Nicholas K. Burgess

Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations

Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The algorithms are based on coupling both p- and h-multigrid (ph-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton-Krylov method. The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton-GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem. It is concluded that the Newton-GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.

Robust Computation of Turbulent Flows Using a Discontinuous Galerkin Method

This work considers the development of a robust discontinuous Galerkin (DG) solver for turbulent aerodynamic flows using the turbulence model of Spalart and Allmaras (SA). Previous work on this subject has demonstrated that applying DG discretizations to turbulent flows can be difficult, due to robustness issues related to non-smooth behavior of the turbulence model variable (or variables). This work presents two options for enhancing solver robustness. The first consists of employing a finite volume discretization with a first-order accurate convection term for the turbulence model, which is a standard practice in the low-order methods context. Computational results show that despite the first-order accurate discretization of the turbulence model there is still benefit to using higher-order discretizations for the mean flow equations and at the very least, discontinuous Galerkin solutions to the Reynolds Averaged Navier-Stokes (RANS) equations are obtained robustly. The second method of robustness enhancement considers modifications to the turbulence model equation. Numerical experiments have shown that the modifications to the turbulence model equation employed in this work are particularly effective at increasing solver robustness. Both robustness enhancement methods are applied to realistic aerodynamic flows including a subsonic turbulent airfoil flow and high-lift configurations at high angles of attack.

Efficient Solution Techniques for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations on Hybrid Anisotropic Meshes

The goal of this paper is to investigate and develop fast and robust solution techniques for high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured meshes. Previous work was focused on the development of hp-multigrid techniques for inviscid flows and the current work concentrates on the extension of these solvers to steady-state viscous flows including the effects of highly anisotropic hybrid meshes. Efficiency and robustness are improved through the use of mixed triangular and quadrilateral mesh elements, the formulation of local order-reduction techniques, the development of a line-implicit Jacobi smoother, and the implementation of a Newton-GMRES solution technique. The methodology is developed for the twoand three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear multigrid schemes. Results are presented for a flat plate boundary layer and for flow over a NACA0012 airfoil and a two-element airfoil. Current results demonstrate convergence rates which are independent of the degree of mesh anisotropy, order of accuracy (p) of the discretization and level of mesh resolution (h). Additionally, preliminary results of on-going work for the extension to the Reynolds Averaged Navier-Stokes(RANS) equations and the extension to three dimensions are given.

An hp-Adaptive Discontinuous Galerkin Solver for Aerodynamic flows on Mixed-Element Meshes

In this work we develop an hp-adaptive discontinuous Galerkin (DG) solver for aerodynamic flows. Previous work has focused on efficient solution techniques for discontinuous Galerkin discretizations. Recent work has focused on improving the robustness and efficiency of our discontinuous Galerkin solver for aerodynamic flows. Herein we propose an hp-adaptive approach which seeks to place degrees of freedom within the domain in the manner most appropriate for the nature of the solution. Near discontinuities the algorithm will refine the mesh while in regions where the solution is smooth it will enrich the discretization order. This has two effects, first of all degrees of freedom are placed where they are needed thus addressing the efficiency of the method and second we avoid attempting to use high-order polynomials to capture solutions which are discontinuous, addressing the robustness of the method. The adaptation procedure is driven via a discrete adjoint-based goaloriented error estimation technique. The method is evaluated using three test cases all of which are steady state flows. Two of these are laminar viscous flows and one is an inviscid transonic flow. In addition the transonic flow has been computed using an artificial diffusion method and the hp-adaptive approach is compared to the artificial diffusion shock capturing method.

Comparison of SU/PG and DG Finite-Element Techniques for the Compressible Navier-Stokes Equations on Anisotropic Unstructured Meshes

In this paper computed results from Steamline Upwind/Petrov-Galerkin and Discontinuous Galerkin finite-element methods are compared for various two-dimensional compressible Navier-Stokes applications. Identical meshes are utilized for each comparison with linear, quadratic, and cubic elements employed. The order of accuracy is assessed for each scheme for viscous flows using the method of manufactured solutions, and results from each scheme are compared to experimental data. Each scheme is notionally of design order, and results from both compare well with experimental data. Both schemes are viable finite-element discretization techniques, and neither applies an unnecessary amount of artificial dissipation.

Progress in High-Order Discontinuous Galerkin Methods for Aerospace Applications

This paper covers progress in developing and applying high-order accurate discretizations based on the Discontinuous Galerkin approach to aerodynamic applications. The paper concentrates on various specific areas including discretization of diusion terms, shock capturing, and multigrid solution methods. The development of a fully conservative Arbitrary Lagrangian Eulerian formulation which obeys the Geometric Conservation Law while maintaining the design order of accuracy of the static mesh discretization is also presented. Additionally, adjoint-based sensitivity analysis for design optimization as well as error estimation and adaptive control is demonstrated. Finally, prospects for future advancements and eciency gains over traditional finite volume methods are discussed in a concluding section.

Computing Shocked Flows with High-order Accurate Discontinuous Galerkin Methods

Shock capturing techniques for high-order methods have become a rich and intense area of research. However, significantly less attention has been paid to the effectiveness of capturing shock waves using high-order discretizations. This work examines the effectiveness of capturing shock waves using high-order discretizations from a robustness and error reduction point of view. Particular attention is paid to the most appropriate method of refinement for a shock wave. It is shown that when one considers exclusively shock capturing accuracy, mesh refinement employing a second-order discretization is significantly more effective at reducing error than order enrichment, despite the sub-cell shock wave resolution of high-order solutions. Furthermore, for flows involving shock waves in combination with smooth features, high-order discretizations are shown to be particularly effective when used as part of an hp-adaptation strategy. hp-adaptation is shown to yield superior efficiency compared with mesh refinement at second-order accuracy alone. To demonstrate the robustness of the proposed approach, hypersonic (M¥ 6:0) applications are considered exclusively. It is demonstrated that hp-adaptation combined with PDE-based artificial viscosity is capable of robustly obtaining low discretization error with accurate and smooth surface heating profiles.