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Article: Constructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution

TitleConstructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution
Authors
Keywords3D self-supporting surface
4D minimal hypersurface of revolution
Hyper-generatrix
Mean curvature
Issue Date2019
PublisherAssociation for Computing Machinery, Inc. The Journal's web site is located at http://tog.acm.org
Citation
ACM Transactions on Graphics, 2019, v. 38 n. 5, p. article no. 144 How to Cite?
AbstractThis 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.
Persistent Identifierhttp://hdl.handle.net/10722/294271
ISSN
2019 Impact Factor: 5.084
2015 SCImago Journal Rankings: 2.552

 

DC FieldValueLanguage
dc.contributor.authorMa, L-
dc.contributor.authorHe, Y-
dc.contributor.authorSun, Q-
dc.contributor.authorZhou, Y-
dc.contributor.authorZhang, C-
dc.contributor.authorWang, W-
dc.date.accessioned2020-11-23T08:28:58Z-
dc.date.available2020-11-23T08:28:58Z-
dc.date.issued2019-
dc.identifier.citationACM Transactions on Graphics, 2019, v. 38 n. 5, p. article no. 144-
dc.identifier.issn0730-0301-
dc.identifier.urihttp://hdl.handle.net/10722/294271-
dc.description.abstractThis 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.-
dc.languageeng-
dc.publisherAssociation for Computing Machinery, Inc. The Journal's web site is located at http://tog.acm.org-
dc.relation.ispartofACM Transactions on Graphics-
dc.rightsACM Transactions on Graphics. Copyright © Association for Computing Machinery, Inc.-
dc.rights©ACM, YYYY. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in PUBLICATION, {VOL#, ISS#, (DATE)} http://doi.acm.org/10.1145/nnnnnn.nnnnnn-
dc.subject3D self-supporting surface-
dc.subject4D minimal hypersurface of revolution-
dc.subjectHyper-generatrix-
dc.subjectMean curvature-
dc.titleConstructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution-
dc.typeArticle-
dc.identifier.emailWang, W: wenping@cs.hku.hk-
dc.identifier.authorityWang, W=rp00186-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1145/3188735-
dc.identifier.scopuseid_2-s2.0-85074420464-
dc.identifier.hkuros319205-
dc.identifier.volume38-
dc.identifier.issue5-
dc.identifier.spagearticle no. 144-
dc.identifier.epagearticle no. 144-
dc.publisher.placeUnited States-

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