Application Of Discontinuous Galerkin Method Research Articles

Development and application of discontinuous Galerkin method for solidification problems in a semitransparent medium-filled cavity

Research on the phase change phenomenon has gained increasing attention due to its extensive engineering applications in the fields of energy storage and release. In the semitransparent medium, the phase change process is accompanied by the transfer of internal radiation energy, which can significantly affect the distributions of temperature and liquid fraction. Based on the enthalpy model, this work develops the discontinuous Galerkin method (DGM) to solve the enthalpy equation (EE) and radiative transfer equation (RTE) that govern the phase change process. By comparing with the analytic solution and other numerical results, the correctness of the proposed DGM framework for solving coupled radiation and phase change problems is validated. Results reveal that the DGM can accurately capture the heat and mass transfer process in solidification problems. The proposed DGM is further extended to analyze the solidification problem within a two-dimensional cavity considering the radiation effect. Effects of Planck number, medium scattering albedo, and refractive index on the temperature and liquid fraction distributions as well as the mushy zone location are investigated. It is found that these parameters have important impacts on the solidification rate. This work provides accurate numerical solutions on phase change problems in the presence of radiation.

Application of discontinuous Galerkin method in supersonic and hypersonic gas flows

Most high-order methods often suffer from two- and three-dimensional unstructured grids, particularly, for supersonic and hypersonic gas flows. We applied a high-order mixed-modal discontinuous Galerkin method with a coupled slope limiter to investigate multidimensional cases of supersonic and hypersonic gas flows. Furthermore, we provide detailed implementations and validations of two- and three-dimensional problems. Examples include double Mach reflection problems and supersonic and hypersonic gas flows around a cylinder. In addition, we provide case studies of a 3D hypersonic vehicle to evaluate its computational performance on complex hypersonic gas flows. This study can be regarded as an attempt to apply the discontinuous Galerkin method to engineering problems.

Application of discontinuous Galerkin method to modeling of two-dimensional flows of a multicomponent ideal gases mixture using local adaptive mesh refinement

In this article a numerical algorithm is developed for solving of gas dynamics equations for a mixture of ideal gases on adaptive locally refined grids. The algorithm is based on discontinuous Galerkin method. To avoid the appearance of non-physical oscillations near the discontinuities, the Barth-Jespersen limiter is used. The numerical algorithm is based on the data structure and algorithms of the p4est library. In present work the numerical simulation of one problem of Richtmyer-Meshkov instability development is considered and the triple point problem is solved using the developed numerical algorithm of high accuracy order. The obtained results are in good agreement with the well-known numerical solutions. The pictures plotted basing on the solution describe in detail the dynamics of the complex flows under consideration.

Application of discontinuous Galerkin method for solving a compressible five-equation two-phase flow model

Abstract In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.

Application of discontinuous Galerkin method to mechanical 2D problem with arbitrary polygonal and very high-order finite elements

Discontinuous Galerkin with finite difference rules (DGFD) is applied to mechanical plane stress state problem. The considered domain is discretized by polygonal mesh. The polygonal elements can be for example a hexagon, pentagon or just quadrangle or triangle. They do not have to be convex and a fish mesh, where the elements have fish shapes, is used. When the elements are rectangular then the orthogonality of Chebyshev basis functions can be utilized. In such a case very high-order approximate solution can be obtained. In this work the approximation order exceeds 10 and reaches 60, which in the latter case means 3600 numbers of degrees of freedom in a single element. The paper is illustrated by a benchmark example in which the exact solution is recovered by DGFD method for various meshes. In the other example the stress concentration is easily recovered by very high-order version of DGFD method.

The Application of Discontinuous Galerkin Methods in Conjugate Heat Transfer Simulations of Gas Turbines

The performance of modern heavy-duty gas turbines is greatly determined by the accurate numerical predictions of thermal loading on the hot-end components. The purpose of this paper is: (1) to present an approach applying a novel numerical technique—the discontinuous Galerkin (DG) method—to conjugate heat transfer (CHT) simulations, develop the engineering-oriented numerical platform, and validate the feasibility of the methodology and tool preliminarily; and (2) to utilize the constructed platform to investigate the aerothermodynamic features of a typical transonic turbine vane with convection cooling. Fluid dynamic and solid heat conductive equations are discretized into explicit DG formulations. A centroid-expanded Taylor basis is adopted for various types of elements. The Bassi-Rebay method is used in the computation of gradients. A coupled strategy based on a data exchange process via numerical flux on interface quadrature points is simply devised. Additionally, various turbulence Reynolds-Averaged-Navier-Stokes (RANS) models and the local-variable-based transition model γ-Reθ are assimilated into the integral framework, combining sophisticated modelling with the innovative algorithm. Numerical tests exhibit good consistency between computational and analytical or experimental results, demonstrating that the presented approach and tool can handle well general CHT simulations. Application and analysis in the turbine vane, focusing on features around where there in cluster exist shock, separation and transition, illustrate the effects of Bradshaw’s shear stress limitation and separation-induced-transition modelling. The general overestimation of heat transfer intensity behind shock is conjectured to be associated with compressibility effects on transition modeling. This work presents an unconventional formulation in CHT problems and achieves its engineering applications in gas turbines.

A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation

Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently (Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483), a simplified two-equation reaction-diffusion system was developed to describe the interaction of two of the key morphogens: the activator and an activator-dependent inhibitor of precartilage condensation formation. A discontinuous Galerkin (DG) finite element method was applied to solve this nonlinear system on complex domains to study the effects of domain geometry on the pattern generated (Zhu et al., Application of Discontinuous Galerkin Methods for reaction-diffusion systems in developmental biology, Journal of Scientific Computing, 2009, v40, pp. 391-418). In this paper, we extend these previous results and develop a DG finite element model in a moving and deforming domain for skeletal pattern formation in the vertebrate limb. Simulations reflect the actual dynamics of limb development and indicate the important role played by the geometry of the undifferentiated apical zone.

A general effective method for combining collocation and DDM: An application of discontinuous Galerkin methods

AbstractDomain decomposition methods (DDM) have received much attention in recent years. They constitute the most effective means of using parallel computing resources to model continuous systems. However, combining collocation procedures with domain decomposition methods presents complications that must be overcome in order to profit from the advantages of parallel computing. The present paper belongs to a line of research in which a theory that constitutes a general and systematic formulation of discontinuous Galerkin methods (dG) is being investigated. Based on it, a new method of collocation of general applicability, TH‐collocation, was recently introduced. For a broad class of symmetric and positive continuous systems, TH‐collocation yields symmetric and positive matrices. This clears the way for applying effectively DDM and parallel computing, in combination with collocation, to such systems. In this paper the general procedure is explained with some detail and then is applied to develop an effective method for processing elliptic equations of second order. This, by the way, overcomes the difficulties encountered in a previous Herrera and Pinder's article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.