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 An Adaptive Discontinuous Galerkin Solver for Aerodynamic Flows written by Nicholas K. Burgess and published by . This book was released on 2011 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work considers the accuracy, efficiency, and robustness of an unstructured high-order accurate discontinuous Galerkin (DG) solver for computational fluid dynamics (CFD). Recently, there has been a drive to reduce the discretization error of CFD simulations using high-order methods on unstructured grids. However, high-order methods are often criticized for lacking robustness and having high computational cost. The goal of this work is to investigate methods that enhance the robustness of high-order discontinuous Galerkin (DG) methods on unstructured meshes, while maintaining low computational cost and high accuracy of the numerical solutions. This work investigates robustness enhancement of high-order methods by examining effective non-linear solvers, shock capturing methods, turbulence model discretizations and adaptive refinement techniques. The goal is to develop an all encompassing solver that can simulate a large range of physical phenomena, where all aspects of the solver work together to achieve a robust, efficient and accurate solution strategy. The components and framework for a robust high-order accurate solver that is capable of solving viscous, Reynolds Averaged Navier-Stokes (RANS) and shocked flows is presented. In particular, this work discusses robust discretizations of the turbulence model equation used to close the RANS equations, as well as stable shock capturing strategies that are applicable across a wide range of discretization orders and applicable to very strong shock waves. Furthermore, refinement techniques are considered as both efficiency and robustness enhancement strategies. Additionally, efficient non-linear solvers based on multigrid and Krylov subspace methods are presented. The accuracy, efficiency, and robustness of the solver is demonstrated using a variety of challenging aerodynamic test problems, which include turbulent high-lift and viscous hypersonic flows. Adaptive mesh refinement was found to play a critical role in obtaining a robust and efficient high-order accurate flow solver. A goal-oriented error estimation technique has been developed to estimate the discretization error of simulation outputs. For high-order discretizations, it is shown that functional output error super-convergence can be obtained, provided the discretization satisfies a property known as dual consistency. The dual consistency of the DG methods developed in this work is shown via mathematical analysis and numerical experimentation. Goal-oriented error estimation is also used to drive an hp -adaptive mesh refinement strategy, where a combination of mesh or h -refinement, and order or p -enrichment, is employed based on the smoothness of the solution. The results demonstrate that the combination of goal-oriented error estimation and hp-adaptation yield superior accuracy, as well as enhanced robustness and efficiency for a variety of aerodynamic flows including flows with strong shock waves. This work demonstrates that DG discretizations can be the basis of an accurate, efficient, and robust CFD solver. Furthermore, enhancing the robustness of DG methods does not adversely impact the accuracy or efficiency of the solver for challenging and complex flow problems. In particular, when considering the computation of shocked flows, this work demonstrates that the available shock capturing techniques are sufficiently accurate and robust, particularly when used in conjunction with adaptive mesh refinement . This work also demonstrates that robust solutions of the Reynolds Averaged Navier-Stokes (RANS) and turbulence model equations can be obtained for complex and challenging aerodynamic flows. In this context, the most robust strategy was determined to be a low-order turbulence model discretization coupled to a high-order discretization of the RANS equations. Although RANS solutions using high-order accurate discretizations of the turbulence model were obtained, the behavior of current-day RANS turbulence models discretized to high-order was found to be problematic, leading to solver robustness issues. This suggests that future work is warranted in the area of turbulence model formulation for use with high-order discretizations. Alternately, the use of Large-Eddy Simulation (LES) subgrid scale models with high-order DG methods offers the potential to leverage the high accuracy of these methods for very high fidelity turbulent simulations. This thesis has developed the algorithmic improvements that will lay the foundation for the development of a three-dimensional high-order flow solution strategy that can be used as the basis for future LES simulations.

Download or read book Improving Shock capturing Robustness for Higher order Finite Element Solvers written by Carlee F. Wagner and published by . This book was released on 2015 with total page 91 pages. Available in PDF, EPUB and Kindle. Book excerpt: Simulation of high speed flows where shock waves play a significant role is still an area of development in computational fluid dynamics. Numerical simulation of discontinuities such as shock waves often suffer from nonphysical oscillations which can pollute the solution accuracy. Grid adaptation, along with shock-capturing methods such as artificial viscosity, can be used to resolve the shock by targeting the key flow features for grid refinement. This is a powerful tool, but cannot proceed without first converging on an initially coarse, unrefined mesh. These coarse meshes suffer the most from nonphysical oscillations, and many algorithms abort the solve process when detecting nonphysical values. In order to improve the robustness of grid adaptation on initially coarse meshes, this thesis presents methods to converge solutions in the presence of nonphysical oscillations. A high order discontinuous Galerkin (DG) framework is used to discretize Burgers' equation and the Euler equations. Dissipation-based globalization methods are investigated using both a pre-defined continuation schedule and a variable continuation schedule based on homotopy methods, and Burgers' equation is used as a test bed for comparing these continuation methods. For the Euler equations, a set of surrogate variables based on the primitive variables (density, velocity, and temperature) are developed to allow the convergence of solutions with nonphysical oscillations. The surrogate variables are applied to a flow with a strong shock feature, with and without continuation methods, to demonstrate their robustness in comparison to the primitive variables using physicality checks and pseudo-time continuation.

Download or read book Shock Capturing for Discontinuous Galerkin Methods written by Eva Casoni Rero and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 Frontiers in Physics Rising Stars written by Alex Hansen and published by Frontiers Media SA. This book was released on 2021-10-04 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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 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 Hybridized Discontinuous Galerkin Methods for Large eddy Simulation of Transitional and Turbulent Flows written by Pablo Fernández and published by . This book was released on 2019 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of computational fluid dynamics (CFD) in the aerospace industry is limited by the inability to accurately and reliably predict complex transitional and turbulent flows. This has become a major barrier to further reduce the costs, times and risks in the design process, further optimize designs, and further reduce fuel consumption and toxic emissions. Large-eddy simulation (LES) is currently the most promising simulation technique to accurately predict transitional and turbulent flows. LES, however, remains computationally expensive and often suffers from accuracy and robustness issues to the extent that it is still not practical for most applications of interest. In this thesis, we develop a series of methods and techniques to improve efficiency, accuracy and robustness of large-eddy simulations with the goal of making CFD a more powerful tool in the aerospace industry. First, we introduce a new class of high-order discretization schemes for the Euler and Navier-Stokes equations, referred to as the entropy-stable hybridized discontinuous Galerkin (DG) methods. As hybridized methods, they are amenable to static condensation and hence to more efficient implementations than standard DG methods. As entropy-stable methods, they are superior to conventional (non-entropy stable) methods for LES of compressible flows in terms of stability, robustness and accuracy. Second, we develop parallel iterative methods to efficiently and scalably solve the nonlinear system of equations arising from the discretization. The combination of hybridized DG methods with the proposed solution method provides excellent parallel scalability up to petascale and, for moderately high accuracy orders, leads to about one order of magnitude speedup with respect to standard DG methods. Third, we introduced a non-modal analysis theory that characterizes the numerical dissipation of high-order discretization schemes, including hybridized DG methods. Non-modal analysis provides critical guidelines on how to define the polynomial approximation space and the Riemann solver to improve accuracy and robustness in LES. Forth, we investigate how to best account for the effect of the subgrid scales (SGS) that, by definition, exist in LES. Numerical and theoretical results show the Riemann solver in the DG scheme plays the role of an implicit SGS model. More importantly, a change in the current best practices for SGS modeling is required in the context of high-order DG methods. And fifth, we present a physics-based shock capturing method for LES of high-Mach-number and high-Reynolds-number flows. The shock capturing method performs robustly from transonic to hypersonic regimes, provides sharp shock profiles, and has a small impact on the resolved turbulent structures. These are all critical ingredients to advance the state-of-the-art of high-order methods for LES, both in terms of methodology and understanding the relationship between the physics and the numerics.

Download or read book Simulation of Multispecies Gas Flows Using the Discontinuous Galerkin Method written by and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Truncation errors and computational cost are obstacles that still hinder large-scale applications of the Computational Fluid Dynamics method. The discontinuous Galerkin method is one of the high-order schemes utilized extensively in recent years, which is locally conservative, stable, and high-order accurate. Besides that, it can handle complex geometries and irregular meshes with hanging nodes. In this document, the nondimensional compressible Euler equations and Reynolds-Averaged Navier-Stokes equations are discretized by discontinuous Galerkin methods with a two-equations turbulence model on both structured and unstructured meshes. The traditional equation of state for an ideal gas model is substituted by a multispecies thermodynamics model in order to complete the governing equations. An approximate Riemann solver is used for computing the convective flux, and the diffusive flux is approximated with some internal penalty based schemes. The temporal discretization of the partial differential equations is either performed explicitly with the aid of Rung-Kutta methods or with semi-implicit methods. Inspired by the artificial viscosity diffusion based limiter for shock-capturing method, which has been extensively studied, a novel and robust technique based on the introduction of mass diffusion to the species governing equations to guarantee that the species mass fractions remain positive has been thoroughly investigated. This contact-surface-capturing method is conservative and a high order of accuracy can be maintained for the discontinuous Galerkin method. For each time step of the algorithm, any trouble cell is first caught by the contact-surface discontinuity detector. Then some amount of mass diffusions are added to the governing equations to change the gas mixtures and arrive at an equilibrium point satisfying some conditions. The species properties are reasonable without any oscillations. Computations are performed for many steady and unsteady flow problems. For general non-mixing fluid flows, the classical air-helium shock bubble interaction problem is the central test case for the high-order discontinuous Galerkin method with a mass diffusion based limiter chosen. The computed results are compared with experimental, exact, and empirical data to validate the fluid dynamic solver.

Download or read book Nodal Discontinuous Galerkin Methods written by Jan S. Hesthaven and published by Springer Science & Business Media. This book was released on 2007-12-18 with total page 507 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.

Download or read book Efficient Implementation of High Order Accurate Numerical Methods on Unstructured Grids written by Wanai Li and published by Springer. This book was released on 2014-05-23 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis focuses on the development of high-order finite volume methods and discontinuous Galerkin methods, and presents possible solutions to a number of important and common problems encountered in high-order methods, such as the shock-capturing strategy and curved boundary treatment, then applies these methods to solve compressible flows.

Download or read book From Shock Capturing to High Order Shock Fitting Using an Unfitted Discontinuous Galerkin Method written by Markus Geisenhofer and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Sub Element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods written by Johannes Markert and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Elements of Gas Dynamics written by H. W. Liepmann and published by Courier Corporation. This book was released on 2013-04-09 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: The increasing importance of concepts from compressible fluid flow theory for aeronautical applications makes the republication of this first-rate text particularly timely. Intended mainly for aeronautics students, the text will also be helpful to practicing engineers and scientists who work on problems involving the aerodynamics of compressible fluids. Covering the general principles of gas dynamics to provide a working understanding of the essentials of gas flow, the contents of this book form the foundation for a study of the specialized literature and should give the necessary background for reading original papers on the subject. Topics include introductory concepts from thermodynamics, including entropy, reciprocity relations, equilibrium conditions, the law of mass action and condensation; one-dimensional gasdynamics, one-dimensional wave motion, waves in supersonic flow, flow in ducts and wind tunnels, methods of measurement, the equations of frictionless flow, small-perturbation theory, transonic flow, effects of viscosity and conductivity, and much more. The text includes numerous detailed figures and several useful tables, while concluding exercises demonstrate the application of the material in the text and outline additional subjects. Advanced undergraduate or graduate physics and engineering students with at least a working knowledge of calculus and basic physics will profit immensely from studying this outstanding volume.

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 Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws written by Rainer Ansorge and published by Springer Science & Business Media. This book was released on 2012-09-14 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: In January 2012 an Oberwolfach workshop took place on the topic of recent developments in the numerics of partial differential equations. Focus was laid on methods of high order and on applications in Computational Fluid Dynamics. The book covers most of the talks presented at this workshop.