University of Leeds mathematicians have devised a formula that is saving manufacturers millions of pounds during tests of product design performance.
Malcolm Bloor and Mike Wilson of the department of applied mathematics have developed a system of partial differential equations that helps computer-aided design (CAD) software to generate models of the object's surface that respond to changing conditions. The design can then be upgraded appropriately to improve performance.
While conventional CAD systems represent the surface shape of an object in simple terms, the geometry created by using partial differential equations takes this a step further by allowing the designer to factor in processes such as heat transfer or air or water resistance.
For example, in designing an aircraft wing, a program using partial differential equations can calculate how a change in surface shape will increase or decrease the drag effect. By mapping the parameters of the object's shape against those of the drag, it can enable the designer to look at a part of it and work out how drag could be minimised in that area.
"Using mathematical analysis for design optimisation in something as big as a ship or an aircraft is dramatically less expensive than the alternative of building, modifying and testing a series of prototypes," Professor Bloor said. "Some of our results have been counter-intuitive: on an aircraft wing, it produced structures at the tip that maintained the distribution of lift along the wing - features that a designer might not have considered."
Researchers are now exploring the potential of partial differential equations in medicine, including work on the geometry of the heart. Research is also being done into using the method to represent the shapes adopted by biological membranes. The Leeds mathematicians are collaborating with colleagues at St James's University Hospital and the University of Strathclyde to find out how the approach can help optimise the design of prosthetic limbs.
Professor Wilson said: "This began more than ten years ago as a piece of speculative research. Since that time, it has been successfully applied in a wide range of down-to-earth areas."