Numerical Initial Value Problems in Ordinary Differential Equations

The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of math ematics and the natural sciences they are central topics in computational geometry with hundreds of papers discussing algorithms and extensions Section discusses the de nition and basic properties in the usual case of point sites in R with the Euclidean metric while section gives basic algorithms Some of the many extensions obtained by varying metric sites environment and constraints are discussed in section Section nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations

Voronoi diagrams and Delaunay triangulations

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Guaranteed-Quality Triangular Meshes

This paper presents a methodology and algorithms of optimizing and smoothing the tool orientation control for 5-axis sculptured surface machining. A searching method in the machining configuration space (C-space) is proposed to find the optimal tool orientation by considering the local gouging, rear gouging and global tool collision in machining. Based on the machined surface error analysis, a boundary search method is developed first to find a set of feasible tool orientations in the C-space to eliminate gouging and collision. By using the minimum cusp height as the objective function, we first determine the locally optimal tool orientation in the C-space to minimize the machined surface error. Considering the adjacent part geometry and the alternative feasible tool orientations in the C-space, tool orientations are then globally optimized and smoothed to minimize the dramatic change of tool orientation during machining. The developed method can be used to automate the planning and programming of tool path generation for high performance 5-axis sculptured surface machining. Computer implementation and examples are also provided in the paper.

Optimizing tool orientations for 5-axis machining by configuration-space search method

In theory, the 5-axis machining of sculptured surfaces offers many advantages over 3-axis machining, including faster material-removal rates, improved surface finish, and the elimination of hand finishing. In practice, 5-axis machining suffers from a number of drawbacks, which are mostly related to gouge avoidance. The paper describes research on algorithms for the generation of gouge-free, nonisoparametric 5-axis tool paths across composite surface patches. The approach includes (a) the tesselation of the parametric surfaces into triangles, (b) algorithms for the placement of a cylindrically shaped cutting tool onto the triangulated surface, and (c) interference detection and tool-position correction.

5-axis machining of sculptured surfaces with a flat-end cutter

The numerically controlled (NC) machining of sculptured surfaces is an error-prone process often requiring several attempts before an error-free NC program is produced. A technique designed to move the NC proof process into software is described. The system outputs a color-coded graphics display of the machined surface that shows out-of-tolerance areas. To gain efficiency, surface curvature and cutting-tool size are used as inputs to a surface discretization algorithm, which guarantees that a user-defined level of simulation accuracy is achieved. The simulation time grows linearly in both desired accuracy and in the number of tool movements. In typical test cases, NC programs for complex automotive body panels were simulated and verified in CPU times that ranged between 5 and 30 minutes. >

Methods for detecting errors in numerically controlled machining of sculptured surfaces

Algorithms for simulation and verification of Numerically Controlled (NC) machining programs are presented. Compared to NC simulation based on conventional solid modeling systems, these models are designed to give approximate results, but with a substantial decrease in computer time. The surfaces of the part are discretized into a Surface Point Set (SPS) with a point spacing dependent on cutting tool size and shape local surface curvature and the desired accuracy of the approximate simulation. The surface-surface intersection calculations of the solid modeling approach are replaced by the intersection of the surface of the tool movement envelope with straight lines emanating from the surface points. The methods are applicable to both 3 and 5 axis machining. Samples test cases are presented, and implementation and efficiency issues are discussed.

Approximate methods for simulation and verification of numerically controlled machining programs

We describe a method for simulating and verifying the correctness of Numerical Control (NC) programs. NC programs contain the sequence of cutting tool movements which machine raw stock into a finished object. Our method is based on a discrete approximation of the object by a set of points. A vector is passed through each of the points and machining is simulated by finding the intersections of tool movements with these vectors. We present a point-selection method and an analysis that shows that the error introduced by the approximation can be made as small as desired. The run time is inversely proportional to the allowable error and the size of the cutting tool, and directly proportional to the distance that the cutting tool moves.

Discrete simulation of NC machining

Voronoi diagrams and variants of Voronoi diagrams have been used for many years to model crystal growth. If the boundary of the growing crystals are circular and all the crystals start at the same time and have the same constant growth rate, then the Voronoi diagram is used to model the growth. If the crystals start at different times, the additively weighted Voronoi diagram is used to model the crystal growth. In this thesis, I propose a new type of Voronoi diagram, called the multiplicatively weighted crystal growth Voronoi diagram, that can be used to model crystal growth when the crystals have different constant growth rates. In this new model, the distance from a site to a point in its region is measured along a shortest path lying entirely within the region. In the multiplicatively weighted crystal growth Voronoi diagram, a growing crystal (or region) may "wrap around" another site's region. When a region wraps around, distances from the site are in part measured along the boundary of the two regions, treating one of the regions as an obstacle, rather than along a straight line that passes through the region.
The worst case size of the multiplicatively weighted crystal growth Voronoi diagram is $\Theta(n\sp2)$. To construct the diagram, techniques from numerical analysis are used to approximate and to intersect curves described by a system of first order differential equations. Numerical methods to approximate a curve construct a polygonal approximation of the curve. One step of the numerical methods constructs an edge of the polygonal approximation. In the new Voronoi diagram, a step may require $\Theta(n)$ constant time operations. Let S be the number of steps required by the numerical method used just to draw the diagram. In the worst case, the algorithm presented in this thesis requires $O(n\sp3)$ intersection calculations plus O(nS lg S) time using $O(n\sp3$ + S) space. A variant of this algorithm requires $O(n\sp3)$ intersection calculations plus $O(nS\sp2$ + $n\sp2S)$ time using $O(n\sp2)$ space.
Also presented are some variants of the new Voronoi diagram. One of these variants uses a convex polygon distance function. The multiplicatively weighted crystal growth Voronoi diagram using a convex polygon distance function does not require numerical methods to construct.

Multiplicatively weighted crystal growth Voronoi diagrams

Given a set S of n sites in the plane, the Voronoi diagram is a partition of the plane into regions, each region consisting of the set of points closer to a particular site in S than to any other site in S [PS85]. The literature on Voronoi diagrams and various extensions to Voronoi diagrams is extensive and has been recently surveyed by Aurenhammer [Aur88]. One extension is the mtdtiplicatively weighted Voronoi diagram in which the distance between a site p and a point X is de (P, x)/W(P), where de (p, X) is the Euclidean distance between p and X. The multiplicatively weighted diagram has size @(n2 ) and @(n2 ) time algorithms to compute it are known [AE84, Aur87]. There are applications of multiplicative weighted diagrams in many fields, including geography and economics [GHJ67, HL79, HJ68]. The standard Voronoi diagram and the additively weighted Voronoi diagram are widely used to model crystal growth [M HM80, GB78]. Modeling crystal growth using the standard Voronoi diagram or the addltively weighted Voronoi diagram assumes all crystals have the same constant growth rate. For many physical systems, this may not be a valid assumption [Guy72]. Frost and Thompson [FT87, FT88] use many variants of the Voronoi diagram to simulate crystal growth on thin films at many different growth rates. When each crystal hss a constant growth rate and the constant is allowed to be different for each crystal, they simulate the crystal growth by using the multiplicatively weighted Voronoi diagram. They note that this simulation is an approximation since the multiplicatively weighted Voronoi diagram produces regions that are not connected. Crystals growing from sites will always pr~ duce connected regions. In this paper, we propose a new model of the multiplicatively weighted Voronoi diagram that can be used to accurately simulate multiplicatively weighted crystal growth. In this new model, the distance from a site to a point in its region is measured along a shortest path lying entirely within the region. In the multiplicatively weighted crystal growth Voronoi diagram, a growing crystal (or region) may “wrap around” another site’s region. When a

Barry F. Schaudt

Papers

Methods for detecting errors in numerically controlled machining of sculptured surfaces

Approximate methods for simulation and verification of numerically controlled machining programs

Discrete simulation of NC machining

Multiplicatively weighted crystal growth Voronoi diagrams

Multiplicatively weighted crystal growth Voronoi diagrams (extended abstract)