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

• 批准号: 1717268
• 负责人: Eitan Grinspun
• 金额: $ 25万
• 依托单位: Columbia 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=1717268&HistoricalAwards=false
• 关键词: AF; Small; Collaborative; Research; Computational

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 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，这些形式可以从一组平面件组装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：不仅可以从胶合板或金属板等丰富的薄平板材料中轻松切割可展几何形状，而且还可以提供一种新颖的几何方法来进行刀具路径规划，提高通过计算机控制的侧铣制造形状的效率和精度。 这样的工艺通过降低成本、增加制造形式的复杂性以及使以前只能通过手工实现的任务自动化（例如，可展开形式的机器人折叠）。 一个基本的问题，解决的研究是自动逼近的任意曲面的少量可开发件-目前这个过程必须进行费力的专家工程师和设计师，严重限制了范围和影响的可开发制造过程。 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，离散微分几何试图从离散算法的角度重新表述经典几何知识。 该领域提供了现代几何理论和需要结合数据和计算的工业/应用程序之间的重要联系，在该项目中培训的学生将有能力为3D制造做出贡献。该项目建立在光滑和离散微分几何的基础上：而不是仅仅将离散网格视为数值近似，统一的目标是开发直接编码分段可展几何最显著特征的数据结构。 两个关键观察结果是：（i）单独的可展平性并不能充分表征离散可展性，通常会导致令人讨厌的“起皱”行为;（ii）分段可展曲面的曲率完全由其面片边界的形状编码，这是一个经常被利用的事实在服装设计中。 这些意见导致两个主要的推力，即（i）表示离散的可展性，自然避免起皱，保证存在离散的“统治线”，和（ii）有效的算法，可展曲面设计的基础上稀疏的曲率描述沿着关键特征曲线。 一个交叉主题是制造的物理考虑，例如，简单几何模型和与物理制品的产生相关的材料/本构特性之间的转换。

项目成果

Developability of Triangle Meshes

• DOI: 10.1145/3197517.3201303
• 发表时间: 2018-08-01
• 期刊: ACM TRANSACTIONS ON GRAPHICS
• 影响因子: 6.2
• 作者: Stein, Oded;Grinspun, Eitan;Crane, Keenan
• 通讯作者: Crane, Keenan

Natural Boundary Conditions for Smoothing in Geometry Processing

• DOI: 10.1145/3186564
• 发表时间: 2017-07
• 期刊: ACM Transactions on Graphics (TOG)
• 作者: Oded Stein;E. Grinspun;M. Wardetzky;Alec Jacobson

A Simple Discretization of the Vector Dirichlet Energy

• DOI: 10.1111/cgf.14070
• 发表时间: 2020-08
• 期刊: Computer Graphics Forum
• 影响因子: 2.5
• 作者: Oded Stein;M. Wardetzky;Alec Jacobson;E. Grinspun

Interactive design of castable shapes using two-piece rigid molds

• DOI: 10.1016/j.cag.2019.03.001
• 发表时间: 2019-05
• 期刊: Comput. Graph.
• 作者: Oded Stein;Alec Jacobson;E. Grinspun

A Smoothness Energy without Boundary Distortion for Curved Surfaces

• DOI: 10.1145/3377406
• 发表时间: 2019-05
• 期刊: ACM Transactions on Graphics (TOG)
• 作者: Oded Stein;Alec Jacobson;M. Wardetzky;E. Grinspun