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AF: Small: Collaborative Research: Computational Representations for Design and Fabrication of Developable Surfaces

AF：小型：协作研究：可展曲面设计和制造的计算表示

基本信息

批准号：
1717320
负责人：
Keenan Crane
金额：
$ 25万
依托单位：
Carnegie-Mellon University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1717320&HistoricalAwards=false
关键词：
AF Small Collaborative Research Computational

项目摘要

This project develops mathematical representations and computer algorithms for three-dimensional forms that can be assembled from a collection of flat pieces without incurring material stress, known as piecewise developable geometry. These will drive new developments in the emerging industry of advanced 3D manufacturing: Not only can developable geometry can easily be cut from a rich array of thin flat materials like plywood or sheet metal, but it can also provide a novel geometric approach to tool path planning that improves the efficiency and accuracy of shapes fabricated via computer-controlled flank milling. Such processes offer a competitive advantage for manufacturing in the US by reducing cost, increasing complexity of fabricated forms, and automating tasks previously only achievable by hand (e.g., robotic folding of developable forms). A fundamental issue addressed by the research is automatic approximation of an arbitrary curved surface by a small number of developable pieces---at present this process must be carried out laboriously by expert engineers and designers, severely limiting the scope and impact of developable manufacturing processes. More broadly, algorithmic models of developable geometry enrich basic understanding in the area of discrete differential geometry, which seeks to reformulate classical geometric knowledge from a discrete, algorithmic point of view. This area provides a crucial link between modern geometric theory and industrial/applied applications that need to incorporate data and computation, and students trained in this project will be well-equipped to contribute in 3d manufacturing. The project builds on foundations from smooth and discrete differential geometry: rather than view discrete meshes as mere numerical approximations, the unifying goal is to develop data structures that directly encode the most salient features of piecewise developable geometry. Two key observations are that (i) flattenability alone is not a sufficient characterization for discrete developability, often leading to nasty "crumpling" behavior and (ii) the curvature of a piecewise developable surface is encoded entirely by the shape of its patch boundaries, a fact often exploited in garment design. These observations lead to two primary thrusts, namely (i) representations for discrete developability that naturally avoid crumpling by guaranteeing the existence of discrete "ruling lines," and (ii) efficient algorithms for developable surface design based on sparse descriptions of curvature along critical feature curves. A cross-cutting theme is physical considerations for fabrication, e.g., translation between simple geometric models and material/constitutive properties relevant to the production of physical artifacts.

该项目开发了三维形式的数学表示和计算机算法，这些三维形式可以由一组扁平部件组装而成，而不会产生材料应力，称为分段可展几何。这些将推动先进 3D 制造新兴行业的新发展：不仅可以轻松地从胶合板或金属板等丰富的薄扁平材料中切割出可开发的几何形状，而且还可以为刀具路径规划提供一种新颖的几何方法，从而提高通过计算机控制的侧面铣削制造的形状的效率和准确性。这些工艺通过降低成本、增加制造形式的复杂性以及自动化以前只能手工完成的任务（例如，机器人折叠可开发形式），为美国的制造业提供了竞争优势。该研究解决的一个基本问题是通过少量可开发部件自动逼近任意曲面——目前这一过程必须由专家工程师和设计师费力地完成，严重限制了可开发制造工艺的范围和影响。更广泛地说，可展几何的算法模型丰富了离散微分几何领域的基本理解，该领域试图从离散的算法角度重新表述经典几何知识。该领域在现代几何理论与需要整合数据和计算的工业/应用应用之间提供了重要的联系，在该项目中接受培训的学生将有能力为 3D 制造做出贡献。该项目建立在平滑和离散微分几何的基础上：统一的目标是开发直接编码分段可开发几何最显着特征的数据结构，而不是将离散网格视为单纯的数值近似。两个关键的观察结果是：(i) 单独的可展性不足以表征离散可展性，通常会导致令人讨厌的“起皱”行为；(ii) 分段可展表面的曲率完全由其面片边界的形状编码，这一事实在服装设计中经常被利用。这些观察结果产生了两个主要推力，即（i）离散可展性的表示，通过保证离散“标尺线”的存在来自然地避免褶皱，以及（ii）基于沿关键特征曲线曲率的稀疏描述的可展曲面设计的有效算法。横切主题是制造的物理考虑因素，例如简单几何模型和与物理制品生产相关的材料/构成属性之间的转换。