Tesis por compendio / [ES] Las averías inesperadas en las máquinas eléctricas rotativas pueden provocar tiempos de inactividad no programados, que pueden tener como consecuencia grandes pérdidas económicas. Esto aplica a las máquinas eléctricas rotativas en general y en particular a los aerogeneradores eólicos, cuya instalación y necesidades de mantenimiento han experimentado un gran crecimiento en los últimos años. Su mantenimiento predictivo evitaría las averías inesperadas y el aumento de costes que se pueden derivar de un mantenimiento preventivo, mediante la medida continua, on-line, de las variables que indican el estado del aerogenerador. Junto con ello, es necesario el desarrollo de técnicas de diagnóstico fiables que necesitan del testeo en aerogeneradores trabajando bajo diferentes condiciones de fallo. Como los ensayos destructivos para el estudio de defectos o fallos no es económicamente factible, se han desarrollado modelos de máquinas eléctricas rotativas trabajando en condiciones de fallo. Así, estos modelos permiten extraer las características de las corrientes de las máquinas bajo condiciones de fallo reduciendo el número de pruebas destructivas. A partir del compendio de artículos científicos y presentaciones en congreso de esta tesis doctoral se han desarrollado las diferentes etapas de implementación del banco de ensayos. En primer lugar, se exploran las diferentes técnicas de modelado de las máquinas eléctricas rotativas mediante una revisión exhaustiva de la literatura técnica disponible hasta el momento. Por otro lado, se pretenden desarrollar técnicas de modelado para ser implementadas en sistemas de prototipado rápido para realizar el diagnóstico online de la máquina. Sin embargo, los modelos analíticos como el presentado, debido a las simplificaciones que se asumen, no pueden modelar con precisión las no idealidades y las no idealidades propias de las máquinas de inducción defectuosas. Los modelos basados en métodos numéricos son más completos, pero requieren una gran capacidad computacional y largos tiempos de simulación. Además, su implementación en sistemas de prototipado rápido, resulta en una tarea muy compleja hasta el momento. Por esta razón, se opta por desarrollar un método de modelado basado en un enfoque híbrido analítico-numérico, que puede ayudar en el desarrollo de pruebas de técnicas de diagnóstico de fallos para ser implementados en dispositivos embebidos, así como para entrenar a sistemas expertos para evaluar la condición de la máquina. No obstante, este modelo desarrollado todavía requiere de importantes capacidades de memoria y tiempos de cómputo, por lo que se aporta un método para optimizar el cálculo de las inductancias de la máquina eléctrica. El estudio de esta técnica, además de una publicación en revista, ha resultado en dos presentaciones en congreso. Por otro lado, se ha optimizado el modelo analítico, aportando un nuevo método basado en el álgebra tensorial que además permite adaptar el modelo híbrido a diferentes tipos de defectos. Finalmente, como alternativa a las técnicas de diagnóstico tradicionales que se basan en el análisis de corriente a través de la transformada de Fourier (FFT), cuyo uso está limitado al diagnóstico en régimen estacionario, así como a las técnicas tiempo-frecuencia que permiten el diagnóstico de fallos bajo condiciones de régimen transitorio pero a un elevado coste computacional, este trabajo presenta una combinación de técnicas de diagnóstico que proporciona alta resolución espectral en todo el rango de carga de la máquina a bajo coste computacional y requisitos de memoria. Para la validación de las técnicas de modelado y la técnica de diagnóstico presentadas se ha implementado un banco de ensayos versátil con la capacidad de reproducir cualquier condición de funcionamiento. / [CA] Les averies inesperades de les màquines elèctriques rotatives poden provocar temps no programats de inactivitat, que esdevindrien en greus pèrdues econòmiques. Aquest raonament és cert per a màquines elèctriques rotatives en general i per als aerogeneradors en particular, la instal·lació dels quals i necessitats de manteniment han experimentat un gran creixement al llarg dels últims anys. El manteniment predictiu evitaria les averies inesperades i el augment de costos derivats de un manteniment preventiu, mitjançant el anàlisi continu i en línia de les magnituds que indiquen l'estat de l'aerogenerador. A més, seria necessari desenvolupar tècniques de diagnosi fiables a través de proves realitzades en aerogeneradors treballant amb distintes condicions de fallada. La realització d'assajos destructius per a l'estudi de defectes o averies no és econòmicament factible. Per tant, s'han desenvolupat models de màquines elèctriques rotatives treballant en condicions de fallada. D'aquesta manera, aquest model permeten extraure les característiques de les corrents absorbides/generades per les màquines en condicions de fallada i permeten reduir el nombre de proves destructives. A partir d'un compendi d'articles científics i presentacions en congrés d'aquesta tesi doctoral s'han desenvolupat les diferents etapes de implementació del banc de proves. En primer lloc s'exploren les diferents tècniques de modelat de màquines elèctriques rotatives mitjançant una revisió exhaustiva de la literatura tècnica disponible fins a la data. D'altra banda, es pretén desenvolupar tècniques de modelat per se implementades en sistemes de prototipat ràpid per realitzar la diagnosi en línia de la màquina. En aquest cas, la primera tècnica de modelat desenvolupada en la tesis es basa en mètodes analítics, obtinguin-se un model molt ràpid que simplifica el procés de càlcul dels paràmetres del model de la màquina elèctrica i és capaç de representar qualsevols tipus i nombre de fallades de simetria del rotor. No obstant això, els models analítics, como el presentat, no poden modelar amb precisió les no idealitats i no linealitats pròpies de les màquines de inducció amb fallada. El models basats en mètodes numèrics son més complets però requereixen una gran capacitat computacional i llargs temps de simulació. A més, la seua implantació en sistemes de prototipat ràpid resulta una tasca complexa i en molts casos inassolible. Per aquesta raó s'ha optat per desenvolupar un mètode de modelat basat en un plantejament híbrid analític-numèric, que pot ajudar al desenvolupament de sistemes per provar tècniques de diagnosi de fallades per se implantats en dispositius embeguts així com per entrenar a sistemes experts per avaluar la condició de la màquina. No obstant això, aquest model encara requereix de importants capacitats de memòria i temps de computació, per tant, en aquesta tesis, s'aporta un mètode per optimitzar el càlcul de les inductàncies de la màquina elèctrica. L'estudi d'aquesta tècnica, a més d'una publicació en revista, ha resultat en dos presentacions en congres. En altra publicació s'ha optimitzat el model analític mitjançant un mètode basat en l'àlgebra tensorial que permet adaptar el model híbrid a altres tipus de fallada. Finalment, com alternativa a les tècniques de diagnòstic tradicionals basades en el anàlisi de la corrent amb la transformada ràpida de Fourier (FFT) limitades al diagnòstic en règim estacionari, així como a les tècniques basades en l'anàlisi temps-freqüència que permeten la diagnosis en règim transitori per a un elevat cost computacional, aquest treball presenta una combinació de tècniques de diagnosi que proporciona alta resolució espectral en totes les zones de treball de la màquina amb un baix cost computacional i de memòria. Per a la validació d'aquestes tècniques de modelat i la tècnica de diagnosi presentada s'ha implementat un banc d'assajos versàtil i amb la capacitat de reproduir qualsevols condició de funcionament. / [EN] Unexpected breakdowns in rotating electrical machines components can lead to unscheduled downtimes and consequently to large economic losses. This relates to electrical machines in general and particularly to wind turbines, whose installation and maintenance needs have raised significantly in recent years. Predictive maintenance would avoid unexpected breakdowns and the associated increased costs from preventive maintenance, by online condition monitoring of the machine. In addition, the development of reliable diagnostic techniques is needed, which requires the testing of wind turbines working under various fault conditions. As destructive testing for fault research purposes is not economically feasible, several rotating electrical machines models running under fault conditions have been developed to investigate the characteristics of faulty machines and have allowed reducing the number of destructive tests. The different stages of the implementation of the test bench have been developed from the compendium of scientific papers and conference presentations of this doctoral thesis. Firstly, the different modeling techniques for rotating electrical machines are explored through an exhaustive review of the technical literature available so far. Moreover, it is aimed at developing modeling techniques valid to be implemented in rapid prototyping systems to perform online diagnosis of the machine. The first modeling technique developed in this thesis is based on analytical methods, obtaining a very fast model which greatly simplifies the process of calculating the parameters of the electrical induction machine model. This model can reproduce any kind and number of rotor asymmetry faults. However, analytical models as the presented one, due to the simplifications assumed, cannot accurately model the inherent non-idealities and non-idealities within faulty induction machines. Models based on numerical methods are more complete but require high computational effort and long simulation times. Moreover, their implementation in rapid prototyping systems is challenging so far. For this reason, a modeling method based on a hybrid analytical- numerical approach is developed, which can contribute to the development of testing fault diagnosis techniques to be implemented in embedded devices, as well as to train expert systems to evaluate the state of the machine. However, this model still requires a significant memory capacity and computation time, so it is provided a method to optimize the computation of the coupling para- meters of the machine. The study of this technique has resulted in a journal publication and two conference presentations. Particularly, it is studied the correct implementation of the modelling technique to obtain a reliable fault diagnosis and it is compared with another method of parameter reduction pro- posed in the technical literature. Besides that, the analytical model has been optimized, providing a new method based on tensor algebra, which also allows the hybrid model to be adapted to different types of defects. Finally, this work shows a combination of diagnostic techniques providing high spectral resolution over the entire machine load range at low computational cost and with negligible memory requirements. This provides an alternative to the traditional diagnostic techniques based on current analysis using Fourier transform (FFT), whose implementation is limited to the diagnosis in the steady state, as well as time-frequency techniques, which allow fault diagnosis under transient regime conditions at high computational cost. For the validation of the presented modeling techniques and diagnostic technique, a versatile test bench has been implemented. This test bench allows to reproduce any operating condition of the machine. / Terrón Santiago, C. (2022). Desarrollo e implementación de un banco de ensayos virtual de aerogeneradores para diferentes regímenes de funcionamiento y condiciones de fallo [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/183782 / Compendio

As IC technologies scale to the nanometer regime, efficient and accurate modeling and analysis of VLSI systems with billions of transistors and interconnects becomes increasingly critical and difficult. VLSI systems impacted by the increasingly high dimensional process-voltage-temperature (PVT) variations demand much more modeling and analysis efforts than ever before, while the analysis of large scale on-chip interconnects that requires solving tens of millions of unknowns imposes great challenges in computer aided design areas. This dissertation presents new methodologies for addressing the above two important challenging issues for large scale on-chip interconnect modeling and analysis: In the past, the standard statistical circuit modeling techniques usually employ principal component analysis (PCA) and its variants to reduce the parameter dimensionality. Although widely adopted, these techniques can be very limited since parameter dimension reduction is achieved by merely considering the statistical distributions of the controlling parameters but neglecting the important correspondence between these parameters and the circuit performances (responses) under modeling. This dissertation presents a variety of performance-oriented parameter dimension reduction methods that can lead to more than one order of magnitude parameter reduction for a variety of VLSI circuit modeling and analysis problems. The sheer size of present day power/ground distribution networks makes their analysis and verification tasks extremely runtime and memory inefficient, and at the same time, limits the extent to which these networks can be optimized. Given today?s commodity graphics processing units (GPUs) that can deliver more than 500 GFlops (Flops: floating point operations per second). computing power and 100GB/s memory bandwidth, which are more than 10X greater than offered by modern day general-purpose quad-core microprocessors, it is very desirable to convert the impressive GPU computing power to usable design automation tools for VLSI verification. In this dissertation, for the first time, we show how to exploit recent massively parallel single-instruction multiple-thread (SIMT) based graphics processing unit (GPU) platforms to tackle power grid analysis with very promising performance. Our GPU based network analyzer is capable of solving tens of millions of power grid nodes in just a few seconds. Additionally, with the above GPU based simulation framework, more challenging three-dimensional full-chip thermal analysis can be solved in a much more efficient way than ever before.

A two-step model order reduction method is investigated in order to overcome problems of certain one-step methods. Not only optimal combinations of one-step reductions are considered but also the selection of a suitable intermediate dimension (ID) is described. Several automated selection methods are presented and their application tested on a gear box model. The implementation is realized using a Matlab-based Software MORPACK. Several recommendations are given towards the selection of a suitable ID, and problems in Model Order Reduction (MOR) combinations are pointed out. A pseudo two-step is suggested to reduce the full system without any modal information. A new node selection approach is proposed to enhance the SEREP approximation of the system’s response for small reduced representations. / Mehrschrittverfahren der Modellreduktion werden untersucht, um spezielle Probleme konventioneller Einschrittverfahren zu lösen. Eine optimale Kombination von strukturmechanischen Reduktionsverfahren und die Auswahl einer geeigneten Zwischendimension wird untersucht. Dafür werden automatische Verfahren in Matlab implementiert, in die Software MORPACK integriert und anhand des Finite Elemente Modells eines Getriebegehäuses ausgewertet. Zur Auswahl der Zwischendimension werden Empfehlungen genannt und auf Probleme bei der Kombinationen bestimmter Reduktionsverfahren hingewiesen. Ein Pseudo- Zweischrittverfahren wird vorgestellt, welches eine Reduktion ohne Kenntnis der modalen Größen bei ähnlicher Genauigkeit im Vergleich zu modalen Unterraumverfahren durchführt. Für kleine Reduktionsdimensionen wird ein Knotenauswahlverfahren vorgeschlagen, um die Approximation des Frequenzganges durch die System Equivalent Reduction Expansion Process (SEREP)-Reduktion zu verbessern.

Die FEM-MKS-Kopplung erfordert Modellordnungsreduktions-Verfahren, die mit kleiner reduzierter Systemdimension das Übertragungsverhalten mechanischer Strukturen abbilden. Rationale Krylov-Unterraum-Verfahren, basierend auf dem Arnoldi-Algorithmen, ermöglichen solche Abbildungen in frei wählbaren, breiten Frequenzbereichen. Ziel ist der Entwurf einer fehlerüberwachten Modelreduktion auf Basis von Krylov-Unterraumverfahren und Anwendung auf ein strukturmechanisches Model. Auf Grundlage der Software MORPACK wird eine Arnoldi-Funktion erster Ordnung um interpolativen Startvektor, Eliminierung der Starrkörperbewegung und Reorthogonalisierung erweitert. Diese Operationen beinhaltend, wird ein rationales, interpolatives SOAR-Verfahren entwickelt. Ein rationales Block-SOAR-Verfahren erweist sich im Vergleich als unterlegen. Es wird interpolative Gleichwichtung verwendet. Das Arnoldi-Verfahren zeichnet kleiner Berechnungsaufwand aus. Das rationale, interpolative SOAR liefert kleinere reduzierte Systemdimensionen für gleichen abgebildeten Frequenzbereich. Die Funktionen werden auf Rahmen-, Getriebegehäuse- und Treibsatzwellen-Modelle angewendet. Zur Fehlerbewertung wird eigenfrequenzbasiert ein H2-Integrationsbereich festgelegt und der übertragungsfunktionsbasierte, relative H2-Fehler berechnet. Es werden zur Lösung linearer Gleichungssysteme mit Matlab entsprechende Löser-Funktionen, auf Permutation und Faktorisierung basierend, implementiert. / FEM-MKS-coupling requires model order reduction methods to simulate the frequency response of mechanical structures using a smaller reduced representation of the original system. Most of the rational Krylov-subspace methods are based on Arnoldi-algorithms. They allow to represent the frequency response in freely selectable, wide frequency ranges. Subject of this thesis is the implementation of an error-controlled model order reduction based on Krylov-subspace methods and the application to a mechanical model. Based on the MORPACK software, a first-order-Arnoldi function is extended by an interpolative start vector, the elimination of rigid body motion and a reorthogonalization. Containing these functions, a rational, interpolative Second Order Arnoldi (SOAR) method is designed that works well compared to a rational Block-SOAR-method. Interpolative equal weighting is used. The first-order-Arnoldi method requires less computational effort compared to the rational, interpolative SOAR that is able to compute a smaller reduction size for same frequency range of interest. The methods are applied to the models of a frame, a gear case and a drive shaft. Error-control is realized by eigenfrequency-based H2-integration-limit and relative H2-error based on the frequency response function. For solving linear systems of equations in Matlab, solver functions based on permutation and factorization are implemented.

Les simulations numériques représentent un outil fondamental pour la conception et l'optimisation de procédés industriels de fabrication tels que le soudage. Malgré le développement impressionnant des méthodes numériques et des moyens de calcul utilisables, la complexité des procédés de fabrication et les nouvelles exigences des industries les plus avancées obligent à repenser les méthodes, les stratégies et les algorithmes de simulation disponibles. Dans cette thèse, de nouvelles méthodes numériques avec une approche de Réduction des Modèles sont proposées, une discipline consolidée qui a fourni des solutions étonnantes dans différentes applications, comme les procédés de fabrication avancés. Tout d'abord, différentes stratégies sont proposées pour la simulation efficace des procédés de soudage conventionnel, à cet effet, l'utilisation de Computational Vademecums est introduite. L’introduction de ces abaques numériques améliorent des méthodes telles que : les Éléments Finis Généralisés pour le calcul thermique, l'approche local-global pour le calcul mécanique et enfin, la construction directe des abaques numériques utiles pour la phase de pré-design. En second lieu, un solveur PGD efficace est présenté pour les simulations thermo-mécaniques de soudage par friction-malaxage. Cette thèse montre comment la réduction des modèles,en plus d'être une fin en soi, peut être un excellent ingrédient pour améliorer l'efficacité des méthodes numériques traditionnelles. Cela représente un grand intérêt pour l'industrie. / Numerical simulations represent a fundamental tool for the design and optimization of industrial manufacturing processes such as welding. Despite the impressive development of the numerical methods and the means of calculation, the complexity of these processes and the new demands of the more advanced industries make it necessary to rethink the available methods, strategies and simulation algorithms. In this thesis, we propose new numerical methods with a Model Order Reduction approach, a consolidated discipline that has provided surprising solutions indifferent applications, such as advanced manufacturing processes. First, different strategies for the efficient simulation of conventional welding processes are proposed. To this end, the use of Computational Vademecums is introduced for the improvement of methods such as the Generalized Finite Element for thermal calculation, the local-global approach for the mechanical calculation or the direct construction of vademecums useful for predesign phases. Then, an efficient PGD solver for thermomechanical simulations for friction stir welding is presented. This thesis shows how Model Reduction, besides being an end, it can be an excellent ingredient to improve the efficiency of traditional numerical methods, with great interest for the industry.

Lab-on-a-Chip-Systeme sind mikrofluidische, portable Systeme mit denen bioanalytische Reaktionen und Auswertungen an kleinen Probenvolumina vor Ort durchführbar sind. In der vorliegenden Arbeit wird eine Entwurfsstrategie für das integrierte, resistive Heizen in einem solchen System auf Basis einer polymerbasierten, modularen Technologieplattform entwickelt. Dabei wird eine Modellierung als Feldmodell, die Ableitung eines reduzierten Makromodells sowie die experimentelle Untersuchung und Verifikation beschrieben. Verschiedene Ansätze für die Abbildung temperaturunabhängiger und -abhängiger elektrisch-thermischer Wandler sind berücksichtigt. Der Einflüsse von Aufbau, Widerstandsverhalten, Randbedingungen, sowie der elektrischen Ansteuerung auf die Temperatur der Biosensorfläche, in der die bioanalytische Reaktion erfolgt, werden dargelegt. / Lab on a chip systems are portable microfluidic systems which enable bioanalytical reactions and the appropriate analysis at the point of need using small sample volumes. In this publication a design strategy for integrated resistive heating in such a polymer based system is developed. The modelling comprises a field model, a derived reduced macro model and the experimental characterisation. Approaches to describe temperature dependent as well as independent electric-thermal converters are taken into account. The effects of the assembly, resistive behaviour, boundary conditions as well as the drive electronics on the temperature of the biosensor are presented.

A two-step model order reduction method is investigated in order to overcome problems of certain one-step methods. Not only optimal combinations of one-step reductions are considered but also the selection of a suitable intermediate dimension (ID) is described. Several automated selection methods are presented and their application tested on a gear box model. The implementation is realized using a Matlab-based Software MORPACK. Several recommendations are given towards the selection of a suitable ID, and problems in Model Order Reduction (MOR) combinations are pointed out. A pseudo two-step is suggested to reduce the full system without any modal information. A new node selection approach is proposed to enhance the SEREP approximation of the system’s response for small reduced representations.:Contents Kurzfassung..........................................................................................iv Abstract.................................................................................................iv Nomenclature........................................................................................ix 1 Introduction........................................................................................1 1.1 Motivation........................................................................................1 1.2 Objectives........................................................................................1 1.3 Outline of the Thesis........................................................................2 2 Theoretical Background.......................................................................3 2.1 Finite Element Method......................................................................3 2.1.1 Modal Analysis...............................................................................4 2.1.2 Frequency Response Function.......................................................4 2.2 Model Order Reduction.....................................................................5 2.3 Physical Subspace Reduction Methods.............................................7 2.3.1 Guyan Reduction...........................................................................7 2.3.2 Improved Reduced System Method...............................................8 2.4 Modal Subspace Reduction Methods...............................................10 2.4.1 Modal Reduction...........................................................................11 2.4.2 Exact Modal Reduction..................................................................11 2.4.3 System Equivalent Reduction Expansion Process.........................13 2.5 Krylov Subspace Reduction Methods...............................................14 2.6 Hybrid Subspace Reduction Methods..............................................15 2.6.1 Component Mode Synthesis........................................................16 2.6.2 Hybrid Exact Modal Reduction......................................................19 2.7 Model Correlation Methods.............................................................21 2.7.1 Normalized Relative Frequency Difference...................................21 2.7.2 Modified Modal Assurance Criterion.............................................22 2.7.3 Pseudo-Orthogonality Check.......................................................22 2.7.4 Comparison of Frequency Response Function.............................23 3 Selection of Active Degrees of Freedom............................................25 3.1 Non-Iterative Methods...................................................................26 3.1.1 Modal Kinetic Energy and Variants..............................................26 3.1.2 Driving Point Residue and Variants..............................................27 3.1.3 Eigenvector Component Product..................................................28 3.2 Iterative Reduction Methods...........................................................29 3.2.1 Effective Independence Distribution.............................................29 3.2.2 Mass-Weighted Effective Independence.......................................32 3.2.3 Variance Based Selection Method.................................................33 3.2.4 Singular Value Decomposition Based Selection Method................34 3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34 3.3 Iterative Expansion Methods...........................................................35 3.3.1 Modal-Geometrical Selection Criterion...........................................36 3.3.2 Triaxial Effective Independence Expansion...................................36 3.4 Measure of Goodness for Selected Active Set..................................39 3.4.1 Determinant and Rank of the Fisher Information Matrix................39 3.4.2 Condition Number of the Partitioned Modal Matrix........................40 3.4.3 Measured Energy per Mode..........................................................40 3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41 3.4.5 Eigenvalue Comparison................................................................41 4 Two-Step Reduction in MORPACK.......................................................42 4.1 Structure of MORPACK.....................................................................42 4.2 Selection of an Intermediate Dimension.........................................43 4.2.1 Intermediate Dimension Requirements........................................44 4.2.2 Implemented Selection Methods..................................................45 4.2.3 Recommended Selection of an Intermediate Dimension...............48 4.3 Combination of Reduction Methods.................................................49 4.3.1 Overview of All Candidates..........................................................50 4.3.2 Combinations with Modal Information.........................................54 4.3.3 Combinations without Modal Information....................................54 5 Applications........................................................................................57 5.1 Gear Box Model...............................................................................57 5.2 Selection of Additional Active Nodes................................................58 5.3 Optimal Intermediate Dimension......................................................64 5.4 Two-Step Model Order Reduction Results........................................66 5.5 Comparison to One-Step Model Order Reduction Methods..............70 5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72 5.7 Proposal of a New Approach for Additional Node Selection..............73 6 Summary and Conclusions...................................................................77 7 Zusammenfassung und Ausblick..........................................................79 Bibliography............................................................................................81 List of Tables..........................................................................................86 List of Figures.........................................................................................88 A Appendix.............................................................................................89 A.1 Results of Two-Step Model Order Reduction.....................................89 A.2 Data CD............................................................................................96 / Mehrschrittverfahren der Modellreduktion werden untersucht, um spezielle Probleme konventioneller Einschrittverfahren zu lösen. Eine optimale Kombination von strukturmechanischen Reduktionsverfahren und die Auswahl einer geeigneten Zwischendimension wird untersucht. Dafür werden automatische Verfahren in Matlab implementiert, in die Software MORPACK integriert und anhand des Finite Elemente Modells eines Getriebegehäuses ausgewertet. Zur Auswahl der Zwischendimension werden Empfehlungen genannt und auf Probleme bei der Kombinationen bestimmter Reduktionsverfahren hingewiesen. Ein Pseudo- Zweischrittverfahren wird vorgestellt, welches eine Reduktion ohne Kenntnis der modalen Größen bei ähnlicher Genauigkeit im Vergleich zu modalen Unterraumverfahren durchführt. Für kleine Reduktionsdimensionen wird ein Knotenauswahlverfahren vorgeschlagen, um die Approximation des Frequenzganges durch die System Equivalent Reduction Expansion Process (SEREP)-Reduktion zu verbessern.:Contents Kurzfassung..........................................................................................iv Abstract.................................................................................................iv Nomenclature........................................................................................ix 1 Introduction........................................................................................1 1.1 Motivation........................................................................................1 1.2 Objectives........................................................................................1 1.3 Outline of the Thesis........................................................................2 2 Theoretical Background.......................................................................3 2.1 Finite Element Method......................................................................3 2.1.1 Modal Analysis...............................................................................4 2.1.2 Frequency Response Function.......................................................4 2.2 Model Order Reduction.....................................................................5 2.3 Physical Subspace Reduction Methods.............................................7 2.3.1 Guyan Reduction...........................................................................7 2.3.2 Improved Reduced System Method...............................................8 2.4 Modal Subspace Reduction Methods...............................................10 2.4.1 Modal Reduction...........................................................................11 2.4.2 Exact Modal Reduction..................................................................11 2.4.3 System Equivalent Reduction Expansion Process.........................13 2.5 Krylov Subspace Reduction Methods...............................................14 2.6 Hybrid Subspace Reduction Methods..............................................15 2.6.1 Component Mode Synthesis........................................................16 2.6.2 Hybrid Exact Modal Reduction......................................................19 2.7 Model Correlation Methods.............................................................21 2.7.1 Normalized Relative Frequency Difference...................................21 2.7.2 Modified Modal Assurance Criterion.............................................22 2.7.3 Pseudo-Orthogonality Check.......................................................22 2.7.4 Comparison of Frequency Response Function.............................23 3 Selection of Active Degrees of Freedom............................................25 3.1 Non-Iterative Methods...................................................................26 3.1.1 Modal Kinetic Energy and Variants..............................................26 3.1.2 Driving Point Residue and Variants..............................................27 3.1.3 Eigenvector Component Product..................................................28 3.2 Iterative Reduction Methods...........................................................29 3.2.1 Effective Independence Distribution.............................................29 3.2.2 Mass-Weighted Effective Independence.......................................32 3.2.3 Variance Based Selection Method.................................................33 3.2.4 Singular Value Decomposition Based Selection Method................34 3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34 3.3 Iterative Expansion Methods...........................................................35 3.3.1 Modal-Geometrical Selection Criterion...........................................36 3.3.2 Triaxial Effective Independence Expansion...................................36 3.4 Measure of Goodness for Selected Active Set..................................39 3.4.1 Determinant and Rank of the Fisher Information Matrix................39 3.4.2 Condition Number of the Partitioned Modal Matrix........................40 3.4.3 Measured Energy per Mode..........................................................40 3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41 3.4.5 Eigenvalue Comparison................................................................41 4 Two-Step Reduction in MORPACK.......................................................42 4.1 Structure of MORPACK.....................................................................42 4.2 Selection of an Intermediate Dimension.........................................43 4.2.1 Intermediate Dimension Requirements........................................44 4.2.2 Implemented Selection Methods..................................................45 4.2.3 Recommended Selection of an Intermediate Dimension...............48 4.3 Combination of Reduction Methods.................................................49 4.3.1 Overview of All Candidates..........................................................50 4.3.2 Combinations with Modal Information.........................................54 4.3.3 Combinations without Modal Information....................................54 5 Applications........................................................................................57 5.1 Gear Box Model...............................................................................57 5.2 Selection of Additional Active Nodes................................................58 5.3 Optimal Intermediate Dimension......................................................64 5.4 Two-Step Model Order Reduction Results........................................66 5.5 Comparison to One-Step Model Order Reduction Methods..............70 5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72 5.7 Proposal of a New Approach for Additional Node Selection..............73 6 Summary and Conclusions...................................................................77 7 Zusammenfassung und Ausblick..........................................................79 Bibliography............................................................................................81 List of Tables..........................................................................................86 List of Figures.........................................................................................88 A Appendix.............................................................................................89 A.1 Results of Two-Step Model Order Reduction.....................................89 A.2 Data CD............................................................................................96

Die FEM-MKS-Kopplung erfordert Modellordnungsreduktions-Verfahren, die mit kleiner reduzierter Systemdimension das Übertragungsverhalten mechanischer Strukturen abbilden. Rationale Krylov-Unterraum-Verfahren, basierend auf dem Arnoldi-Algorithmen, ermöglichen solche Abbildungen in frei wählbaren, breiten Frequenzbereichen. Ziel ist der Entwurf einer fehlerüberwachten Modelreduktion auf Basis von Krylov-Unterraumverfahren und Anwendung auf ein strukturmechanisches Model. Auf Grundlage der Software MORPACK wird eine Arnoldi-Funktion erster Ordnung um interpolativen Startvektor, Eliminierung der Starrkörperbewegung und Reorthogonalisierung erweitert. Diese Operationen beinhaltend, wird ein rationales, interpolatives SOAR-Verfahren entwickelt. Ein rationales Block-SOAR-Verfahren erweist sich im Vergleich als unterlegen. Es wird interpolative Gleichwichtung verwendet. Das Arnoldi-Verfahren zeichnet kleiner Berechnungsaufwand aus. Das rationale, interpolative SOAR liefert kleinere reduzierte Systemdimensionen für gleichen abgebildeten Frequenzbereich. Die Funktionen werden auf Rahmen-, Getriebegehäuse- und Treibsatzwellen-Modelle angewendet. Zur Fehlerbewertung wird eigenfrequenzbasiert ein H2-Integrationsbereich festgelegt und der übertragungsfunktionsbasierte, relative H2-Fehler berechnet. Es werden zur Lösung linearer Gleichungssysteme mit Matlab entsprechende Löser-Funktionen, auf Permutation und Faktorisierung basierend, implementiert.:1. Einleitung 1.1. Motivation 1.2. Einordnung 1.3. Aufbau der Arbeit 2. Theorie 2.1. Simulationsmethoden 2.1.1. Finite Elemente Methode 2.1.2. Mehrkörpersimulation 2.1.3. Kopplung der Simulationsmethoden 2.2. Zustandsraumdarstellung und Reduktion 2.3. Krylov Unterraum Methoden 2.4. Arnoldi-Algorithmen erster Ordnung 2.5. Arnoldi-Algorithmen zweiter Ordnung 2.6. Korrelationskriterien 2.6.1. Eigenfrequenzbezogene Kriterien 2.6.2. Eigenvektorbezogene Kriterien 2.6.3. Übertragungsfunktionsbezogene Kriterien 2.6.4. Fehlerbewertung 2.6.5. Anwendung auf Systeme sehr großer Dimension 3. Numerik linearer Gleichungssysteme 3.1. Grundlagen 3.2. Singularität der Koeffizientenmatrix 3.2.1. Randbedingungen des Systems 3.2.2. Verwendung einer generellen Diagonalperturbation 3.3. Iterative Lösungsverfahren 3.4. Faktorisierungsverfahren 3.4.1. Cholesky-Faktorisierung 3.4.2. LU-Faktorisierung 3.4.3. Fillin-Reduktion durch Permutation 3.4.4. Fazit 3.5. Direkte Lösungsverfahren 3.6. Verwendung externer Gleichungssystem-Löser 3.7. Zusammenfassung 4. Implementierung 4.1. Aufbau von MORPACK 4.2. Anforderungen an Reduktions-Funktionen 4.3. Eigenschaften und Optionen der KSM-Funktionen 4.3.1. Arnoldi-Funktion erster Ordnung 4.3.2. Rationale SOAR-Funktionen 4.4. Korrelationskriterien 4.4.1. Eigenfrequenzbezogen 4.4.2. Eigenvektorbezogen 4.4.3. Übertragungsfunktionsbezogen 4.5. Lösungsfunktionen linearer Gleichungssysteme 4.5.1. Anforderungen und Aufbau 4.5.2. Verwendung der Gleichungssystem-Löser 4.5.3. Hinweise zur Implementierung von Gleichungssystem-Lösern 5. Anwendung 5.1. Versuchsmodelle 5.1.1. Testmodelle kleiner Dimension 5.1.2. Getriebegehäuse 5.1.3. Treibsatzwelle 5.2. Validierung der Reduktionsmethoden an kleinem Modell 5.2.1. Modifizierte Arnoldi-Funktion erster Ordnung 5.2.2. Rationale SOAR-Funktionen 5.2.3. Zusammenfassung 5.3. Anwendung der KSM auf große Modelle 5.3.1. Getriebegehäuse 5.3.2. Treibsatzwelle 5.4. Auswertung 6. Zusammenfassung und Ausblick 6.1. Zusammenfassung 6.2. Ausblick / FEM-MKS-coupling requires model order reduction methods to simulate the frequency response of mechanical structures using a smaller reduced representation of the original system. Most of the rational Krylov-subspace methods are based on Arnoldi-algorithms. They allow to represent the frequency response in freely selectable, wide frequency ranges. Subject of this thesis is the implementation of an error-controlled model order reduction based on Krylov-subspace methods and the application to a mechanical model. Based on the MORPACK software, a first-order-Arnoldi function is extended by an interpolative start vector, the elimination of rigid body motion and a reorthogonalization. Containing these functions, a rational, interpolative Second Order Arnoldi (SOAR) method is designed that works well compared to a rational Block-SOAR-method. Interpolative equal weighting is used. The first-order-Arnoldi method requires less computational effort compared to the rational, interpolative SOAR that is able to compute a smaller reduction size for same frequency range of interest. The methods are applied to the models of a frame, a gear case and a drive shaft. Error-control is realized by eigenfrequency-based H2-integration-limit and relative H2-error based on the frequency response function. For solving linear systems of equations in Matlab, solver functions based on permutation and factorization are implemented.:1. Einleitung 1.1. Motivation 1.2. Einordnung 1.3. Aufbau der Arbeit 2. Theorie 2.1. Simulationsmethoden 2.1.1. Finite Elemente Methode 2.1.2. Mehrkörpersimulation 2.1.3. Kopplung der Simulationsmethoden 2.2. Zustandsraumdarstellung und Reduktion 2.3. Krylov Unterraum Methoden 2.4. Arnoldi-Algorithmen erster Ordnung 2.5. Arnoldi-Algorithmen zweiter Ordnung 2.6. Korrelationskriterien 2.6.1. Eigenfrequenzbezogene Kriterien 2.6.2. Eigenvektorbezogene Kriterien 2.6.3. Übertragungsfunktionsbezogene Kriterien 2.6.4. Fehlerbewertung 2.6.5. Anwendung auf Systeme sehr großer Dimension 3. Numerik linearer Gleichungssysteme 3.1. Grundlagen 3.2. Singularität der Koeffizientenmatrix 3.2.1. Randbedingungen des Systems 3.2.2. Verwendung einer generellen Diagonalperturbation 3.3. Iterative Lösungsverfahren 3.4. Faktorisierungsverfahren 3.4.1. Cholesky-Faktorisierung 3.4.2. LU-Faktorisierung 3.4.3. Fillin-Reduktion durch Permutation 3.4.4. Fazit 3.5. Direkte Lösungsverfahren 3.6. Verwendung externer Gleichungssystem-Löser 3.7. Zusammenfassung 4. Implementierung 4.1. Aufbau von MORPACK 4.2. Anforderungen an Reduktions-Funktionen 4.3. Eigenschaften und Optionen der KSM-Funktionen 4.3.1. Arnoldi-Funktion erster Ordnung 4.3.2. Rationale SOAR-Funktionen 4.4. Korrelationskriterien 4.4.1. Eigenfrequenzbezogen 4.4.2. Eigenvektorbezogen 4.4.3. Übertragungsfunktionsbezogen 4.5. Lösungsfunktionen linearer Gleichungssysteme 4.5.1. Anforderungen und Aufbau 4.5.2. Verwendung der Gleichungssystem-Löser 4.5.3. Hinweise zur Implementierung von Gleichungssystem-Lösern 5. Anwendung 5.1. Versuchsmodelle 5.1.1. Testmodelle kleiner Dimension 5.1.2. Getriebegehäuse 5.1.3. Treibsatzwelle 5.2. Validierung der Reduktionsmethoden an kleinem Modell 5.2.1. Modifizierte Arnoldi-Funktion erster Ordnung 5.2.2. Rationale SOAR-Funktionen 5.2.3. Zusammenfassung 5.3. Anwendung der KSM auf große Modelle 5.3.1. Getriebegehäuse 5.3.2. Treibsatzwelle 5.4. Auswertung 6. Zusammenfassung und Ausblick 6.1. Zusammenfassung 6.2. Ausblick

With the advances in the telecommunications industry, there is a need to solve the in-band full-duplex (IBFD) problem for antenna systems. One premise for solving the IBFD problem is to have strong isolation between transmitter and receiver antennas in an antenna system. To increase isolation, antenna engineers are dependent on simulation software to calculate the isolation between the antennas, i.e., the mutual coupling. Full-wave simulations that accurately calculate the mutual coupling between antennas are timeconsuming, and there is a need to reduce the required time. In this thesis, we investigate how implemented data-driven interpolation methods can be used to reduce the simulation times when applied to frequency domain solvers. Here, we benchmark the four different interpolation methods vector fitting, the Loewner framework, Cauchy interpolation, and a modified version of Nevanlinna-Pick interpolation. These four interpolation methods are benchmarked on seven different antenna frequency responses, to investigate their performance in terms of how many interpolation points they require to reach a certain root mean squared error (RMSE) tolerance. We also benchmark different frequency sampling algorithms together with the interpolation methods. Here, we have predetermined frequency sampling algorithms such as linear frequency sampling distribution, and Chebyshevbased frequency sampling distributions. We also benchmark two kinds of adaptive frequency sampling algorithms. The first type is compatible with all of the four interpolation methods, and it selects the next frequency sample by analyzing the dynamics of the previously generated interpolant. The second adaptive frequency sampling algorithm is solely for the modified NevanlinnaPick interpolation method, and it is based on the free parameter in NevanlinnaPick interpolation. From the benchmark results, two interpolation methods successfully decrease the RMSE as a function of the number of interpolation points used, namely, vector fitting and the Loewner framework. Here, the Loewner framework performs slightly better than vector fitting. The benchmark results also show that vector fitting is less dependent on which frequency sampling algorithm is used, while the Loewner framework is more dependent on the frequency sampling algorithm. For the Loewner framework, Chebyshev-based frequency sampling distributions proved to yield the best performance. / Med de snabba utvecklingarna i telekomindustrin så har det uppstått ett behov av att lösa det så kallad i-band full-duplex (IBFD) problemet. En premiss för att lösa IBFD-problemet är att framgångsrikt isolera transmissionsantennen från mottagarantennen inom ett antennsystem. För att öka isolationen mellan antennerna måste antenningenjörer använda sig av simulationsmjukvara för att beräkna isoleringen (den ömsesidiga kopplingen mellan antennerna). Full-wave-simuleringar som noggrant beräknar den ömsesidga kopplingen är tidskrävande. Det finns därför ett behov av att minska simulationstiderna. I denna avhandling kommer vi att undersöka hur våra implementerade och datadrivna interpoleringsmetoder kan vara till hjälp för att minska de tidskrävande simuleringstiderna, när de används på frekvensdomänslösare. Här prestandajämför vi de fyra interpoleringsmetoderna vector fitting, Loewner ramverket, Cauchy interpolering, och modifierad Nevanlinna-Pick interpolering. Dessa fyra interpoleringsmetoder är prestandajämförda på sju olika antennsystemsvar, med avseende på hur många interpoleringspunkter de behöver för att nå en viss root mean squared error (RMSE)-tolerans. Vi prestandajämför också olika frekvenssamplingsalgoritmer tillsammas med interpoleringsmetoderna. Här använder vi oss av förbestämda frekvenssamplingsdistributioner så som linjär samplingsdistribution och Chebyshevbaserade samplingsdistributioner. Vi använder oss också av två olika sorters adaptiv frekvenssamplingsalgoritmer. Den första sortens adaptiv frekvenssamplingsalgoritm är kompatibel med alla de fyra interpoleringsmetoderna, och den väljer nästa frekvenspunkt genom att analysera den föregående interpolantens dynamik. Den andra adaptiva frekvenssamplingsalgoritmen är enbart till den modifierade Nevanlinna-Pick interpoleringsalgoritmen, och den baserar sitt val av nästa frekvenspunkt genom att använda sig av den fria parametern i Nevanlinna-Pick interpolering. Från resultaten av prestandajämförelsen ser vi att två interpoleringsmetoder framgångsrikt lyckas minska medelvärdetsfelet som en funktion av antalet interpoleringspunkter som används. Dessa två metoder är vector fitting och Loewner ramverket. Här så presterar Loewner ramverket aningen bättre än vad vector fitting gör. Prestandajämförelsen visar också att vector fitting inte är lika beroende av vilken frekvenssamplingsalgoritm som används, medan Loewner ramverket är mer beroende på vilken frekvenssamplingsalgoritm som används. För Loewner ramverket så visade det sig att Chebyshev-baserade frekvenssamplingsalgoritmer presterade bättre.

A common architecture for electric vehicles is to have two electric machines one each on the front and rear axle. Despite the redundancy, this configuration ensures performance. Being energy efficient is equally important for electric vehicles to deliver a sufficiently high range. Hence, operating a single machine at low to medium torque requirement is desirable. A clutch can be implemented on the front axle and its engagement dynamically controlled to reduce the magnetic drag losses in the front machine. With clutch disengaged, the entire torque will be delivered by the rear machine causing it to heat up quickly. As electric machine and inverter losses are also temperature dependent, this work attempts to derive an optimal torque split strategy between the two machines considering thermal effects. An upper-temperature limit for both electric machine and inverter is imposed for component protection. Thermal models for the electric machine, inverter and coolant circuit are simplified using system identification and model order reduction approach. Dynamic optimal torque split is realized by minimizing the energy loss over the entire drive cycle. Dynamic programming is used to investigate the benefits of including thermal losses and to generate a benchmark solution for optimal torque split strategy. Further, two online controllers are developed, one based on non-linear model predictive control and the other being a static controller with added heuristic rules to prevent temperatures of critical components to exceed the limits. A high-fidelity plant model was developed using VSIM as master and GT-Suite thermal model as slave to compare the performance of these controllers. The results show that it is possible to obtain decent thermal performance of electric motor and inverter with one node lumped parameter thermal model and a five-node lumped parameter model for the coolant circuit. Including thermal dynamics in the controller can constraint the temperature within the limits and give an optimal torque split. The benefit of adding temperature-dependent thermal maps is found to be limited to certain operating regions. The static controller with torque split based on instantaneous power loss also performed well for the given configuration. The major contribution to energy saving was obtained by dynamic disengagement of clutch in the form of reduced magnetic drag losses. / En vanlig arkitektur för elfordon är att ha två elmaskiner en vardera på fram- och bakaxeln. Trots redundansen säkerställer denna konfiguration prestanda. Att vara energieffektiv är lika viktigt för att elfordon ska leverera en tillräckligt hög räckvidd. Det är därför önskvärt att driva en enda maskin med lågt till medelhögt vridmoment. En koppling kan implementeras på framaxeln och dess ingrepp kan styras dynamiskt för att minska de magnetiska motståndsförlusterna i den främre maskinen. Med kopplingen urkopplad kommer hela vridmomentet att levereras av den bakre maskinen vilket gör att den snabbt värms upp. Eftersom förluster av elektriska maskiner och växelriktare också är temperaturberoende, försöker detta arbete härleda en optimal vridmomentsdelningsstrategi mellan de två maskinerna med tanke på termiska effekter. En övre temperaturgräns för både elektrisk maskin och växelriktare är införd för komponentskydd. Termiska modeller för den elektriska maskinen, växelriktaren och kylvätskekretsen förenklas med hjälp av systemidentifiering och modellbeställningsreduktion. Dynamisk optimal vridmomentdelning realiseras genom att minimera energiförlusten under hela körcykeln. Dynamisk programmering används för att undersöka fördelarna med att inkludera termiska förluster och för att generera en benchmarklösning för optimal vridmomentsdelningsstrategi. Vidare utvecklas två online-styrenheter, en baserad på icke-linjär modell för prediktiv styrning och den andra är en statisk styrenhet med tillagda heuristiska regler för att förhindra att temperaturer på kritiska komponenter överskrider gränserna. En högfientlig anläggningsmodell utvecklades med VSIM som master och GT-Suite termisk modell som slav för att jämföra prestandan hos dessa styrenheter. Resultaten visar att det är möjligt att erhålla hyfsad termisk prestanda för elmotor och växelriktare med en termisk modell med en nodklumpad parameter och en femnodsmodell med klumpparametrar för kylvätskekretsen. Att inkludera termisk dynamik i regulatorn kan begränsa temperaturen inom gränserna och ge en optimal vridmomentfördelning. Fördelen med att lägga till temperaturberoende termiska kartor har visat sig vara begränsad till vissa driftsområden. Den statiska styrenheten med vridmomentdelning baserad på momentan effektförlust fungerade också bra för den givna konfigurationen. Det största bidraget till energibesparingen erhölls genom dynamisk urkoppling av kopplingen i form av minskade magnetiska motståndsförluster.