Thomas Westermann

Algorithms for interpolation and localization in irregular 2D meshes

Abstract In numerical simulations of power systems; discrete models and mesh-free models are very often used simultaneously, for example, in particle-in-cell (PIC) codes. The aim of this paper is to introduce new techniques for interpolation as well as for localization in irregular fourpoint meshes. These algorithms extend the existing, methods for regular grids to grids consisting of non-equidistant convex four-point meshes. They were developed in order to obtain short CPU times and possible vectorization.

Localization schemes in 2D boundary-fitted grids

A discussion of localization schemes in two-dimensional structured grids consisting of convex four-point meshes is presented. These algorithms are applicable to particle-in-cell codes based on two-dimensional boundary-fitted coordinates in order to localize particles inside the grid. They are fully vectorizable and two of them are directly applicable also to triangular meshes. Since all of them are exact, they avoid an overhead for a special treatment of particles near the boundary as is necessary for the approximate localization proposed by Seldner and Westermann (J. Comp. Phys. 79 (1988)). Hence, they are suitable for complicated geometries with outer and inner curved boundaries. Depending on the vector computer used, a speedup of 3.5 to 8 is achieved for the fastest algorithm.

Calculation of MIG guns for gyrotrons using the BFCPIC code

A quasi-stationary particle-in-cell code based on the use of boundary fitted coordinates has been adapted for use in the simulation of magnetron injection guns for gyrotrons. In addition, a ray tracing version has been written, since this is more convenient for parameter studies. Results calculated with these codes are in good agreement with experiment. The results are also compared with those calculated by other codes. The influence of electrode shape on beam quality, starting from a synthesis program, is investigated.

Use of the BFCPIC and BFCRAY codes to describe space charge limited emission in electron guns for gyrotrons

Numerical modelling of an electron gun in the space charge limited regime requires determining the current density distribution as well as the electric fields and electron trajectories. This is a rather complicated self-consistent problem, since the space charge influences the electric field, which in turn influences the electron trajectories. Previous simulations of magnetron electron guns using the BFCPIC and BFCRAY codes used a simple emission model (constant current density) that is approximately valid for thermionic emission. The code has been modified to include space charge limited emission. Several different ways of doing this are considered.One of the models considered uses Gauss’s law to force the electric field on the emitter to vanish; it was used in the original version of BFCPIC for the simulation of ion diodes. A second is based on the use of Child’s law (locally), which may be more appropriate for extension to fully electromagnetic particle-in-cell (PIC) codes. Calculations were performed with both models, and the results compared with each other and with experiments performed at FZK.

Investigations of diffusional effects in applied‐B ion diodes

In this article, a general study of diffusional effects is made numerically as well as theoretically with the aim to reproduce not only the current scaling of ion diodes but also the divergence of the ion beam and physical features such as virtual cathode movement due to the diamagnetic effect. It has been observed experimentally that the ion current exceeds that quantity which would be expected in the presence of a strong magnetic field. It is demonstrated that with the introduction of a diffusional field Eθ into the stationary 21/2‐dimensional particle‐in‐cell code based on boundary‐fitted coordinates [T. Westermann, Nucl. Instrum. Methods A 263, 271 (1988)] the simulation results agree well with experiment. This ad hoc model is supported theoretically by linear stability analysis of the stationary state.

Investigations on the vacuum current in the magnetic insulated Karlsruhe light ion facility high‐energy linear induction accelerator (KALIF–HELIA)

In order to increase the applied voltage in pulsed power ion diodes, the Karlsruhe light ion facility will be extended by a voltage adder. An important problem with such a device is how the electron loss current can be controlled in the vacuum feed. Based on a static, one‐dimensional analytic model and two‐dimensional particle‐in‐cell (PIC) simulations, a detailed knowledge of the electron flow in the voltage adder is obtained. Time‐dependent simulations support qualitatively the observation of laminar electron flow. The electrons form a band corresponding to the section on which they originate. It is demonstrated that with the introduction of guard rings, appropriately positioned in the feed, the electron loss current can be reduced by more than 50%.

KAD12D-a particle-in-cell code based on finite-volume methods

Pulsed-power diodes have been developed at the Forschungszentrum Karlsruhe and are the objects of extensive experimental as well as numerical investigations. The electrical behavior of the diodes is substantially influenced by a charged particle flow forming a non-neutral plasma inside these devices. A detailed understanding of the fundamental time-dependent phenomena (e.g., the origin of instabilities) caused by this plasma requires the solution of the Maxwell-Lorentz equations for realistic configurations with a very accurate replica of the border of the domain, where several kinds of boundary conditions are imposed. An attractive method to attack this non-linear equations numerically is the particle-in-cell (PIC) technique. As a preliminary to use the PIC approach, the relevant diode domain has to be covered by an appropriate computational mesh. Therefore, we adopt a grid model based on boundary-fitted coordinates resulting in a quadrilateral mesh zone arrangement with regular data structure. The numerical solution of the Maxwell equations in time domain is obtained by using a finite-volume (FV) approach on a non-rectangular quadrilateral mesh in two space dimensions. A very favorable property of these modern FV schemes consists in the fact that they combine inherent robustness at steep gradients with accurate resolution. In the context of self-consistent charged particle simulation in electromagnetic fields the coupling of a high-resolution FV Maxwell solver with the PIC method is a new way of approximation.

Lösen von großen linearen Gleichungssystemen

In Kapitel 27 werden Prozeduren zum Losen von grosen linearen Gleichungssystemen in Maple erstellt. Die Algorithmen konnen bei Bedarf auf andere Programmiersprachen ubernommen bzw. adaptiert werden.

Finite-Elemente-Methode

Wir haben die Methode der finiten Elemente schon in Kapitel 3.7 zum Losen gewohnlicher Randwertprobleme eingefuhrt und angewandt. Das eigentliche Anwendungsgebiet sind jedoch die partiellen Differenzialgleichungen und vorzugsweise die elliptischen Randwertprobleme. Wir zeigen hier die Finite- Elemente-Methode (FE-Methode) vor allem in Kombination mit dem Ritzschen Verfahren. Wie dort beschrieben sind die FE-Verfahren auch mit dem Galerkin-, Least Square oder Kollokations-Ansatz kombinierbar, wobei die Naherungslosung mit lokalen Basisfunktionen, z.B. den Hutfunktionen, aufgebaut wird.

Grundlagen der partiellen Differenzialgleichungen

Viele technische und physikalische Vorgange lassen sich nicht oder nur in Spezialfallen durch eine Koordinate und mit gewohnlichen Differenzialgleichungen beschreiben. Bei mehr als einer unabhangigen Variablen ist das mathematische Modell dann eine partielle Differenzialgleichung. Man denke nur an eine Stromung, welche sich im ganzen Raum ausbreitet.