The primary motivation of this research is to develop and investigate parallel preconditioners for linear elliptic partial differential equations. Three preconditioners are studied: block-Jacobi preconditioner (BJ), a two-level tangential preconditioner (D0), and a three-level preconditioner (D1). Performance and scalability on a distributed memory parallel computer are considered. Communication cost and redundancy are explored as well. After experiments and analysis, we find that the three-level preconditioner D1 is the most efficient and scalable parallel preconditioner, compared to BJ and D0. The D1 preconditioner reduces both the number of iterations and computational time substantially. A new hybrid preconditioner is suggested which may combine the best features of D0 and D1. / Master of Science

La complexification de la modélisation multi-physique conduit d’une part à devoir simuler des systèmes d’équations différentielles ordinaires et d’équations différentielles algébriques de plus en plus grands en nombre d’inconnues et sur des temps de simulation longs. D’autre part l’évolution des architectures de calcul parallèle nécessite d’autres voies de parallélisation que la décomposition de système en sous-systèmes. Dans ce travail, nous proposons de concevoir des méthodes de décomposition de domaine pour la résolution d’EDO en temps. Nous reformulons le problème à valeur initiale en un problème aux valeurs frontières sur l’intervalle de temps symétrisé, sous l’hypothèse de réversibilité du flot. Nous développons deux méthodes, la première apparentée à une méthode de complément de Schur, la seconde basée sur une méthode de type Schwarz dont nous montrons la convergence pouvant être accélérée par la méthode d’Aitken dans le cadre linéaire. Afin d’accélérer la convergence de cette dernière dans le cadre non-linéaire, nous introduisons les techniques d’extrapolation et d’accélération de la convergence des suites non-linéaires. Nous montrons les avantages et les limites de ces techniques. Les résultats obtenus nous conduisent à développer l’accélération de la méthode de type Schwarz par une méthode de Newton. Enfin nous nous intéressons à l’étude de conditions de raccord non-linéaires adaptées à la décomposition de domaine de problèmes non-linéaires. Nous nous servons du formalisme hamiltonien à ports, issu du domaine de l’automatique, pour déduire les conditions de raccord dans le cadre l’équation de Saint-Venant et de l’équation de la chaleur non-linéaire. Après une étude analytique de la convergence de la DDM associée à ces conditions de transmission, nous proposons et étudions une formulation de Lagrangien augmenté sous l’hypothèse de séparabilité de la contrainte. / Complexification of multi-physics modeling leads to have to simulate systems of ordinary differential equations and algebraic differential equations with increasingly large numbers of unknowns and over large times of simulation. In addition the evolution of parallel computing architectures requires other ways of parallelization than the decomposition of system in subsystems. In this work, we propose to design domain decomposition methods in time for the resolution of EDO. We reformulate the initial value problem in a boundary values problem on the symmetrized time interval, under the assumption of reversibility of the flow. We develop two methods, the first connected with a Schur complement method, the second based on a Schwarz type method for which we show convergence, being able to be accelerated by the Aitken method within the linear framework. In order to accelerate the convergence of the latter within the non-linear framework, we introduce the techniques of extrapolation and of acceleration of the convergence of non-linear sequences. We show the advantages and the limits of these techniques. The obtained results lead us to develop the acceleration of the method of the type Schwarz by a Newton method. Finally we investigate non-linear matching conditions adapted to the domain decomposition of nonlinear problems. We make use of the port-Hamiltonian formalism, resulting from the control field, to deduce the matching conditions in the framework of the shallow-water equation and the non-linear heat equation. After an analytical study of the convergence of the DDM associated with these conditions of transmission, we propose and study a formulation of augmented Lagrangian under the assumption of separability of the constraint.

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My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.

This work investigates numerically the process of Teflon ablation using a finite-volume discretization, implicit time integration and a domain decomposition method in three-dimensions. The interest in Teflon stems from its use in Pulsed Plasma Thrusters and in thermal protection systems for reentry vehicles. The ablation of Teflon is a complex process that involves phase transition, a receding external boundary where the heat flux is applied, an interface between a crystalline and amorphous (gel) phase and a depolymerization reaction which happens on and beneath the ablating surface. The mathematical model used in this work is based on a two-phase model that accounts for the amorphous and crystalline phases as well as the depolymerization of Teflon in the form of an Arrhenius reaction equation. The model accounts also for temperature-dependent material properties, for unsteady heat inputs and boundary conditions in 3D. The model is implemented in 3D domains of arbitrary geometry with a finite volume discretization on unstructured grids. The numerical solution of the transient reaction-diffusion equation coupled with the Arrhenius-based ablation model advances in time using implicit Crank-Nicolson scheme. For each time step the implicit time advancing is decomposed into multiple sub-problems by a domain decomposition method. Each of the sub-problems is solved in parallel by Newton-Krylov non-linear solver. After each implicit time-advancing step, the rate of ablation and the fraction of depolymerized material are updated explicitly with the Arrhenius-based ablation model. After the computation, the surface of ablation front and the melting surface are recovered from the scalar field of fraction of depolymerized material and the fraction of melted material by post-processing. The code is verified against analytical solutions for the heat diffusion problem and the Stefan problem. The code is validated against experimental data of Teflon ablation. The verification and validation demonstrates the ability of the numerical method in simulating three dimensional ablation of Teflon.

Orientador: João Antonio Pereira / Banca: Gilberto Pechoto de Melo / Banca: Sérgio Sartori / Resumo: O presente trabalho propõe o estudo e implementação de uma metodologia para a estimação dos parâmetros modais de estruturas utilizando uma técnica de identificação baseada apenas na resposta do modelo, denominada Decomposição no Domínio da Freqüência (DDF). Para tal são abordados os conceitos básicos envolvidos na análise modal, análise modal baseada apenas na resposta e métodos de identificação. A formulação do algoritmo é baseada na decomposição da matriz densidade espectral de potência utilizando a técnica da decomposição em valores singulares (SVD). A decomposição da matriz densidade espectral nas linhas de freqüências correspondentes aos picos de amplitude, permite a estimativa dos modos de vibrar do sistema. Tem-se ainda que, o primeiro vetor singular obtido com a decomposição da matriz densidade espectral, para cada linha de freqüência, na região em torno do modo, contém as respectivas informações daquele modo e o correspondente valor singular leva a uma estimativa da função densidade espectral de um sistema de um grau de liberdade (1GL) equivalente. Neste caso, a matriz densidade espectral de saída é decomposta em um conjunto de sistemas de 1 grau de liberdade. Posteriormente, esses dados são transformados para o domínio do tempo, utilizando a transformada inversa de Fourier, e as razões de amortecimento são estimadas utilizando o conceito de decremento logaritmo. A metodologia é avaliada, numa primeira etapa, utilizando dados simulados e posteriormente utilizando dados experimentais. / Abstract: The present work proposes the study and implementation of a methodology for the estimating of the modal parameters of structures by using the output-only data. The technique called Frequency Domain Decomposition (DDF) identifies the modal parameters without knowing the input. For that, it is discussed the basic concepts involved in identification, modal analysis and output-only modal analysis. The formulation of the algorithm is based on the decomposition of the power spectral density matrix by using the singular values decomposition technique (SVD). The decomposition of the spectral density matrix for the lines of frequency corresponding to the amplitude peaks, allows the estimating of the modes shape of the system. Additionally, the first singular vector obtained with the decomposition of the spectral density matrix, for each line of frequency, in the area around of the peak, contains the respective information of that mode. The corresponding singular value leads to an estimating of the spectral density function of an equivalent system of one degree of freedom. Therefore, the output spectral density matrix is decomposed in a set of one degree of freedom system. Later on, those data are transformed for the time domain by using the inverse Fourier transform and the damping ratios estimated from the crossing times and the logarithm decrement of the corresponding single degree of freedom system correlation function. The methodology is evaluated using simulated and experimental data. / Mestre

The importance of plates, with discontinuities in boundary supports in aeronautical and marine structures, have led to various techniques to solve plate problems with mixed edge support conditions. The domain decomposition method is one of the most effective of these techniques, providing accurate numerical solutions. This method is used to investigate the vibration and buckling of flat, isotropic, thin and elastic plates with mixed edge support conditions. Two practical approaches have been developed as an extension of the domain decomposition method, namely, the primary-secondary domain (PSD) approach and the line-domains (LD) approach. The PSD approach decomposes a plate into one primary domain and one/two secondary domain(s). The LD approach considers interconnecting boundaries as dominant domains whose basic functions take a higher edge restraint from the neighbouring edges. Convergence and comparison studies are carried out on a number of selected rectangular plate cases. Extensive practical plate problems with various shapes, combinations of mixed boundary conditions and different inplane loading conditions have been solved by the PSD and LD approaches. / Master of Engineering (Hons)

In this thesis, we have developed a Subtructured Finite Element / eXtended Finite Element (S-FE/XFE) method. The S-FE/XFE method consists in decomposing the geometry into safe FE-domains and cracked XFE-domains, and solving the interface problem with the Finite Element Tearing and Interconnecting method (FETI).This method allows for handling complex crack configurations in 3D structures with common commercial FE software that do not feature the XFEM. The method is also extended to a mixed dimensional formulation, where the FE-domain is discretised with shell elements while the XFE-domain is modelled with three-dimensional solid elements. This is the so-called S-FE Shell/XFE 3D method. The mixed dimensional formulation is more convenient than a full XFE-3D formulation because it significantly reduces the computational cost and it is more accurate compared to a full shell model because it includes three-dimensional local features such as three-dimensional crack. The compatibility of the displacements through the interface is ensured using the Reissner-Mindlin equation. The method has been extensively validated towards both academic problems and semi-industrial benchmarks in order to demonstrate the benefits of this approach. Among them, the S-FE/XFE method is applied to a crack analysis in a section of a compressor drum of a turbofan engine. The results obtained with the S-FE/XFE method are compared with those obtained with a standard FE computation. Furthermore, two applications of the S-FE shell/XFE 3D approach are proposed. First the load carrying capacity of a section of stiffened panel containing a through-the-thickness crack is investigated (this is the one-bay crack configuration). Second, the ability of the method for handling small surface cracks in large finite element models is addressed by looking at a generic 'large pressure panel' presenting realistic crack configurations.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.

A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov-type upwind finite-volume spatial-discretization scheme, with piecewise limited linear reconstruction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuristic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed.