Jörg Schwerdt

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.

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.

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.

Protecting critical facets in layered manufacturing: implementation and experimental results

In layered manufacturing (LM), 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 process-planning step in LM is choosing a suitable 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. We describe a robust, exact, and efficient implementation of an algorithm that computes a description of a subset of all build directions for which a prescribed facet is not in contact with supports. We also present test results on models obtained from industry, and on collections of random triangles.

Computing the minimum diameter for moving points: an exact implementation using parametric search

Parametric Search, developed by Megiddo [8], is a powerful algorithmic technique that can be used to solve a large variety of geometric optimization problems, see e.g. [1]. Although this technique is ingenious, it is, in general, hard to implement. In this paper, we report on the implementation of (a practical variant of) an algorithm that is based on parametric search, and that is due to Gupta et al. [7]. As far as we know, this is the first implementation of a parametric search algorithm. The algorithm of [7] solves the following problem. We are given a set of n points in the plane that are moving at constant but possibly different velocities. The diameter of the points at time t is the largest Euclidean distance among all pairs of points at time t. Our goal is to compute the time t∗ at which the diameter is minimum. This problem can be solved trivially as follows. For each pair of points, the square of their distance defines a quadratic function in t, i.e., a parabola. The minimum diameter is obtained by finding the lowest point on the upper envelope of these ( n 2 ) parabolas. This gives an algorithm with running time O(n logn), and using O(n) space. It was shown in [7], that the problem can be solved, using parametric search, in O(n log n) time, using O(n) space. The latter algorithm is complicated and not practical; it uses e.g. an optimal parallel sorting algorithm. Therefore, we have implemented a variant of it, which has running time O(n log n). We implemented this algorithm in C++ using LEDA [9, 10] and CGAL [5, 6]. In our first implementation we used the number type double in all numerical computations. For many problem instances the implementation could not find the minimum diameter. During parametric search an inter-

Minimizing the total projection of a set of vectors, with applications to layered manufacturing

In layered manufacturing (LM), a three-dimensional polyhedral solid is built as a stack of two-dimensional slices. Each slice (a polygon) is built by filling its interior with a sequence of parallel line segments (of some small non-zero width), in a process called hatching. A critical step in hatching is choosing a direction which minimizes the number of segments. In this paper, this problem is approximated as the problem of finding a direction which minimizes the total projected length of a certain set of vectors. Efficient algorithms are proposed for the latter problem, using techniques from computational geometry. Experimental and theoretical analyses show that this approach yields results that approximate closely the optimal solution to the hatching problem. Extensions of these results to several related problems are also discussed.

Computing An Optimal Hatching Direction In Layered Manufacturing

In Layered Manufacturing (LM), a prototype of a virtual polyhedral object is built by slicing the object into polygonal layers, and then building the layers one after another. In StereoLithography, a specific LM-technology, a layer is built using a laser which follows paths along equally-spaced parallel lines and hatches all segments on these lines that are contained in the layer. We consider the problem of computing a direction of these lines for which the number of segments to be hatched is minimum, and present an algorithm that solves this problem exactly. The algorithm has been implemented and experimental results are reported for real-world polyhedral models obtained from industry.

Computing Optimal Hatching Directions in Layered Manufacturing

In Layered Manufacturing, a three-dimensional polyhedral solid is built as a stack of two-dimensional slices. Each slice (a polygon) is built by filling its interior with a sequence of parallel line segments, of small non-zero width, in a process called hatching. A critical step in hatching is choosing a direction which minimizes the number of segments. Exact and approximation algorithms are given here for this problem, and their performance is analyzed both experimentally and analytically. Extensions to several related problems are discussed briefly.