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Basics of Discretization Methods
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
The last statement above is best given in terms of the weak form defined in Section 3.7.3. Basically, the governing equation is “projected” into the space of test functions, by taking their inner products. Each such equation is integrated by parts and the weak form obtained is solved numerically. It has been tacitly assumed that the test functions v(x) in the weak form are identical to the basis functions φi(x) used to expand u(x). This is called the Galerkin method and is the most frequently used method in finite-element procedures. When the test functions and the basis functions are not the same, the method is more general and is called a “Petrov–Galerkin” method. It is useful to note that the basis functions need not be continuous functions defined over the entire computational domain. A method which uses such a choice of basis functions (the “discontinuous Galerkin”) is presented in Section 3.7.7 and applied to model differential equations in Chapter 4. The discontinuous Galerkin method bears strong resemblance to finite-volume schemes and has a “local” nature, as opposed to the “global” formulation of the FEM.
Shallow Water Flow Modeling
Published in Saeid Eslamian, Faezeh Eslamian, Flood Handbook, 2022
Franziska Tügel, Ilhan Özgen-Xian, Franz Simons, Aziz Hassan, Reinhard Hinkelmann
In recent years, the discontinuous Galerkin method (DG) has been explored in the context of shallow water flow modeling (Kesserwani et al., 2019; Khan and Lai, 2017). The DG method is a finite-element method (FE) with discontinuous base functions. In contrast to the classical FE methods, e.g., the streamline-upwind Petrov-Galerkin method (SUPG) (Hinkelmann, 2005), the DG method calculates a local flux across cell interfaces and is therefore locally conservative. The main advantage of the DG method is that its extension to a higher order of accuracy has a compact stencil, compared to the FV method. In this chapter, only the FV method is introduced.
Calibration of a nonlinear viscoelasticity model with sensitivity assessment
Published in Per-Erik Austrell, Leif Kari, Constitutive Models for Rubber IV, 2017
In order to give the appropriate variational format in time for subsequent use of a discontinuous Galerkin method of order k, dG(k), we introduce a partition of the considered time span I (0, T) into N time intervals In = (tn−1, tn) of length kn = tn − tn−1. With each In, we associate the appropriate space of continuous viscous stretches, denoted ℚ(In).
An energy-conservative DG-FEM approach for solid–liquid phase change
Published in Numerical Heat Transfer, Part B: Fundamentals, 2023
Bouke Johannes Kaaks, Martin Rohde, Jan-Leen Kloosterman, Danny Lathouwers
Discontinuous Galerkin methods have gained interest over the last decade as an attractive numerical method for computational fluid dynamics, due to its combination of desired features of both the finite volume (FVM) and finite element (FEM) methods, such as local conservation, the possibility for upwinding, an arbitrarily high order of discretization and high geometric flexibility [56–58]. In addition, the high locality of the numerical scheme makes the discontinuous Galerkin method efficient for parallelization [58]. Recent advances in the applicability of DG-FEM methods to computational fluid dynamics include the simulation of turbulent flow with a high-order discontinuous Galerkin method and RANS or LES turbulence modeling [59–63], the development of discontinuous Galerkin methods for low-Mach number flow [64, 65], the simulation of multiphase flows [66, 67] and a DG-FEM multiphysics solver for simulating the Molten Salt Fast Reactor [58]. When coupled to a melting and solidification model, DG-FEM is expected to offer a more reliable capture of nonlinear phase change phenomena as compared to the finite-volume method [52]. Indeed, Schroeder and Lube [53] obtained qualitatively similar results on a mesh that was 14 times coarser as compared to the mesh used in a similar finite volume numerical benchmark study [47]. For these reasons, DG-FEM is an attractive numerical method for modeling solid–liquid phase change problems.
Modeling of Formation and Evolution of Cracks in Zirconium-Based Claddings of Nuclear Fuel Rods Within DIONISIO 3.0
Published in Nuclear Science and Engineering, 2021
Ezequiel Goldberg, Alejandro Soba
The methodologies for damage mechanics mentioned in Sec. I are optimal to be used within the framework of the FEM. It can be observed that the founding works of these methods52–54 start from acknowledging the limitations of the classic theories of fracture mechanics, such as the absence of an initial crack tip or the stress distribution not being essentially two-dimensional (see Sec. I). Starting then from the type of interpolation functions proposed as solutions within the FEM, the idea is to include the detail of the fracture zones. For this, variants of the so-called discontinuous Galerkin method are used in which a generalization of weak formulations that allow discontinuities of the unknowns within the problem domain is applied.53 To do this, integration is restricted to each subdomain, naturally generating limit integral terms for the interfaces that imply jump discontinuities. The role of these terms is to weakly enforce the consistency and continuity of the unknown of the problem, where appropriate. Alternatively, always in the context of finite element formulations of elliptical shape, jump discontinuities are allowed through element boundaries.60