Please use this identifier to cite or link to this item: https://hdl.handle.net/10356/151961
Title: Constructing 3D self-supporting surfaces with isotropic stress using 4D minimal hypersurfaces of revolution
Authors: Ma, Long
He, Ying
Sun, Qian
Zhou, Yuanfeng
Zhang, Caiming
Wang, Wenping
Keywords: Engineering::Computer science and engineering
Issue Date: 2019
Source: Ma, L., He, Y., Sun, Q., Zhou, Y., Zhang, C. & Wang, W. (2019). Constructing 3D self-supporting surfaces with isotropic stress using 4D minimal hypersurfaces of revolution. ACM Transactions On Graphics, 38(5), 144-. https://dx.doi.org/10.1145/3188735
Journal: ACM Transactions on Graphics
Abstract: This article presents a new computational framework for constructing 3D self-supporting surfaces with isotropic stress. Inspired by the self-supporting property of catenary and the fact that catenoid (the surface of revolution of the catenary curve) is a minimal surface, we discover the relation between 3D self-supporting surfaces and 4D minimal hypersurfaces (which are 3-manifolds). Lifting the problem into 4D allows us to convert gravitational forces into tensions and reformulate the equilibrium problem to total potential energy minimization, which can be solved using a variational method. We prove that the hyper-generatrix of a 4D minimal hyper-surface of revolution is a 3D self-supporting surface, implying that constructing a 3D self-supporting surface is equivalent to volume minimization. We show that the energy functional is simply the surface's gravitational potential energy, which in turn can be converted into a surface reconstruction problem with mean curvature constraint. Armed with our theoretical findings, we develop an iterative algorithm to construct 3D self-supporting surfaces from triangle meshes. Our method guarantees convergence and can produce near-regular triangle meshes, thanks to a local mesh refinement strategy similar to centroidal Voronoi tessellation. It also allows users to tune the geometry via specifying either the zero potential surface or its desired volume. We also develop a finite element method to verify the equilibrium condition on 3D triangle meshes. The existing thrust network analysis methods discretize both geometry and material by approximating the continuous stress field through uniaxial singular stresses, making them an ideal tool for analysis and design of beam structures. In contrast, our method works on piecewise linear surfaces with continuous material. Moreover, our method does not require the 3D-to-2D projection, therefore it also works for both height and non-height fields.
URI: https://hdl.handle.net/10356/151961
ISSN: 0730-0301
DOI: 10.1145/3188735
Rights: © 2019 Association for Computing Machinery. All rights reserved.
Fulltext Permission: none
Fulltext Availability: No Fulltext
Appears in Collections:SCSE Journal Articles

Page view(s)

128
Updated on May 20, 2022

Google ScholarTM

Check

Altmetric


Plumx

Items in DR-NTU are protected by copyright, with all rights reserved, unless otherwise indicated.