Cette thèse répond à une problématique de commande frontière d’une conduite acoustique dont l’actionnement est assuré par un réseau d’actionneurs/capteurs co-localisés constituant une peau active. Pour faire face au caractère intrinsèquement multiphysique de ce problème vibro-acoustique, nous avons choisi dans cette thèse d’employer une approche hamiltonienne à ports, approche structurée basée sur la représentation des échanges entre différents domaines énergétiques au sein d’un système et entre différents systèmes. Nous avons proposé une modélisation hamiltonienne à ports de l’équation d’onde interconnectée à la frontière au système d’actionnement distribué, correspondant à une formulation 2D du problème physique. Nous avons développé une méthode de discrétisation spatiale basée sur l’utilisation de différences finies sur plusieurs grilles en quinconce qui préserve la structure hamiltonienne à ports de l’équation d’onde. Cette méthode permet en outre d’interconnecter facilement le système discrétisé avec d’autres sous-systèmes, dans le but de mettre en place un actionnement par exemple. Son principal avantage sur d’autres méthodes préservatives de structure réside dans sa simplicité de mise en œuvre qui découle de l’utilisation de différences finies. Concernant la commande du système vibro-acoustique, nous avons proposé une méthode de synthèse de loi de commande distribuée pour les systèmes régis par deux lois de conservation en 1D. L’originalité de cette méthode réside en le fait qu’elle repose sur le calcul d’invariants structuraux (fonctions de Casimir) exploités afin de modifier la structure du système en boucle fermée. Les conditions d’application sur un système 2D sont étudiées et des résultats numériques valident les lois de commande synthétisées. / This thesis deals with the boundary control of an acoustic by a network of co-localised sensors/actuators which constitutes a smart skin. In order to cope with this multiphysical problem, we chose to place our study in the framework of port-Hamiltonian systems, a structured approach based on the representation of energy exchanges between different energy domains between different systems of subsystems. We proposed a port-Hamiltonian model of the wave equation interconnected through its boundary to the distributed actuation system, which corresponds to a 2D formulation of the physical problem. We developed a spatial discretization method based on the use of finite differences on several staggered grids that preserve the port-Hamiltonian structure of the wave equation. This method also permits to easily interconnect the discretized system with other subsystems, which is convenient for instance for control purposes. Its main advantage over other structure preserving methods is its simplicity of implementation which stems from the use of finite differences. In order to control the vibro-acoustic system, we proposed a control law synthesis method for systems governed by two conservation laws in 1D. The originality of this method lies in the fact that it relies on the computation of structural invariants (Casimir functions) exploited in order to modify the structure of the system in closed loop. The conditions of application of these laws on a 2D system are studied and numerical results validate the synthesized control laws.

Le travail engagé dans cette thèse porte sur l'étude numérique des Interactions Fluide-structure en hydrodynamique. Dans une première partie, une analyse détaillée des méthodes de couplage (schémas décalés) a été effectuée sur un cas académique. Il s'agit de la résolution de l'équation non-linéaire de Burgers dans un domaine mobile, dont I'interface mobile est représentée par un système de type masse ressort. Selon la discrétisation en temps et la linéarisation du problème couplé, on distingue quatre schémas de couplages différents : explicite, semi-implicite, implicite-externe et implicite-interne. Une étude comparative des performances en vitesse de convergence et en temps de calcul de ces schémas a été effectuée. Les performances varient suivant le schéma de couplage utilisé. Le schéma explicite permet un calcul rapide en comparaison des autres schémas. En revanche il n'assure pas la conservation de l'énergie mécanique à I'interface fluide-structure. D'où le problème de stabilité du schéma numérique. Ce problème ne se pose pas pour les algorithmes de couplage implicites, car dans ce cas la conservation de l'énergie à I'interface est assurée. Il s'agit en effet d'une condition de convergence du schéma implicite. Ce schéma requière plus de temps de calcul, mais il est nécessaire pour avoir plus de précision dans les résultats. Par ailleurs, I'analyse des déplacements de I'interface fluide-structure montre que l'écart entre la position de I'interface comme étant le bord mobile du fluide et la position de la structure, dépend principalement du schéma d'actualisation du maillage choisi.Dans une deuxième partie une extension de l'étude des algorithmes de couplage à un problème plus concret d'IFS est effectuée. Un hydrofoil en pilonnement et tangage est ainsi étudié. L'équation de la dynamique de I'hydrofoil est écrite en considérant un centre de rotation situé à une distance non nulle du centre de gravité.Ce qui rend l'équation non-linéaire et introduit un couplage des deux modes pilonnement et tangage) ainsi qu'un amortissement du tangage. La dynamique de I'hydrofoil est étudiée pour différentes configurations : en mouvement libre ou forcé, dans un fluide au repos ou en écoulement. On observe que le mouvement de I'hydrofoil est pseudo périodique amorti. L'évolution des charges hydrodynamiques suit également cette tendance et tend vers un point d'équilibre. L'étude vibratoire montre bien une modification des fréquences propres du système, qui varient suivant que le fluide est au repos ou en écoulement. Le problème est également couplé à l'équation de la position du centre de pression, qui dépend de la position de I'hydrofoil et de l'écoulement. Celle-ci présente une singularité lorsque la portance et la traînée s'annulent simultanément.Enfin Les équations prenant en compte la présence d'un fluide non-homogène à I'interface fluide-structure, du type des écoulements cavitants par poche stationnaire ou auto-oscillante, ont été développés. La méthode consiste à séparer les variables du fluide en écoulement autour d'un hydrofoil immobile d'une part et celles de l'écoulement généré par la vibration de I'hydrofoil d'autre part. Il en résulte un opérateur de masse ajoutée non symétrique en milieu non homogène et un opérateur d'amortissement ajouté dû au taux de variations de masse volumique à l’interface dans le cas auto-oscillant. L'ensemble se traduit par une modulation au cours du temps des fréquences propres et des amplitudes du système. / A numerical study of Fluid Structure Interaction (FSI) in hydrodynamic case is adressed in this thesis. Thirstly, the analysis of coupling methods (staggered schemes) was established to an academic case. It corresponds to the resolution of non linear Burgers equation in a moving domain where the moving interface is assimilated to a mass spring system. According to the time discretisation and linearization of the coupled problem, four coupling scheme can be defined : explicit, semi-implicit, implicit-outer and implicit-inner. A comparative performance study in convergence and computing time were performed. The performance depends on the coupling scheme used. The explicit scheme requires less time compared to the others schemes. However it does not allow the mechanical energy conservation at the interface, inducing the stability issue of the numerical scheme. This instabilities does not arise for the implicit coupling algorithms because the energy conservation at the interface is fulfilled. lndeed, a convergence condition is added for implicit schemes. Even though these schemes require more computing time, they are necessary to get better precision. Inter alia, the fluid-structure interface analysis shows that the gap between the interface taken as the moving boundary and the structure position mostly depends on the actualization scheme of the chosen mesh.In the second part, the coupling algorithm study is extended to physical problem of FSI. A hydrofoil in heave and pitch immersed in a fluid flow is then studied. The equation of hydrofoil movement takes account the distance between the rotation center and the center of gravity. This causes the equation to be nonlinear and introduces a coupling of the two movements (heave and pitch) and a damping of the heave movement. The hydrofoil dynamic is studied for different configurations : forced movements or not, immersed in a fluid at rest or a flowing one. It shows that the hydrofoil movement is pseudo-periodic followed by a damping movement. The hydrodynamic forces tend to follow the same evolution and converge to an equilibrium point. The vibration study clearly shows a frequency modification of the system that depends on the fluid flow (at rest or with an inflow). The problem is also coupled to center of pressure position's equation which depends on the hydrofoil position and the fluid flow. The trend of the position presents a singularity when the lift and drag coefficients vanishes at the same time.Last part, the equation that take into account the inhomogeneous characteristic of the fluid at the fluid-structure interface as well as sheet cavitation in steady or unsteady case, was developed. The method allows the separation of the fluid variables when flowing around the fixed hydrofoil on one hand and the flow generated by the hydrofoil vibration one the other. This introduces an asymmetric added mass operator and an added damping operation due to the variation of the density of the fluid at the interface in unsteady case.The whole system results in a natural frequencies and amplitudes modulation over time.

A esfera é utilizada como domínio computacional na modelagem de diversos fenômenos físicos, como em previsão numérica do tempo. Sua discretização pode ser feita de diversas formas, sendo comum o uso de malha regulares em latitude/longitude. Recentemente, também para melhor uso de computação paralela, há uma tendência ao uso de malhas mais isotrópicas, dentre as quais a icosaédrica. Apesar de já existirem modelos atmosféricos que usam malhas icosaédricas, não há consenso sobre as metodologias mais adequadas a esse tipo de malha. Nos propusemos, portanto, a estudar em detalhe diversos fatores envolvidos no desenvolvimento de modelos atmosféricos globais usando malhas geodésicas icosaédricas. A discretização usual por volumes finitos para divergente de um campo vetorial utiliza como base o Teorema da Divergência e a regra do ponto médio nas arestas das células computacionais. A distribuição do erro obtida com esse método apresenta uma forte relação com características geométricas da malha. Definimos o conceito geométrico de alinhamento de células computacionais e desenvolvemos uma teoria que serve de base para explicar interferências de malha na discretização usual do divergente. Destacamos os impactos de certas relações de alinhamento das células na ordem da discretização do método. A teoria desenvolvida se aplica a qualquer malha geodésica e também pode ser usada para os operadores rotacional e laplaciano. Investigamos diversos métodos de interpolação na esfera adequados a malhas icosaédricas, e abordamos o problema de interpolação e reconstrução vetorial na esfera em malhas deslocadas. Usamos métodos alternativos de reconstrução vetorial aos usados na literatura, em particular, desenvolvemos um método híbrido de baixo custo e boa precisão. Por fim, utilizamos as técnicas de discretização, interpolação e reconstrução vetorial analisadas em um método semi-lagrangiano para o transporte na esfera em malhas geodésicas icosaédricas. Realizamos experimentos computacionais de transporte, incluindo testes de deformações na distribuição do campo transportado, que mostraram a adequação da metodologia para uso em modelos atmosféricos globais. A plataforma computacional desenvolvida nesta tese, incluindo geração de malhas, interpolações, reconstruções vetoriais e um modelo de transporte, fornece uma base para o futuro desenvolvimento de um modelo atmosférico global em malhas icosaédricas. / Spherical domains are used to model many physical phenomena, as, for instance, global numerical weather prediction. The sphere can be discretized in several ways, as for example a regular latitude-longitude grid. Recently, also motivated by a better use of parallel computers, more isotropic grids have been adopted in atmospheric global circulation models. Among those, the icosahedral grids are promising. Which kind of discretization methods and interpolation schemes are the best to use on those grids are still a research subject. Discretization of the sphere may be done in many ways and, recently, to make better use of computational resources, researchers are adopting more isotropic grids, such as the icosahedral one. In this thesis, we investigate in detail the numerical methodology to be used in atmospheric models on icosahedral grids. The usual finite volume method of discretization of the divergence of a vector field is based on the divergence theorem and makes use of the midpoint rule for integration on the edges of computational cells. The error distribution obtained with this method usually presents a strong correlation with some characteristics of the icosahedral grid. We introduced the concept of cell alignment and developed a theory which explains the grid imprinting patterns observed with the usual divergence discretization. We show how grid alignment impacts in the order of the divergence discretization. The theory developed applies to any geodesic grid and can also be used for other operators such as curl and Laplacian. Several interpolation schemes suitable for icosahedral grids were analysed, including the vector interpolation and reconstruction problem on staggered grids. We considered alternative vector reconstruction methods, in particular, we developed a hybrid low cost and good precision method. Finally, employing the discretizations and interpolations previously analysed, we developed a semi-Lagrangian transport method for geodesic icosahedral grids. Several tests were carried out, including deformational test cases, which demonstrated that the methodology is suitable to use in global atmospheric models. The computational platform developed in this thesis, including mesh generation, interpolation, vector reconstruction and the transport model, provides a basis for future development of global atmospheric models on icosahedral grids.

Cette thèse porte sur l'approximation numérique des équations de Saint-Venant et de quelques problèmes qui leur sont reliés. Dans la première partie, nous analysons les propriétés mathématiques et les applications des schémas numériques sur grilles décalées. La robustesse de ces schémas est prouvée sur des applications telles que les équations de Saint-Venant dans un domaine en rotation, en vue des écoulements géostrophiques, ainsi que l'extension de ces équations au cas visqueux. Dans la seconde partie, nous présentons des modèles basés sur les équations de Saint-Venant. Nous commençons par étudier le couplage avec l'équation d'Exner, qui porte sur le transport des sédiments. Nous observons des propriétés de convergence numérique vers la solution exacte dans un cas de solution analytique, et nous constatons un bon accord avec des données expérimentales dans le cas de la rupture de barrage avec fond érodable. Nous continuons par l'étude d'un schéma numérique, basé sur une méthode de volumes finis colocalisés (HLLC) pour l'approximation du modèle de Richard-Gavrilyuk. Ce modèle étend les équations de Saint-Venant au cas des écoulements avec cisaillement. Des tests numériques montrent la validité du schéma / This thesis is devoted to the numerical approximation of the shallow water equations and of some related models. In the first part, we analyze the mathematical properties and the applications of the staggered grid scheme. The robustness of this scheme is validated on various applications such as the rotating shallow water equations for geostrophic flows model and viscous shallow water equations. In the second part, we consider some related models. Firstly focusing on the coupling between the Exner equation and the shallow water equations, modelling bedload sediment transport, we observe in a particular case the numerical convergence of the scheme to the exact solution, as well as a good agreement with the experimental data in the dam-break with erodible bottom test. Secondly, we present a numerical scheme based on the finite volume collocated scheme (HLLC) in order to approximate the Richard-Gavrilyuk model. This model is an extension of the shallow water model, fit for modelling the shear shallow water flows. Some numerical tests provide a validation of the scheme

Aufgrund der großen Bedeutung technischer Gummiwerkstoffe in industriellen Anwendungen ist die Vorhersage des Materialverhaltens ein aktuelles Forschungsgebiet. Insbesondere die Veränderung der Eigenschaften, die durch chemische Prozesse herbeigeführt werden, spielen eine große Rolle, da aufgrund gestiegener Anforderungen an die Haltbarkeit von Bauteilen diese Alterungseffekte verstanden, vorhergesagt und abgeschwächt werden müssen. Im Rahmen dieser Arbeit wird ein mathematisches Modell hergeleitet, das die chemischen Vorgänge vorhersagt und deren Auswirkungen auf wichtige Eigenschaften in einem mechanischen Materialmodell beachtet. Insbesondere der Oxidation durch Luftsauerstoff wird Rechnung getragen. Das mechanische Materialverhalten alternder Gummiwerkstoffe wird mit Hilfe eines neuartigen Ansatzes modelliert. Das in dieser Arbeit vorgestellte Dynamische-Netzwerk-Modell betrachtet die Auswirkungen der chemischen Reaktionen auf das mechanische Verhalten als einen kontinuierlichen Netzwerkumbau durch das Entfernen und neue Einsetzen von Verbindungen zwischen Polymerketten. Basierend auf experimentellen Erkenntnissen werden Hypothesen formuliert, die eine Kopplung zwischen den Oxidationsreaktionen und der Veränderung des mechanischen Verhaltens herstellen. Durch Beachtung von Diffusion und Reaktion kann die lokale Sauerstoffverteilung in großvolumigen Bauteilen berechnet und der Effekt der diffusionslimitierten Oxidation (DLO-Effekt) vorhergesagt werden. Um eine Bestimmung der Modellparameter zu ermöglichen, werden verschiedene Experimente vorgeschlagen, mathematisch modelliert und deren Eignung zur Parameteridentifikation analysiert. Mit einer vergleichsweise geringen Anzahl von Experimenten können sowohl die chemischen als auch die mechanischen Modellparameter sicher identifiziert werden. In dieser Arbeit wird zudem ein sogenannter gestaffelter Lösungsalgorithmus vorgeschlagen, der das Alterungsproblem nach einer mathematischen Entkopplung unterschiedlicher Feldprobleme über geeignete Kopplungsvariablen effizient berechnen kann. Mit Hilfe dieses Algorithmus kann eine Simulation effizient durchgeführt und das Alterungsverhalten komplexer Strukturen vorhergesagt werden. Die Simulation anwendungsnaher Strukturen zeigt, dass die abgeleiteten Modelle und Algorithmen problemlos in einem industriellen Umfeld einsetzbar sind. / Due to the great importance of rubber materials in industrial applications, the prediction of the material behavior is a current research field. Particularly the property changes that are induced by chemical processes play a major role, as these aging effects must be understood, predicted and reduced due to the increased requirements regarding the durability of components. In the context of this thesis, a mathematical model that predicts and considers the chemical processes and their effects on important properties in a mechanical material model is derived. Oxidation through atmospheric oxygen is specifically examined in this work. The mechanical material behavior of aging rubber materials is modeled using a novel approach. The dynamic network model introduced in this paper considers the effects of the chemical reactions on the mechanical behavior as a continuous restructuring of the network by removing and inserting new crosslinks between polymer chains. Based on experimental obervations a coupling between the oxidation reactions and a change of the mechanical material behavior is proposed. By taking into account the diffusion and reaction the local distribution of oxygen in large sized components can be computed. Thus, the effect of diffusion limited oxidation (DLO effect) can be predicted. In order to make an identification of the model parameters possible, different experiments are suggested, modeled mathematically and their suitability for parameter identification is analyzed. Not only the chemical, but also the mechanical model parameters can be identified reliably using a comparably few number of experiments. Moreover, a so-called staggered solution algorithm that can calculate the aging problem efficiently after a mathematical uncoupling of the field problems using a suitable coupling variable is introduced. This algorithm can perform a simulation efficiently and predict the aging behavior of complex structures. The simulation of application oriented structures proofs the applicability of the derived models and algorithms within an industrial environment.

The present rate of fossil fuel burning, the main source of commercial energy, is adversely impacting global climate. Present social and economic practices need to be examined to question their level of energy use and related greenhouse gas emission. Energy used to operate buildings and to transport people is a significant portion of the total energy consumption and reductions in these uses will be crucial to addressing the global issues. This thesis examines the potential for energy use reduction in the performance of office work through the use of Alternative Work Arrangements ("AWAs"). The analysis considers the operating energy of an office building, operating energy of alternative work locations and the transportation energy spent by employees in commuting from home to work. A brief synopsis of the present atmospheric, energy use and workplace trend is presented. The synthesis of these trends is used as a framework to evaluate the impact of AWAs. The work arrangements in the Burnaby Fraser Tax Services Office ('BFTSO'), Surrey, BC are used as a case study to analyze the operating and commuting energy spent by the employees working by different arrangements. Telework is found to be the most energy efficient work arrangement implemented at the BFTSO. Under optimum operating conditions, net energy saving per teleworker can be 34.06 GJ per annum. Net savings in greenhouse gas emission per teleworker can be 1950 kg per annum. For hoteling work arrangement, net energy savings per hoteling employee under optimum conditions can be 32 GJ per annum. Related savings in greenhouse gas emissions can be 1532 kg per annum. The proportion of total operating and total commuting energy for the BFTSO is almost equal. However, the greenhouse gas emission from total commuting energy is three times that from operating energy due to differences in carbon intensity of the fuel mix. Under optimum conditions the implementation of AWAs at the BFTSO can result in 36 percent saving in total operating energy and 43 percent saving in total commuting energy. The magnitude of the savings is location dependent due to differences in climate, fuel mix and transportation patterns. Alternative Work Arrangements have mixed social impact. It can improve the ability of employees to balance their personal and professional life. It can also lead to isolation that can adversely impact morale and work output. The participation rate in AWAs within organizations is currently low, less than 2 percent. However, the implementation of AWA is widespread with most organizations using some form of AWA. To increase the participation rate in AWAs, a comprehensive approach needs to be adopted, considering the professional, personal, economic and social impact of AWAs. Government intervention through supporting regulations and incentives can be a strong catalyst for increasing participation rates in AWAs.

The present rate of fossil fuel burning, the main source of commercial energy, is adversely impacting global climate. Present social and economic practices need to be examined to question their level of energy use and related greenhouse gas emission. Energy used to operate buildings and to transport people is a significant portion of the total energy consumption and reductions in these uses will be crucial to addressing the global issues. This thesis examines the potential for energy use reduction in the performance of office work through the use of Alternative Work Arrangements ("AWAs"). The analysis considers the operating energy of an office building, operating energy of alternative work locations and the transportation energy spent by employees in commuting from home to work. A brief synopsis of the present atmospheric, energy use and workplace trend is presented. The synthesis of these trends is used as a framework to evaluate the impact of AWAs. The work arrangements in the Burnaby Fraser Tax Services Office ('BFTSO'), Surrey, BC are used as a case study to analyze the operating and commuting energy spent by the employees working by different arrangements. Telework is found to be the most energy efficient work arrangement implemented at the BFTSO. Under optimum operating conditions, net energy saving per teleworker can be 34.06 GJ per annum. Net savings in greenhouse gas emission per teleworker can be 1950 kg per annum. For hoteling work arrangement, net energy savings per hoteling employee under optimum conditions can be 32 GJ per annum. Related savings in greenhouse gas emissions can be 1532 kg per annum. The proportion of total operating and total commuting energy for the BFTSO is almost equal. However, the greenhouse gas emission from total commuting energy is three times that from operating energy due to differences in carbon intensity of the fuel mix. Under optimum conditions the implementation of AWAs at the BFTSO can result in 36 percent saving in total operating energy and 43 percent saving in total commuting energy. The magnitude of the savings is location dependent due to differences in climate, fuel mix and transportation patterns. Alternative Work Arrangements have mixed social impact. It can improve the ability of employees to balance their personal and professional life. It can also lead to isolation that can adversely impact morale and work output. The participation rate in AWAs within organizations is currently low, less than 2 percent. However, the implementation of AWA is widespread with most organizations using some form of AWA. To increase the participation rate in AWAs, a comprehensive approach needs to be adopted, considering the professional, personal, economic and social impact of AWAs. Government intervention through supporting regulations and incentives can be a strong catalyst for increasing participation rates in AWAs. / Applied Science, Faculty of / Architecture and Landscape Architecture (SALA), School of / Graduate

La promulgation de la loi de la sécurité financière (LSF) en 2003 a modifié l’architecture institutionnelle et la pratique d’audit en France. Au travers l’adjonction de nouvelles règles imposées par cette loi aux spécificités persistantes depuis 1966 comme le co-commissariat aux comptes et la durée du mandat de six ans, le marché d’audit se montre unique. Cette recherche examine l’impact de cette fusion réglementaire sur la qualité et le coût de l’audit en France. En se basant sur un échantillon composé par 888 observations du SBF 250 sur la période 2005-2010, nous avons révélé l’apport de la rotation des équipes signataires dans la restriction de la discrétion managériale. Outre son avantage dans la promotion de la qualité de l’audit, ce dispositif favorise la négociation à la baisse des honoraires d’audit. Nous avons aussi souligné l’évolution sous forme parabolique de la composante discrétionnaire sur la durée de mandat. En effet, la qualité de l’audit est inférieure pendant la première et la dernière phase de l’engagement. Ce postulat renvoie au renforcement de la compétence dans le temps et l’érosion de l’esprit critique à la fin du mandat. Le décalage, issu de la fusion réglementaire, favorise le maintien du niveau de la qualité élevé sur le marché français. L’association des deux équipes avec des durées auditeur-audité différentes est une contrainte à la discrétion managériale. Toutefois, ce décalage ne doit pas être excessif pour limiter la domination du plus ancien. Cet inconvénient du régime du co-commissariat aux comptes s’est trouvé limité sur le marché français. Nos résultats confirment son efficacité, essentiellement dans le cas d’un collège composé par deux Big 4. / In 2003, the promulgation of the financial security act (LSF) has changed the institutional architecture and audit practice in France. Through the addition of new rules imposed by this act to the persistent specificities since 1966 as the joint audit and tenure of six years, the audit market shows unique. This research examines the impact of this regulatory merger on the quality and the cost of the audit in France. Based on a sample composed by 888 observations of the SBF 250 over the 2005-2010, we found the contribution of rotation audit partners in restricting managerial discretion. Besides its advantage in promoting audit quality, this device supports the negotiation of lower audit fees. We also emphasized the evolution in the parabolic form of the discretionary accruals over tenure. Indeed, the audit quality is lower during the first and last phase of the engagement. This assumption refers to strengthening the skills over time and erosion of critical appraisal at the end of audit tenure. The shift from regulatory fusion helps to maintain the level of high quality on the French market. The combination of the two teams with different listener-audited tenures is a constraint to managerial discretion. However, this shift should not be excessive to limit the oldest’ domination. This drawback of the system of joint audits is limited in the French market. Our results confirm its effectiveness, mainly in the case of two Big 4.

This thesis addresses issues related to the simulation and optimization of the injection molding of short fiber-reinforced plastics (SFRPs). The injection molding process is modeled by a two phase flow problem. The simulation of the two phase flow is accompanied by the solution of the Folgar-Tucker equation (FTE) for the simulation of the moments of fiber orientation densities. The FTE requires the solution of the so called 'closure problem'', i.e. the representation of the 4th order moments in terms of the 2nd order moments. In the absence of fiber-fiber interactions and isotropic initial fiber density, the FTE admits an analytical solution in terms of elliptic integrals. From these elliptic integrals, the closure problem can be solved by a simple numerical inversion. Part of this work derives approximate inverses and analytical inverses for special cases of fiber orientation densities. Furthermore a method is presented to generate rational functions for the computation of arbitrary moments in terms of the 2nd order closure parameters. Another part of this work treats the determination of effective material properties for SFRPs by the use of FFT-based homogenization methods. For these methods a novel discretization scheme, the 'staggered grid'' method, was developed and successfully tested. Furthermore the so called 'composite voxel'' approach was extended to nonlinear elasticity, which improves the approximation of material properties at the interfaces and allows the reduction of the model order by several magnitudes compared to classical approaches. Related the homogenization we investigate optimal experimental designs to robustly determine effective elastic properties of SFRPs with the least number of computer simulations. Finally we deal with the topology optimization of injection molded parts, by extending classical SIMP-based topology optimization with an approximate model for the fiber orientations. Along with the compliance minimization by topology optimization we also present a simple shape optimization method for compensation of part warpage for an black-box production process.:Acknowledgments v Abstract vii Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Nomenclature 3 Chapter 2. Numerical simulation of SFRP injection molding 5 2.1 Introduction 5 2.2 Injection molding technology 5 2.3 Process simulation 6 2.4 Governing equations 8 2.5 Numerical implementation 18 2.6 Numerical examples 25 2.7 Conclusions and outlook 27 Chapter 3. Numerical and analytical methods for the exact closure of the Folgar-Tucker equation 35 3.1 Introduction 35 3.2 The ACG as solution of Jeffery's equation 35 3.3 The exact closure 36 3.4 Carlson-type elliptic integrals 37 3.5 Inversion of R_D-system 40 3.6 Moment tensors of the angular central Gaussian distribution on the n-sphere 49 3.7 Experimental evidence for ACG distribution hypothesis 54 3.8 Conclusions and outlook 60 Chapter 4. Homogenization of SFRP materials 63 4.1 Introduction 63 4.2 Microscopic and macroscopic model of SFRP materials 63 4.3 Effective linear elastic properties 65 4.4 The staggered grid method 68 4.5 Model order reduction by composite voxels 80 4.6 Optimal experimental design for parameter identification 93 Chapter 5. Optimization of parts produced by SFRP injection molding 103 5.1 Topology optimization 103 5.2 Warpage compensation 110 Chapter 6. Conclusions and perspectives 115 Appendix A. Appendix 117 A.1 Evaluation of R_D in Python 117 A.2 Approximate inverse for R_D in Python 117 A.3 Inversion of R_D using Newton's/Halley's method in Python 117 A.4 Inversion of R_D using fixed point method in Python 119 A.5 Moment computation using SymPy 120 A.6 Fiber collision test 122 A.7 OED calculation of the weighting matrix 123 A.8 OED Jacobian of objective and constraints 123 Appendix B. Theses 125 Bibliography 127 / Diese Arbeit befasst sich mit Fragen der Simulation und Optimierung des Spritzgießens von kurzfaserverstärkten Kunststoffen (SFRPs). Der Spritzgussprozess wird durch ein Zweiphasen-Fließproblem modelliert. Die Simulation des Zweiphasenflusses wird von der Lösung der Folgar-Tucker-Gleichung (FTE) zur Simulation der Momente der Faserorientierungsdichten begleitet. Die FTE erfordert die Lösung des sogenannten 'Abschlussproblems'', d. h. die Darstellung der Momente 4. Ordnung in Form der Momente 2. Ordnung. In Abwesenheit von Faser-Faser-Wechselwirkungen und anfänglich isotroper Faserdichte lässt die FTE eine analytische Lösung durch elliptische Integrale zu. Aus diesen elliptischen Integralen kann das Abschlussproblem durch eine einfache numerische Inversion gelöst werden. Ein Teil dieser Arbeit leitet approximative Inverse und analytische Inverse für spezielle Fälle von Faserorientierungsdichten her. Weiterhin wird eine Methode vorgestellt, um rationale Funktionen für die Berechnung beliebiger Momente in Bezug auf die Abschlussparameter 2. Ordnung zu generieren. Ein weiterer Teil dieser Arbeit befasst sich mit der Bestimmung effektiver Materialeigenschaften für SFRPs durch FFT-basierte Homogenisierungsmethoden. Für diese Methoden wurde ein neuartiges Diskretisierungsschema 'staggerd grid'' entwickelt und erfolgreich getestet. Darüber hinaus wurde der sogenannte 'composite voxel''-Ansatz auf die nichtlineare Elastizität ausgedehnt, was die Approximation der Materialeigenschaften an den Grenzflächen verbessert und die Reduzierung der Modellordnung um mehrere Größenordnungen im Vergleich zu klassischen Ansätzen ermöglicht. Im Zusammenhang mit der Homogenisierung untersuchen wir optimale experimentelle Designs, um die effektiven elastischen Eigenschaften von SFRPs mit der geringsten Anzahl von Computersimulationen zuverlässig zu bestimmen. Schließlich beschäftigen wir uns mit der Topologieoptimierung von Spritzgussteilen, indem wir die klassische SIMP-basierte Topologieoptimierung um ein Näherungsmodell für die Faserorientierungen erweitern. Neben der Compliance-Minimierung durch Topologieoptimierung stellen wir eine einfache Formoptimierungsmethode zur Kompensation von Teileverzug für einen Black-Box-Produktionsprozess vor.:Acknowledgments v Abstract vii Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Nomenclature 3 Chapter 2. Numerical simulation of SFRP injection molding 5 2.1 Introduction 5 2.2 Injection molding technology 5 2.3 Process simulation 6 2.4 Governing equations 8 2.5 Numerical implementation 18 2.6 Numerical examples 25 2.7 Conclusions and outlook 27 Chapter 3. Numerical and analytical methods for the exact closure of the Folgar-Tucker equation 35 3.1 Introduction 35 3.2 The ACG as solution of Jeffery's equation 35 3.3 The exact closure 36 3.4 Carlson-type elliptic integrals 37 3.5 Inversion of R_D-system 40 3.6 Moment tensors of the angular central Gaussian distribution on the n-sphere 49 3.7 Experimental evidence for ACG distribution hypothesis 54 3.8 Conclusions and outlook 60 Chapter 4. Homogenization of SFRP materials 63 4.1 Introduction 63 4.2 Microscopic and macroscopic model of SFRP materials 63 4.3 Effective linear elastic properties 65 4.4 The staggered grid method 68 4.5 Model order reduction by composite voxels 80 4.6 Optimal experimental design for parameter identification 93 Chapter 5. Optimization of parts produced by SFRP injection molding 103 5.1 Topology optimization 103 5.2 Warpage compensation 110 Chapter 6. Conclusions and perspectives 115 Appendix A. Appendix 117 A.1 Evaluation of R_D in Python 117 A.2 Approximate inverse for R_D in Python 117 A.3 Inversion of R_D using Newton's/Halley's method in Python 117 A.4 Inversion of R_D using fixed point method in Python 119 A.5 Moment computation using SymPy 120 A.6 Fiber collision test 122 A.7 OED calculation of the weighting matrix 123 A.8 OED Jacobian of objective and constraints 123 Appendix B. Theses 125 Bibliography 127