Cette étude concerne la simulation de systèmes thermiques comportant plusieurs domaines à l’aide d’une méthode d’ordre réduit adaptée au mono-corps. Les modèles modaux réduits sont construits à partir de modes de branche. Ceux-ci sont déterminés à l’aide d’un modèle détaillé et permettent la prise en compte de non-linéarités. La réduction est effectuée par la technique de l’amalgame modal découplé qui permet de façon automatique et immédiate d’obtenir une base réduite qui ne comporte qu’un faible nombre de modes à partir de la base initiale. Le couplage entre les différents domaines s’effectue par l’intermédiaire d’une résistance thermique de contact. On montre la nécessité de faire intervenir un terme supplémentaire de pénalisation de saut de flux qui vient améliorer les résultats. Les simulations numériques effectuées sur des cas tests (microprocesseur et radiateur en 2D, bloc métallique avec cartouches chauffantes en 3D) montrent la pertinence de la méthode. / This study concerns the simulation of thermal systems with multiple fields with a reduced-order method suited to a single body. Reduced models are constructed from modal branch eigenmodes. They are determined using a detailed model and allow the inclusion of non-linearities. The reduction is carried out by the simplified amalgam method which allows an automatic and immediate way to obtain a reduced basis which contains only a small number of modes from the original basis. The coupling between the different areas is carried out through a thermal contact resistance. It shows the need to involve an additional flux jump penalty term to improve the results. The numerical simulations carried out on test cases (microprocessor and radiator in 2D, metal block with hot cartridges in 3D) show the relevance of the method.

<p>Techniques for a posteriori error estimation for finite element approximations of an elliptic partial differential equation are studied.This extends previous work on localized error control in finite element methods for linear elasticity.The methods are then applied to the problem of homogenization of periodic structures. In particular, error estimates for the effective elastic properties are obtained. The usefulness of these estimates is twofold.First, adaptive methods using mesh refinements based on the estimates can be constructed.Secondly, one of the estimates can give reasonable measure of the magnitude ofthe error. Numerical examples of this are given.</p>

Techniques for a posteriori error estimation for finite element approximations of an elliptic partial differential equation are studied.This extends previous work on localized error control in finite element methods for linear elasticity.The methods are then applied to the problem of homogenization of periodic structures. In particular, error estimates for the effective elastic properties are obtained. The usefulness of these estimates is twofold.First, adaptive methods using mesh refinements based on the estimates can be constructed.Secondly, one of the estimates can give reasonable measure of the magnitude ofthe error. Numerical examples of this are given.

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[pt] Métodos estabilizados de elementos finitos são propostos e analisados para problemas de fluidos, com particular ênfase nas equações de Navier-Stokes incomprenssível. Após a apresentação da mecânica dos escoamentos dos fluidos, introduzimos no Capítulo 3, no contexto de problema de Stokes, as dificuldadas numéricas associadas ao método de Galerkin em problemas de fluidos e simulamos em sucesso alguns escoamentos lentos através de formulações finitos para estabilizadas. No capítulo 4, propomos uma nova formulação de elementos finitos para a equação da energia, mais precisamente para o modelo da advecção-difusão do calor. Graças a um novo desenho do parâmetro de estabilidade T, o qual permite adicionar difusão às regiões advectivas e difusivas-dominadas do escoamento de maneira diferemciada, obtivemos um bom desempenho novo método mesmo em situações de altíssimo número de Péclet (10(2) menor que Pe menor que 10 e (6)), conforme ilustram os testes numéricos realizados. Coletando as experiências adquiridas com modelos lineares de Stokes e da advencção-difusão, nos foi possível propor, analisar o erro e testar dois novos métodos estabilizados para o problema de Navier-Stokes transiente. Construídos de maneira a herdar as boas características de estabilidade dos métodos propostos apresentam bom desempenho em escoamentos fortemente advectivos, bem como não necessitam atender a priori à condição de Baduska-Brezzi. Através de um algoritmo preditor/ multi-corretor de integração do termo inercial da equação de movimento, estes ,métodos foram capazes de de simular de maneira precisa escoamentos de interesse em Mecânica(400 menor que Re< menor que 500), captando escoamentos secundários, tais como recirculações de fluido. / [en] Stabilized methods for fluid problems are proposed and analysed with particular emphasis to the incompressible Navier-Stokes equations. We Begin in Chapter 2 introducing the balance equations of fluid Mechanics. Next. In Chapter 3, we discuss the numerical difficulties of the Galerkin method in fluids(in the contexto f the Stokes problem) and performance some succeful simulations of creeping flows, employing stabilized formulations. In Chapter 4, we propose a new finite element formulation for the energy equation, or more preciselly for the advective-diffusive model. Taking advantage of new design of the stability parameter T, which permits to add diffusion to advective and diffusive regions of the flow in a different way, we success to obtain a good performance of the new method in flows with very high Péclet numbers (10(2) lass than Pe lessa than 10(6)), as illustred at numerical testes performed. By collecting the Stokes and advective-diffusive experiences,it was possible to propose, analyse and test two new stabilized methods for the transient Navier-Stokes problem. These methods were built in a way to heritage the good characteristics showed by the stabilized methods introduced for the Stokes and adventive-diffusive models. The new methods propoposed have a good performance in high advective flows, besides there is no need to satisfy the Babuska-Brezzi condition. Employing a predictor/multi-corretor algorithm, we were able to simulate accruratly some useful flows(400 less than Re less than 500), such as fluid recirculations.

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In this thesis a constitutive model is developed for the numerical prediction of UD composite material behaviour under impact loading. Impact induced loading usually results in three dimensional stress states which significantly influences the failure behaviour. The heterogeneous nature of composite materials, in particular, results in a complex failure behaviour which manifests itself in various failure modes. Predicting the onset and evolution of these failure modes requires the use of physically based three dimensional theories for the prediction of the onset of damage and subsequent damage evolution. Furthermore, the use of polymeric matrices in continuous fibre reinforced composites results in a distinct directional strain rate dependent material behaviour which needs to be incorporated in constitutive models for the numerical simulation of impact events. The developed constitutive model relies on the prediction of the onset of damage evolution by the use of physically based three dimensional stress based failure criteria. A special feature of the proposed model is the identification of potential fracture planes. Numerically efficient algorithms for finding such planes are developed thus enabling the implementation into an explicit FE environment which was prohibitive so far. Damage evolution is simulated by degrading the tractions which are acting on the failure mode dependent fracture planes. The damage evolution and consequent energy dissipation is thereby driven by physically based dissipation potentials which consider only stresses which contribute to damage growth. The well known mesh dependent energy dissipation in Continuum Damage Mechanics is reduced by the introduction of an element size dependent parameter into the constitutive equations. An experimental program is conducted to investigate the compressive behaviour of composites. The focus of the study is on the rate dependent failure behaviour. The experiments are designed such that the failure mechanisms can be studied at varying strain rates with identical boundary conditions. This allows for direct conclusions about the strain rate dependent material behaviour. Novel optical measurement techniques are applied across all investigated strain rates thus ensuring an improved observation of the failure modes. The proposed constitutive model is finally verified by modelling of three point beam bending experiments which were performed quasi-statically and at impact velocities. The experimental technique for beam bending at impact loading was therefore improved thus yielding significantly more accurate experimental data.

A thermodynamic framework was presented for the development of powderconstitutive models. The process of powder compaction through Equal ChannelAngular Extrusion (ECAE) at room temperature was modeled using the finiteelement analysis package ABAQUS. The simulation setup was used to conduct aparametric study involving varying the process parameters of ECAE, aimed ataiding the process design.Two powder compaction models, the Gurson model and the Duva and Crowmodel, were used to test their efficacy in modeling this process. Thethermodynamic framework was applied to derive the constitutive equations of theDuva and Crow model. Modeling parameters like friction coefficients, interactionconditions were determined by comparing the simulations for solid billet and anempty can with actual experimental runs for loads, shear angle and workpiece geometry. The simulations using the two powder constitutive models showed nosignificant difference in the stress in the powder during the extrusion.The results obtained from the 3-D simulations were also compared toexperiments conducted to compact copper powder with a size distribution of 10mto 45m. It was found through experiments that the powder does not fullyconsolidate near the outer corner of the workpiece after the first ECAE pass and theresults from the simulations were used to rationalize this phenomenon.Modifications made to the process by applying a back pressure during thesimulations resulted in a uniformly compacted powder region.Further, simulations were carried out by varying the process parameters likethe crosshead velocity, the friction coefficient between the walls of the die and thecan, can dimensions and material, shape of the can cross section etc and the effectof each of these parameters was quantified by doing a sensitivity analysis.

A thermodynamic framework was presented for the development of powderconstitutive models. The process of powder compaction through Equal ChannelAngular Extrusion (ECAE) at room temperature was modeled using the finiteelement analysis package ABAQUS. The simulation setup was used to conduct aparametric study involving varying the process parameters of ECAE, aimed ataiding the process design.Two powder compaction models, the Gurson model and the Duva and Crowmodel, were used to test their efficacy in modeling this process. Thethermodynamic framework was applied to derive the constitutive equations of theDuva and Crow model. Modeling parameters like friction coefficients, interactionconditions were determined by comparing the simulations for solid billet and anempty can with actual experimental runs for loads, shear angle and workpiece geometry. The simulations using the two powder constitutive models showed nosignificant difference in the stress in the powder during the extrusion.The results obtained from the 3-D simulations were also compared toexperiments conducted to compact copper powder with a size distribution of 10mto 45m. It was found through experiments that the powder does not fullyconsolidate near the outer corner of the workpiece after the first ECAE pass and theresults from the simulations were used to rationalize this phenomenon.Modifications made to the process by applying a back pressure during thesimulations resulted in a uniformly compacted powder region.Further, simulations were carried out by varying the process parameters likethe crosshead velocity, the friction coefficient between the walls of the die and thecan, can dimensions and material, shape of the can cross section etc and the effectof each of these parameters was quantified by doing a sensitivity analysis.

We study approximation techniques for incompressible flows with heterogeneous properties. Speci cally, we study two types of phenomena. The first is the flow of a viscous incompressible fluid through a rigid porous medium, where the permeability of the medium depends on the pressure. The second is the ow of a viscous incompressible fluid with variable density. The heterogeneity is the permeability and the density, respectively. For the first problem, we propose a finite element discretization and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence is exponential, we propose a splitting scheme which involves solving only two linear systems. For the second problem, we introduce a fractional time-stepping scheme which, as opposed to other existing techniques, requires only the solution of a Poisson equation for the determination of the pressure. This simpli cation greatly reduces the computational cost. We prove the stability of first and second order schemes, and provide error estimates for first order schemes. For all the introduced discretization schemes we present numerical experiments, which illustrate their performance on model problems, as well as on realistic ones.