Produkte im Maschinenbau gewinnen durch immer weitreichendere Kundenanforderungen zunehmend an Komplexität. Davon angetrieben, werden auch die eingesetzten Softwarewerkzeuge kontinuierlich weiterentwickelt.CAD-Systeme werden beispielsweise heutzutage nicht mehr ausschließlich zur Konstruktion verwendet, sondern beinhalten meist auch weiterführende Module bspw.zur Simulation physikalischer Effekte (CAE) bis hin zur Arbeitsplanung (CAP) und werden in ganzen Softwaresuiten angeboten. Im vergangenen Jahrzehnt sind außerdem Mechanismus- und Bewegungsanalysen fester Bestandteil von CAD-Systemen geworden. Zusatzinformationen, wie bspw. Kinematikbeziehungen und / oder Animationen, die zur Verifikation der Funktionalität und Kollisionsfreiheit unabdingbar sind, können direkt in den CAD-Programen erstellt und überprüft werden. Die aktive Nutzung und Zugänglichkeit dieser CAD-Konstruktionssysteme durch konstruktionsferne Nutzergruppen ist nur bedingt durch so genannte Model-Viewer gegeben, wodurch bspw. die Fehlerbewertung (z.B. Design Review, FMEA) außerhalb der Entwicklungsabteilungen erschwert wird. Abhilfe bietet hier die Nutzung von High-EndTechniken, wie die immersive Darstellung mittels Virtual Reality (VR)-Technologie, im Prozess der Konstruktionsbewertung/Design Review. Dafür arbeiten wir mit der Chemnitzer Firma ARCsolutions GmbH an einer Lösung wie ein (teil-)automatisiert reduziertes Modell, unter Erhaltung im CAD definierter Animationen und Kinematischen Beziehungen automatisiert in die VR gebracht und dort genutzt werden kann.

In this thesis we have studied several aspects of the minimal realization problem for descriptor systems. These aspects include purely theoretical questions such as that about the order of a minimal realization of a general improper rational matrix and problems of a numerical nature, like rounding error analysis of the computing a minimal realization from a nonminimal one. We have also treated the minimal partial realization problem for general descriptor systems with application to model reduction and to generalised eigenvalue problems.

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis.

Produkte im Maschinenbau gewinnen durch immer weitreichendere Kundenanforderungen zunehmend an Komplexität. Davon angetrieben, werden auch die eingesetzten Softwarewerkzeuge kontinuierlich weiterentwickelt.CAD-Systeme werden beispielsweise heutzutage nicht mehr ausschließlich zur Konstruktion verwendet, sondern beinhalten meist auch weiterführende Module bspw.zur Simulation physikalischer Effekte (CAE) bis hin zur Arbeitsplanung (CAP) und werden in ganzen Softwaresuiten angeboten. Im vergangenen Jahrzehnt sind außerdem Mechanismus- und Bewegungsanalysen fester Bestandteil von CAD-Systemen geworden. Zusatzinformationen, wie bspw. Kinematikbeziehungen und / oder Animationen, die zur Verifikation der Funktionalität und Kollisionsfreiheit unabdingbar sind, können direkt in den CAD-Programen erstellt und überprüft werden. Die aktive Nutzung und Zugänglichkeit dieser CAD-Konstruktionssysteme durch konstruktionsferne Nutzergruppen ist nur bedingt durch so genannte Model-Viewer gegeben, wodurch bspw. die Fehlerbewertung (z.B. Design Review, FMEA) außerhalb der Entwicklungsabteilungen erschwert wird. Abhilfe bietet hier die Nutzung von High-EndTechniken, wie die immersive Darstellung mittels Virtual Reality (VR)-Technologie, im Prozess der Konstruktionsbewertung/Design Review. Dafür arbeiten wir mit der Chemnitzer Firma ARCsolutions GmbH an einer Lösung wie ein (teil-)automatisiert reduziertes Modell, unter Erhaltung im CAD definierter Animationen und Kinematischen Beziehungen automatisiert in die VR gebracht und dort genutzt werden kann.

This dissertation concerns the model reduction of linear periodic descriptor systems both in continuous and discrete-time case. In this dissertation, mainly the projection based approaches are considered for model order reduction of linear periodic time varying descriptor systems. Krylov based projection method is used for large continuous-time periodic descriptor systems and balancing based projection technique is applied to large sparse discrete-time periodic descriptor systems to generate the reduce systems. For very large dimensional state space systems, both the techniques produce large dimensional solutions. Hence, a recycling technique is used in Krylov based projection methods which helps to compute low rank solutions of the state space systems and also accelerate the computational convergence. The outline of the proposed model order reduction procedure is given with more details. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders. Model reduction techniques based on balance truncation require to solve matrix equations. For periodic time-varying descriptor systems, these matrix equations are projected generalized periodic Lyapunov equations and the solutions are also time-varying. The cyclic lifted representation of the periodic time-varying descriptor systems is considered in this dissertation and the resulting lifted projected Lyapunov equations are solved to achieve the periodic reachability and observability Gramians of the original periodic systems. The main advantage of this solution technique is that the cyclic structures of projected Lyapunov equations can handle the time-varying dimensions as well as the singularity of the period matrix pairs very easily. One can also exploit the theory of time-invariant systems for the control of periodic ones, provided that the results achieved can be easily re-interpreted in the periodic framework. Since the dimension of cyclic lifted system becomes very high for large dimensional periodic systems, one needs to solve the very large scale periodic Lyapunov equations which also generate very large dimensional solutions. Hence iterative techniques, which are the generalization and modification of alternating directions implicit (ADI) method and generalized Smith method, are implemented to obtain low rank Cholesky factors of the solutions of the periodic Lyapunov equations. Also the application of the solvers in balancing-based model reduction of discrete-time periodic descriptor systems is discussed. Numerical results are given to illustrate the effciency and accuracy of the proposed methods.

When simulating mechanical systems the flexibility of the components often has to be taken into account. This is particularly important for simulations when high detailed information is demanded, e.g. to calculate stresses. To this end the Finite Element Method (FEM) is often used. However the models can become very large, containing millions of degrees of freedom. Solving large linear systems are computationally demanding. Therefore ways of reducing the problem is often sought. These reduction does, however, remove much of the details that was to be investigated. In this thesis this problem is addressed by creating a reduction scheme, using Proper Orthogonal Decomposition (POD), that significantly reduces a problem but still captures much of the details. A novel method for enriching regular POD-based model reduction methods with hierarchically determined enrichment POD-modes is developed. The method is proposed and validated in a FEM application towards dynamical simulation. The enriched method is compared against a regular POD reduction technique. An numerical study is made of a model example of linear elasticity in a gearwheel. The numerical study suggests that the error of displacements is around ten times smaller, on average, when using the enriched basis compared to a reference basis of equal dimensionality consisting of only regular POD modes. Also it is shown that local quantities as the von Mises stress in a gearwheel tooth is preserved much better using the enriched basis. An a posteriori error estimate is proposed and proved for the static case, showing that the error is bound. / När man simulerar mekaniska system så måste man ofta ta hänsyn till de ingående komponenternas flexibilitet. Detta är särskilt viktigt då man gör simuleringar med krav på hög detaljkännedom, såsom mätningar av spänningar i kugghjul etc. Till detta ändamål används ofta en Finit Element Metod (FEM). Dock kan modellerna ofta bli väldigt stora, med över en miljon frihetsgrader. Att lösa linjära system av den storleken är beräkningsmässigt krävande. Därför är det naturligt att försöka reducera problemen. Reduktion innebär dock att information försvinner, i synnerhet de detaljer som skulle beräknas. I detta examensarbete så behandlas problemet genom att skapa en ny metod för reducering av stora finita element modeller. Metoden bygger på tidigare kunskap om Proper Orthogonal Decomposition (POD) som ett sätt att reducera modeller. Den nya metoden reducerar finita ellement modeller samtidigt som den bibehåller hög detalj. En ny metod utvecklas för att berika en vanlig POD-baserad modellreduktion med hjälp av hieraktiskt bestämda berikningsmoder. Metoden beskrivs och testas i en dynamisk FEM-applikation av elasticitet i ett kugghjul i 2 dimensioner. Metoden för berikning jämförs numeriskt med en metod som använder vanlig POD-reduktion. Körningar visar att felet i den berikade metoden är omkring 10 gånger mindre, i genomsnitt, jämfört med en vanlig metod. Det visas också att spänningar bevaras på ett mycket bra sätt med den nya berikningsmetoden. Dessutom så formuleras och bevisas ett a posteriori estimat för statiska lastfall, vilket innebär att felet i metoden är bundet.

In this thesis we have studied several aspects of the minimal realization problem for descriptor systems. These aspects include purely theoretical questions such as that about the order of a minimal realization of a general improper rational matrix and problems of a numerical nature, like rounding error analysis of the computing a minimal realization from a nonminimal one. We have also treated the minimal partial realization problem for general descriptor systems with application to model reduction and to generalised eigenvalue problems.

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.

Today in the scientific and technological world, physical and artificial processes are often described by mathematical models which can be used for simulation, optimization or control. As the mathematical models get more detailed and different coupling effects are required to include, usually the dimension of these models become very large. Such large-scale systems lead to large memory requirements and computational complexity. To handle these large models efficiently in simulation, control or optimization model order reduction (MOR) is essential. The fundamental idea of model order reduction is to approximate a large-scale model by a reduced model of lower state space dimension that has the same (to the largest possible extent) input-output behavior as the original system. Recently, the system-theoretic method Balanced Truncation (BT) which was believed to be applicable only to moderately sized problems, has been adapted to really large-scale problems. Moreover, it also has been extended to so-called descriptor systems, i.e., systems whose dynamics obey differential-algebraic equations. In this thesis, a BT algorithm is developed for MOR of index-1 descriptor systems based on several papers from the literature. It is then applied to the setting of a piezo-mechanical system. The algorithm is verified by real-world data describing micro-mechanical piezo-actuators. The whole algorithm works for sparse descriptor form of the system. The piezo-mechanical original system is a second order index-1 descriptor system, where mass, damping, stiffness, input and output matrices are highly sparse. Several techniques are introduced to reduce the system into a first order index-1 descriptor system by preserving the sparsity pattern of the original models. Several numerical experiments are used to illustrate the efficiency of the algorithm.

Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis.