Jayanth Majhi

On some geometric optimization problems in layered manufacturing

Abstract Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.

Efficient geometric algorithms for workpiece orientation in 4- and 5-axis NC machining

Abstract In 4- and 5-axis NC machines, the time to dismount, recalibrate, and remount the workpiece after each set of accessible faces of the workpiece has been machined can be considerable in comparison to the actual machining time. Unfortunately, the problem of minimizing the number of setups is NP-hard. In this paper, efficient algorithms are given for a greedy heuristic, where the goal is to find an orientation for the workpiece which maximizes the number of faces that can be machined in a single setup—using either a ball-end or a filletend cutter. The algorithms are based on geometric duality, topological sweep, interesting new properties concerning intersection and covering on the unit-sphere, and on techniques for efficiently constructing and searching an arrangement of polygons on the unit-sphere. The results imply that the optimal number of set-ups can be approximated to within a logarithmic factor. Evidence is also provided that it may not be possible to improve substantially on the proposed algorithms.

Minimizing support structures and trapped area in two-dimensional layered manufacturing

Abstract Algorithms are given for the two-dimensional versions of optimization problems arising in layered manufacturing, where a polygonal object is built by slicing its CAD model and manufacturing the slices successively. The problems considered are minimizing (i) the contact-length between the supports and the manufactured object, (ii) the area of the support structures used, and (iii) the area of the so-called trapped regions—factors that affect the cost and quality of the process.

Computing a flattest, undercut free parting line for a convex polyhedron, with application to mold design

Abstract A parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as “flat” as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria.

Multi-criteria geometric optimization problems in layered manufacturing

In Lnycred Manufacturing, the choice of the build direction for the model influences several design criteria, including the number of layers, the volume and contact-area of the support r&ructures, and the surface finish. These, in turn, impact the throughput and cost of the process. In this papcr, efficient geometric algorithms are given to reconcile two or more of these criteria simultaneously, under three formulations of multi-criteria optimization: Finding a build direction which (i) optimizes the criteria sequentially, (ii) optimizes their weighted sum, or (ii) allows the criteria to meet designer-prescribed thresholds. While the algorithms involving “support volume” or %ontact area” apply only to convex models, the solutions for “‘surface finish” and ‘%mmbcr of layers” are applicable to any polyhedral model. Some of the latter algorithms have also been implemented and tested on real-world models obtained from industry. The geometric techniques used include construction and searching of certain arrangements on the unit-sphere, 3dlmcnsional convex hulls, Voronoi diagrams, point location, and hierarchical representations. Additionally, solutions are also provided, for the first time, for the constrahted versions of two fundamental geometric problems, namely polyhedron width and large& empty diik on the unit-sphere.

Protecting critical facets in layered manufacturing

Abstract In layered manufacturing, a three-dimensional polyhedral object is built by slicing its (virtual) CAD model, and manufacturing the slices successively. During this process, support structures are used to prop up overhangs. An important issue is choosing the build direction, as it affects, among other things, the location of support structures on the part, which in turn impacts process speed and part finish. Algorithms are given here that (i) compute a description of all build directions for which a prescribed facet is not in contact with supports, and (ii) compute a description of all build directions for which the total area of all facets that are not in contact with supports is maximum. A simplified version of the first algorithm has been implemented, and test results on models obtained from industry are given.

Computing a Flattest, Undercut-Free Parting Line for a Convex Polyhedron, with Application to Mold Design

A parting line for a convex polyhedron, \(\mathcal{P}\), is a closed curve on the surface of \(\mathcal{P}\). It defines the two pieces of \(\mathcal{P}\) for which mold-halves must be made. An undercut-free parting line is one which does not create recesses or projections in \(\mathcal{P}\) and thus allows easy de-molding of \(\mathcal{P}\). Computing an undercut-free parting line that is as flat as possible is an important problem in mold design. In this paper, an O(n2)-time algorithm is presented to compute such a line, according to a prescribed flatness criterion, where n is the number of vertices in \(\mathcal{P}\).

On Some Geometric Optimization Problems in Layered Manufacturing

Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the degree of stair-stepping on the surfaces of the manufactured object, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object-all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable to convex polyhedra only. Algorithms are also given for optimizing supports for non-convex, simple polygons. The techniques used include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, ray-shooting, visibility, and constrained optimization.

A Decomposition-Based Approach to Layered Manufacturing

Layered Manufacturing allows physical prototypes of 3D parts to be built directly from their computer models, as a stack of 2D layers. This paper proposes a new approach, which decomposes the model into a small number of pieces, builds each separately, and glues them together to generate the prototype. This allows large models to be built in parallel and also reduces the need for so-called support structures. Decomposition algorithms that minimize support requirements are given for convex and non-convex polyhedra. Experiments, on convex polyhedra, show that the approach can reduce support requirements substantially.

Computing the width of a three-dimensional point set: an experimental study

We describe a robust, exact, and efficient implementation of an algorithm that computes the width of a three-dimensional point set. The algorithm is based on efficient solutions to problems that are at the heart of computational geometry: three-dimensional convex hulls, point location in planar graphs, and computing intersections between line segments. The latter two problems have to be solved for planar graphs and segments on the unit sphere, rather than in the two-dimensional plane. The implementation is based on LEDA, and the geometric objects are represented using exact rational arithmetic.