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Title Toward Accurate, Efficient, and Robust Hybridized Discontinuous Galerkin Methods
URL
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Aerospace Engineering
Degree Level doctoral
University/Publisher University of Michigan
Abstract 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.
Subjects/Keywords Computational Fluid Dynamics; High-Order Methods; Discontinous Galerkin Methods; Adjoint-Based Error Estimation; Mesh and Order Adaptation; Aerospace Engineering; Engineering
Contributors Fidkowski, Krzysztof J (committee member); Krasny, Robert (committee member); May, Georg (committee member); Powell, Ken (committee member)
Language en
Country of Publication us
Record ID handle:2027.42/137150
Repository umich
Date Indexed 2020-09-09
Grantor University of Michigan, Horace H. Rackham School of Graduate Studies
Issued Date 2017-01-01 00:00:00
Note [thesisdegreename] PHD; [thesisdegreediscipline] Aerospace Engineering; [thesisdegreegrantor] University of Michigan, Horace H. Rackham School of Graduate Studies; [bitstreamurl] https://deepblue.lib.umich.edu/bitstream/2027.42/137150/1/jdahm_1.pdf;

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adaptation methods used in this work, specifically the adjoint-based algorithms for determining how both the error estimate and the simulation should be adapted in response to this estimate. It also provides further background on mesh adaptation, in which the…

…regions in which adaptation can be most productively applied. By using the adjoint field based on the specified output functional, the error estimation can provide a local adaptation indicator that is focused on reducing the error in the output of interest…

…refinement iteration induces refinement in the neighboring elements. . . . . . . . . . . . . . . . . 93 An example mesh (left) obtained by anisotropic adaptation using metric-based global re-meshing for a viscous Navier-Stokes test case. The…

…diffusion, with application to the inviscid Euler and viscous Navier-Stokes equations. In particular, it addresses highorder discontinuous Galerkin (DG) methods, especially hybridized (HDG) methods, and develops adjoint-based methods for…

…frameworks is developed for combined mesh and order refinement suitable for both DG and HDG discretizations. The first of these xii frameworks uses hanging-node-based mesh adaptation and develops a novel local approach for evaluating the refinement options…

…introduces the adjoint and describes its connection to error estimation, then describes how it is used to create both a global error estimate and localized indicator. 7 With regard to the adaptation itself, in effect the goal is to equidistribute the error…

…output-based hp-adaptation in high-order methods. The specific contributions made by this dissertation are as follows: • Generalization of the existing HDG methods to systems of equations. This involves application of existing DG stabilization methods to…

…direct comparison of DG to both forms of HDG methods in the same computational framework for both the scalar convection-diffusion equation and the Navier-Stokes system. • Development of a novel hanging-node-based hp-adaptation framework for both DG and…

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