A Three-dimensional Particle-in-Cell Methodology on Unstructured Voronoi Grids with Applications to Plasma Microdevices

The development and numerical implementation of a three-dimensional Particle-In-Cell (PIC) methodology on unstructured Voronoi-Delauney tetrahedral grids is presented. Charge assignment and field interpolation weighting schemes of zero- and first-order are formulated based on the theory of long-range constraints for three-dimensional unstructured grids. The algorithms for particle motion, particle tracing, particle injection, and loading are discussed. Solution to Poisson\u27s equation is based on a finite-volume formulation that takes advantage of the Voronoi-Delauney dual. The PIC methodology and code are validated by application to the problem of current collection by cylindrical Langmuir probes in stationary and moving collisionless plasmas. Numerical results are compared favorably with previous numerical and analytical solutions for a wide range of probe radius to Debye length ratios, probe potentials, and electron to ion temperature ratios. A methodology for evaluation of the heating, slowing-down and deflection times in 3D PIC simulations is presented. An extensive parametric evaluation is performed and the effects of the number of computational particles per cell, the ratio of cell-edge to Debye length, and timestep are investigated. The unstructured PIC code is applied to the simulation of Field Emission Array (FEA) cathodes. Electron injection conditions are obtained from a Field Emission microtip model and the simulation domain includes the FEA cathode and anode. Currents collected by the electrodes are compared to theoretical values. Simulations show the formation of the virtual cathode and three-dimensional effects under certain injection conditions. The unstructured PIC code is also applied to the simulation of a micro-Retarding Potential Analyzer. For simple cases the current at the collector plate is compared favorably with theoretical predictions. The simulations show the complex structure of the potential inside the segmented microchannel, the phase space of plasma species and the space-charge effects not captured by the theory

Development of an Unstructured 3-D Direct Simulation Monte Carlo/Particle-in-Cell Code and the Simulation of Microthruster Flows

This work is part of an effort to develop an unstructured, three-dimensional, direct simulation Monte Carlo/particle-in-cell (DSMC/PIC) code for the simulation of non-ionized, fully ionized and partially-ionized flows in micropropulsion devices. Flows in microthrusters are often in the transitional to rarefied regimes, requiring numerical techniques based on the kinetic description of the gaseous or plasma propellants. The code is implemented on unstructured tetrahedral grids to allow discretization of arbitrary surface geometries and includes an adaptation capability. In this study, an existing 3D DSMC code for rarefied gasdynamics is improved with the addition of the variable hard sphere model for elastic collisions and a vibrational relaxation model based on discrete harmonic oscillators. In addition the existing unstructured grid generation module of the code is enhanced with grid-quality algorithms. The unstructured DSMC code is validated with simulation of several gaseous micronozzles and comparisons with previous experimental and numerical results. Rothe s 5-mm diameter micronozzle operating at 80 Pa is simulated and results are compared favorably with the experiments. The Gravity Probe-B micronozzle is simulated in a domain that includes the injection chamber and plume region. Stagnation conditions include a pressure of 7 Pa and mass flow rate of 0.012 mg/s. The simulation examines the role of injection conditions in micronozzle simulations and results are compared with previous Monte Carlo simulations. The code is also applied to the simulation of a parabolic planar micronozzle with a 15.4-micron throat and results are compared with previous 2D Monte Carlo simulations. Finally, the code is applied to the simulation of a 34-micron throat MEMS-fabricated micronozzle. The micronozzle is planar in profile with sidewalls binding the upper and lower surfaces. The stagnation pressure is set at 3.447 kPa and represents an order of magnitude lower pressure than used in previous experiments. The simulation demonstrates the formation of large viscous boundary layers in the sidewalls. A particle-in-cell model for the simulation of electrostatic plasmas is added to the DSMC code. Solution to Poisson\u27s equation on unstructured grids is obtained with a finite volume implementation. The Poisson solver is validated by comparing results with analytic solutions. The integration of the ionized particle equations of motion is performed via the leapfrog method. Particle gather and scatter operations use volume weighting with linear Lagrange polynomial to obtain an acceptable level of accuracy. Several methods are investigated and implemented to calculate the electric field on unstructured meshes. Boundary conditions are discussed and include a formulation of plasma in bounded domains with external circuits. The unstructured PIC code is validated with the simulation of a high voltage sheath formation

TCP-Carson: A loss-event based Adaptive AIMD algorithm for Long-lived Flows

The diversity of network applications over the Internet has propelled researchers to rethink the strategies in the transport layer protocols. Current applications either use UDP without end-to-end congestion control mechanisms or, more commonly, use TCP. TCP continuously probes for bandwidth even at network steady state and thereby causes variation in the transmission rate and losses. This thesis proposes TCP Carson, a modification of the window-scaling approach of TCP Reno to suit long-lived flows using loss-events as indicators of congestion. We analyzed and evaluated TCP Carson using NS-2 over a wide range of test conditions. We show that TCP Carson reduces loss, improves throughput and reduces window-size variance. We believe that this adaptive approach will improve both network and application performance

Numerical Particle-In-Cell studies of Hall thrusters using unstructured grids

In a few decades, space has become a crucial part of our modern society. With the imminent deployment of mega satellite constellations, their number will increase dramatically. These future satellites will be mainly equipped with electric propulsion systems, and in particular Hall thrusters. However, the processes governing the plasma physics within Hall thrusters remain poorly understood, which forces manufacturers to carry out costly and laborious experimental campaigns to certify the finished product. To overcome this difficulty, numerical simulations are essential. They can be based on a Particle-In-Cell (PIC) method, well adapted to the physics of this type of plasma. Indeed, these plasmas present kinetic effects that cannot be accurately described by fluid methods. Due to the cost of PIC simulations and the complex phenomena involved, existing codes in the literature remain limited to academic configurations based on structured meshes. In an effort to overcome these challenges, the AVIP PIC code is developed at CERFACS as a predictive tool capable of modeling industrial configurations. To do this, AVIP PIC works with unstructured meshes, which no other code in the community can currently do. This innovation comes at the cost of a considerable complexity of the code and a substantial optimization work was first done in previous work. Because of its innovative character, the first objective of this thesis was to systematically validate AVIP PIC. Thus, AVIP PIC was first used to participate successfully in an international benchmark on a 2D configuration in the axial-azimuthal plane. During this work, all groups obtained close results with 5% difference at most on the main plasma parameters profiles. An azimuthal plasma oscillation, the electron drift instability, was also observed by all participants with extremely similar characteristics. This instability due to kinetic effects, most probably plays a fundamental role in the anomalous transport of electrons within the engine. Based on this first success, we then used this case to explore and parameterize an active particle control algorithm. By preventing the number of particles from increasing too much, this tool reduces the computational cost and will be very useful in future simulations. Still, in the perspective of code validation, we then studied a simplified 2D configuration in the radial- azimuthal plane of the engine. Indeed, taking into account the presence of the walls can considerably modify the simulated physics of the engine. In particular, we have highlighted a radial-azimuthal instability, also called modified two-stream instability, which is coupled to the electron drift instability mentioned above. A benchmark work, conducted by CERFACS with six international groups, confirmed this result with an excellent agreement, despite the great diversity of the codes involved. Capitalizing on our experience in 2D, we then developed a 3D simulation based on the same geometrical elements and plasma conditions than in the two previous cases. During this study the 3D electron drift instability was identified as well as a possible signature of the radial- azimuth instability. The comparison with the previous 2D configurations seems to show that the 2D simulations tend to create a hotter and denser plasma, which affects the oscillatory phenomena. The general structure of the plasma remains nevertheless similar. Finally tools for the analysis of the code performance have been developed which will prove to be valuable for the development of more advanced 3D configurations

Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Analysis, modeling and numerical simulation of complex plasmas under microgravity conditions

Diese Dissertation hat sich mit dem Prozess der Implementierung numerischer Simulationen auf komplexe Plasmen auseinandergesetzt, aufbauend auf ein Set gekoppelter Partielle Differentialgleichungen. Die Dynamik komplexer Plasmen ist durch die Wechselwirkung ihrer unterschiedlichen Komponenten auf mikroskopischen und mesoskopischen Ebenen charakterisiert worden. Diese Wechselwirkungen resultieren in einer Mischung elektrodynamischer und strömungsdynamischer Effekte. Dieses Differentialgleichungssystem ist mit der Methode der finiten Elemente gelöst worden, die die Verkuppelung verschiedener physikalischer Phänomene in beschränkten Bereichen ermöglicht. Die Sturm-Liouville Theorie ist als mathematisches Gerüst verwendet worden, um Maxwellsche Gleichungen in beschränkten Hohlraumresonatoren mit inhomogenen Randbedingungen zu lösen. Die Profile der elektrischen Energiedichte sind kalkuliert worden, sowohl für den elektrostatischen Fall, als auch für die ersten sechs Eigenresonanzfrequenzen der elektromagnetischen Wellen. Es hat sich herausgestellt, dass die angelegte Hochfrequenz niedriger als die erste Eigenfrequenz der HF-Plasmakammer ist. Es hat sich erwiesen, dass sich die elektromagnetische Energie innerhalb der HF-Plasmakammer unter den Eigenfrequenzen aufspaltet, und dass die Rahmenbedingungen bestimmte Resonanzen erzeugen. Die Form und Verteilung dieser elektromagnetischen Energie korrelieren mit den Eigenfunktionen der respektiven Eigenresonanzfrequenzen. Um eine makroskopische Beschreibung der Dynamik komplexer Plasmen zu erreichen, ist die kinetische Theorie für Modellierung der Strömungsdynamik verwendet worden. Die Kopplung zu den elektromagnetischen Feldern ist auf der kinetischen Ebene durchgeführt worden. Dieses Herangehen überbrückt den Sprung von der mikroskopischen Beschreibung der Boltzmann Gleichung zu einer makroskopischen Beschreibung. Wir haben festgestellt, dass sowohl die dielektrischen Partikel als auch der Dielektrikumfluss einen “Elektrodruck” empfinden. Hohe Gradienten der elektrischen Energiedichte können die komplexen Plasmen zum Wirbeln bringen. Diese Herangehensweise ist neu, denn die gegenwärtige Theorie betrachtet das Neutralgas im Ruhezustand, dabei wird der Reibungswiderstand auf die komplexen Plasmen ausüben. Die beobachteten Wirbel in dem PK-3 Plus Experiment können durch die Stromlinien dieser Gradienten erklärt werden. Wir haben herausgefunden, dass der partikelfreie Raum in dem PK-3 Plus Experiment erklärt werden kann, wenn man sowohl die Elektrostatik als auch die erste Eigenresonanzfrequenz der elektrischen Energiedichte der HF-Plasmakammer berücksichtigt. Dies ist durch ein dreidimensionales Modell visualisiert worden. Dieses Model erklärt auch die Bildung sekundürer Räume, die durch die Einführung metallischer Tastkopfe in die HF-Plasmakammer hervorgebracht werden. Die Hypothese der elektrischen Energiedichte als Quelle der partikelfreien Räume kann durch die Trennung der Partikel in den Plasmakristall-Experimenten geklärt werden. Dielektrophoretische Kräfte stoßen Partikel mit höheren Permittivität (oder größere Partikel, falls alle aus demselben Material sind) in die Richtung der Regionen mit höherer elektrischer Energiedichte. Die Grenze zwischen Partikeln unterschiedlicher Permittivität (oder Größe) ist durch Isoflächen dieser Energiedichte geformt. Die Erklärung dieser Phänomene (die auf der Distribution elektrischer Energiedichte beruht) bietet einen neuen Standpunkt zur aktuellen Theorie, die auf der Reibungskraft der Ionenströmung basiert

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

A Dynamic Coupled Magnetosphere-Ionosphere-Ring Current Model

In this thesis we describe a coupled model of Earth's magnetosphere that consists of the Lyon-Fedder-Mobarry (LFM) global magnetohydrodynamics (MHD) simulation, the MIX ionosphere solver and the Rice Convection Model (RCM). We report some results of the coupled model using idealized inputs and model parameters. The algorithmic and physical components of the model are described, including the transfer of magnetic field information and plasma boundary conditions to the RCM and the return of ring current plasma properties to the LFM. Crucial aspects of the coupling include the restriction of RCM to regions where field-line averaged plasma-beta <=1, the use of a plasmasphere model, and the MIX ionosphere model. Compared to stand-alone MHD, the coupled model produces a substantial increase in ring current pressure and reduction of the magnetic field near the Earth. In the ionosphere, stronger region-1 and region-2 Birkeland currents are seen in the coupled model but with no significant change in the cross polar cap potential drop, while the region-2 currents shielded the low-latitude convection potential. In addition, oscillations in the magnetic field are produced at geosynchronous orbit with the coupled code. The diagnostics of entropy and mass content indicate that these oscillations are associated with low-entropy flow channels moving in from the tail and may be related to bursty bulk flows and bubbles seen in observations. As with most complex numerical models, there is the ongoing challenge of untangling numerical artifacts and physics, and we find that while there is still much room for improvement, the results presented here are encouraging. Finally, we introduce several new methods for magnetospheric visualization and analysis, including a fluid-spatial volume for RCM and a field-aligned analysis mesh for the LFM. The latter allows us to construct novel visualizations of flux tubes, drift surfaces, topological boundaries, and bursty-bulk flows