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

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

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

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

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Stability-preserving model reduction for linear and nonlinear systems arising in analog circuit applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 221-229).Despite the increasing presence of RF and analog components in personal wireless electronics, such as mobile communication devices, the automated design and optimization of such systems is still an extremely challenging task. This is primarily due to the presence of both parasitic elements and highly nonlinear elements, which makes simulation computationally expensive and slow. The ability to generate parameterized reduced order models of analog systems could serve as a first step toward the automatic and accurate characterization of geometrically complex components and subcircuits, eventually enabling their synthesis and optimization. This thesis presents techniques for reduced order modeling of linear and nonlinear systems arising in analog applications. Emphasis is placed on developing techniques capable of preserving important system properties, such as stability, and parameter dependence in the reduced models. The first technique is a projection-based model reduction approach for linear systems aimed at generating stable and passive models from large linear systems described by indefinite, and possibly even mildly unstable, matrices. For such systems, existing techniques are either prohibitively computationally expensive or incapable of guaranteeing stability and passivity. By forcing the reduced model to be described by definite matrices, we are able to derive a pair of stability constraints that are linear in terms of projection matrices.(cont.) These constraints can be used to formulate a semidefinite optimization problem whose solution is an optimal stabilizing projection framework. The second technique is a projection-based model reduction approach for highly nonlinear systems that is based on the trajectory piecewise linear (TPWL) method. Enforcing stability in nonlinear reduced models is an extremely difficult task that is typically ignored in most existing techniques. Our approach utilizes a new nonlinear projection in order to ensure stability in each of the local models used to describe the nonlinear reduced model. The TPWL approach is also extended to handle parameterized models, and a sensitivity-based training system is presented that allows us to efficiently select inputs and parameter values for training. Lastly, we present a system identification approach to model reduction for both linear and nonlinear systems. This approach utilizes given time-domain data, such as input/output samples generated from transient simulation, in order to identify a compact stable model that best fits the given data. Our procedure is based on minimization of a quantity referred to as the 'robust equation error', which, provided the model is incrementally stable, serves as up upper bound for a measure of the accuracy of the identified model termed 'linearized output error'. Minimization of this bound, subject to an incremental stability constraint, can be cast as a semidefinite optimization problem.by Bradley Neil Bond.Ph.D

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described