Computational Geometry for Design and Manufacture
Computational Geometry for Design and Manufacture
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A Geometric model of the sculptured surface of a vehicle is represented in terms of bicubic patches in the PATRAN modeling system. In Technology Engineering, a patch is subdivided into finite elements. The experimental data consists of pressure distribution at the centroids of these elements. It is desired to formulate the pressure distribution at arbitrary points on the vehicle by using this discrete data. The pressure points, which are theoretically on the surface of the vehicle, involve round off errors and thus may be offset from the surface. The problem becomes that of co-relating the pressure points and the patches on which they lie, determining the parametric values for the pressure points, and then defining the pressure distribution functions for the patches. The determination of the parametric values of the pressure points is a complex inverse problem. A new algorithm which is simpler, faster and more general than those using Newton-Raphson type techniques is designed to accomplish this. This algorithm uses divide-and-conquer paradigm and is applicable to both smooth and non-smooth surfaces.This algorithm for parametric coordinates (U,V) of a point on the closest surface patch at a minimum distance from the given point is stated as follows: Identify the point, P. Identify the list of surface patches. Pick a surface Patch from the list of patches. Evaluate the grid of N2 parametrically equally spaced points on the patch. Find a point, P1, from these N2 grid points which is the closest to the point P. Keep track of the point P1, the pair (U1, V1) of its parametric coordinates, the distance D1 between the points P and P1, the Patch, and create a subpatch surrounding the point P1. Evaluate the grid of N2 parametrically equally spaced points on the subpatch. Find a point, P2, from these N2 grid points which is the closest to the point P. Keep track of the point P2, the pair (U2, V2) of its parametric coordinates, the distance D2 between the points P and P2, the Patch, and create a subpatch surrounding the point P2. If abs(D2-D1) tolerance, then replace D1 by D2, point P1 by P2, (U1,V1) by (U2,V2). Repeat this step on the subpatch until abs(D2-old D1) 2 in this process and the corresponding point P2, the parameter values (U2, V2) and the Patch. Repeat the above process until the entire list of Patches has been processed.Once the parametric coordinates for the pressure points have been computed, sixteen points (ANSI Y14.26M definition of a bicubic patch) on the patch are identified at the sixteen parametric points equally spaced in both parametric directions. The algorithm for pressure distribution may be stated as follows: For each of the sixteen points on the patch, find N points from the specified pressure points which are closest to the patch point. Keep track of the distances Di's between the patch point and the N pressure points. Let M be a smoothing parameter to be used in defining the weights for the pressures at the N pressure points, Pi's. Define the pressure at the patch point by the formula.Applying this formula to each of the sixteen patch points, the pressure distribution at an arbitrary point on the patch can be found by replacing the geometry matrix by the matrix of pressure values. Finally, the pressure distribution on the surface is expressed in terms of the pressure distribution on the patches.Acknowledgement. The author acknowledges the support of McDonnell Douglas Corporation where this work was performed.