Michael Woopen

Adjoint‐based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods

Summary We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection–diffusion problems, including the compressible Euler and Navier–Stokes equations. The hybridization of finite element discretizations has the main advantage that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the computational mesh. Consequently, solving for these DOFs involves the solution of a potentially much smaller system. This not only reduces storage requirements but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement and secondly, to investigate the efficiency of the global error estimate. Copyright

A Hybridized Discontinuous Galerkin Method for Three-Dimensional Compressible Flow Problems

We present a hybridized discontinuous Galerkin method for three-dimensional flow problems. As an implementation technique hybridization is a classic paradigm for dual-mixed finite element discretizations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the numerical mesh. Solving for these thus involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The accuracy of the method has been validated with a scalar convection-diffusion test case. Results are shown for external, compressible flow.

Adjoint-Based Hp-Adaptation for a Class of High-Order Hybridized Finite Element Schemes for Compressible Flows

We present a robust and efficient hp-adaptation methodology, building on a class of hybridized finite element schemes for (nonlinear) convection-diffusion problems, including compressible Euler and Navier-Stokes equations. Using a discrete-adjoint approach, sensitivities with respect to output functionals of interest are computed to drive the adaptation. The theoretical framework is embedded in a unified formulation of a large class of hybridized, adjoint consistent schemes. From the error distribution given by the adjointbased error estimator, hor p-refinement is chosen based on the smoothness of the solution which can be quantified by some smoothness indicators. Numerical results are shown for a scalar convection-diffusion case, and also inviscid subsonic, transonic, and laminar flow around the NACA0012 airfoil to demonstrate the viability of the hp-adaptivity in reducing the error in the target functional.

A Hybridized DG/Mixed Scheme for Nonlinear Advection-Diffusion Systems, Including the Compressible Navier-Stokes Equations

We present a novel discretization method for nonlinear convection-diffusion equations and, in particular, for the compressible Navier-Stokes equations. The method is based on a Discontinuous Galerkin (DG) discretization for convection terms, and a mixed method using H(div) spaces for the diffusive terms. Furthermore, hybridization is used to reduce the number of globally coupled degrees of freedom. The method reduces to a DG scheme for pure convection, and to a mixed method for pure diffusion, while for the intermediate case the combined variational formulation requires no additional parameters. We formulate and validate our scheme for nonlinear model problems, as well as compressible flow problems. Furthermore, we compare our scheme to a recently developed Hybridized DG scheme with respect to formulation and convergence behavior.

Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme

We present an efficient adaptation methodology on anisotropic meshes for the recently developed hybridized discontinuous Galerkin scheme for (nonlinear) convection-diffusion problems, including compressible Euler and Navier-Stokes equations. The methodology extends the refinement strategy of Dolejsi [8] based on an interpolation error estimate to incorporate an adjoint-based error estimate. For each element, we set the area using the adjoint-based error estimate, and we seek the anisotropy, of the element, which gives the smallest interpolation error in the L-norm (q ∈ [1,∞)). For hp-adaptation, the local polynomial degree is also chosen in such a way that the configuration element shape and the polynomial degree, gives the smallest interpolation error in the L-norm. Numerical results are shown for a scalar convection-diffusion case with a strong boundary layer, as well as for inviscid subsonic, transonic and supersonic and viscous subsonic flow around the NACA0012 airfoil, to demonstrate the effectiveness of the adaptation methodology.

A Unifying Computational Framework for Adaptive High-Order Finite Element Methods

We present a comprehensive overview of our computational framework for adaptive high-order finite element methods, including discontinuous Galerkin (DG) methods and their hybridized counterparts (HDG). Besides covering the numerical methods, we grant their actual implementation a prominent position in this paper. Finally, we apply our framework to a variety of flow problems, including laminar, and turbulent flow in both twoand three-dimensional domains.

An Anisotropic Adjoint-Based hp-Adaptive HDG Method for Compressible Turbulent Flow

We present an anisotropic adjoint-based hp-adaptive hybridized discontinuous Galerkin method for turbulent compressible ow. We use the Reynolds-averaged Navier-Stokes equations complemented with the k-! turbulence model. By means of hybridization, we can formulate the resulting set of algebraic equations only in terms of the degrees of freedom on the skeleton of the computational mesh, i.e. the element interfaces. As a result, both storage requirements and computational time can be reduced. The anisotropic mesh-renement strategy involves adjoint-based error estimation, high-order anisotropydetection and Riemannian metrics. In order to improve the eciency of our adaptive routine even further, we couple it with local p-adaptation.

An HDG Method for Unsteady Compressible Flows

Recent gain of interest in discontinuous Galerkin (DG) methods shows their success in computational fluid dynamics. One potential drawback is the high number of globally coupled unknowns. By means of hybridization, this number can be significantly reduced. The hybridized DG (HDG) method has proven to be beneficial especially for steady flows. In this work we apply it to a time-dependent flow problem with shocks. Due to its inherently implicit structure, time integration methods such as diagonally implicit Runge-Kutta (DIRK) methods present themselves as natural candidates. Furthermore, as the application of flux limiting to HDG is not straightforward, an artificial viscosity model is applied to stabilize the method.

A Hybridized Discontinuous Galerkin Method for Unsteady Flows with Shock-Capturing

We present a hybridized discontinuous Galerkin (HDG) solver for the time-dependent compressible Euler and Navier-Stokes equations. In contrast to discontinuous Galerkin (DG) methods, the number of globally coupled degrees of freedom is usually tremendously smaller for HDG methods, as these methods can rely on hybridization. However, applying the method to a time-dependent problem amounts to solving a differential-algebraic nonlinear system of equations (DAE), rendering the problem extremely stiff. This implies that implicit time discretization has to be used. Suited methods for the treatment of these DAEs are, e.g., diagonally implicit Runge-Kutta (DIRK) methods, or the backward differentiation formulas (BDF). In order to solve a wide range of problems in an efficient manner, we employ adaptive time stepping using an embedded error estimator. Additionally, we investigate the use of artificial viscosity for shock-capturing in this setting, and we propose a new strategy of coarsening the mesh using non-standard polygonal elements.

A Hybridized Discontinuous Galerkin Method for Turbulent Compressible Flow

We present a hybridized discontinous Galerkin method for two-dimensional turbulent compressible ow. More precisely, we use the Reynolds-averaged Navier-Stokes equations complemented with the k-! turbulence model devised by Wilcox [30] and modied by Bassi et al. [5] for high-order methods. By means of hybridization, we can formulate the resulting set of algebraic equations only in terms of the degrees of freedom on the skeleton of the computational mesh, i.e. the element interfaces. As a result, both storage requirements and computational time can be reduced. We discuss turbulent, nearly incompressible ow along a at plate, and turbulent transonic ow around the RAE 2822 airfoil.