What is the best material?

New mathematical methods make aeroplanes up to 30% lighter. Boom expected in the automotive industry

Elegant bridges, efficient engines, and powerful aeroplanes are all wonders of engineering. But mathematics also plays a large role in their construction. Engineers were already using computers in the Fifties to design building structures or vehicles. "Now is time for the changing of the guard", says Jochem Zowe, Professor of Applied Mathematics at the University of Erlangen-Nuremberg. "We are making great progress towards the automatic design of buildings and the mathematically optimum design of components." The goal of the engineers is to reduce weight as far as possible by using new materials, and thus to reduce the costs of bridges, buildings, cars or aeroplanes, while at the same time making them better able to withstand events such as earthquakes or air turbulence. The vision of the mathematicians is a computer which produces the shape and composition of the components together with complete plans in accordance with the engineer's specifications.

Only if the design and the selection of materials are coordinated perfectly right from the start is it possible to achieve significant savings. At the International Congress of Mathematicians, Jochem Zowe is presenting the latest research on "material optimisation". The title of his talk (in German) is: "What is the best material?" (Wednesday, 19 August, 5.15 p.m., Venue: TU Main Building, Strasse des 17. Juni 135, 10623 Berlin, Room H 2032)

In the case of gear wheels, for example, the situation is relatively simple. The core should remain fairly elastic, but at the same time the teeth must be hard if they are not to wear away too quickly. A research group in Berlin at the Weierstrass Institute of Applied Analysis and Stochastics has developed a computer method for simulating the hardening of gear teeth using laser beams. This makes it possible to determine the best treatment without the need for expensive and time-consuming experimentation.

Jochem Zowe and his co-researchers are working on larger objects, such as ribs, spars, flaps, and wings of aeroplanes. The components, made of metal or modern composite materials are intended to prevent aeroplanes from beginning to vibrate. Zowe's computation method can show which shape the components should have in order to be as stiff as possible for a given material input. "The computer shows the external form of the components, but also their internal structure", explains Zowe. Sometimes the results are counter-intuitive. "It might be best to have holes in a component. At first that does not seem to be as stable, but it can be better under certain circumstances, for example if forces which would otherwise concentrate there can be diverted."

"The intuition of even the best engineers fails when confronted with more complex structures, such as those which are possible with composite materials", says Herbert Hörnlein, optimisation expert of DASA (Daimler Benz Aerospace AG) in Munich. Hörnlein has been working together with Zowe's research group since 1994 in a collaborative project funded by the German Ministry of Research. The DASA company already has planes in the air which incorporate mathematically designed spars. "Progress in material optimisation is unbelievable", says Hörnlein. Since optimisation already begins at the design phase, it is possible to reduce weight by up to 30%, without compromising lift or stiffness of the aeroplane. The reduced weight can in turn lead to enormous reductions in fuel consumption. "Within a few years material optimisation will have made a breakthrough in ship design and the automotive industry", says Hörnlein. Without integrated material optimisation it will not be possible to get car fuel consumption down below the magic 3 litres for 100 km.

However, technical innovation can only be as good as the basic research on which it is based. And Jochen Zowe has been invited to the International Congress of Mathematicians less because of economic successes, and more because of his mathematical research. Material optimisation is basically a simple problem, involving formulas and laws to describe the physical properties of materials and their deformation which were already known in the nineteenth century. It is the sheer number of the equations involved which makes the problem so complex. A component has first to be divided up into thousands of tiny cells. The computer can then begin to calculate an optimum elastic tensor to describe the elastic properties of each of these surfaces. Thousands of equations are produced, with possibly hundreds of thousands of conditions and constraints which also have to be taken into consideration. The job of mathematics is to simplify the problem, but without distorting the final result. The use of computers is only appropriate if the engineers can carry out the calculations themselves and obtain reliable results within a few minutes. Zowe's achievement has been to develop such calculation procedures, in cooperation with groups from Israel Institute of Technology (Haifa) and the Technical University Denmark (Lyngby). The mathematical task was to develop optimisation strategies such as semi-definite programming and to combine these with finite element methods.

Zowe attracted attention some years ago, when his group presented a computer program which was able to design steel bridges without human assistance. "The engineer just defines the outline structure of the girder bridge", explains Zowe, "and the computer then determines the sub-structure design with the optimum stiffness. The engineer only has to decide at the end whether the design looks good, or whether the computer should make another suggestion."

"Many engineers think that using mathematics is like reading tea leaves", says Jochem Zowe. Herbert Hörnlein also experiences reservations amongst the engineers at his company. "The engineers must learn to develop a feeling for these new computer methods", he demands. Zowe himself finds some of the new possibilities "highly speculative", particularly in material optimisation. For example, computers can calculate how the thickness of material should be spread over the component. But nobody yet knows how to build such optimised components. "Perhaps with composite materials involving carbon fibre it might be possible to realise these ideas in the near future", forecasts Zowe.