Aravind Balan

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.

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.

Mesh Adaptation and Optimization for Discontinuous Galerkin Methods Using a Continuous Mesh Model

We present a method for anisotropic mesh adaptation and optimization for high-order Discontinuous Galerkin (DG) Schemes. Given the total number of degrees of freedom, we propose a metric-based method, which aims to globally optimize the mesh with respect to the L norm of the error. This is done by minimizing a suitable error model associated with the approximation space. Advantages of using a metric based method in this context are several. Firstly, it facilitates changing and manipulating the mesh in a general nonisotropic way. Secondly, defining a suitable continuous interpolation operator allows us to use an analytic optimization framework which operates on the metric field, rather than the discrete mesh. We present the formulation of the method as well as numerical experiments in the context of convection-diffusion systems.

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.

A Stable Spectral Difference Method for Triangles

Numerical schemes using piecewise continuous polynomials are very popular for high order approximation of conservation laws. While the most widely used numerical scheme under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity of formulation and implementation. However, recently it has been shown that the scheme is not unconditionally linearly stable on triangles. In this paper we present an alternate formulation of the scheme, featuring a new flux interpolation technique using Raviart-Thomas spaces, which proves stable under a similar linear analysis in which the standard scheme failed. We demonstrate viability of the concept by showing linear stability both in the semi-discrete sense and for time stepping schemes of the SSP Runge-Kutta type. Furthermore, we present a convergence study using the linear advection equation, as well as case studies in compressible flow simulation using the Euler equations.

Time‐Relaxation Methods for High‐Order Discretization of Compressible Flow Problems

We present recent developments in solution methods for steady compressible flow simulation in conjunction with high‐order spatial discretization methods. Lack of efficient solution methods has often been identified as a major deficiency of high‐order techniques in this setting. Our schemes aim to combine the advantages of several solution techniques, such as multigrid methods, and Newton‐Krylov methods.