{"@context":"http://iiif.io/api/presentation/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/manifest.json","@type":"sc:Manifest","label":"Accurate, Semi-Implicit Methods with Mesh Adaptivity for Mean Curvature and Surface Diffusion Flows Using Triangulated Surfaces","metadata":[{"label":"dc.description.sponsorship","value":"This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree."},{"label":"dc.format","value":"Monograph"},{"label":"dc.format.medium","value":"Electronic Resource"},{"label":"dc.identifier.uri","value":"http://hdl.handle.net/1951/59616"},{"label":"dc.language.iso","value":"en_US"},{"label":"dc.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.abstract","value":"Geometric partial differential equations, such as mean-curvature flow and surface diffusion, are challenging to solve numerically due to their strong non-linearity and stiffness, when solved explicitly. Solving these high-order PDEs using explicit methods would require very small time steps to achieve stability, whereas using implicit methods would result in complex nonlinear systems of equations that are expensive to solve. In addition, accurate spatial discretizations of these equations pose challenges in their own rights, especially on triangulated surfaces. We propose new methods for mean curvature flow and surface diffusion using triangulated surfaces. Our method uses a weighted least-squares approximation for improved accuracy and stability, and semi-implicit schemes for time integration for larger time steps and higher efficiency. If mesh element quality is initially poor, or becomes poor through evolution under mean curvature flow or surface diffusion, we utilize mesh adaptivity to improve mesh quality and proceed further in evolution. Numerical experiments and comparisons demonstrate that our method can achieve second-order accuracy for both mean-curvature flow and surface diffusion, while being much more accurate and stable than using explicit schemes or alternative methods."},{"label":"dcterms.available","value":"2015-04-24T14:45:35Z"},{"label":"dcterms.contributor","value":"Qin, Hong"},{"label":"dcterms.creator","value":"Clark, Bryan L."},{"label":"dcterms.dateAccepted","value":"2015-04-24T14:45:35Z"},{"label":"dcterms.dateSubmitted","value":"2015-04-24T14:45:35Z"},{"label":"dcterms.description","value":"Department of Applied Mathematics and Statistics"},{"label":"dcterms.extent","value":"84 pg."},{"label":"dcterms.format","value":"Monograph"},{"label":"dcterms.identifier","value":"Clark_grad.sunysb_0771E_10871"},{"label":"dcterms.issued","value":"2012-05-01"},{"label":"dcterms.language","value":"en_US"},{"label":"dcterms.provenance","value":"Made available in DSpace on 2013-05-22T17:34:20Z (GMT). No. of bitstreams: 1\nClark_grad.sunysb_0771E_10871.pdf: 1046958 bytes, checksum: cf76a8d0f4290e70ec8b674eb0ab1e22 (MD5)\n  Previous issue date:    1"},{"label":"dcterms.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.subject","value":"discrete mesh, finite element method, general finite difference method, mean curvature flow, surface diffusion, surface laplacian"},{"label":"dcterms.title","value":"Accurate, Semi-Implicit Methods with Mesh Adaptivity for Mean Curvature and Surface Diffusion Flows Using Triangulated Surfaces"},{"label":"dcterms.type","value":"Dissertation"},{"label":"dc.type","value":"Dissertation"}],"description":"This manifest was generated dynamically","viewingDirection":"left-to-right","sequences":[{"@type":"sc:Sequence","canvases":[{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json","@type":"sc:Canvas","label":"Page 1","height":1650,"width":1275,"images":[{"@type":"oa:Annotation","motivation":"sc:painting","resource":{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/95%2F85%2F31%2F95853107143963514732126145867379338536/full/full/0/default.jpg","@type":"dctypes:Image","format":"image/jpeg","height":1650,"width":1275,"service":{"@context":"http://iiif.io/api/image/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/95%2F85%2F31%2F95853107143963514732126145867379338536","profile":"http://iiif.io/api/image/2/level2.json"}},"on":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json"}]}]}]}