The goal of the present doctoral research is to create a theoretical framework and develop a numerical implementation for a shell finite element that can potentially achieve higher performance (i.e. combination of speed and accuracy) than current Continuum-based (CB) shell finite elements (FE), in particular in applications related to soft biological tissue dynamics. Specifically, this means complex and irregular geometries, large distortions and large bending deformations, and anisotropic incompressible hyperelastic material properties. The critical review of the underlying theories, formulations, and capabilities of the existing CB shell FE revealed that a general nonlinear CB shell FE with the abovementioned capabilities needs to be developed. Herein, we propose the theoretical framework of a new such CB shell FE for dynamic analysis using the total and the incremental updated Lagrangian (UL) formulations and explicit time integration. Specifically, we introduce the geometry and the kinematics of the proposed CB shell FE, as well as the matrices and constitutive relations which need to be evaluated for the total and the incremental UL formulations of the dynamic equilibrium equation. To verify the accuracy and efficiency of the proposed CB shell element, its large bending and distortion capabilities, as well as the accuracy of three different techniques presented for large strain analysis, we implemented the element in Matlab and tested its application in various geometries, with different material properties and loading conditions. The new high performance and accuracy element is shown to be insensitive to shear and membrane locking, and to initially irregular elements.

Nous nous intéressons dans cette thèse aux équations de Navier-Stokes pour un fluide visqueux incompressible. Dans la première partie, nous étudions le cas d’un fluide homogène. Rappelons que la grande question de la régularité globale en dimension 3 est plus ouverte que jamais : on ne sait pas si la solution de l’équation correspondant à un état initial suffisamment régulier mais arbitrairement loin du repos, va perdurer indéfiniment dans cet état (régularité globale) ou exploser en temps fini(singularité). Une façon d’aborder le problème est de supposer cette éventuelle rupture de régularité et d’envisager les différents scenarii possibles. Après un rapide survol de la structure propre aux équations de Navier-Stokes et des résultats connus à ce jour (chapitre 1), nous nous intéressons(chapitre 2) à l’existence locale (en temps) de solutions dans des espaces de Sobolev qui ne sont pas invariants d’échelle. Partant d’une donnée initiale qui produit une singularité, on prouve l’existence d’une constante optimale qui minore le temps de vie de la solution. Cette constante, donnée parla méthode rudimentaire du point fixe, fournit ainsi un bon ordre de grandeur sur le temps de vie maximal de la solution. Au chapitre 3, nous poursuivons les investigations sur le comportement de telles solutions explosives à la lumière de la méthode des éléments critiques.Dans le seconde partie de la thèse, nous sommes intéressés à un modèle plus réaliste du point de vue de la physique, celui d’un fluide incompressible à densité variable. Ceci est modélisé par les équations de Navier-Stokes incompressible et inhomogènes. Nous avons étudié le caractère globalement bien posé de ces équations dans la situation d’un fluide évoluant dans un tore de dimension 3, avec des données initiales appartenant à des espaces critiques et sans hypothèse de petitesse sur la densité. / This thesis is concerned with incompressible Navier-Stokes equations for a viscous fluid. In the first part, we study the case of an homogeneous fluid. Let us recall that the big question of the global regularity in dimension 3 is still open : we do not know if the solution associated with a data smooth enough and far from the immobile stage will last over time (global regularity) or on the contrary will stop living in finite time and blow up (singularity). The goal of this thesis is to study this regularity break. One way to deal witht his question is to assume that such a phenomen on occurs and to study differents scenarii. The chapter 1 is devoted to a recollection of well-known results. In chapter 2, we are interesting in the local (in time) existence of a solution in some Sobolev spaces which are not invariant under the natural sclaing of Navier-Stokes. Starting with a data generating a singularity, we can prove there exists an optimal lower boundary of the lifes pan of such a solution. In this way, the lower boundary provided by the elementary procedure of fixed-point, gives the correctorder of magnitude. Then, we keep on investigations about the behaviour of regular solution near the blow up, thanks to the method of critical elements (chapter 3).In the second part, we are concerned with a more relevant model, from a physics point of view : the inhomogeneous Navier-Stokes system. We deal with the global well poseness of such a model for a inhomogeneous fluid, evolving on a tor us in dimension 3, with critical data and without smallnes sassumption on the density.

Investigation of material behaviour in a squeeze flow geometry provides an impor- tant technique in rheology and it is relevant also from the technological point of view (some types of dampers, compression moulding). To our best knowledge, the sque- eze flow has not been solved for fluids-like materials with pressure-dependent material moduli. In the main scope of the present thesis, an incompressible fluid whose visco- sity strongly depends on the pressure is studied in both the perfect-slip and the no-slip squeeze flow. It is shown that such a material model can provide interesting departures compared to the classical model for viscous (Navier-Stokes) fluid even on the level of analytical solutions, which are obtained using some physically relevant simplificati- ons. Numerical simulation of a free boundary problem for the no-slip squeeze flow is then developed in the thesis using body-fitted curvilinear coordinates and spectral collocation method. An interesting behaviour is expected especially in the corners of the computational domain where the stress singularities are normally located. Unfor- tunately, numerical results reveal some fundamental drawbacks related to the physical model and its possible improvement is discussed at the end of the thesis.

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Due to high specific strength and stiffness as well as outstanding energy-absorption characteristics, sandwich structures are extensively used in aircraft, aerospace, automobile, and marine industries. With the objective of finding lightweight blast-resistant sandwich structures for protecting infrastructure, we have found, for a fixed areal mass density, one- or two-core doubly-curved sandwich shell's (plate's) geometries and materials and fiber angles of unidirectional fiber-reinforced face sheets for it to have the maximum first failure load under quasistatic (blast) loads. The analyses employ a third-order shear and normal deformable plate/shell theory (TSNDT), the finite element method (FEM), a stress recovery scheme (SRS), the Tsai-Wu failure criterion and the Nest-Site selection (NeSS) optimization algorithm, and assume the materials to be linearly elastic. For a sandwich shell under the spatially varying static pressure on the top surface, the optimal non-symmetric one-core (two-core) design improves the first failure load by approximately 33% (27%) and 50% (36%) from the corresponding optimal symmetric design with clamped and simply-supported edges, respectively. For a sandwich plate under blast loads, it is found that the optimal one-core design is symmetric about the mid-surface with thick face sheets, and the optimal two-core design has a thin middle face sheet and thick top and bottom face sheets. Furthermore, the transverse shear stresses (in-plane transverse axial stresses) primarily cause the first failure in a core (face sheet). For the computed optimal design under a blast load, we also determined the collapse load by using the progressive failure analysis that degrades all elasticities of the failed material point to very small values. The collapse load of the clamped (simply-supported) sandwich structure is approximately 15%–30% (0%–17%) higher than its first failure load. Incompressible materials such as rubbers, polymers, and soft tissues that can only undergo volume preserving deformations have numerous applications in engineering and biomedical fields. Their vibration characteristics are important for using them as wave reflectors at interfaces with a fiber-reinforced sheet. In this work we have numerically analyzed free vibrations of plates made of a linearly elastic incompressible rubber-like material (Poison's ratio = 0.5) by using a TSNDT for incompressible materials and the mixed FEM. The displacements at nodes of a 9-noded quadrilateral element and the hydrostatic pressure at four interior nodes are taken as unknowns. Computed results are found to match well with the corresponding either analytical or numerical ones obtained with the commercial FE software Abaqus and the 3-dimensional linear elasticity theory. The analysis discerns plate's in-plane vibration modes. It is found that a simply supported plate admits more in-plane modes than the corresponding clamped and clamped-free plates. / Doctor of Philosophy / A simple example of a sandwich structure is a chocolate ice cream bar with the chocolate layer replaced by a stiff plate. Another example is the packaging material used to protect electronics during shipping and handling. The intent is to find the composition and the thickness of the "chocolate layer" so that the ice cream bar will not shatter when dropped on the floor. The objective is met by enforcing the chocolate layer with carbon fibers and then finding fiber materials, their alignment, ice cream or core material, and its thickness to resist anticipated loads with a prescribed level of certainty. Thus, a sandwich structure is usually composed of a soft thick core (e.g., foam) bonded to two relatively stiff thin skins (e.g., made of steel, fiber-reinforced composite) called face sheets. They are lightweight, stiff, and effective in absorbing mechanical energy. Consequently, they are often used in aircraft, aerospace, automobile, and marine industries. The load that causes a point in a structure to fail is called its first failure load, and the load that causes it to either crush or crumble is called the ultimate load. Here, for a fixed areal mass density (mass per unit surface area), we maximize the first failure load of a sandwich shell (plate) under static (dynamic) loads by determining its geometric dimensions, materials and fiber angles in the face sheets, and the number (one or two) of cores. It is found that, for a non-uniformly distributed static pressure applied on the central region of a sandwich shell's top surface, an optimal design that has different materials for the top and the bottom face sheets improves the first failure load by nearly 30%-50% from that of the optimally designed structure with identical face sheets. For the structure optimally designed for the first failure blast load, the ultimate failure load with all of its edges clamped (simply supported) is about 15%-30% (0%-17%) higher than its first failure load. This work should help engineers reduce weight of sandwich structures without sacrificing their integrity and save on materials and cost. Rubberlike materials, polymers, and soft tissues are incompressible since their volume remains constant when they are deformed. Plates made of incompressible materials have a wide range of applications in everyday life, e.g., we hear because of vibrations of the ear drum. Thus, accurately predicting their dynamic behavior is important. A first step usually is determining natural frequencies, i.e., the number of cycles of oscillations per second (e.g., a human heart beats at about 1 cycle/sec) completed by the structure in the absence of any externally applied force. Here, we numerically find natural frequencies and mode shapes of rubber-like material rectangular plates with different supporting conditions at the edges. We employ a plate theory that reduces a 3-dimensional (3-D) problem to a 2-D one and the finite element method. The problem is challenging because the incompressibility constraint requires finding the hydrostatic pressure as a part of the problem solution. We show that the methodology developed here provides results that match well with the corresponding either analytical or numerical solutions of the 3-D linear elasticity equations. The methodology is applicable to analyzing the dynamic response of composite structures with layers of incompressible materials embedded in it.

Developments are reported of unstructured-mesh methods for simulating stratified, turbulent and shear flows. The numerical model employs nonoscillatory forward in-time integrators for anelastic and incompressible flow PDEs, built on Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) and a preconditioned conjugate residual elliptic solver. Finite-volume spatial discretisation adopts an edge-based data structure. Tetrahedral-based and hybrid-based median-dual options for unstructured meshes are developed, enabling flexible spatial resolution. Viscous laminar and detached eddy simulation (DES) flow solvers are developed based on the edge-based NFT MPDATA scheme. The built-in implicit large eddy simulation (ILES) capability of the NFT scheme is also employed and extended to fully unstructured tetrahedral and hybrid meshes. Challenging atmospheric and engineering problems are solved numerically to validate the model and to demonstrate its applications. The numerical problems include simulations of stratified, turbulent and shear flows past obstacles involving complex gravity-wave phenomena in the lee, critical-level laminar-turbulence transitioning and various vortex structures in the wake. Qualitative flow patterns and quantitative data analysis are both presented in the current study.

Ce travail est consacré à la résolution e?cace de problèmes d’optimisation de forme ou de contrôle d’écoulements. Le couplage entre la pénalisation, permettant d’imposer des conditions aux bords sur maillage cartésien, et la méthode Level-Set, autorisant une représentation d’obstacles non-paramétrique et un suivi d’interface précis, est implémenté. En première partie, un problème inverse modèle, puis une optimisation géométrique en régime de Stokes, sont traités itérativement. Une attention particulière est portée à la solution des EDP près des zones pénalisées, et une montée en ordre est réalisée. Divers préconditionnements du gradient de forme sont aussi discutés a?n d’améliorer la convergence. La seconde partie est dédiée à la simulation directe d’écoulements au voisinage d’un actionneur dans le cadre d’un contrôle par jets pulsés exercé sur le corps d’Ahmed. L’étude locale montre l'in?uence de paramètres comme la fréquence de pulsation ou l’allure des pro?ls de vitesse en sortie sur la qualité de l’action. En guise de synthèse, une optimisation de la forme de l’actionneur du chapitre deux est pratiquée sous contraintes topologiques et dans un cadre simpli?é, à l’aide du couplage Level-Set/pénalisation préalablement introduit. L’objectif du problème inverse posé est de modi?er la géométrie intérieure du MEMS pour obtenir un pro?l de vitesses désiré en sortie de jet. / This work deals with e?cient numerical solving of problems linked with shape optimization or ?ow control. The combination between penalization, that allows to impose boundary conditions while avoiding the use of body-?tted grids, and Level-Set methods, which enable a natural non-parametric representation of the geometries to be optimized, is implemented. In the ?rst part, a model inverse problem, and an application pertaining to optimal design in Stokes ?ows, are treated with an iterative algorithm. Special care is devoted to the solution of the PDE’s in the vicinity of the penalized regions. The discretization accuracy is increased. Various gradient preconditionings aiming at improving the convergence are also discussed. The second part is dedicated to direct numerical simulation of ?ows in the neighborhood of an actuator, in the context of active control by pulsed jets used on the Ahmed body. The local study emphasizes the in?uence of various parameters on the action quality, in particular the pulsation frequency, or the aspect of exit velocity pro?les. As a synthesis, shape optimization is performed on the actuator of chapter two, thanks to the previously introduced coupling between Level-Set and penalization. The framework is simpli?ed and topological constraints are imposed. The inverse problem we set intends to modify the MEMS inner geometry to retrieve a given jet pro?le on the exit section.

O objetivo desta dissertação de mestrado é estudar técnicas numéricas para simular escoamentos incompressíveis multifásicos e implementar uma ferramenta computacional utilizando malhas hierárquicas e discretizações por diferenças finitas. São apresentados a formulação matemática e o desenvolvimento do método numérico, levando em consideração o caráter multifásico do escoamento. Foi adotado o modelo de força superficial contínua e a representação da interface foi feita pelo método de acompanhamento de fronteira. São expostos todos os testes realizados durante o desenvolvimento da ferramenta para checar cada etapa do método. Finalmente, testes visando verificar o código foram feitos e os resultados obtidos foram considerados satisfatórios para a verificação da ferramenta aqui desenvolvida. / The objective of this masters degree essay is to study numerical techniques to simulate incompressible multiphase flows and to implement a computational tool using hierachical meshes and discretizations by finite diferences. We introduce the mathematical formulation and the development of the numerical method, for the multiphase flow problem. A continuum surface force model is employed with the interface representation by the front tracking method. We show all tests performed to verify each stage of the methods development. Finally, results obtained in classical benchmark flow tests show good agreement with previous published results, corroborating the validity of this newly developed numerical tool.

Foram feitas simulações de um jato plano livre e incompressível usando o modelo de uma equação Spalart-Allmaras padrão e um modelo Spalart-Allmaras modificado através da alteração da escala de comprimento turbulenta. Sabe-se da literatura que no caso de jatos livres o modelo Spalart-Allmaras não consegue predizer adequadamente os resultados observados experimentalmente. Os resultados das simulações foram comparados com experimentos da literatura através de perfis de velocidade e da taxa de expansão do jato. Como esperado, os resultados obtidos das simulações utilizando o modelo Spalart-Allmaras padrão foram considerados insatisfatórios, porém o modelo Spalart-Allmaras modificado teve uma melhor concordância com os resultados experimentais. Além disso, o modelo Spalart-Allmaras modificado foi usado para simular os casos do escoamento sobre uma placa plana sem gradiente de pressão e o escoamento em um degrau com separação e gradiente adverso de pressão. Quando comparados com resultados experimentais da literatura e com resultados obtidos usando o modelo padrão, os resultados do modelo modificado obtidos para ambos os casos foram muito satisfatórios, concluindo-se que a modificação da escala de comprimento permite obter uma maior generalidade para o modelo Spalart-Allmaras. / Simulations of a plane and incompressible free jet using the standard Spalart-Allmaras model and a Spalart-Allmaras model modified by changing the turbulent length scale were carried out. It is known from literature that, in the case of the free jet, the Spalart-Allmaras model fails to adequately predict the experimentally observed results. The results of our simulations were compared with published experiments using the velocity profiles and the jet spreading rate. As expected, the results of simulations using the standard Spalart-Allmaras model were considered unsatisfactory while the modified Spalart-Allmaras model had a better agreement with the experimental results. Furthermore, the modified Spalart-Allmaras model was used to simulate the cases of flow over a flat plate with no pressure gradient and flow through a backward facing step with separation and adverse pressure gradient. When compared with experimental results from the literature and with results obtained using the standard model, the results of the modified model for both cases were very satisfactory, allowing the conclusion that the change in the length scale provided a greater generality for the Spalart-Allmaras model.