Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

Design for Additive Manufacturing: A Method to Explore Unexplored Regions of the Design Space

Additive Manufacturing (AM) technologies enable the fabrication of parts and devices that are geometrically complex, have graded material compositions, and can be customized. To take advantage of these capabilities, it is important to assist designers in exploring unexplored regions of design spaces. We present a Design for Additive Manufacturing (DFAM) method that encompasses conceptual design, process selection, later design stages, and design for manufacturing. The method is based on the process-structure-property-behavior model that is common in the materials design literature. A prototype CAD system is presented that embodies the method. Manufacturable ELements (MELs) are proposed as an intermediate representation for supporting the manufacturing related aspects of the method. Examples of cellular materials are used to illustrate the DFAM method.Mechanical Engineerin

Combined 3D thinning and greedy algorithm to approximate realistic particles with corrected mechanical properties

The shape of irregular particles has significant influence on micro- and macro-scopic behavior of granular systems. This paper presents a combined 3D thinning and greedy set-covering algorithm to approximate realistic particles with a clump of overlapping spheres for discrete element method (DEM) simulations. First, the particle medial surface (or surface skeleton), from which all candidate (maximal inscribed) spheres can be generated, is computed by the topological 3D thinning. Then, the clump generation procedure is converted into a greedy set-covering (SCP) problem. To correct the mass distribution due to highly overlapped spheres inside the clump, linear programming (LP) is used to adjust the density of each component sphere, such that the aggregate properties mass, center of mass and inertia tensor are identical or close enough to the prototypical particle. In order to find the optimal approximation accuracy (volume coverage: ratio of clump's volume to the original particle's volume), particle flow of 3 different shapes in a rotating drum are conducted. It was observed that the dynamic angle of repose starts to converge for all particle shapes at 85% volume coverage (spheres per clump < 30), which implies the possible optimal resolution to capture the mechanical behavior of the system.Comment: 34 pages, 13 figure

Geodesic grassfire for computing mixed-dimensional skeletons

Skeleton descriptors are commonly used to represent, understand and process shapes. While existing methods produce skeletons at a fixed dimension, such as surface or curve skeletons for a 3D object, often times objects are better described using skeleton geometry at a mixture of dimensions. In this paper we present a novel algorithm for computing mixed-dimensional skeletons. Our method is guided by a continuous analogue that extends the classical grassfire erosion. This analogue allows us to identify medial geometry at multiple dimensions, and to formulate a measure that captures how well an object part is described by medial geometry at a particular dimension. Guided by this analogue, we devise a discrete algorithm that computes a topology-preserving skeleton by iterative thinning. The algorithm is simple to implement, and produces robust skeletons that naturally capture shape components. Under Revie