La première partie de cette thèse (articles I et II) est consacrée à l'étude du comportement asymptotique des solutions d'un système dynamique du second ordre avec dissipation évanescente. Le système dynamique est étudié dans sa version continue et dans sa version discrète via un algorithme.La deuxième partie de cette thèse (articles III à VI) est consacrée à l'étude de plusieurs algorithmes de type proximal. Nous montrons que ces algorithmes convergent vers des solutions de certains problèmes de minimisation. Dans chaque cas, une application est donnée dans le cadre de la décomposition de domaine pour les EDP. / The first part of this thesis is devoted to the study of the asymptotic behavior of solutions of a second order dynamic system with vanishing dissipation. The dynamic system is studied in its continuous version and in its discrete version via an algorithm.The second part is about the study of several proximal-type algorithms. We show that these algorithms converge to solutions of some minimization problems. In each case, an application is given in the area of domain decomposition for PDE's.

Diese Dissertation befasst sich mit Modellrechnungen zur Dynamik vom photoinduzierten Elektrontransfer und Exzitontransfer in Systemen mit vielen Freiheitsgraden. Außerdem trägt diese Arbeit zu einigen theoretischen und numerischen Aspekten der Redfield-Theorie bei. Betrachtet werden der ultraschnelle Elektrontransfer im Farbstoff Betain-30, die Elektroninjektion von einem Chromophormolekül ins Leitungsband von einem Halbleiter, sowie die Exziton-Ausbreitung in einem modellhaften ringförmigen System mit 18 Lokalisierungszentren. Zuerst wird der Einfluss der elektronischen Kopplung auf die Dissipationsterme der Redfield-Gleichung untersucht. Es wird gezeigt, dass bei bestimmten Potenzialkonfigurationen die Vernachlässigung der elektronischen Kopplung (die soganannte diabatische Dämpfungsnäherung) dazu führt, dass das System nicht in das thermische Gleichgewicht mit dem Wärmebad relaxiert. Jedoch verliert diese Näherung ihre Gültigkeit nicht für kleine elektronische Kopplung in einer ganzen Reihe von Fällen, z.B. im Marcus-invertierten Bereich. Die Transfermechanismen, welche jenseits dieser Näherung auftreten, werden mit Hilfe der Störungstheorie erster Ordnung in der elektronischen Kopplung detailliert untersucht. Weiterhin werden direkte Verfahren zur genauen numerischen Lösung zeitlokaler Mastergleichungen implementiert und getestet. Die Effizienz dieser Methoden wird am Beispiel von einem eindimensionalen Elektrontransfer-Modell bestimmt. Desweiteren wird noch ein neues stochastisches Verfahren zur Propagation von Dichtematrizen entwickelt und in den Simulationen verwendet. Der ultraschnelle photoinduzierte Elektrontransfer in Betain-30 wird sowohl mit einer einzelnen Reaktionsmode als auch mit zwei Reaktionsmoden modelliert. Anhand der reduzierten Dichtematrix lässt sich die Gesamtpolarisation berechnen und somit ist es möglich, ein Pump-Probe-Experiment zu simulieren. Die Rechenergebnisse werden mit experimentellen Daten verglichen.

Title: The study of the association behaviour of the amphiphilic copolymers in solutions containing low molar compounds by means of computer simulations. Author: Mgr. Karel Šindelka Department: Faculty of Science, Charles University Supervisor: Doc. Ing. Zuzana Limpouchová, Csc. Abstract This doctoral thesis focuses on the study of electrostatic self- and co-assembly in complex polymer solutions containing polyelectrolyte (PE) block copolymers together with surfactants, neutral homopolymers, or oppositely charged PEs using the dissipative particle dynamics (DPD). It was shown that the electro- static self-assembly depends not only on the cooperative interactions of oppo- sitely charged PE chains, but also on the amphiphilicity of PE species or on the polymer block compatibility, among other properties. PEs with incompatible blocks create well-defined core-shell structures, while large ill-defined crew-cut aggregates form from PEs with compatible blocks In non-stoichiometric mixtures of PEs with incompatible blocks, co-assembled nanoparticles are smaller than in stoichiometric mixtures and are charged. The destabilization of larger aggregates depends on how the PE charge surplus is introduced: the effect is strongest when the density of the surplus PE charge on the PE chains is increased and weakest when the...

Energy storage systems with many Lithium Ion battery cells per string require sophisticated balancing hardware due to individual cells having manufacturing inconsistencies, different self discharge rates, internal resistances and temperature variations. For capacity maximization, safe operation, and extended lifetime, battery balancing is required. Redistributive Non-Dissipative balancing further improves the pack capacity and efficiency over a Dissipative approach where energy is wasted as heat across shunt resistors. Redistribution techniques dynamically shuttle charge to and from weak cells during operation such that all of the stored energy in the stack is utilized. This thesis identifies and develops different balancing control methods. These methods include a unconstrained optimization problem using a Linear Quadratic Regulator (LQR) and a constrained optimization problem using Model Predictive Control (MPC). These methods are benchmarked against traditional rule based (RB) balancing. The control systems are developed using MATLAB/Simulink and validated experimentally on a multiple transformer individual cell to stack topology. The implementation uses a DC2100A Demo-board from Linear Technology with bi-directional flyback converters to transfer the energy between the cells. The results of this thesis show that the MPC control method has the highest balancing efficiency and minimum balancing time. / Thesis / Master of Applied Science (MASc)

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[EN] This Thesis is focused on the development and implementation of efficient numerical methods for the acoustic modelling and design of noise control devices in the exhaust system of combustion engines. Special attention is paid to automotive perforated dissipative silencers, in which significant differences are likely to appear in their acoustic behaviour, depending on the temperature variations within the absorbent material. Also, material heterogeneities can alter the silencer attenuation performance. Therefore, numerical techniques considering all these features are required to guarantee the accuracy of the results. A literature review is carried out, mainly related to one-dimensional models, as well as to acoustic models for absorbent materials and perforated surfaces. However, plane wave model limitations make indispensable using alternative multidimensional methods. In addition, the possibility of using new acoustic elements is explored. These elements have as an objective being a potential alternative to the fibrous absorbent materials, which can have a negative impact on health. The Thesis considers the use of microperforated and sintered surfaces. The latter have, in some cases, a nearly constant acoustic impedance, whose value depends, among others, on the thickness and porosity of the plates. To avoid the limitations of plane wave models, a finite element (FE) approach is proposed for the acoustic analysis of dissipative silencers including a perforated duct with uniform axial mean flow and an outer chamber with a heterogeneous distribution of the absorbent material. On the other hand, property variations can be also produced by temperature gradients. In this case, a hybrid FE model has been derived for perforated dissipative silencers including: (1) Thermal gradients in the central duct and the chamber; (2) A perforated passage carrying non-uniform axial mean flow. A FE approach has been implemented to solve the pressure-based wave equation for a non-moving heterogeneous medium, associated with the chamber. Also, the governing equation in the central duct has been written and solved in terms of an acoustic velocity potential to allow the presence of an axially inhomogeneous flow. The coupling between both regions has been carried out by means of a perforated duct and its acoustic impedance, adapted here to include absorbent material heterogeneities and mean flow effects. It has been found that the presence of non-homogeneities can have a significant influence on the acoustic attenuation of a silencer and should be included in the theoretical models. Optimization techniques for industrial noise control devices are relevant, since they lead to the production of elements with better characteristics. Evolutionary algorithms are emergent techniques able to obtain a solution, even in those problems in which the traditional optimization have difficulties. Optimization techniques are combined with the FE method to achieve the maximum attenuation in the frequency range of interest. A multichamber silencer optimization problem is defined and several analyses are carried out to obtain the most suitable configuration for each application. Under certain assumptions of axial uniformity, several techniques have been considered to reduce the computational effort of a full 3D FE analysis for dissipative silencers with temperature gradients and mean flow. These are based on a decomposition of the acoustic field into transversal and axial modes within each silencer subdomain, and a matching procedure of the modal expansions at the silencer area changes through the continuity conditions of the acoustic fields. The relative computational efficiency and accuracy of predictions for the matching techniques are studied, including point collocation at nodes and Gauss points and also mode-matching with weighted integration. All of them provide accurate predictions of the attenuation and improve the computational cost of a FE calculation / [ES] Esta Tesis se centra en el desarrollo e implementación de métodos numéricos eficientes para el diseño y modelado de componentes de la línea de escape en motores de combustión interna. Merecen especial atención los silenciadores disipativos perforados de automóviles, ya que su comportamiento acústico puede sufrir variaciones importantes debidas a las variaciones de temperatura en el material absorbente, así como a las heterogeneidades de la fibra. Por tanto, se requieren técnicas numéricas que consideren estos casos para garantizar la precisión de los resultados. Se lleva a cabo una revisión bibliográfica que recoge los modelos de onda unidimensionales, así como modelos acústicos de materiales absorbentes y superficies perforadas. Sin embargo, las limitaciones de los primeros hacen indispensable el uso de modelos multidimensionales. Además se explora la posibilidad de usar nuevos elementos acústicos, cuyo objetivo es ser una alternativa potencial a los materiales absorbentes, que pueden tener un efecto negativo sobre la salud. La Tesis considera el uso de superficies microperforadas y sinterizadas. Estas últimas en algunos casos presentan una impedancia casi constante, cuyo valor depende, entre otras cosas, del espesor y la porosidad de las placas. Para evitar las limitaciones de los modelos de onda plana, se propone un enfoque en elementos finitos (EF) para el análisis acústico de silenciadores disipativos que incluyen un conducto con flujo medio axial uniforme y una cámara externa con una distribución heterogénea de material absorbente. Por otro lado, la variación de las propiedades también puede producirse por gradientes térmicos. En este caso, se propone una formulación híbrida de EF para silenciadores disipativos perforados que incluye: (1) Gradientes térmicos en el conducto central y la cámara; (2) Un conducto perforado que canaliza flujo medio axial no uniforme. Se ha implementado una formulación de EF para resolver la ecuación de ondas en términos de presión para el medio estacionario heterogéneo asociado a la cámara. Además, la ecuación asociada al conducto central, expresada en términos de potencial de velocidad acústica, permite la presencia de flujo axial no uniforme. El acoplamiento entre ambas regiones se ha realizado mediante un conducto perforado y su impedancia acústica y se ha adaptado para incluir la citada falta de homogeneidad. Se ha visto que las heterogeneidades pueden influir notablemente en la atenuación acústica de un silenciador, debiéndose incluir en los modelos teóricos. Las técnicas de optimización para componentes industriales de control de ruido son importantes, ya que producen elementos con mejores características. Los algoritmos evolutivos son técnicas emergentes capaces de obtener una solución, incluso cuando la optimización tradicional tiene dificultades. Las técnicas de optimización se combinan con el MEF para conseguir la máxima atenuación posible en el rango de frecuencias de interés. Se ha definido un problema de optimización de un silenciador multicámara y se han llevado a cabo varios análisis para obtener la configuración más adecuada para cada caso. Bajo ciertas hipótesis de uniformidad axial, se han considerado varias técnicas para reducir el coste computacional de un análisis 3D completo para silenciadores disipativos con gradientes de temperatura y flujo medio. Éstas se basan en la descomposición del campo acústico en modos axiales y transversales dentro de cada subdominio, y un procedimiento de acoplamiento de las expansiones modales en los cambios de sección del silenciador mediante las condiciones de continuidad de los campos acústicos. Se estudia la eficiencia computacional y precisión de las predicciones de las técnicas de acoplamiento, incluyendo colocación puntual en nodos y puntos de Gauss, así como ajuste modal. Todos ellos proporcionan predicciones precisas de la atenuación mejorando el coste / [CA] Aquesta Tesi es centra en el desenvolupament i implementació de mètodes numèrics eficients per al disseny i modelatge de components de la línia d'escapament en motors de combustió interna. Mereixen especial atenció els silenciadors dissipatius perforats d'automòbils, ja que el seu comportament acústic pot patir variacions importants degudes a les variacions de temperatura en el material absorbent, així com a les heterogeneïtats de la fibra. Per tant, es requereixen tècniques numèriques que considerin aquests casos per garantir la precisió dels resultats. Es porta a terme una revisió bibliogràfica que recull els models d'ona unidimensionals, així com models acústics de materials absorbents i superfícies perforades. No obstant això, les limitacions dels primers fan indispensable l'ús de models multidimensionals. A més s'explora la possibilitat d'usar nous elements acústics amb l'objectiu que siguen una alternativa potencial als materials absorbents, que poden tenir un efecte negatiu sobre la salut. La Tesi considera l'ús de superfícies microperforades i sinteritzades. Aquestes últimes en alguns casos presenten una impedància gairebé constant. El seu valor depèn, entre altres coses, del gruix i la porositat de les plaques. Per evitar les limitacions dels models d'ona plana, es proposa un enfocament amb elements finits (EF) per a l'anàlisi acústic de silenciadors dissipatius que inclouen un conducte amb flux mig axial uniforme i una càmera externa amb una distribució heterogènia de material absorbent. D'altra banda, la variació de les propietats també es pot produir per gradients tèrmics. En aquest cas, es proposa una formulació híbrida d'EF per silenciadors dissipatius perforats que inclou: (1) Gradients tèrmics en el conducte central i la càmera; (2) Un conducte perforat que canalitza flux mig axial no uniforme. S'ha implementat una formulació d'EF per resoldre l'equació d'ones en termes de pressió per al medi estacionari heterogeni associat a la càmera. A més, l'equació associada al conducte central, expressada en termes de potencial de velocitat acústica, permet la presència de flux axial no uniforme. L'acoblament entre les dues regions s'ha realitzat mitjançant un conducte perforat i la seva impedància acústica i s'ha adaptat per incloure la esmentada falta d'homogeneïtat. S'ha vist que les heterogeneïtats poden influir notablement en l'atenuació acústica d'un silenciador i s'han d'incloure en els models teòrics. Les tècniques d'optimització per a components industrials de control de soroll són importants, ja que produeixen elements amb millors característiques. Els algoritmes evolutius són tècniques emergents capaces d'obtenir una solució, fins i tot quan l'optimització tradicional té dificultats. Les tècniques d'optimització es combinen amb el mètode d'elements finits (MEF) per aconseguir la màxima atenuació possible en el rang de freqüències d'interès. S'ha definit un problema d'optimització d'un silenciador multicàmera i s'han dut a terme diverses anàlisis per obtenir la configuració més adequada per a cada cas. Sota certes hipòtesis d'uniformitat axial, s'han considerat diverses tècniques per reduir el cost computacional d'una anàlisi 3D complet per silenciadors dissipatius amb gradients de temperatura i flux mig. Aquestes es basen en la descomposició del camp acústic en modes axials i transversals dins de cada subdomini, i un procediment d'acoblament de les expansions modals en els canvis de secció del silenciador mitjançant les condicions de continuïtat dels camps acústics. S'estudia l'eficiència computacional i precisió de les prediccions de les tècniques d'acoblament, incloent col·locació puntual en nodes i punts de Gauss, així com ajust modal. Tots ells proporcionen prediccions precises de l'atenuació millorant el cost computacional d'EF. / Sánchez Orgaz, EM. (2016). Advanced numerical techniques for the acoustic modelling of materials and noise control devices in the exhaust system of internal combustion engines [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/64090

Les deux principales prédictions de la théorie quantique des champs en espace-temps courbe, à savoir la radiation de Hawking et la production de paires de particules ayant lieu dans un espace-temps non stationnaire, n'ont jamais été testé expérimentalement et impliquent toutes deux des processus à ultra haute énergie. En conséquence, de telles prédictions doivent être considérées prudemment. En utilisant l'analogie avec des systèmes de matière condensée mise en avant par Unruh, leur analogue pourrait être testé en laboratoire. Par ailleurs, dispersion et dissipation sont toujours présentes dans de tels systèmes, ce qui régularise la théorie à courte distances. Lors d'expériences destinées à tester les prédictions citées ci-dessus, le bruit thermique modifiera le résultat. En effet, il existe une compétition entre l'émission stimulée dudit bruit thermique et l'émission spontanée issue du vide quantique. Afin de mesurer la radiation de Hawking analogue et de l'analogue des productions de paires (souvent appelé effet Casimir dynamique), il est alors nécessaire de calculer les conséquence de la dispersion et de la dissipation, ainsi que d'identifier des observables permettant de certifier que l'amission spontanée a eu lieu. Dans cette thèse, nous analyserons d'abord les effets de la dispersion et de la dissipation à la fois sur la radiation de Hawking et sur la production de paires de particules. Afin d'obtenir des résultats explicites, nous travaillerons avec l'espace-temps de de Sitter. Les symétries de la théorie nous permettront d'obtenir des résultats exacts. Ceux-ci seront alors appliqués aux trous noirs grâce aux ressemblances entre la région proche du trou noir et l'espace de de Sitter. Afin d’introduire de la dissipation, nous considérerons un modèle exactement soluble permettant de modéliser n'importe quel taux de dissipation. Dans un tel modèle, le champ est couplé de manière linéaire à un environnement contenant un ensemble dense de degrés de liberté. Dans un tel contexte, nous étudierons l'intrication des particules produites. Ensuite, nous considérerons des systèmes de matière condensée spécifiques, à savoir les condensats de Bose et les polaritons. Nous analyserons les effets de la dissipation sur l'intrication de l’effet Casimir dynamique. Enfin, nous étudieront de manière générique l'intrication de la radiation de Hawking en présence de dispersion pour des systèmes analogues. / The two main predictions of quantum field theory in curved space-time, namely Hawking radiation and cosmological pair production, have not been directly tested and involve ultra high energy configurations. As a consequence, they should be considered with caution. Using the analogy with condensed matter systems put forward by Unruh, their analogue versions could be tested in the lab. Moreover, the high energy behavior of these systems is known and involved dispersion and dissipation, which regulate the theory at short distances. When considering experiments which aim to test the above predictions, the thermal noise will contaminate the outcome. Indeed, there will be a competition between the stimulated emission from thermal noise and the spontaneous emission out of vacuum. In order to measure the quantum analogue Hawking radiation, or the analogue pair production also called dynamical Casimir effect, one should thus compute the consequences of ultraviolet dispersion and dissipation, and identify observables able to establish that the spontaneous emission took place. In this thesis, we first analyze the effects of dispersion and dissipation on both Hawking radiation and pair particle production. To get explicit results, we work in the context of de Sitter space. Using the extended symmetries of the theory in such a background, exact results are obtained. These are then transposed to the context of black holes using the correspondence between de Sitter space and the black hole near horizon region. To introduce dissipation, we consider an exactly solvable model producing any decay rate. In such a model, the field is linearly coupled to an environment containing a dense set of degrees of freedom. We also study the quantum entanglement of the particles so produced. In a second part, we consider explicit condensed matter systems, namely Bose Einstein condensates and exciton-polariton systems. We analyze the effects of dissipation on entanglement produced by the dynamical Casimir effect. As a final step, we study the entanglement of Hawking radiation in the presence of dispersion for a generic analogue system.

Le métabolisme énergétique est le principal facteur déterminant de la viabilité cellulaire. Les maladies présentent de nombreuses anomalies métaboliques et énergétiques. En effet, les cellules altérées proviennent de procédés exergoniques et émettent de la chaleur vers leur environnement proche. De nombreux processus irréversibles peuvent se produire en modifiant le taux de production d'entropie. Ce niveau représente une quantité thermodynamique qui mesure ces processus irréversibles. Le niveau d'entropie est augmenté par plusieurs anomalies métaboliques et thermodynamiques dans les tumeurs cérébrales, les processus inflammatoires et les maladies neurodégénératives. Les travaux de recherche de cette thèse ont démontré et mis en évidence l'existence d'une diaphonie entre la voie canonique WNT/beta-caténine et le PPAR gamma qui joue un rôle majeur dans la reprogrammation du métabolisme de l'énergie cellulaire entre la phosphorylation oxydative, la glycolyse aérobie et la glycolyse anaérobie, dont le point d'équilibre de cette diaphonie entre ces voies moléculaires varie selon les maladies. Ces maladies sont des structures dissipatives, qui échangent de l'énergie ou de la matière avec leur environnement. Ce sont des systèmes ouverts, loin de l'équilibre thermodynamique qui opèrent sous un régime non linéaire évoluant vers des états non stationnaires. La thermodynamique loin de l'équilibre est une notion axée sur les rythmes circadiens. En effet, les rythmes circadiens participent directement à la régulation de cette diaphonie étudiée. Celle-ci représente une cible innovante dans le cadre l'imagerie moléculaire pour le diagnostic positif et différentiel de ces maladies. / Energy metabolism is the primary determinant of cellular viability. Diseases are the sites of numerous metabolic and energetic production abnormalities. Indeed, the altered cells are derived from exergonic processes and emit heat that flows to the surrounding environment. Many irreversible processes can occur through changing the rate of entropy production. This rate represents a thermodynamic quantity that measures these irreversible processes. Entropy rate is increased by several metabolic and thermodynamics abnormalities in brain tumors, inflammatory processes and neurodegenerative diseases. The research works of this thesis have demonstrated and highlighted the existence of a crosstalk between canonical WNT/beta-catenin pathway and PPAR gamma which plays a major role in the reprogramming of cellular energy metabolism between oxidative phosphorylation, aerobic glycolysis and anaerobic glycolysis, of which the equilibrium point of crosstalk between these molecular pathways varies according to tumor, inflammatory and neurodegenerative diseases. These diseases are dissipative structures, that exchange energy or matter with their environment. They are open systems, far-from the thermodynamic equilibrium that operate under non-linear regime evolving to non-stationary states. Far-from-equilibrium thermodynamics are notions driven by circadian rhythms. Indeed, circadian rhythms directly participate in regulating the crosstalk of the studied molecular pathways. This crosstalk represents an innovative therapeutic target, and molecular data usable for molecular imaging in both positive and differential diagnosis of these diseases.

<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>

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. 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. 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. 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. 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.