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，这些形式可以从扁平碎片集合中组装而成，而不会产生材料应力，称为分段可开发的几何形状。这些将推动高级3D制造业新兴行业的新发展：不仅可以从丰富的薄材料（如胶合板或钣金）中轻松切割可开发的几何形状，而且还可以为工具路径计划提供一种新颖的几何方法，从而提高了通过计算机控制的额定级铣削来提高形状的效率和准确性。这样的过程通过降低成本，增加制造形式的复杂性以及以前只能通过手动实现（例如，可开发形式的机器人折叠）来自动化任务，从而为美国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数可开发的部分自动近似任意弯曲的表面 - 目前，这一过程必须由专家工程师和设计师努力地进行，严重限制了可开发制造过程的范围和影响。更广泛的是，可开发几何形状的算法模型丰富了离散差异几何领域中的基本理解，该几何形状试图从离散的，算法的角度从经典的几何知识进行重新制定经典的几何知识。该领域提供了现代几何理论与需要合并数据和计算的工业/应用应用程序之间的重要联系，并且在该项目中培训的学生将有足够的能力在3D制造中做出贡献。该项目建立在平滑和离散的微分几何形状的基础上：而不是将离散网格视为数值近似，而是统一目标是开发直接编码分段可开发几何形状最显着特征的数据结构。有两个关键的观察结果是（i）单独的扁平性不是离散可开发性的足够表征，通常会导致令人讨厌的“弯曲”行为，并且（ii）（ii）分段可开发表面的曲率完全由其斑块边界的形状编码，这是在服装设计中经常被利用的事实。这些观察结果导致了两个主要的推力，即（i）具有离散可开发性的表示形式，它们自然地避免通过保证离散的“统治线”的存在，以及（ii）基于沿关键特征曲线的曲率描述的稀疏描述的可开发表面设计的有效算法。一个跨切割主题是用于制造的物理考虑，例如，简单的几何模型与与物理伪像的生产相关的材料/本构特性之间的翻译。