Type
Text
Type
Dissertation
Advisor
Jiao, Xiangmin | Glimm, James | Samulyak, Roman | Chen, Shikui.
Date
2017-05-01
Keywords
Applied mathematics | curved boundary, curved elements, finite element methods, high order, neumann boundaries, weighted residual method
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/11401/77088
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
High-order numerical methods for PDE discretizations have attracted significant interests for scientific and engineering applications in recent years. For engineering problems with complex geometries, achieving high-order convergence is decidedly challenging, especially with curved boundaries. The existing high-order finite element methods based on isoparametric elements require the definition of curved volumetric elements to represent the geometry accurately and ensure the validity of their variational formulations. However, these high-order elements have much stricter mesh quality requirements due to the possibilities of internal foldings of the elements, which are very hard to detect. In addition, poor mesh quality may also lead to potential loss of the completeness of the basis functions. Some recently proposed alternatives such as isogeometric analysis and NURBS-enhanced FEM can achieve high-order convergence but are very complicated and also have even stricter requirement on mesh quality. In our recent work, we have developed the adaptive extended stencil finite element method (AES-FEM), which has less dependent on mesh quality. In this dissertation, we extend the AES-FEM to achieve high order convergence on geometries with curved boundaries and Neumann boundary conditions. AES-FEM uses high-degree polynomial basis functions to accurately discretize the PDE. In the interior, AES-FEM uses piecewise linear test functions for simplicity and efficiency. For elements adjacent to the curved boundary, we construct new superparametric elements, whose test functions are piecewise linear in the parametric space but curved in the real space to capture the curved geometry accurately. We construct these superparametric elements using simplices with curved faces and edges defined by the curved geometry. As another contribution, we propose a new strategy for enforcing Neumann boundary conditions for weighted residual methods in the variational formulations, which only require integrating the Neumann boundary conditions over small regions on the boundary. The method is consistent with the variational problem and simplifies some of the implementation, and it allows enforcing Neumann boundary conditions even for boundaries with discontinuous normal directions. We present the method both for AES-FEM as well as generalized finite difference (GFD) methods. We present results of our method applied to second-order elliptic problems on curved boundaries. | 113 pages
Recommended Citation
Delaney, Tristan Joseph, "High-Order Adaptive Extended Stencil Finite Element Methods for Applications with Curved Boundaries" (2017). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2925.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2925