A global Arnoldi method for the model reduction of second-order structural dynamical systems

Abstract In this paper we consider the reduction of second-order dynamical systems with multiple inputs and multiple outputs (MIMO) arising in the numerical simulation of mechanical structures. In commercial software for the kind of application considered here, modal reduction is commonly used to obtain a reduced system with good approximation abilities of the original transfer function in the lower frequency range. In recent years new methods to reduce dynamical systems based on (block) versions of Krylov subspace methods emerged. This work concentrates on the reduction of second-order MIMO systems by the global Arnoldi method, an efficient extension of the standard Arnoldi algorithm for MIMO systems. In particular, a new model reduction algorithm for second order MIMO systems is proposed which automatically generates a reduced system of given order approximating the transfer function in the lower range of frequencies. It is based on the global Arnoldi method, determines the expansion points iteratively and the number of moments matched per expansion point adaptively. Numerical examples comparing our results to modal reduction and reduction via the block version of the rational Arnoldi method are presented

Modelling and analysis of vibration on power electronic module structure and application of model order reduction

Modelling and analysis of vibration of an IGBT power electronic module (PEM) structure were undertaken. PEM structure considered in this study was without molding compound and wirebonds. The most critical resonant frequency was identified by modal analysis. At the critical frequency of 1345Hz, for the vertical displacement of the base excitation, subsequent stress distribution on the PEM structure was analysed. Concurrent vibration and thermo-mechanical fatigue loads on the reliability of PEM structure solder interconnects were also estimated by widely used linear damage superposition approach. It was concluded that the at critical resonant frequency the vibration induced damage is more severe than the thermo-mechanical fatigue loading. In addition, a quarter car model (QCM) was used to mimic the dynamic interaction between the rough road surface and an electric vehicle (EV) in order to analyse the road surface roughness induced excitation on the PEM structure in the engine compartment. Stress and strain distribution on the PEM structure due to road surface roughness were analysed. Furthermore, three Krylov subspace based model order reduction (MOR) techniques were applied to the resulting dynamic system in vibration analysis. Due to the limits on computing resources, a submodel was utilized for MOR analysis. Within the three MOR techniques, Passive Reduced order Interconnect Macromodeling Algorithm (PRIMA) MOR technique performs better than the other techniques. Computational time ratio between reduced system iteration and the full system iteration is 1:53

Reduced-order modeling of power electronics components and systems

This dissertation addresses the seemingly inevitable compromise between modeling fidelity and simulation speed in power electronics. Higher-order effects are considered at the component and system levels. Order-reduction techniques are applied to provide insight into accurate, computationally efficient component-level (via reduced-order physics-based model) and system-level simulations (via multiresolution simulation). Proposed high-order models, verified with hardware measurements, are, in turn, used to verify the accuracy of final reduced-order models for both small- and large-signal excitations. At the component level, dynamic high-fidelity magnetic equivalent circuits are introduced for laminated and solid magnetic cores. Automated linear and nonlinear order-reduction techniques are introduced for linear magnetic systems, saturated systems, systems with relative motion, and multiple-winding systems, to extract the desired essential system dynamics. Finite-element models of magnetic components incorporating relative motion are set forth and then reduced. At the system level, a framework for multiresolution simulation of switching converters is developed. Multiresolution simulation provides an alternative method to analyze power converters by providing an appropriate amount of detail based on the time scale and phenomenon being considered. A detailed full-order converter model is built based upon high-order component models and accurate switching transitions. Efficient order-reduction techniques are used to extract several lower-order models for the desired resolution of the simulation. This simulation framework is extended to higher-order converters, converters with nonlinear elements, and closed-loop systems. The resulting rapid-to-integrate component models and flexible simulation frameworks could form the computational core of future virtual prototyping design and analysis environments for energy processing units

MODEL ORDER REDUCTION OF NONLINEAR DYNAMIC SYSTEMS USING MULTIPLE PROJECTION BASES AND OPTIMIZED STATE-SPACE SAMPLING

Model order reduction (MOR) is a very powerful technique that is used to deal with the increasing complexity of dynamic systems. It is a mature and well understood field of study that has been applied to large linear dynamic systems with great success. However, the continued scaling of integrated micro-systems, the use of new technologies, and aggressive mixed-signal design has forced designers to consider nonlinear effects for more accurate model representations. This has created the need for a methodology to generate compact models from nonlinear systems of high dimensionality, since only such a solution will give an accurate description for current and future complex systems.The goal of this research is to develop a methodology for the model order reduction of large multidimensional nonlinear systems. To address a broad range of nonlinear systems, which makes the task of generalizing a reduction technique difficult, we use the concept of transforming the nonlinear representation into a composite structure of well defined basic functions from multiple projection bases.We build upon the concept of a training phase from the trajectory piecewise-linear (TPWL) methodology as a practical strategy to reduce the state exploration required for a large nonlinear system. We improve upon this methodology in two important ways: First, with a new strategy for the use of multiple projection bases in the reduction process and their coalescence into a unified base that better captures the behavior of the overall system; and second, with a novel strategy for the optimization of the state locations chosen during training. This optimization technique is based on using the Hessian of the system as an error bound metric.Finally, in order to treat the overall linear/nonlinear reduction task, we introduce a hierarchical approach using a block projection base. These three strategies together offer us a new perspective to the problem of model order reduction of nonlinear systems and the tracking or preservation of physical parameters in the final compact model

Krylov-subspace based model reduction for simulation of machine tools

Die vorliegende Arbeit hat die Reduktion von Strukturmodellen, wie sie bei der Simulation von Werkzeugmaschinen zum Einsatz kommt, zum Thema. Dabei steht die Entwicklung neuer mathematischer Reduktionsverfahren, die auf Krylov-Unterräumen basieren im Fokus der Arbeit. Daneben wird auch die Bedeutung der Modellreduktion als wichtiges Hilfsmittel zur Gewährleistung einer effizienten Simulation im Gesamtentwicklungsprozess von Werkzeugmaschinen dargestellt. Für die Reduktion von Strukturmodellen werden bislang vorwiegend die sogenannten modalen Verfahren verwendet. Die mit diesen Verfahren reduzierten Modelle können im statischen Bereich einen erheblichen Fehler aufweisen. In der vorliegenden Arbeit werden neue Reduktionsverfahren, die auf der Grundlage mathematischer Methoden zur Modellreduktion aufbauen und eine Automatisierung des Modellreduktionsprozesses ermöglichen, entwickelt und validiert. Die neu entwickelten Verfahren basieren dabei auf einer angepassten iterativen und adaptiven Auswahl der für die mathematischen Reduktionsmethoden benötigten Parameter. Zudem basieren diese Verfahren auf globale Krylov-Unterräume und der Verwendung der globalen Arnoldi-Methode, die sich durch eine hohe Effizienz bei der Berechnung der Reduktion von Systemen mit mehreren Ein- und Ausgängen auszeichnet. Mit Hilfe einer geeigneten Methode zur Abschätzung des Approximationsfehlers des reduzierten Systems erlauben die neu entwickelten Verfahren eine automatische Modellreduktion ohne Benutzerinteraktion während des Reduktionsprozesses.The subject of the present work is the reduction of structural models, as used in the simulation of machine tools. The focus of the work is the development of new mathematical reduction procedures, based on Krylov-subspaces and distinguished by a feasible automation. In addition, however, the model reduction as an important tool for an efficient simulation in the overall development process of machine tools is presented. For the reduction of structure models usually the so-called modal method based on the solution of a eigenvalue problem is used. Reduced models obtained with the modal method can have considerable errors in the static area. In this work new reduction procedures, based on mathematical reduction methods, for automated reduction of finite element models are developed and validated. The new procedures are based on a adapted approach to choose optimal parameter for the mathematical reduction methods. Moreover the new procedures are based on global Krylov-subspaces and the global Arnoldi-method, characterized by high efficiency in the calculation of the reduction of systems with multiple inputs and outputs. By using a suitable method for estimation of the approximation error, a automated reduction process with any user interaction is suggested

Krylov Subspace Model Order Reduction for Nonlinear and Bilinear Control Systems

The use of Krylov subspace model order reduction for nonlinear/bilinear systems, over the past few years, has become an increasingly researched area of study. The need for model order reduction has never been higher, as faster computations for control, diagnosis and prognosis have never been higher to achieve better system performance. Krylov subspace model order reduction techniques enable this to be done more quickly and efficiently than what can be achieved at present. The most recent advances in the use of Krylov subspaces for reducing bilinear models match moments and multimoments at some expansion points which have to be obtained through an optimisation scheme. This therefore removes the computational advantage of the Krylov subspace techniques implemented at an expansion point zero. This thesis demonstrates two improved approaches for the use of one-sided Krylov subspace projection for reducing bilinear models at the expansion point zero. This work proposes that an alternate linear approximation can be used for model order reduction. The advantages of using this approach are improved input-output preservation at a simulation cost similar to some earlier works and reduction of bilinear systems models which have singular state transition matrices. The comparison of the proposed methods and other original works done in this area of research is illustrated using various examples of single input single output (SISO) and multi input multi output (MIMO) models

Theoretical and practical aspects of linear and nonlinear model order reduction techniques

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 133-142).Model order reduction methods have proved to be an important technique for accelerating time-domain simulation in a variety of computer-aided design tools. In this study we present several new techniques for model reduction of the large-scale linear and nonlinear systems. First, we present a method for nonlinear system reduction based on a combination of the trajectory piecewise-linear (TPWL) method with truncated-balanced realizations (TBR). We analyze the stability characteristics of this combined method using perturbation theory. Second, we describe a linear reduction method that approximates TBR model reduction and takes advantage of sparsity of the system matrices or available accelerated solvers. This method is based on AISIAD (approximate implicit subspace iteration with alternate directions) and uses low-rank approximations of a system's gramians. This method is shown to be advantageous over the common approach of independently approximating the controllability and observability gramians, as such independent approximation methods can be inefficient when the gramians do not share a common dominant eigenspace. Third, we present a graph-based method for reduction of parameterized RC circuits. We prove that this method preserves stability and passivity of the models for nominal reduction. We present computational results for large collections of nominal and parameter-dependent circuits. Finally, we present a case study of model reduction applied to electroosmotic flow of a marker concentration pulse in a U-shaped microfluidic channel, where the marker flow in the channel is described by a three-dimensional convection-diffusion equation. First, we demonstrate the effectiveness of the modified AISIAD method in generating a low order models that correctly describe the dispersion of the marker in the linear case; that is, for the case of concentration-independent mobility and diffusion constants.(cont) Next, we describe several methods for nonlinear model reduction when the diffusion and mobility constants become concentration-dependent.by Dmitry Missiuro Vasilyev.Ph.D

Momenten-Abgleich-Verfahren in der Modellreduktion von elektromagnetischen Problemstellungen

In this thesis, the application of moment matching based model order reduction techniques to first- and second-order model problems of Maxwell's equations in semiconductor structures is considered. Apart from preserving the specific structure of Maxwell's equations in the reduced order model, we provide a new Greedy-type expansion point selection strategy based on the adaptive-order rational Arnoldi method. Moreover, we give an overview on the appropriate treatment of the discrete divergence conditions for moment matching based model order reduction. With respect to the offline stage of model order reduction, we introduce a specific framework of moment matching methods allowing for the efficient computation of a reduced order model. In detail, we consider a modification of the adaptive-order rational Arnoldi method avoiding the complete recomputation of the orthonormal vector sequences for subsequently computed reduced order models. Apart from employing an algebraic two-level approach for solving sequences of shifted linear systems, we have also discussed the application of the recycling SQMR method in moment matching based model order reduction. In the latter case, we typically benefit from exploiting the fact that the discretized first- and second-order Maxwell's equations offer a specific J-symmetry.Die zugrundeliegende Arbeit beinhaltet die Anwendung der Modellreduktion mittels Momenten-Abgleich-Verfahren auf Maxwell-Gleichungen erster bzw. zweiter Ordnung aus dem Anwendungsgebiet der Halbleiterstrukturen. Abgesehen von der Erhaltung der speziellen Struktur der Maxwell-Gleichungen im reduzierten Modell, wird eine neue Greedy-artige Entwicklungspunktauswahl basierend auf dem adaptiven rationalen Arnoldi-Verfahren eingeführt. Darüber hinaus geben wir einen Überblick über die geeignete Behandlung der diskreten Divergenz-Bedingungen für Momenten-Abgleich-Verfahren in der Modellreduktion. Im Hinblick auf die Offline-Phase der Modellreduktion, werden wir im weiteren Verlauf ein effizientes Framework für Momenten-Abgleich-Verfahren einführen, die eine effiziente Berechnung einer Folge reduzierter Modelle erlaubt. Insbesondere werden wir dabei eine Modifikation des adaptiven rationalen Arnoldi-Verfahrens vorstellen, die eine vollständige, wiederholte Berechnung der Sequenzen orthonormaler Vektoren für aufeinanderfolgende reduzierte Modelle vermeidet. Abgesehen von der Anwendung eines algebraischen Zwei-Level-Verfahrens für die Lösung geshifteter linearer Gleichungssysteme, haben wir darüber hinaus die Anwendung des recycling SQMR Verfahrens innerhalb der Modellreduktion mittels Momenten-Abgleich-Verfahren betrachtet. Im letzteren Fall profitieren wir in der Regel von der Tatsache, dass die diskretisierten Maxwell-Gleichungen erster bzw. zweiter Ordnung eine spezielle J-Symmetrie aufweisen