Potted analysis scores on artistic merit

Tom Shelley investigates a new method of analysis that shows great promise for both artistic product design and automatic optimisation Automatically optimised designs can now be produced much faster and more efficiently, thanks to an analytical method of generating geometry. It is based on those elegant partial differential equations familiar to all engineers in their mathematics courses. It applies them in a unique way, purely to generate geometry, followed by traditional finite element methods and a second dose of traditional analysis to quickly reach a conclusion. So far it has been applied to optimising yacht hulls, engine manifolds, wings for supersonic airliners, human heart valves and, believe it or not, yoghurt pots. The method is the brainchild of Professor Mike Wilson, chairman of the University of Leeds School of Mathematics, and some of his co-researchers. He says it all started back in 1988 while seeking a better method of blending intersecting surfaces in such a way as to achieve smooth transitions and better appearances. The usual method of using computers to create and manipulate complex curved surfaces is to break them down into a lot of very small patches, joined at their edges. Since this requires a large amount of computing power, Professor Wilson thought a great deal of effort could be saved by handling such problems analytically, using what are termed ‘Fourth-order elliptical partial differential equations’. (Elliptical equations are solved in terms of boundary conditions and, therefore, have closed solutions. Hyperbolic equations are exemplified by wave equations and have starting conditions but then proceed off, in theory, to infinity.) Solutions to PDEs such as the celebrated Laplace equation, which is found to underlie the basis of much of engineering, can be found in terms of infinite series. In recent years engineers have tended to shy away from such methods, because of their intellectual difficulty and the ready availability of commercial software based on purely digital methods. “However,” says Professor Wilson, “we can generate geometry by looking for analytic approximations close to the true solution which fulfils the boundary conditions.” Such an approach may be intellectually complicated but it is very computer efficient. Instead of tens or hundreds of thousands of patches, even very complicated shapes can be generated from two or possibly three patches, with boundary conditions that specify which direction surfaces connected to the patches should take off in. The software then automatically generates a smooth intervening surface. The technique can be used artistically to create new product shapes, such as wine glasses or packaging containers, and does so very well. But while such capabilities should not be scoffed at, since they are often the deciding factor in consumer purchase decisions, the main intent of the technology is to improve engineering performance. To investigate possible improvements in design, the user takes one of the generic shapes developed by the researchers, and then adjusts the boundary conditions to generate shape variations. There is currently a large library of shapes, including propellers and aerofoils, but should none be available the team is pretty good at coming up with new ones. Optimisation comes by subsequently applying traditional methods, such as finite element analysis or computational fluid dynamics, to the shape or the fluid flowing round it, or any other quantity or measurand acting on it or within it. The next clever step is to decide the direction in which the shape should be changed in order to achieve optimum performance. In most optimising processes, shapes are varied fairly arbitrarily and in many possible ways, leading to very long computation times even with modern computers. A week is not uncommon even for relatively simple design problems. And when such processes are completed, holes or other features may appear in the completed configuration which may or may not be acceptable. But with the new method, it is only necessary to change one or two parameters to vary the shape in a controlled manner. “I wouldn’t say the optimisation process is simple,” says Professor Wilson, the implication being that there may be quite a few maxima and minima, only one of which will be the best one. It may, therefore, be necessary “to jump about a bit in the design space” rather than assume a rising or falling overall performance figure is leading to the desired ideal solution. Nonetheless, the method is extremely powerful and the team has successfully investigated challenges such as finding the best compromise between ideal wing shapes for low supersonic drag and high subsonic lift for NASA Langley. A yacht hull has been optimised by a research student for minimal wave drag and a two stroke engine manifold optimised for efficient combustion for an Italian manufacturer with European Union support. Much of the study concerned different configurations for the transfer port. The team has in the past, used both Vectis, from Ricardo, and Flow3D, from Fluent, for the CFD part of the analysis. The most recent studies have concentrated on achieving optimal performance in damaged human hearts with the aid of surgery and prosthetics, and optimising the design of yoghurt ports. This latter project has been supported by EPSRC and, while it may seem trivial, points the way to a reduction in material requirement of 30% while maintaining the original strength. A similar approach could, in theory, be applied to vessels in process plant, but consumer product packaging is the potentially more valuable market. So far, none of this has been offered commercially, although several of the studies have been conducted on a paid consulting basis. The efficiency of the geometry generating method can be tested by users over the Internet, accessing a demonstration in the form of a Java Applet on the research group’s website. A wide range of shapes can be generated at surprising speed by varying different parameters. The present business model envisages prospective users working with the research group initially on a consulting basis and then possibly licensing the software when the necessary expertise has been acquired. Alternatively, the team is interested in talking to software companies about possible alliances. Development is currently being undertaken on Silicon Graphics hardware but the code potentially runs on any system. (More information: University of Leeds) Design Pointers: The technique is a much more computer efficient method for generating complex shapes It can be used to generate artistic product design concepts and to investigate different shapes with a view to optimising performance It has so far been used in aerospace, marine and packaging studies