A fast method for estimating discrete field values in early engineering design

Much of the analysis done in engineering design involves the solution of partial differential equations (PDEs) that are subject to initial-value or boundary-value conditions; generically these are called “field problems.” Finite-element and finite-difference methods (FEM, FDM) are the predominant solution techniques, but these are often too expensive or too tedious to use in the early phases of design. What’s needed is a fast method to compute estimates of field values at a few critical points that uses simple and robust geometric tools. This paper describes such a method. It is based on an old technique-integrating PDEs through stochastic (Monte Carlo) sampling-that is accelerated through the use of ray representations. In the first (pre-processing) stage, the domain (generally a mechanical part) is coherently sampled to produce a ray-rep. The second stage involves the usual stochastic sampling of the field, which is now enhanced by exploiting the semi-discrete character of ray-reps. The method is relatively insensitive to the complexity of the shape being analyzed, and it has adjustable precision. Its mechanics and advantages are illustrated by using Laplace’s equation as an example.