Boundary integral equations in Kinetic Plasma Theory

In this thesis, we use boundary integral equations (BIE) as a powerful tool to gain new insights into the dynamics of plasmas. On the theoretical side, our work provides new results regarding the oscillation of bounded plasmas. With the analytical computation of the frequencies for a general ellipsoid we contribute a new benchmark for numerical methods. Our results are validated by an extensive numerical study of several three-dimensional problems, including a particle accelerator with complex geometry and mixed boundary conditions. The use of Boundary Element Methods (BEM) reduces the dimension of the problem from three to two, thus drastically reducing the number of unknowns. By employing hierarchical methods for the computation of the occurring nonlocal sums and integral operators, our method scales linearly with the number of particles and the number of surface triangles, where the error decays exponentially in the expansion parameter. Furthermore, our method allows the pointwise evaluation of the electric field without loss of convergence order. As we are able to compute the occurring boundary integrals analytically, we can precisely predict the electric field near the boundary. This property makes our method exceptionally well suited for the numerical simulation of plasma sheaths near irregular boundaries or of plasma-surface interaction such as etching of semiconductors.In der vorliegenden Arbeit nutzen wir Randintegralgleichungen als ein mächtiges Werkzeug, um neue Einsichten in die Dynamik von Plasmen zu gewinnen. Auf theoretischer Seite entwickelt diese Arbeit neue Resultate bezüglich der Oszillation beschränkter Plasmen. Durch die ana- lytische Berechnung der Frequenzen im Fall eines allgemeinen Ellipsoids stellen wir ein neues Testbeispiel für numerische Methoden bereit. Unsere Resultate werden durch umfangreiche numerische Untersuchen dreidimensionaler Beispiele validiert, etwa einen Partikelbeschleuniger mit komplexer Geometrie und gemischten Randwerten. Mithilfe der Randelementmethode reduziert sich die Dimension des Problems von drei auf zwei, womit sich die Anzahl der Un- bekannten drastisch reduziert. Dank der Nutzung hierarchischer Methoden zur Berechnung der auftauchenden nichtlokalen Summen und Integraloperatoren skaliert unsere Methode linear mit der Anzahl der Partikel und der Anzahl der Oberflächendreiecken, wobei der Fehler exponen- tiell im Entwicklungsparameter abfällt. Des Weiteren erlaubt unsere Methode die Berechnung des elektrischen Felds ohne Verringerung der Konvergenzordnung. Da wir die auftretenden Randintegrale analytisch berechnen können, können wir präzise Aussagen über das elektrische Feld nahe des Rands treffen. Dank dieser Eigenschaft ist unsere Methode außergewöhnlich gut geeignet, um Plasmaränder nahe irregulärer Ränder oder Plasma-Oberflächen-Interaktionen, etwa das Ätzen von Halbleitern, zu simulieren

A massively parallel semi-Lagrangian solver for the six-dimensional Vlasov-Poisson equation

This paper presents an optimized and scalable semi-Lagrangian solver for the Vlasov-Poisson system in six-dimensional phase space. Grid-based solvers of the Vlasov equation are known to give accurate results. At the same time, these solvers are challenged by the curse of dimensionality resulting in very high memory requirements, and moreover, requiring highly efficient parallelization schemes. In this paper, we consider the 6d Vlasov-Poisson problem discretized by a split-step semi-Lagrangian scheme, using successive 1d interpolations on 1d stripes of the 6d domain. Two parallelization paradigms are compared, a remapping scheme and a classical domain decomposition approach applied to the full 6d problem. From numerical experiments, the latter approach is found to be superior in the massively parallel case in various respects. We address the challenge of artificial time step restrictions due to the decomposition of the domain by introducing a blocked one-sided communication scheme for the purely electrostatic case and a rotating mesh for the case with a constant magnetic field. In addition, we propose a pipelining scheme that enables to hide the costs for the halo communication between neighbor processes efficiently behind useful computation. Parallel scalability on up to 65k processes is demonstrated for benchmark problems on a supercomputer

hyper.deal: An efficient, matrix-free finite-element library for high-dimensional partial differential equations

This work presents the efficient, matrix-free finite-element library hyper.deal for solving partial differential equations in two to six dimensions with high-order discontinuous Galerkin methods. It builds upon the low-dimensional finite-element library deal.II to create complex low-dimensional meshes and to operate on them individually. These meshes are combined via a tensor product on the fly and the library provides new special-purpose highly optimized matrix-free functions exploiting domain decomposition as well as shared memory via MPI-3.0 features. Both node-level performance analyses and strong/weak-scaling studies on up to 147,456 CPU cores confirm the efficiency of the implementation. Results of the library hyper.deal are reported for high-dimensional advection problems and for the solution of the Vlasov--Poisson equation in up to 6D phase space.Comment: 33 pages, 18 figure

Matrix-free finite-element computations at extreme scale and for challenging applications

For numerical computations based on finite element methods (FEM), it is common practice to assemble the system matrix related to the discretized system and to pass this matrix to an iterative solver. However, the assembly step can be costly and the matrix might become locally dense, e.g., in the context of high-order, high-dimensional, or strongly coupled multicomponent FEM, leading to high costs when applying the matrix due to limited bandwidth on modern CPU- and GPU-based hardware. Matrix-free algorithms are a means of accelerating FEM computations on HPC systems, by applying the effect of the system matrix without assembling it. Despite convincing arguments for matrix-free computations as a means of improving performance, their usage still tends to be an exception at the time of writing of this thesis, not least because they have not yet proven their applicability in all areas of computational science, e.g., solid mechanics. In this thesis, we further develop a state-of-the-art matrix-free framework for high-order FEM computations with focus on the preconditioning and adopt it in novel application fields. In the context of high-order FEM, we develop means of improving cache efficiency by interleaving cell loops with vector updates, which we use to increase the throughput of preconditioned conjugate gradient methods and of block smoothers based on additive Schwarz methods; we also propose an algorithm for the fast application of hanging-node constraints in 3D for up to 137 refinement configurations. We develop efficient geometric and polynomial multigrid solvers with optimized transfer operators, whose performance is experimentally investigated in detail in the context of locally refined meshes, indicating the superiority of global-coarsening algorithms. We apply the developed solvers in the context of novel stage-parallel implicit Runge–Kutta methods and demonstrate the benefit of stage–parallel solvers in decreasing the time to solution at the scaling limit. Novel challenging application fields of matrix-free computations include high-dimensional computational plasma physics, solid-state-sintering simulations with a high and dynamically changing number of strongly coupled components, and coupled multiphysics problems with evaluation and integration at arbitrary points. In the context of these fields, we detail computational challenges, propose modified versions of the standard matrix-free algorithms for high-performance computing, and discuss preconditioning-related topics. The efficiency of the derived algorithms on the node level and at extreme scales is demonstrated experimentally on SuperMUC-NG, one of Germany’s leading supercomputers, with up to 150k processes and by solving systems of up to 5 × 1012 unknowns. Such problem sizes would not be conceivable for equivalent matrix-based algorithms. The major achievements of this thesis allow to run larger simulations faster and more efficiently, enabling progress and new possibilities for a range of application fields in computational science

Fluid modeling and simulation of the electron population in Hall Effect Thrusters with complex magnetic topologies

Mención Internacional en el título de doctorLa propulsión eléctrica es una tecnología consolidada, utilizada por vehículos espaciales para llevar a cabo maniobras no atmosféricas. Este tipo de motores cohete ha estado presente en numerosas aplicaciones en las últimas décadas y sus usos van desde el mantenimiento de la posición orbital de satélites comerciales a transferencias interplanetarias en misiones de exploración. La mayor ventaja de los numerosos tipos de propulsores eléctricos es su capacidad de proporcionar un determinado impulso a un coste de propelente reducido, en comparación con otros tipos de propulsión. El desarrollo de los motores de plasma, la clase más común de propulsor eléctrico, se ha visto impedido en mayor medida que los cohetes químicos, por ejemplo, debido a la complejidad de la interacción de los fenómenos físicos y a dificultades asociadas con las campañas experimentales. En las últimas dos décadas se ha introducido el uso de simulaciones numéricas para ayudar a la caracterización de estos aparatos. A pesar de que el diseño asistido por ordenador juega aún un papel muy reducido, el incremento de recursos computacionales y la creciente exactitud de los modelos físicos han permitido a estas simulaciones describir numerosos mecanismos físicos, explorar el espacio de diseño de estos aparatos y complementar los ensayos experimentales. Esta tesis está centrada en el estudio numérico de la población de electrones en descargas de plasma poco colisionales, bajo la influencia de campos eléctricos y magnéticos. El trabajo realizado ha contribuido al desarrollo de una nueva herramienta de simulación híbrida, cuasi-neutra, bidimensional y axisimétrica, denominada HYPHEN; su naturaleza híbrida se debe al tratamiento por separado de las especies pesadas, descritas a través de un conocido método de partículas, y de la población de electrones, descrita como un fluido. Una de nuestras mayores contribuciones es la introducciÃsn de un modelo anisotrÃspico de dos temperaturas, que permite capturar los efectos de la falta de uniformidad del campo magnético sobre el transporte de electornes. Esta función abre el camino para la caracterización de nuevos propulsores electromagnéticos. Actualemente, el código está orientado hacia la simulación de las regiones del canal y de la pluma cercana en motores de efecto Hall, en los que se enfoca esta tesis. Parte del trabajo se ha dedicado a dotar al código de las capacidades necesarias para la simulación de topologías magnéticas complejas. El presente documento detalla la motivación detrás de HYPHEN, su metodología de diseño y la influencia de trabajos previos. Se ha prestado una especial atención al modelo fluido propuesto, detallando el uso de una malla alineada con el campo magnético para el tratamiento numérico de la población confinada de electrones, para la cual se han utilizado diversos métodos ad-hoc de discretización temporal y espacial. Varios modelos auxiliares también se han descrito, con el objetivo de caracterizar la respuesta de la capa límite del plasma y de los distintos procesos colisionales en el seno del mismo. Se presenta también el estudio de los aspectos numéricos del modelo fluido, incluyendo la sensibilidad a condiciones iniciales, a los valores del paso temporal, el refinamiento de la malla, etc. Finalmente, HYPHEN se ha testeado para la configuración de un conocido motor Hall. Los resultados demuestran que las propiedades físicas y las actuaciones obtenidas son comparables con resultados provenientes de estudios experimentales. Bajo este contexto, se ha llevado a cabo un estudio paramétrico para determinar la dependencia de la respuesta del motor con algunos de los parámetros más relevantes del modelo, tales como el transporte anómalo de electrones o la fracción de termalización de la capa límite, y con los diferentes modelos colisionales.Electric propulsion is an established technology used for non-atmospheric spacecraft maneuvering. This type of rockets have been present in numerous applications in the last decades, and their uses range from station keeping of commercial satellites to interplanetary transfers in deep space exploration missions. While electric propulsion thrusters are multi-faceted, presenting numerous and distinct types, their best selling point is the capability to deliver a given impulse at much lower propellant cost, in comparison to other types of propulsion. The maturation of plasma thrusters, the most common type of electric propulsion devices, has faced more limitations than chemical rockets, for example, due to the complexity of the physical interactions at play, and the difficulties associated with experimental campaigns. Over the past two decades, numerical simulations were introduced as a novel tool in the characterization of these devices. While true computer-aided-design is not yet a reality, the increment of computational resources and the heightened fidelity of the physical models have allowed to describe numerous physical mechanisms, explore the design space of these devices and complement experimental testing. This thesis focuses on the numerical study of the electron population in weakly collisional plasma discharges, under the influence of applied magnetic and electric fields. The work has been a primary contribution in the development of a new, quasi-neutral, two-dimensional, axisymmetric, hybrid simulation tool, called HYPHEN. Its hybrid nature responds to the different treatment of the heavy species populations, described through a well known discrete-particle approach, and the electron population, described as a fluid. One of our main contributions has been the introduction of a two-temperature anisotropic approach, which allows capturing of the magnetic non-uniformity effects over electron transport; this feature paves the way for the characterization of some novel electromagnetic propulsion technologies. Presently, the code is oriented to the simulation of the channel and near-plume regions in Hall effect thrusters, which have been the main focal point of the thesis. Dedicated efforts have been directed to providing the capabilities for the simulation of the plasma under complex magnetic field topologies. The manuscript details the motivation and design methodology behind HYPHEN, as well as the influence of previous work. Special attention has been given to the particularities of the proposed fluid model; this includes the use of a magnetic field aligned mesh for the numerical treatment of the electron population under magnetic confinement, for which ad-hoc spatial and temporal discretization methods have been proposed. Additional ancillary physical models have also been developed, characterizing the response of plasma boundary layers and the various collisional processes in the plasma. The numerical aspects of the model have been investigated, including the sensitivity to initial conditions, time-step values, mesh refinement, etc. Finally, HYPHEN has been tested in the context of a representative Hall-thruster configuration. The results were found to be in line with experimentally reported thruster performances and plasma discharge quantities. Additionally, a parametric investigation has been carried out in order to investigate the dependency of the thruster response with the most relevant model parameters, such as the anomalous electron transport or the boundary layer thermalization fraction, and the different collisional models.This work has been partially supported by the CHEOPS project, that received funding from the European Union’s Horizon 2020 research and innovation program, under grant agreement No. 730135. Additional support came from Project ESP2016-75887, funded by the National research and development program of Spain.Programa Oficial de Doctorado en Plasmas y Fusión NuclearPresidente: José Javier Honrubia Checa.- Secretario: Mario Merino Martínez.- Vocal: Paul-Quentin Elia

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions