10 research outputs found
Numerical methods for parabolic equations
Bei der Beschreibung physikalischer Systeme tritt die Diffusionslgeichung hĂ€ufig auf. Aufgrund der KomplexitĂ€t dieser Systeme ist es meistens nicht möglich sie analytisch zu lösen, weshalb numerische Verfahren angewendet werden mĂŒssen. AbhĂ€ngig von dem Problem ist es wichtig, effiziente Algorithmen zu entwickeln, um die Rechenzeit in einem vertretbaren Rahmen zu halten.
Traditionelle Methoden, wie etwa der VorwĂ€rts Euler Algorithmus, die ublicherweise fĂŒr das Lösen der Diffusionsgleichung verwendet werden, schrĂ€nken bekanntlich die Zeitschrittweite sehr stark ein, so dass sie in diesem Sinn als unbrauchbar gelten mĂŒssen. In dieser Diplomarbeit wird eine neue Klasse von Methoden, genannt Smoothed Essentially Non-Oscillatory (SENO), konstruiert, welche diese Grenzen signifikant erweitern.
Bevor dieser Algorithmus zusammen mit den dazu notwendigen numerischen Methoden eingefĂŒhrt wird, wird die Diffusionsgleichung eingehend untersucht. Dazu werden mehrere analytische Eigenschaften der Gleichung und ihrer Lösungen aufgefĂŒhrt. Mehrere Beweise fĂŒr den Spezialfall der eindimensionalen WĂ€rmeleitungsgleichung illustrieren diese charakteristischen Eigenschaften.
Die Grundidee fĂŒr die SENO Methoden ist die Diffusionsgleichung als Erhaltungssatz zu sehen um darauf Weighted Essentially Non-Oscillatory (WENO) Methoden anzuwenden, die die örtliche Diskretisierung berechnen. ZusĂ€tzliche Verbesserungen werden durch das Lösen eines Spline Approximations Problems auf Basis der Methode der kleinsten Quadrate erzielt, was ein GlĂ€tten der Lösung bewirkt. Ein zusĂ€tzlicher konditionierungs Schritt, bestehend aus einer gewichteten Mittelung Ă€hnlich der der exakten Lösung, verschafft eine zusĂ€tzliche Verbesserung. AuĂerdem ist es vorteilhaft gewisse analytische Eigenschaften der Lösung miteinzubeziehen. FĂŒr die Zeitintegration wird ein RungeâKutta Verfahren zweiter Ordnung verwendet, welches unter dem Namen Heunâs Methode bekannt ist.
Um die Effizienz des neuen Algorithmus zu demonstrieren werden mehrere Simulationen in einer und zwei Raumdimensionen gezeigt und analysiert. Ein Teil der Simulationen zeigt die komplette Evolution des Algorithmus anhand zweier speziell ausgewĂ€hlter Anfangsbedingungen in 2D. Diese werden auĂerdem herangezogen um die OptimalitĂ€t gewisser Parameter zu zeigen, welche fĂŒr die SENO Methoden verwendet werden. ZusĂ€tzliche Simulationen werden abschlieĂend benutzt um mögliche Richtungen zukĂŒnftiger Forschung aufzuzeigen.The diffusion equation shows up in the description of a multitude of physical systems. Due to their complexity most of them cannot be solved analytically and thus numerical methods have to be employed. Depending on the problem the computation time can be several weeks resulting in the need for efficient algorithms.
The traditional methods, e.g. the Forward Time Central Space algorithm, for solving the diffusion equation are well known for their strict time-stepping restrictions. In this thesis a new class of methods, named Smoothed Essentially Non-Oscillatory (SENO) schemes, will be implemented. By means of numerical simulations it will be shown that these restrictions are surpassed considerably.
Before giving specific information about the SENO algorithm and the numerical methods involved the diffusion equation is examined in detail. For this several analytical properties of the equation and its solutions are discussed and illustrative proofs are given for the special case of the one dimensional heat equation.
The main idea of the SENO methods is to treat the diffusion equation as a conservation law. This leads to the possibility of using Weighted Essentially Non-Oscillatory (WENO) methods to calculate the spatial discretization. Further enhancements are obtained by a least squares spline approximation which provides a smoothing effect. The most significant feature is that the smoothing only takes place inside the truncation error interval which can be calculated analytically. An additional preconditioning step that consists of a weighted average, similar to the one used for the exact solution, provides even more improvements. Furthermore it is advantageous to incorporate certain analytical properties of the solution into the numerical method. The time integration will be performed by a second order RungeâKutta algorithm commonly known as Heunâs method.
To demonstrate the effectiveness of the algorithm several simulations in one and two dimensions will be shown and analyzed. Included is a complete series of simulations that illustrate the evolution of the algorithm. Two representative initial conditions in 2D are utilized for this series. They are also used for showing the optimality of certain parameters of the SENO method. Additional simulations are conducted to point out possible directions of future research
Boundary Graph Neural Networks for 3D Simulations
The abundance of data has given machine learning considerable momentum in
natural sciences and engineering. However, the modeling of simulated physical
processes remains difficult. A key problem is the correct handling of geometric
boundaries. While triangularized geometric boundaries are very common in
engineering applications, they are notoriously difficult to model by machine
learning approaches due to their heterogeneity with respect to size and
orientation. In this work, we introduce Boundary Graph Neural Networks (BGNNs),
which dynamically modify graph structures to address boundary conditions.
Boundary graph structures are constructed via modifying edges, augmenting node
features, and dynamically inserting virtual nodes. The new BGNNs are tested on
complex 3D granular flow processes of hoppers and rotating drums which are
standard components of industrial machinery. Using precise simulations that are
obtained by an expensive and complex discrete element method, BGNNs are
evaluated in terms of computational efficiency as well as prediction accuracy
of particle flows and mixing entropies. Even if complex boundaries are present,
BGNNs are able to accurately reproduce 3D granular flows within simulation
uncertainties over hundreds of thousands of simulation timesteps, and most
notably particles completely stay within the geometric objects without using
handcrafted conditions or restrictions
AN INVESTIGATION INTO WALL BOUNDARY CONDITIONS AND THREE-DIMENSIONAL TURBULENT FLOWS USING SMOOTHED PARTICLE HYDRODYNAMICS
Study of differential operators in the context of the semi-analytical wall boundary conditions
Unified semi-analytical wall boundary conditions in SPH: analytical extension to 3-D
International audienc
The elephant in the room: the role of time in expatriate adjustment
This conceptual article explores the role of temporal dynamics in the study of expatriate adjustment. We introduce the dimensions and the domains of adjustment and discuss the dynamics between them, as well as the dynamics between antecedents, state and consequences of adjustment. Issues such as the role of time lags, duration and rate of change as well as reciprocal causation are discussed. We address the consequences of these issues for theory building in the area of expatriate adjustment and the implications for methodological choices. We conclude with specific recommendations for the future research of expatriate adjustment that recognise the nature of adjustment as a process evolving over time and that we hope will enhance the rigour and relevance of this area of research
In Situ Stability Studies of Platinum Nanoparticles Supported on RutheniumâTitanium Mixed Oxide (RTO) for Fuel Cell Cathodes
Using a variety of in situ techniques, we tracked the structural stability and concomitantly the electrocatalytic oxygen reduction reaction (ORR) of platinum nanoparticles on rutheniumâtitanium mixed oxide (RTO) supports during electrochemical accelerated stress tests, mimicking fuel cell operating conditions. High-energy X-ray diffraction (HE-XRD) offered insights in the evolution of the morphology and structure of RTO-supported Pt nanoparticles during potential cycling. The changes of the atomic composition were tracked in situ using scanning flow cell measurements coupled to inductively coupled plasma mass spectrometry (SFC-ICP-MS). We excluded Pt agglomeration, particle growth, dissolution, or detachment as cause for the observed losses in catalytic ORR activity. Instead, we argue that Pt surface poisoning is the most likely cause of the observed catalytic rate decrease. Data suggest that the gradual growth of a thin oxide layer on the Pt nanoparticles due to strong metalâsupport interaction (SMSI) is the most plausible reason for the suppressed catalytic activity. We discuss the implications of the identified catalyst degradation pathway, which appear to be specific for oxide supports. Our conclusions offer previously unaddressed aspects related to oxide-supported metal particle electrocatalysts frequently deployed in fuel cells, electrolyzers, or metalâair batteries
Anisotropy of Pt nanoparticles on carbon- and oxide-support and their structural response to electrochemical oxidation probed by in situ techniques
Identifying the structural response of nanoparticleâsupport ensembles to the reaction conditions is essential to determine their structure in the catalytically active state as well as to unravel the possibledegradation pathways. In this work, we investigate the (electronic) structure of carbon- and oxide-supported Pt nanoparticles during electrochemical oxidation byin situX-ray diffraction, absorptionspectroscopy as well as the Pt dissolution rate byin situmass spectrometry. We prepared ellipsoidal Pt nanoparticles by impregnation of the carbon and titanium-based oxide support as well as spherical Pt nanoparticles on an indium-based oxide support by a surfactant-assisted synthesis route. Duringelectrochemical oxidation, we show that the oxide-supported Pt nanoparticles resist (bulk) oxideformation and Pt dissolution. The lattice of smaller Pt nanoparticles exhibits a size-induced latticecontraction in the as-prepared state with respect to bulk Pt but it expands reversibly during electrochemical oxidation. This expansion is suppressed for the Pt nanoparticles with a bulk-like relaxedlattice. We could correlate the formation of d-band vacancies in the metallic Pt with Pt lattice expansion. PtOxformation is strongest for platelet-like nanoparticles and we explain this with a higher fraction of exposed Pt(100) facets. Of all investigated nanoparticleâsupport ensembles, the structural response of RuO2/TiO2-supported Pt nanoparticles is the most promising with respect to their morpho-logical and structural integrity under electrochemical reaction conditions