Numerical simulation of electromagnetic fields in complex multi-cavity superconducting radio frequency resonators

This thesis deals with the computation of electromagnetic fields in complex, superconducting resonators, as well as the efficient calculation of the losses of such resonances by the couplers and beampipes. A perturbation approach is used to efficiently assemble the resulting nonlinear eigenvalue problem, which is then solved by using the Newton method. Using the proposed methods, current research questions for the Third Harmonic Module of the FLASH accelerator, the bERLinPro mainlinear accelerator and the BESSY VSR cavity-design are be answered.Diese Arbeit beschäftigt sich mit der Berechnung elektromagnetischer Felder in komplexen, supraleitenden Resonatoren sowie der effizienten Berechnung der Verluste solcher Resonanzen durch die Koppler und Strahlrohre. Ein Störungsansatz wird verwendet, um das resultierende nichtlineare Eigenwertproblem effizient zusammenzusetzen, das dann mit der Newton-Methode gelöst wird. Mit den vorgeschlagenen Methoden werden aktuelle Forschungsfragen für das Third Harmonic Modul des FLASH-Beschleunigers, des bERLinPro Haupt-Linearbeschleunigers und des BESSY VSR Resonator-Designs beantwortet

Uncertainty Quantification for Maxwell's Eigenproblem based on Isogeometric Analysis and Mode Tracking

The electromagnetic field distribution as well as the resonating frequency of various modes in superconducting cavities used in particle accelerators for example are sensitive to small geometry deformations. The occurring variations are motivated by measurements of an available set of resonators from which we propose to extract a small number of relevant and independent deformations by using a truncated Karhunen-Lo\`eve expansion. The random deformations are used in an expressive uncertainty quantification workflow to determine the sensitivity of the eigenmodes. For the propagation of uncertainty, a stochastic collocation method based on sparse grids is employed. It requires the repeated solution of Maxwell's eigenvalue problem at predefined collocation points, i.e., for cavities with perturbed geometry. The main contribution of the paper is ensuring the consistency of the solution, i.e., matching the eigenpairs, among the various eigenvalue problems at the stochastic collocation points. To this end, a classical eigenvalue tracking technique is proposed that is based on homotopies between collocation points and a Newton-based eigenvalue solver. The approach can be efficiently parallelized while tracking the eigenpairs. In this paper, we propose the application of isogeometric analysis since it allows for the exact description of the geometrical domains with respect to common computer-aided design kernels, for a straightforward and convenient way of handling geometrical variations and smooth solutions

Gradient-Based Eigenvalue Optimization for Electromagnetic Cavities with Built-in Mode Matching

Shape optimization with respect to eigenvalues of a cavity plays an important role in the design of new resonators or in the optimization of existing ones. In our paper, we propose a gradient-based optimization scheme, which we enhance with closed-form shape derivatives of the system matrices. Based on these, we can compute accurate derivatives of eigenvalues, eigenmodes and the cost function with respect to the geometry, which significantly reduces the computational effort of the optimizer. We demonstrate our work by applying it to the 9-cell TESLA cavity, for which we tune the design parameters of the computational model to match the design criteria for devices in realistic use cases. Since eigenvalues may cross during the shape optimization of a cavity, we propose a new algorithm based on an eigenvalue matching procedure, to ensure the optimization of the desired mode in order to also enable successful matching along large shape variations

Compact State-Space Models for Complex Superconducting Radio-Frequency Structures Based on Model Order Reduction and Concatenation Methods

The modeling of large chains of superconducting cavities with couplers is a challenging task in computational electrical engeneering. The RF properties of the arising segments are described by state-space equations. Their model order is reduced and the reduced-order models are concatenated in accordance with the topology of the complete structure. The scheme enables the investigation of radio-frequency properties of large structures without the application of supercomputers

NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc

[EN] SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.This work was partially funded by the Spanish Agencia Estatal de Investigacion AEI http://ciencia.gob.es under grants TIN2016-75985-P AEI and PID2019-107379RB-I00 AEI (including European Commission FEDER funds).Campos, C.; Roman, JE. (2021). NEP: A Module for the Parallel Solution of Nonlinear Eigenvalue Problems in SLEPc. ACM Transactions on Mathematical Software. 47(3):1-29. https://doi.org/10.1145/3447544S12947

Methods for the design and analysis of higher-order mode couplers applied to superconducting accelerating structures

Higher-order modes (HOMs) may affect beam stability and refrigeration requirements of superconducting proton linacs such as the SPL which is being studied at CERN. One option being considered to limit these effects is the use of coaxial HOM couplers. In this work, potentially dangerous modes are analyzed and corresponding damping requirements derived. The design process of coaxial HOM couplers is examined under new aspects. Several prototypes are elaborated and compared in terms of mode damping, thermal loads, structural deformations, mechanical tolerances, and multipacting.Moden höherer Ordnung (HOMs) können beträchtlich die Strahldynamik und Kühlanforderungen supraleitender Linearbeschleuniger, wie dem am CERN untersuchten SPL, beeinflussen. Koaxiale HOM Koppler sind eine Möglichkeit, um Auswirkungen entsprechender Moden zu begrenzen. Im Rahmen dieser Arbeit wurden potentiell gefährliche Moden analysiert und Dämpfungsanforderungen abgeleitet. Der Kopplerentwurf wurde unter neuen Gesichtspunkten aufgegriffen. Verschiedene Prototypen wurden bezüglich Modendämpfung, thermisches Verhaltens, strukturmechanischer Verformungen, Toleranzen und Multipactings verglichen

Chaos and Localisation

This thesis investigates quantum transport in the energy space of two paradigm systems of quantum chaos theory. These are highly excited hydrogen atoms subject to a microwave field, and kicked atoms which mimic the delta-kicked rotor model. Both of these systems show a complex dynamical evolution arising from the interaction with an external time-periodic driving force. In particular two quantum phenomena, which have no counterpart on the classical level, are studied: the suppression of classical diffusion, known as dynamical localisation, and quantum resonances as a regime of enhanced transport for the delta-kicked rotor. The first part of the thesis provides new support for the quantitative analogy between energy transport in strongly driven highly excited atoms and particle transport in Anderson-localised solids. A comprehensive numerical analysis of the atomic ionisation rates shows that they obey a universal power-law distribution, in agreement with Anderson localisation theory. This is demonstrated for a one-dimensional model as well as for the real three-dimensional atom. We also discuss the implications of the universal decay-rate distributions for the asymptotic time-decay of the survival probability of the atoms. The second part of the thesis clarifies the effect of decoherence, induced by spontaneous emission, on the quantum resonances which have been observed in a recent experiment with delta-kicked atoms. Scaling laws are derived, based on a quasi-classical approximation of the quantum evolution. These laws describe the shape of the resonance peaks in the mean energy of an experimental ensemble of kicked atoms. Our analytical results match perfectly numerical computations and explain the initially surprising experimental observations. Furthermore, they open the door to the study of the competing effects of decoherence and chaos on the stability of the time evolution of kicked atoms. This stability may be characterised by the overlap of two identical initial states which are subject to different time evolutions. This overlap, called fidelity, is investigated in an experimentally accessible situation.In dieser Arbeit untersuchen wir quantalen Transport im Energieraum anhand zweier Paradebeispiele der Quantenchaostheorie: hoch angeregte Wasserstoffatome im Mikrowellenfeld, und gekickte Atome, die das Modellsystem des delta-gekickten Rotors simulieren. Beide Systeme unterliegen aufgrund des aeusseren, zeitlich periodischen Antriebs einer komplexen Zeitentwicklung. Insbesondere werden zwei Quantenphenomaene untersucht, die kein klassisches Analogon besitzen: die Unterdrueckung klassischer Diffusion, bekannt unter dem Schlagwort dynamischer Lokalisierung, und die Quantenresonanzen als dynamisches Regime, das sich durch beschleunigten Transport im delta-gekickten Rotor auszeichnet. Der erste Teil der Arbeit belegt auf neue Weise die quantitative Analogie zwischen dem Energietransport in stark getriebenen, hoch angeregten Atomen und dem Teilchentransport im Anderson-lokalisierten Festkoerper. Eine umfassende numerische Analyse der atomaren Ionisationsraten zeigt in Uebereinstimmung mit der Lokalisierungstheorie nach Anderson, dass die Ratenverteilungen einem universellen Potenzgesetz unterliegen. Dies wird sowohl fuer ein eindimensionales Modell als auch fuer das reale dreidimensionale Atom demonstriert. Ausserdem werden die Konsequenzen aus der universellen Verteilung der Ionisationsraten fuer die asymptotische Zeitabhaengigkeit der Ueberlebenswahrscheinlichkeit der Atome diskutiert. Der zweite Teil der Arbeit klaert den Einfluss von Dekohaerenz -- induziert durch Spontanemission -- auf die kuerzlich im Experiment mit delta-gekickten Atomen beobachteten Quantenresonanzen. Wir leiten Skalierungsgesetze ab, die auf einer quasiklassischen Naeherung der Quantendynamik beruhen und die Form von Resonanzpeaks beschreiben, welche in der mittleren Energie eines atomaren Ensembles im Experiment beobachtet wurden. Unsere analytischen Resultate stimmen mit numerischen Rechnungen ausgezeichnet ueberein und erklaeren die zunaechst ueberraschenden experimentellen Befunde. Darueberhinaus weisen sie den Weg zur Untersuchung des wechselseitig konkurrierenden Einflusses von Dekohaerenz und Chaos auf die Stabilitaet der quantenmechanischen Zeitentwicklung gekickter Atome. Die Stabilitaet laesst sich mittels des Ueberlapps zweier anfaenglich gleicher, aber unterschiedlich propagierter Zustaende charakterisieren. Dieser Ueberlapp, bekannt als ,,Fidelity'', wird hier fuer eine experimentell realisierbare Situation untersucht