works great, the simulations are wonderful, but the idea doesn't survive first contact with patient data.
Not to mention pilots.
Aide-toi, le Ciel t'aidera - Jeanne D'Arc.
Backups are for wimps. Real men post their data in comments and have slashdot mirror it
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.
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
Cosner, RR and Roetman, EL, "Application of Computational Fluid Dynamics to Air Vehicle Design and Analysis", IEEE Aerospace Proceedings, 2: 129-42 (2000).