Type
Text
Type
Dissertation
Advisor
Jiao, Xiangmin | Glimm, James | Mahadevan, Vijay S. | Samulyak, Roman | .
Date
2016-12-01
Keywords
Applied mathematics
Department
Department of Applied Mathematics and Statistics
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/77421
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
In large scale simulations of complex partial differential equations (PDE’s) using finite element methods (FEM), mesh generation, remeshing and linear solver are the most vital steps to obtain accurate solutions. All these areas have been explored quite extensively. We seek to develop an integrated framework for these steps. During simulation it is often desirable to start with a relatively coarse mesh and then refine the mesh accordingly (since in most cases the criteria for mesh resolution is not known a priori). The mesh hierarchy generated from mesh refinement could be utilized by efficient linear solvers like geometric multigrid methods (GMG), which can theoretically deliver optimal time complexity. Thus, it would be advantageous to use hierarchical mesh refinement to achieve high-order of accuracy and computational efficiency. One effective approach is to refine the mesh uniformly. Successive uniform refinement can not only increase the accuracy of solution but also generate a natural hierarchy which could be further used by GMG. We develop parallel uniform refinement-based algorithms to generate multi-degree, multi-dimensional and multi-level meshes from coarse unstructured meshes, based on the array-based half-facet (AHF) data structure. We demonstrate its applicability to a multigrid finite element solver and the capability is developed under the parallel mesh framework “Mesh Oriented dAtaBase†(MOAB). Meanwhile, we make effort to extend this framework to adaptive mesh refinement (AMR) which delivers solution more efficiently by increasing the computational effort near interesting features of the solutions. AMR has gradually become a vital step in large-scale numerical simulations. We develop a data structure called Hierarchical AHF to support both refinement and coarsening effectively. A key aspect of the refinement algorithm is the positioning of the new vertices on curved boundaries. Using linear point projection scheme for the new vertices compromises the accuracy of the geometry and in turn that of the finite element solver. To address this issue, we develop a discrete geometry module in MOAB that provides high-order point projection schemes. To improve the robustness of this method on coarse mesh, we propose two extensions: first, we introduce a Hermite-style weighted-least squares formulation, to take account of both point locations as well as surface normals in the surface reconstruction; second we introduce a new blending technique to ensure G0 continuity along sharp ridges and corners, while assuring high-order accuracy. | 146 pages
Recommended Citation
Zhao, Xinglin, "Mesh Refinement and High-order Reconstruction for Finite Element Methods on Unstructured Meshes" (2016). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 3236.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/3236