Simplicial Ricci flow: an example of a neck pinch singularity in 3D
We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with topology S³. SRF was recently introduced as an unstructured mesh formulation of Hamilton’s Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geom...
| Main Authors: | , , , , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Published: |
International Press of Boston
2018
|
| Online Access: | http://hdl.handle.net/1721.1/119822 |
| _version_ | 1811088213481095168 |
|---|---|
| author | Alsing, Paul M. Miller, Warner A. Corne, Matthew Gu, David Lloyd, Seth Ray, Shannon Yau, Shing-Tung |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Alsing, Paul M. Miller, Warner A. Corne, Matthew Gu, David Lloyd, Seth Ray, Shannon Yau, Shing-Tung |
| author_sort | Alsing, Paul M. |
| collection | MIT |
| description | We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with topology S³. SRF was recently introduced as an unstructured mesh formulation of Hamilton’s Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The S² cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice wellcentered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model. |
| first_indexed | 2024-09-23T13:58:30Z |
| format | Article |
| id | mit-1721.1/119822 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:58:30Z |
| publishDate | 2018 |
| publisher | International Press of Boston |
| record_format | dspace |
| spelling | mit-1721.1/1198222022-10-01T18:18:33Z Simplicial Ricci flow: an example of a neck pinch singularity in 3D Alsing, Paul M. Miller, Warner A. Corne, Matthew Gu, David Lloyd, Seth Ray, Shannon Yau, Shing-Tung Massachusetts Institute of Technology. Department of Mechanical Engineering Lloyd, Seth We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with topology S³. SRF was recently introduced as an unstructured mesh formulation of Hamilton’s Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The S² cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice wellcentered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model. 2018-12-21T19:28:58Z 2018-12-21T19:28:58Z 2015-01 2018-12-13T14:51:38Z Article http://purl.org/eprint/type/JournalArticle 2328-8876 2328-8884 http://hdl.handle.net/1721.1/119822 Alsing, Paul M. et al. “Simplicial Ricci Flow: An Example of a Neck Pinch Singularity in 3D.” Geometry, Imaging and Computing 1, 3 (2014): 303–331 © 2014 International Press of Boston http://dx.doi.org/10.4310/GIC.2014.V1.N3.A1 Geometry, Imaging and Computing Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf International Press of Boston arXiv |
| spellingShingle | Alsing, Paul M. Miller, Warner A. Corne, Matthew Gu, David Lloyd, Seth Ray, Shannon Yau, Shing-Tung Simplicial Ricci flow: an example of a neck pinch singularity in 3D |
| title | Simplicial Ricci flow: an example of a neck pinch singularity in 3D |
| title_full | Simplicial Ricci flow: an example of a neck pinch singularity in 3D |
| title_fullStr | Simplicial Ricci flow: an example of a neck pinch singularity in 3D |
| title_full_unstemmed | Simplicial Ricci flow: an example of a neck pinch singularity in 3D |
| title_short | Simplicial Ricci flow: an example of a neck pinch singularity in 3D |
| title_sort | simplicial ricci flow an example of a neck pinch singularity in 3d |
| url | http://hdl.handle.net/1721.1/119822 |
| work_keys_str_mv | AT alsingpaulm simplicialricciflowanexampleofaneckpinchsingularityin3d AT millerwarnera simplicialricciflowanexampleofaneckpinchsingularityin3d AT cornematthew simplicialricciflowanexampleofaneckpinchsingularityin3d AT gudavid simplicialricciflowanexampleofaneckpinchsingularityin3d AT lloydseth simplicialricciflowanexampleofaneckpinchsingularityin3d AT rayshannon simplicialricciflowanexampleofaneckpinchsingularityin3d AT yaushingtung simplicialricciflowanexampleofaneckpinchsingularityin3d |