GPU-accelerated discontinuous Galerkin methods on hybrid meshes

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM

Numerical studies of the fluid and optical fields associated with complex cavity flows

Numerical solutions for the flowfield about several cavity configurations have been computed using the Reynolds averaged Navier-Stokes equations. Comparisons between numerical and experimental results are made in two dimensions for free shear layers and a rectangular cavity, and in three dimensions for the transonic aero-window problem of the Stratospheric Observatory for Infrared Astronomy (SOFIA). Results show that dominant acoustic frequencies and magnitudes of the self excited resonant cavity flows compare well with the experiment. In addition, solution sensitivity to artificial dissipation and grid resolution levels are determined. Optical path distortion due to the flow field is modelled geometrically and is found to match the experiment. The fluid field was computed using a diagonalized scheme within an overset mesh framework. An existing code, OVERFLOW, was utilized with the additions of characteristic boundary condition and output routines required for reduction of the unsteady data. The newly developed code is directly applicable to a generalized three dimensional structured grid zone. Details are provided in a paper included in Appendix A

Algebraic generation of single domain computational grid for twin screw machines. Part I. Implementation

Special attention is required for generation of computational grids in highly deforming working chambers of twin screw machines for 3D CFD calculations. Two approaches for customised grid generation are practically available. The first is an algebraic grid generation and the second is a differential decomposition method. This paper reports on new developments in the algebraic approach that has the advantages associated with both algebraic and differential methods. Two control functions are introduced for regularisation of the initial algebraic distribution. One is based on an analytical control function in transformed coordinate system while the other uses background blocking structure in order to guide the initial algebraic distribution towards a single computational mesh. This paper presents implementation and grid characteristics of these new functions. Developed grids have been tested and results from flow calculations on a dry air compressor have been validated in part II of the paper [29]. It was possible to achieve two distinct characteristics desirable in a twin screw rotor domain mesh. Firstly, it is possible to independently control grid refinement in the interlobe region thereby providing better accuracy in representation of the leakage gaps. Secondly and most importantly, it is possible now to eliminate the non-conformal interface between the two rotor domains thereby producing a single domain structured grid for the rotors, while still maintaining the fully hexahedral cell topology. An improvement in the global orthogonality of the cells was achieved. Despite of a decrement in the Face warp quality, aspect ratio of cells retained similar scale

Interactive volume ray tracing

Die Visualisierung von volumetrischen Daten ist eine der interessantesten, aber sicherlich auch schwierigsten Anwendungsgebiete innerhalb der wissenschaftlichen Visualisierung. Im Gegensatz zu Oberflächenmodellen, repräsentieren solche Daten ein semi-transparentes Medium in einem 3D-Feld. Anwendungen reichen von medizinischen Untersuchungen, Simulation physikalischer Prozesse bis hin zur visuellen Kunst. Viele dieser Anwendungen verlangen Interaktivität hinsichtlich Darstellungs- und Visualisierungsparameter. Der Ray-Tracing- (Stahlverfolgungs-) Algorithmus wurde dabei, obwohl er inhärent die Interaktion mit einem solchen Medium simulieren kann, immer als zu langsam angesehen. Die meisten Forscher konzentrierten sich vielmehr auf Rasterisierungsansätze, da diese besser für Grafikkarten geeignet sind. Dabei leiden diese Ansätze entweder unter einer ungenügenden Qualität respektive Flexibilität. Die andere Alternative besteht darin, den Ray-Tracing-Algorithmus so zu beschleunigen, dass er sinnvoll für Visualisierungsanwendungen benutzt werden kann. Seit der Verfügbarkeit moderner Grafikkarten hat die Forschung auf diesem Gebiet nachgelassen, obwohl selbst moderne GPUs immer noch Limitierungen, wie beispielsweise der begrenzte Grafikkartenspeicher oder das umständliche Programmiermodell, enthalten. Die beiden in dieser Arbeit vorgestellten Methoden sind deshalb vollständig softwarebasiert, da es sinnvoller erscheint, möglichst viele Optimierungen in Software zu realisieren, bevor eine Portierung auf Hardware erfolgt. Die erste Methode wird impliziter Kd-Baum genannt, eine hierarchische und räumliche Beschleunigungstruktur, die ursprünglich für die Generierung von Isoflächen reguläre Gitterdatensätze entwickelt wurde. In der Zwischenzeit unterstützt sie auch die semi-transparente Darstellung, die Darstellung von zeitabhängigen Datensätzen und wurde erfolgreich für andere Anwendungen eingesetzt. Der zweite Algorithmus benutzt so genannte Plücker-Koordinaten, welche die Implementierung eines schnellen inkrementellen Traversierers für Datensätze erlauben, deren Primitive Tetraeder beziehungsweise Hexaeder sind. Beide Algorithmen wurden wesentlich optimiert, um eine interaktive Bildgenerierung volumetrischer Daten zu ermöglichen und stellen deshalb einen wichtigen Beitrag hin zu einem flexiblen und interaktiven Volumen-Ray-Tracing-System dar.Volume rendering is one of the most demanding and interesting topics among scientific visualization. Applications include medical examinations, simulation of physical processes, and visual art. Most of these applications demand interactivity with respect to the viewing and visualization parameters. The ray tracing algorithm, although inherently simulating light interaction with participating media, was always considered too slow. Instead, most researchers followed object-order algorithms better suited for graphics adapters, although such approaches often suffer either from low quality or lack of flexibility. Another alternative is to speed up the ray tracing algorithm to make it competitive for volumetric visualization tasks. Since the advent of modern graphic adapters, research in this area had somehow ceased, although some limitations of GPUs, e.g. limited graphics board memory and tedious programming model, are still a problem. The two methods discussed in this thesis are therefore purely software-based since it is believed that software implementations allow for a far better optimization process before porting algorithms to hardware. The first method is called implicit kd-tree, which is a hierarchical spatial acceleration structure originally developed for iso-surface rendering of regular data sets that now supports semi-transparent rendering, time-dependent data visualization, and is even used in non volume-rendering applications. The second algorithm uses so-called Plücker coordinates, providing a fast incremental traversal for data sets consisting of tetrahedral or hexahedral primitives. Both algorithms are highly optimized to support interactive rendering of volumetric data sets and are therefore major contributions towards a flexible and interactive volume ray tracing framework

The algorithm for generation of structured grids in deformed volumes of revolution

For the volume of revolution deformed by another volume of revolution, a grid generation algorithm is suggested. The algorithm is designed for multimaterial hydrodynamic simulation and for solving other physical and engineering problems. The algorithm represents the non-stationary procedure generating three-dimensional structured grids in domains with moving boundaries. The algorithm is developed within the variational approach for constructing optimal curvilinear grids. The volume of revolution is obtained by the rotation about the axis through 180° of a plane generatrix curve consisting of straight line segments and arcs of circles. The non-stationary algorithm is the iterative process at each stage of which the deformation of a grid and then its optimization are carried out. The deformation of a grid is implemented within the geometrical approach, and the optimization of a grid within the variational approach. Iterations are continued until the necessary deformation is reached. The algorithm is realized in the computer code written in C++. © Published under licence by IOP Publishing Ltd

Curvilinear virtual elements for contact mechanics

The virtual element method (VEM) for curved edges with applications to contact mechanics is outlined within this work. VEM allows the use of non-matching meshes at interfaces with the advantage that these can be mapped to a simple node-to-node contact formulation. To account for exact approximation of complex geometries at interfaces, we developed a VEM technology for contact that considers curved edges. A number of numerical examples illustrate the robustness and accuracy of this discretization technique. The results are very promising and underline the advantages of the new VEM formulation for contact between two elastic bodies in the presence of curved interfaces

Solution Adaptive Isotropic And Anisotropic Mesh Refinement Using General Elements

Two refinement techniques to generate solution adaptive meshes have been developed. Both techniques utilize arbitrary polyhedra (general elements) to constrain the propagation of refinement. A face-based approach that produces isotropic refinement and a combined element- and edge-based approach that produces anisotropic refinement are presented. Refinement is triggered through sensors that use a shock detection algorithm or error estimation based on the smoothness of the reconstructed solution variables. The basic algorithms as well as specific implementation issues are presented. The advantages and disadvantages of the different methods are discussed and illustrated through a set of synthetic and realistic test cases. It is shown that general elements can be employed effectively in solution adaptive meshes generated using refinement

Navier-Stokes Simulations of Flows About Complex Configurations Using Domain Decomposition Techniques

An algorithm is developed to obtain numerical simulations of flows about complex configurations composed of multiple and nonsimilar components with arbitrary geometries. The algorithm uses a hybridization of the domain decomposition techniques for grid generation and to reduce the computer memory requirement. Three dimensional, Reynolds-averaged, unsteady, compressible, and complete Navier-Stokes equations are solved on each of the subdomains by a fully-vectorized, finite-volume, upwind-biased, approximately-factored, and multigrid method. The effect of Reynolds stresses is incorporated through an algebraic turbulence model with several modifications for interference flows. The present algorithm combines the advantages of an efficient, geometrically conservative, minimally and automatically dissipative algorithm with advantages and flexibility of domain decomposition techniques. The algorithm is used to simulate supersonic flows over two-dimensional profiles and a body of revolution at high angles of attack. This study is performed to examine the suitability of the baseline solution algorithm and gain a better understanding of this class of flows. The grid overlapping is tested by obtaining the solution of a supersonic flow over a blunt-nose-cylinder at high angles of attack using a composite of overlapped grids. This solution compares very well with the solution of the same flowfield obtained with no overlapping and the experimental data. The multigrid algorithm used for this case shows substantial savings in the computational time. To accomplish one of the main objectives of this study, the algorithm is then applied to simulate the supersonic flow over an ogive-nose-cylinder near and inside a cavity. The cylinder is attached to an offset L-shaped sting when placed above the cavity opening. The results of the time-accurate computations depict these complex flows and help understanding interference effects. The unsteady nature of these flowfields and the interaction of the cavity shear layer with the cylinder are simulated. These cases illustrate two significantly different and important interference characteristics for a store separating from its parent body. Unsteadiness of the cavity flow has a more pronounced effect on the normal forces acting on the cylinder when the cylinder is placed inside the cavity. A clearer understanding of the flow between the base of the cylinder and the cavity rear face is gained by eliminating the offset sting when the cylinder is inside the cavity. The time averaged surface pressures compare favorably with the wind tunnel data, despite the averaging time period for the computations being three orders of magnitude smaller than that of the experimental measurements. The results of the present computations contribute to the much needed database for the internal store carriage and separation