The principal objective of this investigation is to develop a nonlinear continuum based finite element formulation to examine dynamic response of flexible riser structures with large displacement and large rotation. Updated Lagrangian incremental approach together with the 2nd Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor is employed to derive the nonlinear finite element formulation. The 2nd Piola-Kirchhoff stress and the Green-Lagrange strain tensors are energy conjugates. These two Lagrangian tensors are not affected by rigid body rotations. Thus, they are used to describe the equilibrium equation of the body independent of rigid rotations. While the current configuration in Updated Lagrangian incremental approach is unknown, the resulting equation becomes strongly nonlinear and has to be modified to a linearized form. The main contribution of this work is to obtain a modified linearization method during development of incremental Updated Lagrangian formulation for large displacement and large rotation analysis of riser structures. For this purpose, the Green-Lagrange strain and the 2nd Piola-Kirchhoff stress tensors are decomposed into two second-order six termed functions of through-thethickness parameters. This decomposition makes it possible to explicitly account for the nonlinearities in the direction along the riser thickness, as well. It is noted that using this linearization scheme avoids inaccuracies normally associated with other linearization schemes. The effects of buoyancy force, riser-seabed interaction as well as steady-state current loading are considered in the finite element solution for riser structure response. An efficient riser problem fluid-solid interaction Algorithm is also developed to maintain the quality of the mesh in the vicinity of the riser surface during riser and fluid mesh movements. To avoid distortions in the fluid mesh two different approaches are proposed to modify fluid mesh movement governing elasticity equation matrices values; 1) taking the element volume into account 2) taking both element volume and distance between riser centre and element centre into account. The formulation has been implemented in a nonlinear finite element code and the results are compared with those obtained from other schemes reported in the literature.

A preliminary investigation of the added properties called added mass, added damping and added stiffness have been performed for a Kaplan turbine. The magnitude of dimensionless numbers have been used in order to classify the interaction of the fluid and the solid. The classification is done to bring clarity in which of the added properties are of importance for the system. The diameter of the runner and the hub have been calculated using the power output and the head for a Kaplan turbine. These dimensions have been used to determine the magnitude of the dimensionless numbers along with the velocity of the fluid. It turned out that all added properties affect the turbine, however, the magnitude of them are quite different. The magnitude of the added mass and the added damping are greater than the added stiffness, which often is neglected. The added mass can be determined if the natural frequencies of the structure in air and in water are known. The difference in natural frequencies can be used to determine the added mass factor and thereby the added mass of the system. The added damping can be determined by the change in damping ratio for different surrounding fluids. This was done using the simulation software ANSYS Workbench v.17.1, where two different types of simulation were used, ”acoustic coupled simulation” and ”two way coupled simulation”. The complexity of the geometry of the Kaplan turbine was simplified to a disc and a shaft. The result for the added mass was validated using results from an experiment [1]. The added damping could be determined, but not validated. The different types of simulation have been compared and it turned out that the added mass could be determined using ”acoustic coupled simulation” and ”two way coupled simulation”, but the added damping could only be determined using the ”two way coupled simulation”. / En preliminär undersökning av de adderade egenskaperna kallade, adderad massa, adderad dämpning och adderad styvhet har utförts för en Kaplan turbin. Magnituden av dimensionslösa tal har använts för att klassificera interaktionen av fluiden och soliden. Klassificeringen görs för att bringa klarhet i vilka av de adderade egenskaperna är av betydelse för systemet. Diametrarna för löphjulet och navet har beräknats utifrån effekt och fallhöjd för en Kaplan turbin. Dessa längder har använts för att bestämma magnituden av de dimensionslösa talen tillsammans med fluidens hastighet. Det visade sig att alla adderade egenskaper påverkar turbinen, men omfattningen av dem är helt annorlunda. Magnituden av den adderade massan och den adderade dämpningen är större än den adderade styvheten, som ofta försummas. Den adderade massan kan bestämmas om de naturliga frekvenserna av strukturen i luft och vatten är kända. Skillnaden i egenfrekvenser kan användas för att bestämma faktorn av den adderade massan och därigenom den adderade massan. Den adderade dämpningen kan bestämmas genom ändringen i dämpningsförhållande för olika omgivande fluider. Detta gjordes med hjälp av simuleringsprogrammet ANSYS Workbench v.17.1, där två olika typer av simulering användes, ”acoustic coupled simulation” och ”two way coupled simulation”. Komplexiteten i geometrin för en Kaplan turbin förenklades till en skiva och en axel. Resultatet för den adderade massan validerades med resultat från ett experiment [1]. Den adderade dämpningen kunde bestämmas, men inte valideras. De olika typerna av simulering har jämförts och det visade sig att den adderade massan kan bestämmas med hjälp av både ”acoustic coupled simulation” och ”two way coupled simulation”, men den adderade dämpningen kunde endast bestämmas med hjälp av ”two way coupled simulation”.

Un modèle mathématique d'interfaces diffuses pour l'interaction de N solides élasto-plastiques a été construit. C'est une extension du modèle développé par Favrie & Gavrilyuk (2012) pour l'interaction d'un fluide et d'un solide. En dépit du grand nombre d'équations présentes dans ce modèle, deux propriétés remarquables ont été démontrées : ce modèle est hyperbolique (quelles que soient les déformations admissibles) et il vérifie le second principe de la thermodynamique. En dépit du grand nombre d'équations présentes dans ce modèle, deux propriétés remarquables ont été démontrées: ce modèle est hyperbolique (quelles que soient les déformations admissibles) et il vérifie le second principe de la thermodynamique. L'énergie interne de chaque solide est prise sous forme séparable: c'est la somme d'une énergie hydrodynamique qui ne dépend que de la densité et de l'entropie, et d'une énergie de cisaillement. L'équation d'état de chaque solide est telle que si nous prenons le module de cisaillement du solide égale à zéro, on retrouve les équations de la mécanique des fluides. Ce modèle permet, en particulier, de:- prédire les déformations de solides élasto-plastiques en petites déformations et en très grandes déformations.- prédire l'interaction d'un nombre arbitraire de solides élasto-plastiqueset de fluides. L'aptitude de ce modèle à résoudre des problèmes complexes a été démontrée. Sans être exhaustif, on peut citer:-le phénomène d'écaillage dans les solides.- La fracturation et la fragmentation dynamique dans les solides. / A mathematical model of diffuse interface for the interaction of N elasto-plastic solidS was built. It is an extension of the model developed by Favrie & Gavrilyuk (2012) for a fluid-solid interaction. Despite the large number of equations present in this model, two remarkable properties have been demonstrated: it is hyperbolic for any admissible deformations and satisfies the second principle of thermodynamics. In this model, the internal energy of each solid is taken in separable form: it is the sum of a hydrodynamic energy (which depends only on the density and entropy) and shear energy. The equation of state of each solid is such that if we take the shear modulus of the solid vanishes, we find the equations of fluid mechanics. This model allows, in particular:- predict the deformation of elastic-plastic solids in small and very large deformations.- predict the interaction of an arbitrary number of elasto-plastic solids and fluids.The ability of this model to solve complex problems has been demonstrated. Without being exhaustive, one can mention:- the spall phenomenon in solids.- fracturing and fragmentation in solids.

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This doctoral research project aims the study of finite element methods discretized in dynamic meshes in order to simulate fluid-solid interaction and multiphase flow phenomena, particularly flows involving phenomena that are most significant in microfluidic and biofluidic applications. The equations that model multiphase flow will be treated in an arbitrary Lagrangian-Eulerian framework, when required, with several types of boundary conditions at the interfaces, depending on the nature of the phases. The numerical challenges found in this application range from the correct representation of the interface between fluids, passing through geometric challenges in the maintainability of the computer mesh, to the challenges posed by microscales. Special attention is given to squimer models, by means of a general formulation of the swimming problem as well as the steps to transform a standard fluid-solid model to a squirming model. / Este projeto de pesquisa de doutorado visa o estudo de métodos de elementos finitos discretizados em malhas dinâmicas, com o objetivo de simular fenómenos de interação fluídoestrutura e escoamentos multifásicos, em especial escoamentos que envolvem aplicações em microfluídica e biofluídica. As equações que modelam escoamentos multifásicos serão tratadas em um referencial lagrangeano-euleriano arbitrário, quando requerido, com differentes tipos de condições de contorno nas interfaces, dependendo da natureza das fases. Os desafios numéricos impostos por esta aplicação vão desde a correta representação das interfaces, passando por desafios geométricos na manutenibilidade da malha computacional, até os desafios impostos pelas escalas microscópicas. Atenção especial é dada aos modelos de squimer, por meio de uma formulação geral do problema de natação, bem como os passos para transformar um modelo fluido-sólido padrão em um modelo de squirming.

In order to reach climate and environmental goals, Sweden is increasing the implementation of intermittent renewable energy sources such as wind and solar power to the electricity grid. The increase of intermittent energy sources is rising power regulation requirement towards hydropower, which increasingly exposes the hydraulic turbines to high loads and fluctuating hydraulic forces. These conditions affect the turbine’s structural and rotor dynamic behavior, leading to fatigue in turbine components. Identifying the parameters that affect the dynamics of the water turbine is an essential part of analyzing and, if possible, avoiding these situations. Furthermore, accurate rotor dynamic models are necessary to design for a robust hydropower unit and improve the estimate of wear on turbine components. Added parameters (added mass, polar moment of inertia, and damping) are hydrodynamic effects occurring due to interaction between structural vibrations and surrounding fluid. Added parameters can modify the turbine’s natural frequencies and consequently its dynamic behavior. Therefore, it is of interest to study and quantify the impact of these parameters on the turbine for accurate rotor dynamic modeling and turbine design. The added parameters have been investigated by conducting experiments on a model Kaplan runner, for which the project has been divided into two consecutive parts. First, experiments were performed in a test rig, in which the runner was excited in a lateral movement to determine added mass and linear damping. Secondly, experiments were performed in a test rig similar to the first, except the runner was excited in a torsional movement to determine added polar moment of inertia and torsional damping. Force and displacement have been measured during both movements, with the runner placed in air and thereafter in quiescent water. The added parameters were quantified by comparing measurements conducted with the runner in air against those conducted in water. By varying the excitation frequency and amplitude, added parameters have been analyzed against excitation frequency, velocity, and acceleration to determine dependent variables. The dimensionless added mass ratio, γma, was investigated within a range of acceleration of 0.07m/s2 to 5.00 m/s2 and in an excitation frequency of 2-9 Hz. Results exhibited a frequency-dependent added mass ratio, leading to a mass addition variation of 1.00-1.49 times the test rig mass with a mean γma of 1.22. Similarly, the dimensionless added polar moment of inertia, γIp, was investigated within a range of angular acceleration between 2.4 rad/s2 to 29.6 rad/s2 and in an excitation frequency range of 2-10 Hz. The mean added polar inertia ratio, γIp, was obtained as 1.09 times the polar moment of inertia of the test rig, corresponding to an increase in polar inertia of about 9%, compared to the total dry polar inertia of the test rig. Results showed that the added polar inertia ratio varied by approximately 1.8% within the studied frequency range. Thus, no frequency dependence could be determined. Due to measurement uncertainties and limitations of the test rigs, added linear damping and torsional damping could not be determined in either of the existing test rigs (lateral and torsional movement).

Cette thèse est consacrée à l’analyse mathématique du problème du mouvement d’un nombre fini de corps rigides homogènes au sein d’un fluide visqueux incompressible homogène. Les fluides visqueux sont classés en deux catégories: les fluides newtoniens et les fluides non newtoniens. En premier lieu, nous considérons le système formé par les équations de Navier Stokes incompressible couplées aux lois de Newton pour décrire le mouvement de plusieurs disques rigides dans un fluide newtonien visqueux homogène dans l’ensemble de l’espace R^2. Nous montrons que ce problème est bien posé jusqu’à l’apparition de la première collision. Ensuite, nous éliminons tous les types de contacts pouvant survenir si le domaine fluide reste connexe à tout moment. Avec cette hypothèse, le système considéré est globalement bien posé. Dans la deuxième partie de cette thèse, nous montrons la non-unicité des solutions faibles au problème d’interaction fluide-solide 3D, dans le cas d’un fluide newtonien, après collision. Nous montrons qu’il existe des conditions initiales telles que nous pouvons étendre les solutions faibles après le temps pour lequel le contact a eu lieu de deux manières différentes. Enfin, dans la dernière partie, nous étudions le mouvement bidimensionnel d’un nombre fini de disques immergés dans une cavité remplie d’un fluide viscoélastique tel que des solutions polymériques. Les équations de Navier Stokes incompressible sont utilisées pour modéliser le solvant, dans lesquelles un tenseur de contrainte élastique supplémentaire apparaît comme un terme source. Dans cette partie, nous supposons que le tenseur de contrainte supplémentaire satisfait la loi différentielle d’Oldroyd ou sa version régularisée. Dans les deux cas, nous prouvons l’existence et l’unicité des solutions fortes locales en temps du problème considéré. / This thesis is devoted to the mathematical analysis of the problem of motion of afinite number of homogeneous rigid bodies within a homogeneous incompressible viscous fluid. Viscous fluids are classified into two categories: Newtonian fluids, and non-Newtonian fluids. First, we consider the system formed by the incompressible Navier-Stokes equations coupled with Newton’s laws to describe the movement of several rigid disks within a homogeneous viscous Newtonian fluid in the whole space R^2. We show the well-posedness of this system up to the occurrence of the first collision. Then we eliminate all type of contacts that may occur if the fluid domain remains connected at any time. With this assumption, the considered system is well-posed globally in time. In the second part of this thesis, we prove the non-uniqueness of weak solutions to the fluid-rigid body interaction problem in 3D in Newtonian fluid after collision. We show that there exist some initial conditions such that we can extend weak solutions after the time for which contact has taken place by two different ways. Finally, in the last part, we study the two-dimensional motion of a finite number of disks immersed in a cavity filled with a viscoelastic fluid such as polymeric solutions. The incompressible Navier–Stokes equations are used to model the flow of the solvent, in which the elastic extra stress tensor appears as a source term. In this part, we suppose that the extra stress tensor satisfies either the Oldroyd or the regularized Oldroyd constitutive differential law. In both cases, we prove the existence and uniqueness of local-in-time strongsolutions of the considered moving-boundary problem.

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields are considered in a general anisotropic case. An elastic structure is assumed to be a bounded homogeneous anisortopic body occupying domain $\Omega^+\sub\R^3$ , where the thermoelastic field is defined, while in the physically anisotropic unbounded exterior domain $\Omega^-=\R^3\\ \overline{\Omega^+}$ there is defined the scalar field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface $\delta\Omega^{+-}$. Uniqueness and existence theorems, for the non-resonance case, are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations ($\Psi DEs$) . The invertibility of the corresponding matrix pseudodifferential operators ($\Psi DO$) in appropriate functional spaces is shown on the basis of generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case, the co-kernels of the $\Psi DOs$ are analysed and the efficent conditions of solvability of the transmission problems are established.

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields are considered in a general anisotropic case. An elastic structure is assumed to be a bounded homogeneous anisortopic body occupying domain $\Omega^+\sub\R^3$ , where the thermoelastic field is defined, while in the physically anisotropic unbounded exterior domain $\Omega^-=\R^3\\ \overline{\Omega^+}$ there is defined the scalar field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface $\delta\Omega^{+-}$. Uniqueness and existence theorems, for the non-resonance case, are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations ($\Psi DEs$) . The invertibility of the corresponding matrix pseudodifferential operators ($\Psi DO$) in appropriate functional spaces is shown on the basis of generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case, the co-kernels of the $\Psi DOs$ are analysed and the efficent conditions of solvability of the transmission problems are established.