Supermath For The Real World

Above all else, engineers are practical. If developing the perfect camera or oil refinery takes too long, they settle for a design that's "good enough." Increasingly, though, this no longer suffices. In companies driven by competition to wring the last ounce of efficiency from manufacturing and to design products to far tighter tolerances, engineers are being forced to dip into a new mathematics toolbox. And the ultimate result, experts believe, will revolutionize engineering just as quantum mechanics has transformed physics.

The new tools are called nonlinear equations, and the name says it all. These equations are used for precisely describing the behavior of things with an unpredictable facet. That's nearly everything--from the workings of car engines to the actions of DNA molecules. Even baking a cake is nonlinear: Turning up the oven's temperature twice as high won't bake the cake twice as fast. And with some industrial recipes, such as those for making drugs and plastics, a tiny change in ingredients or processing conditions can mean a huge difference in the finished product. Nonlinear math can help explain such lopsided effects. In short, it "allows you to describe things the way they work in the real world," says David Kinderlehrer, director of the Center for Nonlinear Analysis at the University of Minnesota.

Engineers have shunned nonlinear equations until now because getting good answers is hideously difficult. Since you can't be sure how any change will affect the outcome, you have to plug in every conceivable variation and solve the equations thousands or millions of times. Once, even a simple nonlinear problem was a job for a supercomputer--and the toughest problems are still unsolvable: They would take decades of continuous number-crunching. But now that desktop workstations can outrun yesterday's supercomputers, mathematicians have the means to tackle a much wider range of industrial challenges. And for many of those still too hot to handle, programmers are writing shortcuts to approximations that make "good enough" a lot better. "Great strides are being made in techniques for solving large, complex problems," says James L. Phillips, manager of mathematics and engineering analysis at Boeing Co.

SHRINK-WRAP. He should know. Boeing has been using such nonlinear techniques as computational fluid dynamics (CFD) since the 1970s. CFD involves carving up an airframe's design into thousands or millions of interconnected shapes called finite elements: The more there are, the more accurate the results will be. The resulting computer model is covered with a "mesh," making it look like it's shrink-wrapped in graph paper. Then the computer simulates air flowing over each element and integrates all the answers to determine how well the plane will fly. The latest CFD software, which costs a fraction of earlier versions, even generates the mesh automatically, saving days of engineering time. As a result, Boeing figures it now pays to revamp aged planes. An upcoming edition of the 30-year-old 737 will be lighter, carry more payload, and use less fuel.

The big news is that nonlinear math is spreading to many other industries. General Motors Corp. is crash-testing finite-element models instead of real cars--and has perfected nonlinear software for designing interior panels. "We'll be able to explore design alterations much more easily than before," says James C. Cavendish, principal research scientist at GM's research center.

IBM, meanwhile, has turned to CFD to improve its hard-disk drives. The read/write head that skims over the disk at 30 mph creates so much aerodynamic pressure that engineers couldn't figure how to shrink the gap between the components to less than one millimeter. With nonlinear equations, they got it down to one micrometer--a thousandth of a millimeter. This boosts storage capacity, since the closer proximity lets the head read and write smaller magnetic spots.

Interest goes beyond high tech. Steelmakers have commissioned studies of new furnace designs. And at Los Alamos National Laboratory, Mobil Oil Corp. is funding development of software that simulates the movement of oil through porous rock--to help improve extraction techniques.

For virtually any product, there now are nonlinear tools for tinkering with designs and getting accurate feedback in a matter of hours. Engineers are thus spending more time honing designs--often trying hundreds of alternatives, says Luis G. Reyna, manager of analysis and modeling at IBM's Thomas J. Watson Research Laboratory. That's important because 70% to 90% of a product's total cost is fixed during design.

Nonlinear math is also uncovering new approaches to so-called optimization problems--finding the best way to run a factory, schedule a fleet of trucks and drivers, or manage a stock portfolio. Here, even finding a good-enough solution is tough. But David L. Jensen, manager of mathematical optimization at IBM's Research Div., is adapting quadratic optimization to quickly calculate the optimum balance between risk and profit for a portfolio.

CELL GAME. Jensen is typical of the new-generation mathematicians weaned on software. He taps on his laptop, and a nearby workstation spits out a torrent of numbers. "A portfolio manager can do data collection on the laptop," he says, "then go to a machine with higher horsepower to do calculations." Still, optimizing a portfolio of 500 financial instruments in a matter of seconds is no snap. For now, Jensen has to pare the number of variables to a bare minimum.

Nonlinear math is also opening the door to technologies that were unmanageable a few years ago. It's crucial to data networks that use so-called asynchronous transfer mode (ATM) to transmit data at billions of bits per second--up to 100 times existing speeds. Because the data move so fast, ATM systems must be virtually glitch-free--no more than one error per trillion bits, vs. one per thousand in conventional networks. Nonlinear optimization provided the answer, says Debasis Mitra, head of AT&T Bell Laboratories' Mathematics of Networks & Systems Dept.

Biotech may get an even bigger boost. Scientists at Los Alamos hope to get a quantitative handle on the human immune system by adapting nonlinear factory simulations. Makes sense. In much the way a factory manager revises production for a rush order or when some equipment clogs, the immune system continually reallocates its cells between building antibodies and cell replication.

RESISTANCE. Other researchers are trying to develop models for the behavior of DNA molecules. Scientists believe the "folding" action of DNA, in which spiral-staircase molecules coil and uncoil, is one key to understanding how DNA works--and could point to new proteins that inhibit cancer. Tamar Schlick, associate professor of mathematics at New York University, is building a DNA model based on nonlinear equations of motion. Eventually, she hopes to solve enough permutations to simulate a full second in the life of DNA.

For engineers, nonlinear math is still something of a pain. While mathematicians are good at crunching thousands of nonlinear equations in a jiffy, the front-end preparation remains tedious. Setting up a problem takes so long, says GM's Cavendish, that "the question is how to deliver results quickly enough to have an impact on the design process." Moreover, a finite-element mesh is so complex that finding errors in it is next to impossible--and scientists are just starting to develop software that can flag errors on the screen. Beyond that, inertia is holding back the changeover to the new math. "People seem to prefer poor models they understand well to good models they understand poorly," says Andrew Conn, a Bell Laboratories mathematician.

Still, as computers grow more powerful, nonlinear engineering is likely to displace older, cruder methods. Engineers at Eastman Kodak Co. were shaken recently when Kodak began hiring mathematicians to back them up. A few more embarrassments like that and the old engineering dogma will soon give way--like the straw that broke the camel's back. That, incidentally, is a classic example of nonlinearity.


Major industries are benefiting from

software based on nonlinear math.


Engineers can simulate the

aerodynamics of entire air-frames instead of just wings, improving performance and cutting costs.


Geneticists are getting a better

handle on the complex behavior of DNA. Eventually, biology's most important discoveries may stem from computer, not test-tube, experiments.


Designers can shape stronger,

safer car bodies.


Traders are using "nonlinear

optimization" algorithms to help maximize profits and minimize risk.


Chemical and oil producers are

devising more efficient produc-tion processes, and manufacturers are designing products to tighter tolerances.

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