Numerical methods for parabolic equations

Bei der Beschreibung physikalischer Systeme tritt die Diffusionslgeichung häufig auf. Aufgrund der Komplexität dieser Systeme ist es meistens nicht möglich sie analytisch zu lösen, weshalb numerische Verfahren angewendet werden müssen. Abhängig von dem Problem ist es wichtig, effiziente Algorithmen zu entwickeln, um die Rechenzeit in einem vertretbaren Rahmen zu halten. Traditionelle Methoden, wie etwa der Vorwärts Euler Algorithmus, die ublicherweise für das Lösen der Diffusionsgleichung verwendet werden, schränken bekanntlich die Zeitschrittweite sehr stark ein, so dass sie in diesem Sinn als unbrauchbar gelten müssen. 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 eingeführt wird, wird die Diffusionsgleichung eingehend untersucht. Dazu werden mehrere analytische Eigenschaften der Gleichung und ihrer Lösungen aufgeführt. Mehrere Beweise für den Spezialfall der eindimensionalen Wärmeleitungsgleichung illustrieren diese charakteristischen Eigenschaften. Die Grundidee für die SENO Methoden ist die Diffusionsgleichung als Erhaltungssatz zu sehen um darauf Weighted Essentially Non-Oscillatory (WENO) Methoden anzuwenden, die die örtliche Diskretisierung berechnen. Zusätzliche Verbesserungen werden durch das Lösen eines Spline Approximations Problems auf Basis der Methode der kleinsten Quadrate erzielt, was ein Glätten der Lösung bewirkt. Ein zusätzlicher konditionierungs Schritt, bestehend aus einer gewichteten Mittelung ähnlich der der exakten Lösung, verschafft eine zusätzliche Verbesserung. Außerdem ist es vorteilhaft gewisse analytische Eigenschaften der Lösung miteinzubeziehen. Für die Zeitintegration wird ein Runge–Kutta Verfahren zweiter Ordnung verwendet, welches unter dem Namen Heun’s 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 ausgewählter Anfangsbedingungen in 2D. Diese werden außerdem herangezogen um die Optimalität gewisser Parameter zu zeigen, welche für die SENO Methoden verwendet werden. Zusätzliche Simulationen werden abschließend benutzt um mögliche Richtungen zukünftiger 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 Runge–Kutta algorithm commonly known as Heun’s 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

Unified semi-analytical wall boundary conditions in SPH: analytical extension to 3-D

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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 Ruthenium−Titanium 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 ruthenium–titanium 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 metal–support 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 metal–air 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 nanoparticle–support 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 nanoparticle–support 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