Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods.
Degree: PhD, Aerospace Engineering, 2017, University of Michigan
Computational science, including computational fluid dynamics (CFD), has become an indispensible tool for scientific discovery and engineering design, yet a key remaining challenge is to simultaneously ensure accuracy, efficiency, and robustness of the calculations. This research focuses on advancing a class of high-order finite element methods and develops a set of algorithms to increase the accuracy, efficiency, and robustness of calculations involving convection and diffusion, with application to the inviscid Euler and viscous Navier-Stokes equations. In particular, it addresses high-order discontinuous Galerkin (DG) methods, especially hybridized (HDG) methods, and develops adjoint-based methods for simultaneous mesh and order adaptation to reduce the error in a scalar functional of the approximate solution to the discretized equations. Contributions are made in key aspects of these methods applied to general systems of equations, addressing the scalability and memory requirements, accuracy of HDG methods, and efficiency and robustness with new adaptation methods.
First, this work generalizes existing HDG methods to systems of equations, and in so doing creates a new primal formulation by applying DG stabilization methods as the viscous stabilization for HDG. The primal formulation is shown to be even more computationally efficient than the existing methods. Second, by instead keeping existing viscous stabilization methods and developing a new convection stabilization, this work shows that additional accuracy can be obtained, even in the case of purely convective systems. Both HDG methods are compared to DG in the same computational framework and are shown to be more efficient.
Finally, the set of adaptation frameworks is developed for combined mesh and order refinement suitable for both DG and HDG discretizations. The first of these frameworks uses hanging-node-based mesh adaptation and develops a novel local approach for evaluating the refinement options. The second framework intended for simplex meshes extends the mesh optimization via error sampling and synthesis (MOESS) method to incorporate order adaptation.
Collectively, the results from this research address a number of key issues that currently are at the forefront of high-order CFD methods, and particularly to output-based hp-adaptation for DG and HDG methods.
Advisors/Committee Members: Fidkowski, Krzysztof J (committee member), Krasny, Robert (committee member), May, Georg (committee member), Powell, Ken (committee member).
Subjects/Keywords: Computational Fluid Dynamics; High-Order Methods; Discontinous Galerkin Methods; Adjoint-Based Error Estimation; Mesh and Order Adaptation; Aerospace Engineering; Engineering
to Zotero / EndNote / Reference
APA (6th Edition):
Dahm, J. (2017). Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/137150
Chicago Manual of Style (16th Edition):
Dahm, Johann. “Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods.” 2017. Doctoral Dissertation, University of Michigan. Accessed April 16, 2021.
MLA Handbook (7th Edition):
Dahm, Johann. “Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods.” 2017. Web. 16 Apr 2021.
Dahm J. Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods. [Internet] [Doctoral dissertation]. University of Michigan; 2017. [cited 2021 Apr 16].
Available from: http://hdl.handle.net/2027.42/137150.
Council of Science Editors:
Dahm J. Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods. [Doctoral Dissertation]. University of Michigan; 2017. Available from: http://hdl.handle.net/2027.42/137150