A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow

We present a comparison between hybridized and non-hybridized discontinuous Galerkin methods in the context of target-based hp-adaptation for compressible flow problems. The aim is to provide a critical assessment of the computational efficiency of hybridized DG methods. 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 computational mesh. Consequently, solving for these degrees of freedom 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. Using a discrete-adjoint approach, sensitivities with respect to output functionals are computed to drive the adaptation. From the error distribution given by the adjoint-based error estimator, h- or p-refinement is chosen based on the smoothness of the solution which can be quantified by properly-chosen smoothness indicators. Numerical results are shown for subsonic, transonic, and supersonic flow around the NACA0012 airfoil. hp-adaptation proves to be superior to pure h-adaptation if discontinuous or singular flow features are involved. In all cases, a higher polynomial degree turns out to be beneficial. We show that for polynomial degree of approximation p=2 and higher, and for a broad range of test cases, HDG performs better than DG in terms of runtime and memory requirements

Verification of Unstructured Grid Adaptation Components

Adaptive unstructured grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Known differences and previously unknown differences in grid adaptation components and their integrated processes are identified here for study. Unstructured grid adaptation tools are verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. Laminar flow over a delta wing and turbulent flow over an ONERA M6 wing are verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and a moment. The scalar problems illustrate known differences in a grid adaptation component implementation and a previously unknown interaction between components. The wing adaptation cases in the current study document a clear improvement to existing grid adaptation procedures. The stage is set for the infusion of verified grid adaptation into production fluid flow simulations

Harnessing van der Waals CrPS4 and Surface Oxides for unique pre-set field induced Exchange Bias in Fe3GeTe2

Two-dimensional van der Waals (vdW) heterostructures are an attractive platform for studying exchange bias due to their defect free and atomically flat interfaces. Chromium thiophosphate (CrPS4), an antiferromagnetic material, possesses uncompensated magnetic spins in a single layer, rendering it a promising candidate for exploring exchange bias phenomena. Recent findings have highlighted that naturally oxidized vdW ferromagnetic Fe3GeTe2 exhibits exchange bias, attributed to the antiferromagnetic coupling of its ultrathin surface oxide layer (O-FGT) with the underlying unoxidized Fe3GeTe2. Anomalous Hall measurements are employed to scrutinize the exchange bias within the CrPS4/(O-FGT)/Fe3GeTe2 heterostructure. This analysis takes into account the contributions from both the perfectly uncompensated interfacial CrPS4 layer and the interfacial oxide layer. Remarkably, a distinct and non-monotonic exchange bias trend is observed as a function of temperature below 140 K. Intriguingly, a pre-set field-induced exchange bias suggests that the predominant phase in the polycrystalline surface oxide is ferrimagnetic Fe3O4. Moreover, the exchange bias induced by the ferrimagnetic Fe3O4 is significantly modulated by the presence of the van der Waals antiferromagnetic CrPS4 layer, forming a heterostructure, along with additional iron oxide phases within the oxide layer. These findings underscore the intricate and unique nature of exchange bias in van der Waals heterostructures, highlighting their potential for tailored manipulation and control

Verification of Unstructured Grid Adaptation Components

Adaptive unstructured grid techniques have made limited impact on production analysis workflows where the control of discretization error is critical to obtaining reliable simulation results. Recent progress has matured a number of independent implementations of flow solvers, error estimation methods, and anisotropic grid adaptation mechanics. Known differences and previously unknown differences in grid adaptation components and their integrated processes are identified here for study. Unstructured grid adaptation tools are verified using analytic functions and the Code Comparison Principle. Three analytic functions with different smoothness properties are adapted to show the impact of smoothness on implementation differences. A scalar advection-diffusion problem with an analytic solution that models a boundary layer is adapted to test individual grid adaptation components. The scalar problems illustrate known differences in a grid adaptation component implementation and a previously unknown interaction between components. Laminar flow over a delta wing is verified with multiple, independent grid adaptation procedures to show consistent convergence to fine-grid forces and pitching moment

Adjoint-based hphp-adaptivity on anisotropic meshes for high-order compressible flow simulations

High-order numerical methods such as Discontinuous Galerkin, Spectral Difference, and Flux Reconstruction etc, which use polynomials that are local to each mesh element to represent the solution field, are becoming increasingly popular in solving convection-dominated flows. This is due to their potential in giving accurate results more efficiently than lower order methods such as the classical Finite Volume methods. In most engineering applications, we are more interested in some specific scalar quantities rather than the full flow details. In the case of aerodynamic flow simulations, these quantities can be lift or drag coefficient. To get accurate values for such target functional quantities, adjoint-based error estimators, along with a high-order solver, have been found to be quite useful. They can identify the mesh elements that contribute the most to the error, and adapting these elements should result in a more accurate target functional. To adapt a mesh element, one can either do mesh refinement (h-adaptation) or polynomial space enrichment (p-adaptation) or both (hp-adaptation). Of these, hp-adaptation offers the most efficient way for adaptation, since one can locally choose between mesh refinement or polynomial space enrichment based on what is more efficient in resolving the local solution features. We present efficient adjoint-based hp-adaptation methodologies on isotropic and anisotropic meshes for the recently developed high order Hybridized Discontinuous Galerkin scheme for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. hp-adaptation on isotropic meshes is based on the spatial error distribution for a given target functional given by the adjoint error estimator and the solution regularity given by a regularity indicator. For anisotropic meshes, we extend the refinement strategy based on an interpolation error estimate, due to Dolejsi, by incorporating an adjoint-based error estimate. Using the two error estimates we determine the size and the shape of the triangular mesh elements on the desired mesh to be used for the subsequent adaptation steps. This is done using the concept of mesh-metric duality, where the metric tensors can encode information about mesh elements, which can be passed to a metric-conforming mesh generator to generate the required anisotropic mesh. The effectiveness of the adaptation methodology is demonstrated using numerical results: for a scalar convection-diffusion case with a strong boundary layer; inviscid subsonic, transonic and supersonic flows and viscous subsonic flow around a NACA0012 airfoil