Quadric-of-revolution (quador) beams and their applications to lattice structures

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Gupta, Ashish
Rossignac, Jarek
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Additive Manufacturing (AM) or 3D printing presents the potential to fabricate parts with novel mechanical properties by allowing control over the internal microstructure of these parts. Lattices used for designing these microstructures are often defined by 3D arrangements of balls and beams. Attempting to discover optimal structures with desired mechanical properties, engineers explore variations: of the overall shape of these arrangements; of the dimensions and placements of the balls; of the profile shape of the beams and of the lattice connectivity that they form. Current Solid Modeling technology only provides limited support for such explorations for several reasons. In this thesis, we propose to address two problems: (1) The challenge of designing and modeling beams with curved profiles and of computing their boundary representation reliably and accurately. (2) The challenge of precisely modeling highly complex lattices with profiled beams and of efficiently computing their mass properties. To address the first challenge: 1. We propose a family of beams, which we call the \define{quador beams}, that are bound by a quadric-of-revolution (quador) surface abutting tangentially to the two balls. We propose geometric constructions for quador beams, that have simple mathematical expressions and provides intuitive control of its shape. 2. We propose three analytically exact representations, Constructive Solid Geometry (CSG), Constructive Solid Trimming (CST), and Boundary Representation (Brep) of a lattice with quador beams. We propose compact data structures to store these representations and query the lattice. 3. We propose a numerically robust approach to compute the topology of the Brep a lattice of quador beams. Our approach avoids computing square roots (or any other irrational values) and involves only rational numbers and basic arithmetic operations (addition, subtraction, multiplication and division) on them. To address the second challenge: 1. We propose a class of lattices, which we call the \define{Steady Lattices}. A steady lattice consists of a three-directional tensor of cells of balls and beams. We use an extremely concise representation of the lattice: The balls and beams of a template cell, three transformations, and 3 repetition counts. 2. We propose closed-form expressions that exploit steadiness for accelerated computation of mass properties (e.g., surface area, volume, center-of-mass) of a steady lattice. Our work lays the foundation for designing enormously large (trillions of beams) and complex (bent, graded and complex connectivity of beams) lattice microstructures for 3D printed parts and forms the backbone of the state-of-the-art geometric applications developed for DARPA's TRAansformative DESign (TRADES) project.
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