Consider a Spherical Cow - Conservation of Geometry in Analysis: Implications for Computational Methods in Engineering

Friday, May 14, 2004 - 1:30pm - 2:20pm
Keller 3-180
Thomas Hughes (The University of Texas at Austin)
Consider the spherical cow is the punch line of a mathematics joke. The joke itself is not so important to the subject of this work but the message is, namely, that simplifications of geometry are often made to facilitate analysis. Let us take the engineering design process as an example. There are estimated to be of the order of a million analyses a day performed in engineering design offices throughout the world. Engineering designs are encapsulated in Computer Aided Design (CAD) systems. Up to manufacturing tolerances, these systems exactly represent the geometry of designs. The analysis process begins with CAD geometry but the predominate method of analysis, finite elements, requires a different representation of geometry. This creates two problems: 1) The need to generate the geometric description suitable for the finite element method; and 2) the geometric errors that are produced in the process. The first problem, mesh generation, is attributed to taking over 80% of all analysis time in major engineering industries such as shipbuilding, aerospace and automotive. It has become the major bottleneck in engineering analysis. The second problem is very important in certain situations, such as, for example, the buckling of thin shells, which exhibit strong geometric imperfection sensitivity. Since approximating the geometry for analysis purposes is costly, time consuming, and potentially creates significant errors, it raises the question, why do we do it? It would seem beneficial to conserve the exact CAD geometry in analysis, up to, of course, features that we definitely want to remove. This work takes the point of view that conserving geometry is an important conservation law that should be satisfied. We pursue this idea and see where it takes us. It suggests a very different analytical structure but one in which mesh generation may be dramatically simplified. Some simple computations in structural analysis are presented which indicate the ideas are viable and we argue why we feel that developing a complete mathematical convergence theory should be straightforward.