Three-dimensional magnetohydrodynamic phenomena in circular pipe flow

Ein numerischer Rechencode für die Simulation magnetohydrodynamischer (MHD) Flüssigmetallströmungen in Blankets zukünftiger Fusionsreaktoren wurde erweitert, um komplexe MHD-Phänomene in Geometrien mit gekrümmten Wänden wie z.B. kreisförmigen Rohren zu untersuchen. In diesen MHD-Strömungen wechselwirken induzierte elektrische Ströme mit dem Magnetfeld, welches das Fusionsplasma einschließt, was letztendlich zu starken Lorentz-Kräften, erheblicher Fluidumverteilung und hohen Druckverlusten in der Flüssigmetallströmung führt. Die Anwendung eines schiefheitskorrigierten Green-Gauss- oder eines Least-Squares-Verfahrens zur Berechnung des elektrischen Potentialgradienten zeigt, dass die höhere Genauigkeit dieser Diskretisierungsmethoden für die Lösung der schlecht konditionierten Gleichungen für das elektrische Potential und die elektrische Stromdichte zwingend erforderlich ist. Die detaillierten Untersuchungen auf strukturierten und unstrukturierten Rechengittern spiegeln die große Herausforderung wider, nicht-geradlinige Geometrien adäquat aufzulösen, wobei ein Hauptproblem die Diskretisierung extrem dünner und gekrümmter MHD-Grenzschichten darstellt, ohne dabei die Anzahl der Gitterpunkte im Strömungskern übermäßig zu erhöhen. Die neuen robusteren Methoden führen so zu einer erheblichen Verbesserung der Genauigkeit der Ergebnisse und damit der Leistungsfähigkeit des Codes. Sie ermöglichen zum ersten Mal die Verwendung unstrukturierter Rechengitter, selbst wenn das Magnetfeld sehr stark ist. Der weiterentwickelte Code wird für dreidimensionale Simulationen von Flüssigmetallrohrströmungen in einem inhomogenen Magnetfeld sowie von ein- und austretenden MHD-Strömungen in Strömungskanaleinsätze, die der Druckverlustreduzierung dienen, angewandt. Diese Beispiele wurden ausgewählt, da experimentelle Ergebnisse sowohl für den Druck als auch für das elektrische Oberflächenpotential als Validierungsdaten zur Verfügung stehen. Die Simulationen bieten detailreiche Einblicke in physikalische Phänomene und zeigen Druckbelastungen in Strömungskanaleinsätzen, die mit experimentellen Methoden schwer zugänglich sind. In beiden untersuchten Fällen führen axiale Potentialgradienten zu großräumigen dreidimensionalen Rezirkulationsschleifen elektrischer Ströme, die sich auf Druckverteilung und Geschwindigkeitsprofil auswirken. Ein Vergleich von experimentellen und numerischen Ergebnissen entlang der Rohroberfläche zeigt eine gute Übereinstimmung. Die Simulationen offenbaren zudem, dass ein Kräfteausgleich hauptsächlich zwischen der Lorentzkraft und der Druckkraft besteht. Nur ein kleiner Restanteil der elektromagnetischen Kraft, der als magnetodynamische Kraft eingeführt wird, gleicht sich über Trägheits- und Reibungskräfte aus. Die in der vorliegenden Arbeit gewonnenen Erkenntnisse tragen zur Verbesserung des verwendeten numerischen Codes bei und erweitern seine Anwendbarkeit für zukünftige Anforderungen der Blanketentwicklung und Fusionsforschung. Dies betrifft insbesondere Anwendungen des Codes für komplexere Geometrien und ganze Blanket-Module

GPU driven finite difference WENO scheme for real time solution of the shallow water equations

The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3

Vandermonde Neural Operators

Fourier Neural Operators (FNOs) have emerged as very popular machine learning architectures for learning operators, particularly those arising in PDEs. However, as FNOs rely on the fast Fourier transform for computational efficiency, the architecture can be limited to input data on equispaced Cartesian grids. Here, we generalize FNOs to handle input data on non-equispaced point distributions. Our proposed model, termed as Vandermonde Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently compute forward and inverse Fourier transforms, even on arbitrarily distributed points. We present numerical experiments to demonstrate that VNOs can be significantly faster than FNOs, while retaining comparable accuracy, and improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.Comment: 21 pages, 10 figure

Doctor of Philosophy

dissertationVolumetric parameterization is an emerging field in computer graphics, where volumetric representations that have a semi-regular tensor-product structure are desired in applications such as three-dimensional (3D) texture mapping and physically-based simulation. At the same time, volumetric parameterization is also needed in the Isogeometric Analysis (IA) paradigm, which uses the same parametric space for representing geometry, simulation attributes and solutions. One of the main advantages of the IA framework is that the user gets feedback directly as attributes of the NURBS model representation, which can represent geometry exactly, avoiding both the need to generate a finite element mesh and the need to reverse engineer the simulation results from the finite element mesh back into the model. Research in this area has largely been concerned with issues of the quality of the analysis and simulation results assuming the existence of a high quality volumetric NURBS model that is appropriate for simulation. However, there are currently no generally applicable approaches to generating such a model or visualizing the higher order smooth isosurfaces of the simulation attributes, either as a part of current Computer Aided Design or Reverse Engineering systems and methodologies. Furthermore, even though the mesh generation pipeline is circumvented in the concept of IA, the quality of the model still significantly influences the analysis result. This work presents a pipeline to create, analyze and visualize NURBS geometries. Based on the concept of analysis-aware modeling, this work focusses in particular on methodologies to decompose a volumetric domain into simpler pieces based on appropriate midstructures by respecting other relevant interior material attributes. The domain is decomposed such that a tensor-product style parameterization can be established on the subvolumes, where the parameterization matches along subvolume boundaries. The volumetric parameterization is optimized using gradient-based nonlinear optimization algorithms and datafitting methods are introduced to fit trivariate B-splines to the parameterized subvolumes with guaranteed order of accuracy. Then, a visualization method is proposed allowing to directly inspect isosurfaces of attributes, such as the results of analysis, embedded in the NURBS geometry. Finally, the various methodologies proposed in this work are demonstrated on complex representations arising in practice and research

Fluctuations of the vortex line density in turbulent flows of quantum fluids

We present an analytical study of fluctuations of the Vortex Line Density (VLD) $$ in turbulent flows of quantum fluids. Two cases are considered. The first one is the counterflowing (Vinen) turbulence, where the vortex lines are disordered, and the evolution of quantity $\mathcal{L}(t)$ obeys the Vinen equation. The second case is the quasi-classic turbulence, where vortex lines are believed to form the so called vortex bundles, and their dynamics is described by the HVBK equations. The latter case, is of a special interest, since a number of recent experiments demonstrate the $\omega ^{-5/3}$ dependence for spectrum VLD, instead of $\omega ^{1/3}$ law, typical for spectrum of vorticity. In nonstationary situation, in particular, in the fluctuating turbulent flow there is a retardation between the instantaneous value of the normal velocity and the quantity $\mathcal{L}$. This retardation tends to decrease in the accordance with the inner dynamics, which has a relaxation character. In both cases the relaxation dynamics of VLD is related to fluctuations of the relative velocity, however if for the Vinen case the rate of temporal change for $\mathcal{L}(t)$ is directly depends on $\delta \mathbf{v}_{ns}$, for the HVBK dynamics it depends on $\nabla \times \delta \mathbf{v}_{ns}$. As a result, for the disordered case the spectrum $<\delta \mathcal{L}(\omega) \delta \mathcal{L}(-\omega)>$ coincides with the spectrum $\omega ^{-5/3}$. In the case of the bundle arrangement, the spectrum of the VLD varies (at different temperatures) from $\omega ^{1/3}$ to $\omega ^{-5/3}$ dependencies. This conclusion may serve as a basis for the experimental determination of what kind of the turbulence is implemented in different types of generation.Comment: 8 pages, 29 reference

Characterization and Simulation of Discrete Fracture Networks in Unconventional Shale Reservoirs

Fracture characterization and simulation of complex fracture networks are investigated with the emphasis on better and faster approaches to generate fractures by conforming to available data resources, and on accurate, robust, and efficient techniques to grid and discretize complex fracture networks. Three fracture characterization techniques such as fractal-based, microseismic-constrained, and outcrop-based are presented. Natural fractures are generated either stochastically from fractal-based theory, or constrained by microseismic information, or from outcrop maps. Hydraulic fractures are computed from a fast proxy model for fracture propagation that incooperates material balance and lab-measured conductivity data. Then, optimization-based unstructured gridding and discretization technique is developed to handle complex fracture networks with extensively fracture clustering, nonorthogonal and low-angle fracture intersections, and nonuniform fracture aperture distributions. Moreover, through fracture simulation, sensitivity analysis of natural fracture related parameters, nonuniform fracture aperture, and unstructured gridding related parameters on well production performance are investigated, which are followed by well testing behaviors and CO2 EOR of complex fracture networks. This work presents an integrated workflow to model discrete fractures in unconventional shale reservoirs, together with detailed illustrations of each critical component using both synthetic and field application examples