Download or read book Discontinuous Galerkin Finite Element Methods with Shock capturing for Nonlinear Convection Dominated Models written by Hamdullah Yücel and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: In this paper, convection-diffusion-reaction models with nonlinear reaction mechanisms, which are typical problems of chemical systems, are studied by using the upwind symmetric interior penalty Galerkin (SIPG) method. The local spurious oscillations are minimized by adding an artificial viscosity diffusion term to the original equations. A discontinuity sensor is used to detect the layers where unphysical oscillations occur. Finally, the proposed method is tested on various single- and multi-component problems.
Download or read book High order Hybridizable Discontinuous Galerkin Method for Viscous Compressible Flows written by Mostafa Javadzadeh Moghtader and published by . This book was released on 2017 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Fluid Dynamics (CFD) is an essential tool for engineering design and analysis, especially in applications like aerospace, automotive and energy industries. Nowadays most commercial codes are based on Finite Volume (FV) methods, which are second order accurate, and simulation of viscous compressible flow around complex geometries is still very expensive due to large number of low-order elements required. One the other hand, some sophisticated physical phenomena, like aeroacoustics, vortex dominated flows and turbulence, need very high resolution methods to obtain accurate results. High-order methods with their low spatial discretization errors, are a possible remedy for shortcomings of the current CFD solvers. Discontinuous Galerkin (DG) methods have emerged as a successful approach for non-linear hyperbolic problems and are widely regarded very promising for next generation CFD solvers. Their efficiency for high-order discretization makes them suitable for advanced physical models like DES and LES, while their stability in convection dominated regimes is also a merit of them. The compactness of DG methods, facilitate the parallelization and their element-by-element discontinuous nature is also helpful for adaptivity. This PhD thesis focuses on the development of an efficient and robust high-order Hybridizable Discontinuous Galerkin (HDG) Finite Element Method (FEM) for compressible viscous flow computations. HDG method is a new class of DG family which enjoys from merits of DG but has significantly less globally coupled unknowns compared to other DG methods. Its features makes HDG a possible candidate to be investigated as next generation high-order tools for CFD applications. The first part of this thesis recalls the basics of high-order HDG method. It is presented for the two-dimensional linear convection-diffusion equation, and its accuracy and features are investigated. Then, the method is used to solve compressible viscous flow problems modelled by non-linear compressible Navier-Stokes equations; and finally a new linearized HDG formulation is proposed and implemented for that problem, all using high-order approximations. The accuracy and efficiency of high-order HDG method to tackle viscous compressible flow problems is investigated, and both steady and unsteady solvers are developed for this purpose. The second part is the core of this thesis, proposing a novel shock-capturing method for HDG solution of viscous compressible flow problems, in the presence of shock waves. The main idea is to utilize the stabilization of numerical fluxes, via a discontinuous space of approximation inside the elements, to diminish or remove the oscillations in the vicinity of discontinuity. This discontinuous nodal basis functions, leads to a modified weak form of the HDG local problem in the stabilized elements. First, the method is applied to convection-diffusion problems with Bassi-Rebay and LDG fluxes inside the elements, and then, the strategy is extended to the compressible Navier-Stokes equations using LDG and Lax-Friedrichs fluxes. Various numerical examples, for both convection-diffusion and compressible Navier-Stokes equations, demonstrate the ability of the proposed method, to capture shocks in the solution, and its excellent performance in eliminating oscillations is the vicinity of shocks to obtain a spurious-free high-order solution.
Download or read book High Order Methods for Computational Physics written by Timothy J. Barth and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining chal lenges facing the field of computational fluid dynamics. In structural me chanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the com putation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order ac curacy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence sug gests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Cen ter. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18,1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25,1998 at the NASA Ames Research Center in the United States.
Download or read book Adaptive Discontinuous Galerkin Methods for Non linear Reactive Flows written by Murat Uzunca and published by Birkhäuser. This book was released on 2016-05-17 with total page 111 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence.As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface.
Download or read book Discontinuous Galerkin Methods written by Bernardo Cockburn and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.
Download or read book Runge Kutta Discontinuous Galerkin Methods for Convection dominated Problems written by Bernardo Cockburn and published by . This book was released on 2000 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Entropy stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock capturing for Hyperbolic Systems of Conservation Laws written by Andreas Eduard Hiltebrand and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Mathematics and Advanced Applications ENUMATH 2013 written by Assyr Abdulle and published by Springer. This book was released on 2014-11-25 with total page 759 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gathers a selection of invited and contributed lectures from the European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) held in Lausanne, Switzerland, August 26-30, 2013. It provides an overview of recent developments in numerical analysis, computational mathematics and applications from leading experts in the field. New results on finite element methods, multiscale methods, numerical linear algebra and discretization techniques for fluid mechanics and optics are presented. As such, the book offers a valuable resource for a wide range of readers looking for a state-of-the-art overview of advanced techniques, algorithms and results in numerical mathematics and scientific computing.
Download or read book Advanced Numerical Approximation of Nonlinear Hyperbolic Equations written by B. Cockburn and published by Springer. This book was released on 2006-11-14 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the texts of the four series of lectures presented by B.Cockburn, C.Johnson, C.W. Shu and E.Tadmor at a C.I.M.E. Summer School. It is aimed at providing a comprehensive and up-to-date presentation of numerical methods which are nowadays used to solve nonlinear partial differential equations of hyperbolic type, developing shock discontinuities. The most effective methodologies in the framework of finite elements, finite differences, finite volumes spectral methods and kinetic methods, are addressed, in particular high-order shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon a-posteriori error analysis.
Download or read book Computational Science and Its Applications ICCSA 2024 written by Osvaldo Gervasi and published by Springer Nature. This book was released on with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Discontinuous Galerkin Finite Element Method for the Nonlinear Hyperbolic Problems with Entropy based Artificial Viscosity Stabilization written by Valentin Zingan and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p> 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.
Download or read book Monotonicity Preserving Shock Capturing Techniques for Finite Elements written by Alba Hierro Fabregat and published by . This book was released on 2016 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main object of study of this thesis is the development of artificial diffusion shock capturing techniques for continuous and discontinuous Galerkin (cG and dG) approximations of the convection-diffusion problem. Special emphasis is given to the fulfillment of the Discrete Maximum Principle (DMP). Two artificial diffusion techniques are proposed for the transport problem in cG. They scale the corresponding artificial viscosity according to the variation of the gradient of the discrete solution between elements and one of them is proven to be monotonicity preserving. Both methods are used in combination with linear stabilization to enhance its performance; in particular a novel symmetric projection stabilization technique based on a local Scott-Zhang projector is proposed. The weighting of such detector in order to preserve the monotonicity properties "including entropy stability for 1D" of the underlying methods is faced. Both shock capturing techniques are shown to outperform other methods in the literature for different sets of numerical tests. In the dG case a novel definition of the DMP has been provided. One of the gradient jump shock detectors previously used for cG methods has been adapted to this new paradigm and proved to enjoy the DMP property in the one dimensional case. A possible extension to the multidimensional case is proposed. A DMP-enjoying multidimensional dG method for the convection-diffusion equation is obtained by means of graph-viscosity techniques. The method perturbs the entries of the problem matrix to enforce some properties that lead to a DMP. Appropriate shock detectors are used to weight the perturbation of the problem matrix and the lumping of the Mass matrix, avoiding an excessive smearing of the final solution. Finally an hp-adaptive technique is proposed to solve the steady convection-diffusion problem. A novel troubled-cell detector based on the evolution of the gradient of the discrete solution along the refinement process is proposed. This troubled-cell detector is able to detect the shock layers in which linear order is enforced. Moreover the application of the artificial viscosity is restricted to such regions. At the same time, high order polynomials are reached through p-refinement in the smooth regions of the solutions. The performance of all the methods has been tested by means of various numerical tests and the results obtained are provided and commented in the document.
Download or read book Galerkin Finite Element Methods for Parabolic Problems written by V. Thomee and published by Springer. This book was released on 2006-11-14 with total page 243 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Discontinuous Galerkin Finite Element Method for Hamilton Jacobi Equations written by Changqing Hu and published by . This book was released on 1998 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Finite Element Methods for Flow Problems written by Jean Donea and published by John Wiley & Sons. This book was released on 2003-06-02 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: Die Finite-Elemente-Methode, eines der wichtigsten in der Technik verwendeten numerischen Näherungsverfahren, wird hier gründlich und gut verständlich, aber ohne ein Zuviel an mathematischem Formalismus abgehandelt. Insbesondere geht es um die Anwendung der Methode auf Strömungsprobleme. Alle wesentlichen aktuellen Forschungsergebnisse wurden in den Band aufgenommen; viele davon sind bisher nur verstreut in der Originalliteratur zu finden.
Download or read book Robust and Accurate Shock capturing in Discontinuous Galerkin Discretizations written by Jae Hwan Choi and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Fluid Dynamics (CFD) has become a critical component in analyzing fluid flows and designing industrial products. Among various numerical methods in CFD, second-order numerical schemes have been widely used in both industry and academia. Second-order methods are robust enough to use on complex geometries and usually provide a sufficient amount of accuracy in flow simulations. However, second-order accurate solutions may not be sufficient for many aerodynamic applications such as vortex flows, Large Eddy Simulations (LES), and aeroacoustics problems. As a consequence, researchers have sought high-order numerical methods to simulate complex flows with low dissipation over the past few decades. Many approaches have been suggested including Finite Difference (FD), Finite Volume (FV), and Finite Element (FE) frameworks for CFD. In the group of high-order methods, discontinuous Galerkin (DG) methods have become popular in academia because of their distinctive benefits. For DG methods, high-order accuracy in flow solutions can be easily achieved by just adding more degrees of freedom in each element. Furthermore, DG methods are well suited to modern computer hardware, even on GPUs, due to high arithmetic intensity and the locality of operations. Despite their numerous benefits, DG methods are not widely adopted because of some remaining challenges, especially in industry. One of these difficulties is shock-capturing. Similarly to other numerical methods in CFD, DG methods also suffer from spurious oscillations if discontinuities arise during flow simulations. The accuracy of solutions will degrade significantly, or solutions may diverge unless these discontinuities are captured appropriately. Therefore, a shock-capturing capability becomes necessary for DG methods to simulate compressible flows with shocks. In this work, robust and accurate shock-capturing approaches for DG methods will be demonstrated. To precisely capture various strengths of shocks, a simple shock-detector is first proposed for DG discretizations, which only relies on local flow information. Additionally, filtering strengths are precalculated to avoid parameter tuning procedures and are optimized to achieve maximum accuracy while capturing shocks. The proposed methods are then applied to two- and three-dimensional canonical problems to demonstrate the shock-capturing capabilities of the proposed methods.
Download or read book The Finite Element Method Theory Implementation and Applications written by Mats G. Larson and published by Springer Science & Business Media. This book was released on 2013-01-13 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.
