**Title: Quadric-Of-Revolution (QUADOR) beams and their applications to lattice structures**

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Ashish Gupta

School of Interactive Computing

College of Computing

Georgia Institute of Technology

Date: Thursday, July 2, 2020

Time: 10:00 am EST

Location: https://us02web.zoom.us/j/86250641438?pwd=dmNsYndkWkRBL1IyS0R6alBOb0dvUT09

** **NOTE**: those who are not on the committee but plan to attend, may send (ashish.gupta@gatech.edu) their email ids, so that, if needed, we can invite them to another online platform.

**Note: this defense is remote-only due to the institute's guidelines on COVID-19**

**Committee:**

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Prof. Jarek Rossignac (Advisor, School of Interactive Computing, Georgia Tech)

Prof. Greg Turk (School of Interactive Computing, Georgia Tech)

Prof. Concettina Guerra (School of Interactive Computing, Georgia Tech)

Prof. Athanassios Economou (School of Architecture, Georgia Tech)

Dr. Suraj Musuvathy (nTopology Inc.)

**Abstract**

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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: (1) Each beam connects two balls. It is often desired to ensure that the surface of the beam connects with tangential continuity to the surfaces of the two balls that it joins. Furthermore, it is often desired to use beams with curved profiles and not just those bounded by straight circular cone sections. The round-off errors resulting from computing the intersections of such beams, and hence the precise boundary representation of the lattice prohibit doing so reliably when the beams are relatively small compared to the overall size of the lattice. (2) Microstructure lattices have extremely high complexity (possibly billions of beams). Precise models of such structures are not supported by the current generation of commercial modelers. Furthermore, it is computationally prohibitive to compute mass properties of such lattices by iterating over its balls and beams.

In this thesis, we propose to address both of these problems: (1) The challenge of designing and modeling beams with curved profiles and of computing their surface 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:

- We propose a family of beams, which we call the
**quador beams**, that are bound by a**quadric-of-revolution (quador)**surface abutting tangentially to the two balls. This family includes cone beams. We propose geometric constructions for quador beams, that have simple mathematical expressions and provides intuitive control of its shape. - We propose three analytically exact representations,
**Constructive Solid Geometry (CSG), Constructive Solid Trimming (CST),**and**Boundary Representation (BRep)**of a lattice with quador beams. Each of these representations is more suitable than others in efficiently performing certain geometric queries on a lattice, e.g. CSG for Point Membership Classification (PMC), CST for ray intersection, and BRep for minimum distance query. We propose compact data structures to store these three lattice representations, describe efficient methods to compute them, and provide associated operators to efficiently query the lattice. - We propose a numerically robust approach to compute the topology of the Brep a lattice of quador beams. In this approach, we infer the topology of the Brep of the lattice from the Brep of a
**polyhedral complex**. This complex is formed by a set of linear half-spaces, each of which trims a quador (or a ball) surface of the lattice. 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:

- We propose a class of lattices, which we call the
**Steady Lattices**. A steady lattice consists of a three-directional tensor of cells of balls and beams that each connects two balls in the same cell or two different cells. We discuss rows of balls and beams in each direction. In a steady lattice, all rows in at least one direction are steady, by which we mean that each cell of balls in a row and the beams incident on its balls are related to the previous cell by the same similarity transformation. We use an extremely concise representation of the lattice: The balls and beams of a template cell, three transformations, and 3 repetition counts. Steady lattices are easy to compute and are amenable to fast and robust queries. - We propose closed-form expressions that exploit steadiness for computing mass properties (e.g., surface area, volume, center-of-mass) of a steady lattice without iterating over all of its elements.

Our proposals lay the foundation for designing enormously large and complex lattice microstructures for 3D printed parts. Several of our proposals form the backbone of the state-of-the-art geometric applications developed for DARPA's TRAansformative DESign (TRADES) project. Part of the research reported in this thesis is published in the following papers: