Constructing 3D Self-Supporting Surfaces with Isotropic Stress Using 4D Minimal Hypersurfaces of Revolution
| Autor: | Caiming Zhang, Long Ma, Ying He, Qian Sun, Yuanfeng Zhou, Wenping Wang |
|---|---|
| Přispěvatelé: | School of Computer Science and Engineering |
| Rok vydání: | 2019 |
| Předmět: |
Surface (mathematics)
Minimal surface Mean curvature Computer science Mathematical analysis Isotropy 020207 software engineering 02 engineering and technology Computing Methodologies Computer Graphics and Computer-Aided Design Potential energy Finite element method Piecewise linear function Catenary Computer Graphics 0202 electrical engineering electronic engineering information engineering Computer science and engineering [Engineering] 020201 artificial intelligence & image processing Polygon mesh Surface of revolution Centroidal Voronoi tessellation Surface reconstruction Energy functional |
| Zdroj: | ACM Transactions on Graphics. 38:1-13 |
| ISSN: | 1557-7368 0730-0301 |
| DOI: | 10.1145/3188735 |
| Popis: | 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. Ministry of Education (MOE) This project was partially supported by Singapore Ministry of Education Tier 1 & Tier 2 Grants, National Natural Science Foundation of China Grants (No. 61702363, No. 61772312, No. 61802228), NSFC Joint Fund with Zhejiang Integration of Informatization and Industrialization under Key Project (U1609218) and the Key Research and Development Project of Shandong Province (2017GGX10110). |
| Databáze: | OpenAIRE |
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