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Diese Dissertation befasst sich mit der Analyse von semi-expliziten Systemen aus stochastischen Differentialgleichungen (SDEs) gekoppelt mit stochastischen partiellen Differentialgleichungen (SPDEs) und algebraischen Gleichungen (AEs) mit möglicherweise stochastischen Anteilen in den Operatoren. Diese Systeme spielen eine entscheidende Rolle bei der Modellierung von realen Anwendungen, wie zum Beispiel elektrischen Schaltkreisen und Gasnetzwerken. Der Hauptbeitrag dieser Arbeit besteht darin, einen Rahmen bereitzustellen, in dem diese semiexpliziten Systeme auch bei stochastischen Einflüssen in den algebraischen Randbedingungen eine eindeutige Lösung haben. Wir führen einen numerischen Ansatz für solche Systeme ein und schlagen eine neue Möglichkeit vor, um Konvergenzergebnisse von driftimpliziten Methoden für SDEs auf stochastische Differential-Algebraische Gleichungen (SDAEs) zu erweitern. Dies ist wichtig, da viele Methoden für SDEs gut entwickelt sind, aber im Allgemeinen nicht für SDAEs in Betracht gezogen werden. Darüber hinaus untersuchen wir praktische Anwendungen in der Schaltkreis- und Gasnetzwerksimulation und diskutieren die dabei auftretenden Herausforderungen und Einschränkungen. Insbesondere stellen wir dabei auch einen Modellierungsansatz für Gasnetzwerke bestehend aus Rohren und algebraischen Komponenten vor. Abschließend testen wir in beiden Anwendungsfeldern die numerische Konvergenz anhand konkreter Beispiele mit verschiedenen Arten von stochastischer Modellierung. / This dissertation delves into the analysis of semi-explicit systems of stochastic differential equations (SDEs) coupled with stochastic partial differential equations (SPDEs) and algebraic equations (AEs) with possibly noise-driven operators. These systems play a crucial role in modeling real-world applications, such as electrical circuits and gas networks. The main contribution of this work is to provide a setting in which these semi-explicit systems have a unique solution even with stochastic influences in the algebraic constraints. We introduce a numerical approach for such systems and propose a new approach for extending convergence results of drift-implicit methods for SDEs to stochastic differential-algebraic equations (SDAEs). This is important, as many methods are well-developed for SDEs but generally not considered for SDAEs. Furthermore, we examine practical applications in circuit and gas network simulation, discussing the challenges and limitations encountered. In particular, we provide a modeling approach for gas networks consisting of pipes and algebraic components. To conclude, we test numerical convergence in both application settings on concrete examples with different types of stochastic modeling.

Diese Arbeit befasst sich mit Differential-algebraischen Gleichungen (DAEs). DAEs spielen eine wichtige Rolle in der Modellierung, der Simulation und der Optimierung von Netzwerken und gekoppelten Problemen in vielen Anwendungsgebieten. Es werden in Bezug auf die Modellierung und die numerische Simulation von DAEs bereits bestehende Ergebnisse diskutiert und erweitert. Des Weiteren wird die globale eindeutige Lösbarkeit und die Sensitivität der Lösungen mit Hinsicht auf Störungen der DAEs untersucht. Häufig wird die Modellierung von multiphysikalischen Anwendungen durch die Kopplung mehrerer einzelner DAE Systeme realisiert. Diese Herangehensweise kann hochdimensionale DAEs erzeugen, welche aufgrund von Instabilitäten nicht von klassischen numerischen Methoden, simuliert werden können. Angesichts dieser Herausforderungen werden drei Ziele formuliert: Erstens wird ein globales Lösungstheorem formuliert und bewiesen, welches auf gekoppelte Systeme angewandt werden kann, um deren Kopplungsansatz mathematisch zu rechtfertigen. Zweitens werden numerische Methoden vorgestellt, welche unter wesentlich schwächeren Strukturannahmen stabil sind und sich daher für die Simulation von gekoppelten Systemen eignen. Drittens wird eine Strategie präsentiert, die es ermöglicht, explizite Methoden auf gekoppelte Systeme anzuwenden. Um diese Ziele zu erreichen, braucht man ein Entkopplungsverfahren für DAEs, welches die folgenden drei Eigenschaften erfüllt: Die Komplexität des Entkopplungsverfahrens sollte nicht die Komplexität der DAE überschreiten. Das Entkopplungsverfahren sollte Eigenschaften wie Symmetrie, Monotonie und positive Definitheit erhalten. Das Entkopplungsverfahren sollte durch einen Schritt-für-Schritt Ansatz mit unabhängigen Schritten realisiert werden. Sowohl das Konzept des Tractability Index als auch das des Strangeness Index liefert kein solches Entkopplungsverfahren. Daher wird hier ein neues Index Konzept eingeführt, das diesen Anforderungen entspricht. / This thesis addresses differential-algebraic equations (DAEs). They play an important role in the modeling, simulation and optimization of networks and coupled problems in various applications. The main application in this thesis are coupled problems in electric circuit simulation. We discuss and extend existing results regarding the modeling and numerical simulation of DAEs. Furthermore, we investigate the global unique solvability and the sensitivity of solutions with respect to perturbations of DAEs. Nowadays the modeling of multi-physical applications is often realized by coupling systems of DAEs together with the help of additional coupling terms. This approach may yield high dimensional DAEs which cannot be simulated, due to instabilities, by standard numerical methods. Regarding these challenges we formulate three objectives: First we provide a global solvability theorem which can be applied to coupled systems to mathematically justify their coupling approach. Second we introduce numerical methods which are stable without needing any structural assumptions. Third we provide a way to apply explicit methods to coupled systems to be able to handle the size of the coupled systems by parallelizing the algorithms. To achieve these objectives, we need a decoupling procedure which fulfills the following three properties: The complexity of the decoupling procedure has to reflect the complexity of the DAE, i.e. the decoupling procedure should be state-independent if possible. The decoupling procedure should preserve properties like symmetry, monotonicity and positive definiteness. The decoupling procedure should be realized by a step-by-step approach with independent stages. Both the Tractability Index concept and the Strangeness Index concept do not provide such a decoupling procedure. For this reason we introduce a new index concept which provides such a decoupling procedure.

Industrial robot manipulators are general-purpose machines used for industrial automation in order to increase productivity, flexibility, and quality. Other reasons for using industrial robots are cost saving, and elimination of heavy and health-hazardous work. Robot motion control is a key competence for robot manufacturers, and the current development is focused on increasing the robot performance, reducing the robot cost, improving safety, and introducing new functionalities. Therefore, there is a need to continuously improve the models and control methods in order to fulfil all conflicting requirements, such as increased performance for a robot with lower weight, and thus lower mechanical stiffness and more complicated vibration modes. One reason for this development of the robot mechanical structure is of course cost-reduction, but other benefits are lower power consumption, improved dexterity, safety issues, and low environmental impact. This thesis deals with three different aspects of modeling and control of flexible, i.e., elastic, manipulators. For an accurate description of a modern industrial manipulator, the traditional flexible joint model, described in literature, is not sufficient. An improved model where the elasticity is described by a number of localized multidimensional spring-damper pairs is therefore proposed. This model is called the extended flexible joint model. This work describes identification, feedforward control, and feedback control, using this model. The proposed identification method is a frequency-domain non-linear gray-box method, which is evaluated by the identification of a modern six-axes robot manipulator. The identified model gives a good description of the global behavior of this robot. The inverse dynamics control problem is discussed, and a solution methodology is proposed. This methodology is based on a differential algebraic equation (DAE) formulation of the problem. Feedforward control of a two-axes manipulator is then studied using this DAE approach. Finally, a benchmark problem for robust feedback control of a single-axis extended flexible joint model is presented and some proposed solutions are analyzed.

<p>Industrial robot manipulators are general-purpose machines used for industrial automation in order to increase productivity, flexibility, and quality. Other reasons for using industrial robots are cost saving, and elimination of heavy and health-hazardous work. Robot motion control is a key competence for robot manufacturers, and the current development is focused on increasing the robot performance, reducing the robot cost, improving safety, and introducing new functionalities. Therefore, there is a need to continuously improve the models and control methods in order to fulfil all conflicting requirements, such as increased performance for a robot with lower weight, and thus lower mechanical stiffness and more complicated vibration modes. One reason for this development of the robot mechanical structure is of course cost-reduction, but other benefits are lower power consumption, improved dexterity, safety issues, and low environmental impact.</p><p>This thesis deals with three different aspects of modeling and control of flexible, i.e., elastic, manipulators. For an accurate description of a modern industrial manipulator, the traditional flexible joint model, described in literature, is not sufficient. An improved model where the elasticity is described by a number of localized multidimensional spring-damper pairs is therefore proposed. This model is called the extended flexible joint model. This work describes identification, feedforward control, and feedback control, using this model.</p><p>The proposed identification method is a frequency-domain non-linear gray-box method, which is evaluated by the identification of a modern six-axes robot manipulator. The identified model gives a good description of the global behavior of this robot.</p><p>The inverse dynamics control problem is discussed, and a solution methodology is proposed. This methodology is based on a differential algebraic equation (DAE) formulation of the problem. Feedforward control of a two-axes manipulator is then studied using this DAE approach.</p><p>Finally, a benchmark problem for robust feedback control of a single-axis extended flexible joint model is presented and some proposed solutions are analyzed.</p>

Consider the problem of solving a large system of linear algebraic equations, using the Krylov subspace methods. In order to find the solution efficiently, the system often needs to be preconditioned, i.e., transformed prior to the iterative scheme. A feature of the system that often enables fast solution with efficient preconditioners is the structural sparsity of the corresponding matrix. A recent development brought another and a slightly different phe- nomenon called the data sparsity. In contrast to the classical (structural) sparsity, the data sparsity refers to an uneven distribution of extractable information inside the matrix. In practice, the data sparsity of a matrix ty- pically means that its blocks can be successfully approximated by matrices of low rank. Naturally, this may significantly change the character of the numerical computations involving the matrix. The thesis focuses on finding ways to construct Cholesky-based preconditioners for the conjugate gradi- ent method to solve systems with symmetric and positive definite matrices, exploiting a combination of the data and structural sparsity. Methods to exploit the data sparsity are evolving very fast, influencing not only iterative solvers but direct solvers as well. Hierarchical schemes based on the data sparsity concepts can be derived...

[pt] Esse trabalho apresenta uma abordagem baseada em programação matemática para a solução de problemas de valor inicial de contorno constitutivo elastoplástico. Considerando a plasticidade associativa, as equações constitutivas locais, em sua forma discreta, são formuladas como problemas de otimização cônica. Especificamente, é demonstrado que métodos implícitos de retorno a superfície para os critérios mais conhecidos da literatura, como o de Rankine, von Mises, Tresca, Drucker-Prager e Mohr Coulomb, podem ser expressos como problemas de otimização cônica de segunda ordem e semidefinida. Além disso, um novo método numérico para a determinação do operador elastoplástico consistente, baseado na derivada paramétrica de primeira ordem das soluções ótimas, é proposto. / [en] This work presents a mathematical programming approach for elastoplastic constitutive initial boundary value problems. Considering associative plasticity, the local discrete constitutive equations are formulated as conic programs. Specifically, it is demonstrated that implicit return-mapping schemes for well-known yield criteria, such as the Rankine, von Mises, Tresca, Drucker-Prager, and Mohr-Coulomb criteria, can be expressed as secondorder and semidefinite conic programs. Additionally, a novel scheme for the numerical evaluation of the consistent elastoplastic tangent operator, based on a first-order parameter derivative of the optimal solutions, is proposed.

This research effort develops a comprehensive computational framework to support the parametric optimal design of uncertain dynamical systems. Uncertainty comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; not accounting for uncertainty may result in poor robustness, sub-optimal performance and higher manufacturing costs. Contemporary methods for the quantification of uncertainty in dynamical systems are computationally intensive which, so far, have made a robust design optimization methodology prohibitive. Some existing algorithms address uncertainty in sensors and actuators during an optimal design; however, a comprehensive design framework that can treat all kinds of uncertainty with diverse distribution characteristics in a unified way is currently unavailable. The computational framework uses Generalized Polynomial Chaos methodology to quantify the effects of various sources of uncertainty found in dynamical systems; a Least-Squares Collocation Method is used to solve the corresponding uncertain differential equations. This technique is significantly faster computationally than traditional sampling methods and makes the construction of a parametric optimal design framework for uncertain systems feasible. The novel framework allows to directly treat uncertainty in the parametric optimal design process. Specifically, the following design problems are addressed: motion planning of fully-actuated and under-actuated systems; multi-objective robust design optimization; and optimal uncertainty apportionment concurrently with robust design optimization. The framework advances the state-of-the-art and enables engineers to produce more robust and optimally performing designs at an optimal manufacturing cost. / Ph. D.

La recherche et les questions abordées dans cette thèse sont de type inverse : la reconstitution d'une source ponctuelle ou la complétion d'une donnée à la limite inconnue à l'extrémité du domaine dans les modèles paraboliques de transport de contaminants. La modélisation mathématique des problèmes de pollution des eaux fait intervenir deux traceurs, l'oxygène dissous (OD) et la demande biochimique en oxygène (DBO) qui est la quantité d'oxygène nécessaire à la biodégradation de la matière organique. En effet, au cours des procédés d'autoépuration, certaines bactéries aérobies jouent un rôle principal. Ces micro-organismes décomposent les matières organiques polluantes en utilisant l'oxygène dissous dans le milieu. Afin de compenser ces données manquantes, les champs, solutions du problème, sont observés directement ou indirectement. Les problèmes inverses qui en résultent sont quasi certainement mal-posés voire même sévèrement mal-posés pour la plupart. Dans cette thèse, nous proposons justement une analyse aussi poussée que possible sur la question de l'identifiabilité pour les deux problèmes inverses décrits ci-dessus. Nous avons démontré un résultat d'unicité pour des sources fixes dans le cas d'observations décalées. La réalité pour l'observation est nuancée et l'idéal n'est pas acquis ; des mesures directes sur la DBO sont difficiles à obtenir. En revanche collecter des données sur l'OD est possible en temps réel et avec un faible coût. La DBO est donc observée de façon indirecte, grâce au couplage dans le système de Streeter et Phelps, l'information passe de l'OD à la DBO. Pour ce problème aussi, nous avons produit un résultat d'unicité pour la reconstruction de la source ou puits ponctuel qui serait présent dans l'équation de transport sur l'OD. Nous avons ensuite examiné des questions annexes à l'identifiabilité telles que le degré d'instabilité des équations à résoudre. De ce type d'informations dépendent le comportement des méthodes numériques et des algorithmes de calcul à utiliser. / The research and the questions approached on this thesis are inverse type : the reconstruction of point-wise source or the data completion problem in parabolic models of transport of contaminants. The mathematical modelling of the problems of water pollution includes two tracers, the dissolved oxygen (DO) and the biochemical demand in oxygen (BDO) which is the quantity of oxygen necessary for the biodegradation of organic matter. Indeed, during the biodegradation process, aerobic bacteria play a leading part. These micro-organisms decompose polluting organic matters by using the dissolved oxygen in the middle. To compensate these missing data, fields, solutions of the problem, are observed directly or indirectly. The resulting inverse problems are ill-posed. Their mathematical study rises big complications and their numerical treatment isn't easy. We demonstrated a uniqueness result for fixed sources in the case of moved observations. The reality for the observation is qualified and the ideal is not acquired; direct measures on the BOD are difficult to obtain. On the Other hand to collect data on the DO is possible in real time With a moderate cost. The BOD is thus observed in indirect way, thanks to the coupling in the system of Streeter and Phelps, the information passes from the DO to the BOD. For this problem, we produced a uniqueness result for the reconstruction of source. Then, we examined the degree of instability of the equation to be solved. The behaviour of numerical methods depend on this type of information.

<p>A time-discrete formulation of the variational principle of mechanics is used to provide a consistent theoretical framework for the construction and analysis of low order integration methods. These are applied to mechanical systems subject to mixed constraints and dry frictional contacts and impacts---machines. The framework includes physics motivated constraint regularization and stabilization schemes. This is done by adding potential energy and Rayleigh dissipation terms in the Lagrangian formulation used throughout. These terms explicitly depend on the value of the Lagrange multipliers enforcing constraints. Having finite energy, the multipliers are thus massless ghost particles. The main numerical stepping method produced with the framework is called SPOOK.</p><p>Variational integrators preserve physical invariants globally, exactly in some cases, approximately but within fixed global bounds for others. This allows to product realistic physical trajectories even with the low order methods. These are needed in the solution of nonsmooth problems such as dry frictional contacts and in addition, they are computationally inexpensive. The combination of strong stability, low order, and the global preservation of invariants allows for large integration time steps, but without loosing accuracy on the important and visible physical quantities. SPOOK is thus well-suited for interactive simulations, such as those commonly used in virtual environment applications, because it is fast, stable, and faithful to the physics.</p><p>New results include a stable discretization of highly oscillatory terms of constraint regularization; a linearly stable constraint stabilization scheme based on ghost potential and Rayleigh dissipation terms; a single-step, strictly dissipative, approximate impact model; a quasi-linear complementarity formulation of dry friction that is isotropic and solvable for any nonnegative value of friction coefficients; an analysis of a splitting scheme to solve frictional contact complementarity problems; a stable, quaternion-based rigid body stepping scheme and a stable linear approximation thereof. SPOOK includes all these elements. It is linearly implicit and linearly stable, it requires the solution of either one linear system of equations of one mixed linear complementarity problem per regular time step, and two of the same when an impact condition is detected. The changes in energy caused by constraints, impacts, and dry friction, are all shown to be strictly dissipative in comparison with the free system. Since all regularization and stabilization parameters are introduced in the physics, they map directly onto physical properties and thus allow modeling of a variety of phenomena, such as constraint compliance, for instance.</p><p>Tutorial material is included for continuous and discrete-time analytic mechanics, quaternion algebra, complementarity problems, rigid body dynamics, constraint kinematics, and special topics in numerical linear algebra needed in the solution of the stepping equations of SPOOK.</p><p>The qualitative and quantitative aspects of SPOOK are demonstrated by comparison with a variety of standard techniques on well known test cases which are analyzed in details. SPOOK compares favorably for all these examples. In particular, it handles ill-posed and degenerate problems seamlessly and systematically. An implementation suitable for large scale performance and accuracy testing is left for future work.</p>