Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	蒋才桂教授，西安交通大学
Inviter:
Title:	Geometric Modeling from Flat Sheet Material
Time & Venue:	2022.01.04 18：30-19：30 腾讯会议：703-690-115
Abstract:	In this presentation, I will talk about our recent work on geometric modeling based on planar materials which consists of two major parts. 1 Quad-Mesh Based Isometric Mappings and Developable Surfaces. We discretize isometric mappings between surfaces as correspondences between checkerboard patterns derived from quad meshes. This method captures the degrees of freedom inherent in smooth isometries and enables a natural definition of discrete developable surfaces. This definition, which is remarkably simple, leads to a class of discrete developables which is much more flexible in applications than previous concepts of discrete developables. In this paper, we employ optimization to efficiently compute isometric mappings, conformal mappings and isometric bending of surfaces. We perform geometric modeling of developables, including cutting, gluing and folding. The discrete mappings presented here have applications in both theory and practice: We propose a theory of curvatures derived from a discrete Gauss map as well as a construction of watertight CAD models consisting of developable spline surfaces. 2 Shape-morphing mechanical metamaterials. Small-scale cut and fold patterns imposed on sheet material enable its morphing into threedimensional shapes. This manufacturing paradigm has been receiving much attention in recent years and poses challenges in both fabrication and computation. It is intimately connected with the interpretation of patterned sheets as mechanical metamaterials, typically of negative Poisson ratio. We here present an affirmative solution to a fundamental geometric question, namely the targeted programming of a shape morph. We use optimization to compute kirigami patterns that realize a morph between shapes, in particular between a flat sheet and a surface in space. The shapes involved can be arbitrary; in fact we are able to approximate any mapping between shapes whose principal distortions do not exceed certain bounds. This amounts to a solution of the so-called inverse problem for kirigami cut and fold patterns. The methods we employ include a differential-geometric interpretation of the morph, besides drawing on recent progress in geometric computing