Browsing by Subject "Discontinuous Galerkin methods"

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Coupling of a nonlinear dispersive water wave model with sediment transport and seabed morphodynamic models for application in near-shore areas
(2021-04-30) Kazhyken, Kazbek; Dawson, Clinton N.; Videman, Juha H; Gamba, Irene M; Heimbach, Patrick; Bui-Thanh, Tan
Sediment transport and bed morphodynamic processes in near-shore regions pose significant risks to coastal infrastructure and environment, such as compromised structural integrity of coastal structures due to excessive scouring, roles of sediment deposits as sinks, vessels, and sources of dangerous pollutants, natural habitat degradation due to beach and shoreline erosion. Therefore, a mathematical modeling of these processes carries a clear engineering and scientific relevance. The sediment transport and bed morphodynamic processes are primarily forced by water waves and motion, which, in their turn, are affected by changes in the sediment bed surface driven by the sediment transport processes. Therefore, any mathematical modeling of these hydro-sediment-morphodynamic processes involves a coupling between a hydrodynamic model, which describes water waves and motion, and a sediment transport and bed evolution model, which resolves changes in the sediment bed surface driven by sediment erosion, transport, and deposition rates. One of the most widespread mathematical models used to resolve these processes is formed by the shallow water hydro-sediment-morphodynamic equations (SHSM), where the nonlinear shallow water equations (NSWE) are coupled with a sediment advection model and the sediment continuity Exner equation. If the shallowness parameter is defined as μ=H₀²/L₀², where H₀ and L₀ are the depth and length scales of the water flow, respectively, then NSWE provide a depth-averaged hydrodynamic model that is Ο(μ) consistent with the incompressible Euler equations. Therefore, SHSM provide a suitable computationally efficient model for the shallow water flow regimes where μ<<1. One of the drawbacks of using NSWE is the model's lack of capacity to capture wave dispersion effects that can be remedied by replacing NSWE with a nonlinear dispersive wave hydrodynamic model. One such model is formed by the Green-Naghdi equations (GN), a depth-averaged hydrodynamic model that is Ο(μ²) consistent with the incompressible Euler equations. In the presented work a new dispersive wave hydro-sediment-morpho-dynamic model is presented, where NSWE in the hydrodynamic part of SHSM is replaced with a single parameter variation of GN introduced by Bonneton et al. The new model is Ο(μ²) consistent with the incompressible Euler equations in its hydrodynamic part, and can be applied in coastal areas where wave dispersion effects are prevalent. The model is further subdivided into two classes: (1) a model that does not take into account the suspended load transport and considers the effects of the bed load transport only, (2) a model that resolves the effects of both the suspended and bed load transport. The first model is referred to as a dispersive wave hydro-morphodynamic model, and the second one as a dispersive wave hydro-sediment-morphodynamic model. The models are treated numerically with Strang operator splitting, and discontinuous Galerkin finite element methods. For the first model two modes of discretization are proposed: (1) the decoupled mode where GN and the Exner equations are solved separately, (2) the coupled mode where both of the equations are solve simultaneously at each time step. For the second model a fully coupled mode of discretization is employed. The developed numerical solution algorithms are validated with benchmark numerical examples. The results of the validation simulations indicate that the developed model has the ability to accurately resolve hydrodynamics of regular and solitary waves along with their dispersive properties, and sediment transport and bed morphodynamic processes provided that the empirical models for the suspended and bed load transport are properly calibrated. The results indicate that the model has the potential to be used in studies of coastal morphodynamics. As an example application of the model, the Ria Formosa lagoon is selected for a simulation of hydro-sediment-morphodynamic processes. An unstructured finite element mesh representation of the western circulation cell of the lagoon is generated, and data to parametrize sediment transport, bottom friction, and tidal wave models is gathered. The example application indicates that the developed dispersive wave model can be used to simulate the processes in coastal areas with complex irregular geometries. Moreover, the terms of the model that are Ο(μ²) consistent with the incompressible Euler equations introduce new features into the flow parameters that have the potential to significantly affect the resulting sediment transport and bed morphodynamic process simulations
Discontinuous Galerkin methods for spectral wave/circulation modeling
(2013-08) Meixner, Jessica Delaney; Dawson, Clinton N.
Waves and circulation processes interact in daily wind and tide driven flows as well as in more extreme events such as hurricanes. Currents and water levels affect wave propagation and the location of wave-breaking zones, while wave forces induce setup and currents. Despite this interaction, waves and circulation processes are modeled separately using different approaches. Circulation processes are represented by the shallow water equations, which conserve mass and momentum. This approach for wind-generated waves is impractical for large geographic scales due to the fine resolution that would be required. Therefore, wind-waves are instead represented in a spectral sense, governed by the action balance equation, which propagates action density through both geographic and spectral space. Even though wind-waves and circulation are modeled separately, it is important to account for their interactions by coupling their respective models. In this dissertation we use discontinuous-Galerkin (DG) methods to couple spectral wave and circulation models to model wave-current interactions. We first develop, implement, verify and validate a DG spectral wave model, which allows for the implementation of unstructured meshes in geographic space and the utility of adaptive, higher-order approximations in both geographic and spectral space. We then couple the DG spectral wave model to an existing DG circulation model, which is run on the same geographic mesh and allows for higher order information to be passed between the two models. We verify and validate coupled wave/circulation model as well as analyzing the error of the coupled wave/circulation model.
A discontinuous Petrov-Galerkin methodology for incompressible flow problems
(2013-08) Roberts, Nathan Vanderkooy; Demkowicz, Leszek; Moser, Robert deLancey
Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations.
Hybridized discontinuous Galerkin methods for magnetohydrodynamics
(2018-11-02) Shannon, Stephen James; Bui-Thanh, Tan; Arbogast, Todd; Demkowicz, Leszek; Ghattas, Omar; Shadid, John; Waelbroeck, François
Discontinuous Galerkin (DG) methods combine the advantages of classical finite element and finite volume methods. Like finite volume methods, through the use of discontinuous spaces in the discrete functional setting, we automatically have local conservation, an essential property for a numerical method to behave well when applied to hyperbolic conservation laws. Like classical finite element methods, DG methods allow for higher order approximations with compact stencils. For time-dependent problems with implicit time stepping and for steady-state problems, DG methods give a larger globally coupled linear system than continuous Galerkin methods (especially for three dimensional problems and low polynomial orders). The primary motivation of the hybridized (or hybridizable) discontinuous Galerkin (HDG) methods is to reduce the number of globally coupled unknowns in DG methods when implicit time stepping or direct-to-steady-state solutions are desired. This is accomplished by the introduction of new “trace unknowns” defined on the mesh skeleton, the definition of one-sided numerical fluxes, and the enforcement of local conservation. This results in a globally coupled linear system where the local “volume unknowns” can be eliminated in a Schur complement procedure, resulting in a reduced globally coupled system in terms of only the trace unknowns. Magnetohydrodynamics (MHD) is the study of the flow of electrically conducting fluids under the influence of magnetic fields. The MHD equations are used to describe important physical phenomena including laboratory plasmas (plasma confinement in fusion energy devices), astrophysical plasmas (solar coronas, planetary magnetospheres) and liquid metal flows (metallurgy processes, the Earth’s molten core, cooling for nuclear reactors). Incompressible MHD, which is the main focus of this work, is relevant in low Lundquist number liquid metals, in high Lundquist number, large guide field fusion plasmas, and in low Mach number compressible flows. The equations of MHD are highly nonlinear, and are characterized by physical phenomena spanning wide ranges of length and time scales. For numerical methods, this presents challenges in both spatial and temporal discretization. In terms of temporal discretization, fully implicit numerical methods are attractive in their robustness; they allow for stable, high-order time integration over long time scales of interest.
Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical solar cells
(2016-08) Harmon, Michael David; Martínez Gamba, Irene, 1957-; Ren, Kui; Dawson, Clint; Ghattas, Omar; Henkelman, Graeme
Large-scale utilization of photovoltaic (PV) devices, or solar cells, has been hampered for years due to high costs and lack of energy storage mechanisms. Photoelectrochemical solar cells (PECs) are an attractive alternative to conventional solid state PV devices because they are able to directly convert solar energy into hydrogen fuel. The hydrogen fuel can then be used at a later time to generate electricity. Photoelectrochemical solar cells are able to produce fuel through chemical reactions at the interface of a semiconductor and electrolyte when the device is illuminated. In this dissertation, we focus on the modeling and numerical simulation of charge transport in both the semiconductor and electrolyte region as well as their interaction through a reactive interface using the drift-diffusion-Poisson equations. The main challenges in constructing a numerical algorithm that produces reliable simulations of PECs are due to the highly nonlinear nature of the semiconductor and electrolyte systems as well as the nonlinear coupling between the two systems at the interface. In addition, the evolution problem under consideration is effectively multi-scale in the sense that the evolution of the system in the semiconductor and the corresponding one in the electrolyte evolve at different time scales due to the quantitative scaling differences in their relevant physical parameters. Furthermore, regions of stiffness caused by boundary layer formation where sharp transitions in densities and electric potential occur near the interface and pose severe constraints on the choice of discretization strategy in order to maintain numerical stability. In this thesis we propose, implement and analyze novel numerical algorithms for the simulation of photoelectrochemical solar cells. Spatial discetizations of the drift-diffusion-Poisson equations are based on mixed finite element methods and local discontinuous Galerkin methods. To alleviate the stiffness of the equations we develop and analyze Schwarz domain decomposition methods in conjunction with implicit-explicit (IMEX) time stepping routines. We analyze the numerical methods and prove their convergence under mesh refinement. Finally, we present results from numerical experiments in order to develop a strategy for optimizing solar cell design at the nano-scale.