Discontinuous Galerkin Finite Element Model Research Articles

A discontinuous Galerkin finite element model for compound flood simulations

Many tropical cyclones, e.g., Hurricane Harvey (2017), have lead to significant rainfall and resulting runoff. When the runoff interacts with storm surge, the resulting floods can be greatly amplified and lead to effects that cannot be correctly modeled by simple superposition of its distinctive sources. In an effort to develop accurate numerical simulations of runoff, surge, and compounding floods, we develop a local discontinuous Galerkin method for modified shallow water equations. In this modification, nonzero sources to the continuity equation are included to incorporate rainfall into the model using parametric rainfall models from literature as well as hindcast data. The discontinuous Galerkin spatial discretization is accompanied with a strong stability preserving explicit Runge Kutta time integrator. Hence, temporal stability is ensured through the Courant–Friedrichs–Lewy (CFL) condition and we exploit the embarrassingly parallel nature of the developed method using MPI parallelization.We demonstrate the capabilities of the developed method though a sequence of physically relevant numerical tests, including small scale test cases based on laboratory measurements and large scale experiments with Hurricane Harvey in the Gulf of Mexico. The results highlight the conservation properties and robustness of the developed method and show the potential of compound flood modeling using our approach.

Ultimate Bearing Capacity Analysis of Manned Submersible Based on the Genetic Algorithm Discontinuous and Galerkin Finite Element Method

The pressure hull of deep manned submersible is the most basic component to ensure its intended function. It is necessary to study the influence of initial geometric defects on the bearing capacity of pressure hull of manned submersible with different depths. According to the idea of discontinuous Galerkin finite element method, the theoretical model is constructed and the corresponding algorithm is designed, and the genetic algorithm is combined with discontinuous Galerin finite element method to establish the inverse method to obtain the ultimate bearing capacity of manned submersible. First, the discontinuous Galerkin finite element model is constructed, the inversion model is also established through combing the discontinuous Galerkin finite element method and genetic algorithm, and then the corresponding solution algorithm is designed. Moreover, then, the ultimate bearing analysis of manned submersible for different deep is carried out based on the inversion model combing discontinuous Galerkin finite element method and genetic algorithm. The effect of defect parameters on ultimate bearing capacity of manned submersible is obtained.

An implicit wetting–drying algorithm for the discontinuous Galerkin method: application to the Tonle Sap, Mekong River Basin

The accurate simulation of wetting–drying processes in floodplains and coastal zones is a challenge for hydrodynamic modelling, especially for long time simulations. Indeed, dedicated numerical procedures are generally time-consuming, instabilities can occur at the wet/dry front, rapid transition of wet/dry interface and mass conservation are not always ensured. We present the extension of an existing wetting–drying algorithm in two space dimensions and its application to a real case. The wetting–drying algorithm is implemented in Second-generation Louvain-la-Neuve Ice-ocean Model ( www.slim-ocean.be ), a discontinuous Galerkin finite element model solving the shallow water equations in a fully implicit way. This algorithm consists in applying a threshold value of fluid depth for a thin layer and a blending parameter in order to guarantee positive values of the water depth, while preserving local mass conservation and the well balanced property at wet/dry interfaces. The technique is first validated against standard analytical test cases (Balzano 1, Balzano 3 and Thacker test cases) and is subsquently applied in a realistic domain, the Tonle Sap Lake in the Mekong River Basin, where the water level can vary by about 10 m between the dry and the wet season.

Discontinuous Galerkin modeling of the Columbia River’s coupled estuary-plume dynamics

The Columbia River (CR) estuary is characterized by high river discharge and strong tides that generate high velocity flows and sharp density gradients. Its dynamics strongly affects the coastal ocean circulation. Tidal straining in turn modulates the stratification in the estuary. Simulating the hydrodynamics of the CR estuary and plume therefore requires a multi-scale model as both shelf and estuarine circulations are coupled. Such a model has to keep numerical dissipation as low as possible in order to correctly represent the plume propagation and the salinity intrusion in the estuary. Here, we show that the 3D baroclinic discontinuous Galerkin finite element model SLIM 3D is able to reproduce the main features of the CR estuary-to-ocean continuum. We introduce new vertical discretization and mode splitting that allow us to model a region characterized by complex bathymetry and sharp density and velocity gradients. Our model takes into account the major forcings, i.e. tides, surface wind stress and river discharge, on a single multi-scale grid. The simulation period covers the end of spring-early summer of 2006, a period of high river flow and strong changes in the wind regime. SLIM 3D is validated with in-situ data on the shelf and at multiple locations in the estuary and compared with an operational implementation of SELFE. The model skill in the estuary and on the shelf indicate that SLIM 3D is able to reproduce the key processes driving the river plume dynamics, such as the occurrence of bidirectional plumes or reversals of the inner shelf coastal currents.

A baroclinic discontinuous Galerkin finite element model for coastal flows

Numerical modelling of coastal flows is a challenging topic due to complex topography of the coastal zone, rapid flow dynamics and large density variations. Such phenomena are best simulated with unstructured grid models due to their highly flexible spatial discretisation. This article presents a three-dimensional discontinuous Galerkin finite element marine model. Discontinuous Galerkin spatial discretisation is combined with an explicit mode splitting time integration scheme. Slope limiters are introduced to guarantee high quality of the tracer fields in the presence of strong gradients. Free surface movement is accounted for by means of an Arbitrary Lagrangian Eulerian (ALE) moving mesh method. Water volume and tracers are conserved. The conservation properties and baroclinic adjustment under gravity are tested with numerical benchmarks. Finally, the model is applied to the Rhine river plume in an idealised setting.

A fully implicit wetting–drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary

Resolving the shoreline undulation due to tidal excursion is a crucial part of modelling water flow in estuaries and coastal areas. Nevertheless, maintaining positive water column depth and numerical stability has proved out to be a very difficult task that requires special attention. In this paper we propose a novel wetting–drying method in which the position of the sea bed is allowed to fluctuate in drying areas. The method is implemented in a Discontinuous Galerkin Finite Element Model (DG-FEM). Unlike most methods in the literature our method is compatible with fully implicit time-marching schemes, thus reducing the overall computational cost significantly. Moreover, global and local mass conservation is guaranteed which is crucial for long-term environmental applications. In addition consistency with tracer equation is also ensured. The performance of the proposed method is demonstrated with a set of test cases as well as a real-world application to the Scheldt Estuary. Due to the implicit time integration, the computational cost in the Scheldt application is reduced by two orders of magnitude. Although a DG-FEM implementation is presented here, the wetting–drying method is applicable to a wide variety of shallow water models.

A discontinuous Galerkin finite element model for river bed evolution under shallow flows

The accurate representation of morphodynamic processes and the ability to propagate changes in the riverbed over a wide range of space and time scales make the design and implementation of appropriate numerical schemes challenging. In particular, requirements of accuracy and stability for medium and long term simulations are difficult to meet. In this work, the derivation, design, and implementation of a discontinuous Galerkin finite element method (DGFEM) for sediment transport and bed evolution equations are presented. Numerical morphodynamic models involve a coupling between a hydrodynamic flow solver which acts as a driving force and a bed evolution model which accounts for sediment flux and bathymetry changes. A space DGFEM is presented based on an extended approach for systems of partial differential equations with non-conservative products, in combination with two intertwined Runge–Kutta time stepping schemes for the fast hydrodynamic and slow morphodynamic components. The resulting numerical scheme is verified by comparing simulations against (semi-)analytical solutions. These include the evolution of an initially symmetric, isolated bedform; the formation and propagation of a step in a straight channel due to a sudden overload of sediment discharge; the propagation of a travelling diffusive sediment wave in a straight channel; and, the evolution of an initially flat bed in a channel with a contraction. Finally, a comparison is made between a numerical simulation and field data of a trench excavated in the main channel of the Paraná river near Paraná city, Argentina.