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 minimi7,ing 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 t.o convex polyhedra only. Algorithms arc a1so given for optimizing supports for non-convex, simple polygons. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, balfplanc range searching, ray-shooting, visibility, and constrained optimization.

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

“…•Portions of this work appear, in preliminary form, in [11]. The present paper and a companion paper [12] expand upon and improve the results presented in [11].…”

Manufacturing processes

“…The classes of objects that can be built by these processes are also related to those buildable via NC-machining and casting. In [MJSG99], an algorithm is given to minimize the maximum stairstep error ( [BB95]) over all facets of a polyhedron in O(n log n) time and to minimize the sum of the stairstep errors on all facets in O(n 2 ) time; the first algorithm even allows facets to be weighted to indicate their relative importance with respect to surface finish. Also given are O(n 2 )-time algorithms to minimize the volume and (independently) the contact-area of supports for a convex polyhedron.…”

“…Three formulations for reconciling the criteria are considered: optimizing the criteria sequentially, optimizing a weighted combination of the criteria, and allowing the criteria to meet designer-specified thresholds. The methods in [MJSG99,MJSS01] use well-known techniques from computational geometry, such as spherical arrangements, convex hulls, and Voronoi Diagrams, in conjunction with constrained optimization methods such as Lagrangian Multipliers. In [AD00], an approximation algorithm is given for minimizing the contact-area for a convex polyhedron.…”

Global Optimization for a Class of Nonlinear Sum of Ratios Problem

“…The sum of ratios problem has attracted considerable attention in the literature because of its large number of practical applications in various fields of study, including transportation planning, government contracting, economics, and finances [1][2][3][4][5][6]. And from a research point of view, the sum of ratios problem poses significant theoretical and computational challenges.…”