Integrating Topology and Numerics at CAD Interfaces

University Of Connecticut, Storrs CT

Investigators

Abstract

9985802 A typical geometric model for Computer-Aided Design (CAD) is represented as a finite collection of compact 2-manifolds, joinedalong boundaries formed by their intersection sets. Hence, these representations critically depend upon intersection algorithms. To minimize data volume and improve algorithmic performance, the intersection sets are only approximated. The consequences of these approximation errors may be far from trivial and can lead to `gaps' or `cracks' between surfaces that are understood to be adjoining. Unfortunately, the prevalent practice is that intersection algorithms do not give any error bounds on these approximations. This research focuses upon the central issue of the interdependencies between topology of a model and the approximations within numerical algorithms. Any formal notion of topology for computational representations of geometric sets must specifically acknowledge the role of approximation and finite arithmetic. This project will develop formal definitions and computational representations for needed approximation bounds. Computational experiments will be performed to test the communication of the critical approximation information across programmatic interfaces. The expected result is a formal framework for reasoning about the propagation of topological tolerances within large software systems. Three recent workshops (OPAAL '97 in New Orleans, CAD/CFD '99 at Davis, and MSRI '99 at Berkeley) have directed attention to the need for rigorous mathematical foundations for computer-aided design. Particularly troublesome are regions where surfaces should meet, as computational models are limited to only approximating such surface joinings. A computational model of a coffee cup will necessarily have small cracks along its base and near its handle. If fluid were to be placed in this cup in a computer simulation, the fluid would leak out of the cup. Of course, similar cracks will exist in a computational model of an airplane; for example, where the wing joins the fuselage. The performance characteristics of this joint during flight are crucial to aircraft safety. Fluid dynamic simulations are performed on airplane models to verify critical engineering performance characteristics. Even small errors in the geometric model can cause large errors in these engineering simulations. These simulations hold the promise of significant cost savings by permitting creation of a reliable airplane directly from the electronic model. However, to attain that goal, the interface between modeling and simulation must accept the inevitability of modeling `cracks' and still produce valid engineering analyses. This project will create new mathematical formulations and computational representations as a necessary step for reliable engineering simulations. Recent economic studies done for NIST indicate that these engineering problems cost about 1 billion dollars annually, for the automotive industry alone. Hence, there is opportunity for significant practical economic impact from this new theory, as it will improve results from complex engineering simulations which must be executed within high-performance computing environments. Utilization of these high performance computing resources will be improved and the more reliable engineering estimates will also lead to direct savings in material and manufacturing costs for major industries.