Numerical Initial Value Problems in Ordinary Differential Equations

We present an efficient method for detecting collisions between complex solid objects. The method features a stable processing time and low sensitivity to the complexity of contact between objects. The algorithm handles both concave and convex objects; however, the best performance is achieved when at least one object is convex in the proximity of the collision zone (our techniques check the required convexity property as a byproduct of the calculations). The method achieves real-time performance when calculations are supported by the standard functionality of graphics hardware available on high-end workstations.

Fast collision detection between complex solids using rasterizing graphics hardware

An important class of surface intersection problems involves the sectioning of finite parametric polynomial patches by an unbounded algebraic surface. The section curve may be represented precisely by a high order algebraic curve F(u,ν) = 0 in the parameter space of the patch. It is then desired to evaluate the curve, i.e., to identify each of its open segments and closed loops, singular features such as cusps or self-intersections, and to generate ordered sequences of points along each segment or loop. Concepts from algebraic curve theory are employed to determine a set of characteristic points for the section curve. These comprise all curve points on the parameter domain boundary, all turning points where the curve tangent is parallel to u = 0 or ν = 0, and all singular points. The characteristic points dissect the section curve into a set of monotonic branches. Each characteristic point is assigned a link multiplicity, giving the number of branches entering or leaving that point. The number of monotonic curve branches is then uniquely determined by the sum of the link multiplicities. To complete the section curve evaluation, it is necessary to trace the curve branches between characteristic points. Two methods are described to identify and generate ordered point sequences along each branch: (1) locating curve points on an isoparametric grid and employing a heuristic sorting procedure; (2) marching along branches in small steps by local power-series expansions. The identification of all characteristic points coupled with the power series curve-tracing procedure provides an essentially deterministic method for evaluating parametric surface sections. This is a substantial improvement over current heuristic numerical algorithms.

The characterization of parametric surface sections

Sheet metal forming dies constructed of laminations offer advantages over more conventional tooling fabrication methods (e.g. CNC-machining) in terms of tooling accessibility, reduced limitations on die geometry and faster fabrication with harder die materials. Furthermore, the recently introduced Profiled Edge Lamination (PEL) tooling method improves upon other lamination-based tooling methods. Adoption of this promising rapid tooling method by industry is being hindered by the lack of formal analysis, design principles, and manufacturing requirements needed to construct dies in such a manner. Therefore, the propensity for delamination of the die is discussed and preventive measures are suggested. The basic machining instructions, i.e., an array of points and directional vectors for each lamination, are outlined for both compound and planar profiled-edge bevels. Laser, AWJ and flute-edge endmilling are experimentally identified as the most promising methods for machining bevels. Development of a stand-alone PEL fabrication machine is suggested over retrofitting commercially-available 5-axis machines. Finally, the general procedure for creating PEL dies is implemented in the construction of a matched set of sheet metal forming tools. These tools are used to successfully stamp a sheet metal part out of draw-quality steel.

Rapid Tooling for Sheet Metal Forming Using Profiled Edge Laminations—Design Principles and Demonstration

Computer graphics and CAD literature—A keyword-indexed bibliography

A new method for numerically computing the intersection of implicitly defined surfaces provides the capability to display intersections that are impossible to find analytically.

An Algorithm for Locating and Displaying the Intersection of Two Arbitrary Surfaces

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An Algorithm for Locating and Displaying the Intersection of Two Arbitrary Surfaces