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PRACTICAL APPLICATIONS

Numerical Methods

Donald Schwendeman, associate professor of mathematics, uses numerical methods—usually by devising algorithms—to solve mathematical problems that arise in scientific and engineering applications. He’s especially interested in fluid flows and shock-wave propagation. Shock waves are compression waves in which the properties of a medium—say, the air pressure around an airplane wing—change abruptly.   

“I have used numerical methods to compute the motion of the shock waves. Algorithms based on these methods can be picked up and used by people in the medical community or by engineers,” Schwendeman says. “There are many applications.”   

Among them is the shock-wave lithotripter, which uses focused shock waves to break up kidney stones. Other problems Schwendeman has worked on include the design of optimal critical airfoils on aircraft and mathematical modeling of chemical-mechanical planarization of microelectronic materials.   

Schwendeman is the organizer of the annual Mathematical Problems in Industry workshops, which started at Rensselaer 15 years ago. For the last two years, they have been held at the University of Delaware, but the idea has remained consistent: Representatives from four to six industries present real-life problems to a panel of mathematicians from universities around the country and overseas. For five days, the academics meet with the industrial representatives, study the problem, and issue a report on progress made during the workshop. Topics have ranged from “Design of Computer Hard Drive Slider Bearings,” submitted by IBM, to “Predictive Models for Financial Businesses,” which came from Citibank.   

“Those workshops are very interesting. The subjects are all over the place and you get to work in a sort of communal setting. You realize that mathematics can be applied to a lot of situations you might not have thought of before,” Schwendeman says.    

 

 

 

 

 

 

Recently, Schwendeman has been working on a problem for a papermaking company in upstate New York.

“It involves squeezing the water out of the slurry before it becomes paper pulp, and before that becomes paper, the finished product. This has been a terrific project, with all sorts of side issues and applications. It’s been fun. I suppose that’s a bizarre notion for a lot of people,” he says.

Ashwani Kapila, professor of mathematics, is a theoretician of explosions, but despite his government security clearance he seldom actually sees something getting blown up.   

“Merely watching an explosion go off is not interesting in itself. What does fascinate me, as a mathematical modeler, is the amount of information a well-designed experiment can deliver, even in the hostile environment of an explosion,” says Kapila, a Rensselaer faculty member since 1976.   

Explosions are complex events—complex in their physics and chemistry, and in their mathematical underpinnings.   

“I started out looking at gas explosions—home cooking gas, for instance. These are relatively simple phenomena. The materials are homogenous, their chemical properties are relatively well understood, they are mechanically simple. Then, colleagues got me interested in condensed-phase explosions—explosive powders, for example. Being heterogenous, these are more complex. There is lots of energy, large molecules, and very fine grains. How do we study a problem that begins at the level of microns or tenths of microns, a very fine scale, and leads very quickly to the macro scale level of centimeters?” Kapila says.

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