Dans cette thèse nous proposons de développer un système d’imagerie ultrasonore innovante de micro- défauts basé sur l’utilisation conjointe de techniques d’acoustique non linéaire et du concept de "transducteur à cavité chaotique". Ce transducteur correspond à la combinaison d’une céramique piézoélectrique collée sur une cavité de forme chaotique et du principe de retournement temporel. La faisabilité et les performances de ce nouveau système sont explorées par des simulations numériques. Des paramètres optimaux d’utilisation pour une implémentation expérimentale sont proposés. Une grande partie des travaux menés dans le cadre de cette thèse se concentre sur le développement d’outils numériques permettant l’amélioration de telles techniques d’imagerie. Un schéma d’éléments finis de type Galerkin Discontinu (GD) est étendu à l’élastodynamique non linéaire. Un type de zone absorbante parfaitement adaptée, appelée "Nearly Perfectly Matched Layer" (NPML) a aussi été développé. Dans le cas de matériaux orthotropes, comme des problèmes de stabilité apparaissent, un mélange de NPML et de zone atténuante, dont on contrôle la proportion respective, est introduit afin de stabiliser les NPML. Une validation expérimentale du concept de "transducteur à cavité chaotique" pour la focalisation dans un milieu solide, réverbérant ou non, en utilisant une seule source est réalisée. Les méthodes de retournement temporel et de filtre inverse sont présentées et comparées. La démonstration expérimentale qu’un "transducteur à cavité chaotique" peut être utilisé conjointement avec les méthodes d’inversion d’impulsion afin de réaliser une image de non linéarités localisées est présentée / In this thesis we propose the development of an innovative micro-damage imaging system based on a combination of Nonlinear Elastic Wave Spectroscopy techniques and “chaotic cavity transducer” concept. It consists of a combination of a PZT ceramic glued to a cavity of chaotic shape with the time reversal principle. The feasibility and capabilities of these new ideas is explored by numerical simulations, and optimal operational parameters for experimental implementation are suggested based on the modelling support. A large part of the research work conducted in this thesis is concentrated on the development of numerical simulation tools to help the improvement of such nonlinear imaging methods. A nodal Discontinuous Galerkin Finite Element Method (DG-FEM) scheme is extended to nonlinear elasto-dynamic including source terms. A Perfectly Matched Layer absorbing boundary condition well adapted to the DG-FEM scheme, called Nearly Perfectly Matched Layer (NPML), is also developed. In the case of orthotropic material as stability problems appear, a mixture of NPML and sponge layer, with a controllable ratio of these two kinds of absorbing layers, is introduced. The experimental validation of “chaotic cavity transducer” to focalize in reverberant and non-reverberant solid media with only one source is made. Classical time reversal, inverse filter and 1 Bit time reversal process are discussed and compared. The experimental demonstration of the use of a “chaotic cavity transducer”, in combination with the pulse inversion and 1-bit methods, to obtain an image of localized nonlinearity is made. This opens the possibility for high resolution imaging of nonlinear defects

Efficient, accurate numerical simulation of coupled heat transfer and fluid dynamics systems continues to be a challenge. Direct numerical simulation (DNS) packages like FLU- ENT exist and are sufficient for design and predicting flow in a static system, but in larger systems where input parameters can change rapidly, the cost of DNS increases prohibitively. Major obstacles include handling the scales of the system accurately - some applications span multiple orders of magnitude in both the spatial and temporal dimensions, making an accurate simulation very costly. There is a need for a simulation method that returns accurate results of multi-scale systems in real time. To address these challenges, the Multi- Resolution Discontinuous Galerkin (MRDG) method has been shown to have advantages over other reduced order methods. Using multi-wavelets as the local approximation space provides an inherently efficient method of data compression, while the unique features of the Discontinuous Galerkin method make it well suited to composition with wavelet theory. This research further exhibits the viability of the MRDG as a new approach to efficient, accurate thermal system simulations. The development and execution of the algorithm will be detailed, and several examples of the utility of the MRDG will be included. Comparison between the MRDG and the "vanilla" DG method will also be featured as justification of the advantages of the MRDG method.

Este trabalho apresenta uma metodologia, baseada no método de Galerkin, para quantificar a incerteza no problema de flexão estocástica da viga de Euler-Bernoulli repousando em fundação de Pasternak. A incerteza nos coeficientes de rigidez da viga e da fundação é representada por meio de processos estocásticos parametrizados. A limitação em probabilidade dos parâmetros randômicos e a escolha adequada do espaço de soluções aproximadas, necessárias à posterior demonstração de unicidade e existência do problema, são consideradas por meio de hipóteses teóricas. O espaço de soluções aproximadas de dimensão finita é construído pelo produto tensorial entre espaços (determinístico e randômico), obtendo-se um espaço denso no espaço das soluções teóricas. O esquema de Wiener-Askey dos polinômios do caos generalizados é utilizado na representação do processo estocástico de deslocamento da viga. O método dos elementos finitos estocásticos é apresentado e empregado na solução numérica de exemplos selecionados. Os resultados, em termos de momentos estatísticos, são comparados aos obtidos por meio de simulações de Monte Carlo. / This study presents a methodology, based on the Galerkin method, to quantify the uncertainty in the stochastic bending problem of an Euler-Bernoulli beam resting on a Pasternak foundation. The uncertainty in the stiffness coefficients of the beam and foundation is represented by parametrized stochastic processes. The probability limitation on the random parameters and the choice of an appropriated approximate solution space, necessary for the subsequent demonstration of uniqueness and existence of the problem, are considered by means of theoretical hypothesis. The finite dimensional space of approximate solutions is built by tensor product between spaces (deterministic and randomic), obtaining a dense space in the theoretical solution space. The Wiener-Askey scheme of generalizes chaos polynomials is used to represent the stochastic process of the beam deflection. The stochastic finite element method is presented and employed in the numerical solution of selected examples. The results, in terms of statistical moments, are compared to results obtained through Monte Carlo simulations.

The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement. The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind. This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.

In the last half a century, Computational Fluid Dynamics (CFD) has been established as an important complementary part and some times a significant alternative to Experimental and Theoretical Fluid Dynamics. Development of efficient computational algorithms for digital simulation of fluid flows has been an ongoing research effort in CFD. An accurate numerical simulation of compressible Euler equations, which are the gov-erning equations of high speed flows, is important in many engineering applications like designing of aerospace vehicles and their components. Due to nonlinear nature of governing equations, such flows admit solutions involving discontinuities like shock waves and contact discontinuities. Hence, it is nontrivial to capture all these essential features of the flows numerically. There are various numerical methods available in the literature, the popular ones among them being the Finite Volume Method (FVM), Finite Difference Method (FDM), Finite Element Method (FEM) and Spectral method. Kinetic theory based algorithms for solving Euler equations are quite popular in finite volume framework due to their ability to connect Boltzmann equation with Euler equations. In kinetic framework, instead of dealing directly with nonlinear partial differential equations one needs to deal with a simple linear partial differential equation. Recently, FEM has emerged as a significant alternative to FVM because it can handle complex geometries with ease and unlike in FVM, achieving higher order accuracy is easier. High speed flows governed by compressible Euler equations are hyperbolic partial differential equations which are characterized by preferred directions for information propagation. Such flows can not be solved using traditional FEM methods and hence, stabilized methods are typically introduced. Various stabilized finite element methods are available in the literature like Streamlined-Upwind Petrov Galerkin (SUPG) method, Galerkin-Least Squares (GLS) method, Taylor-Galerkin method, Characteristic Galerkin method and Discontinuous Galerkin Method. In this thesis a novel stabilized finite element method called as Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) method is formulated. Both explicit and implicit versions of KSUPG scheme are presented. Spectral stability analysis is done for explicit KSUPG scheme to obtain the stable time step. The advantage of proposed scheme is, unlike in SUPG scheme, diffusion vectors are obtained directly from weak KSUPG formulation. The expression for intrinsic time scale is directly obtained in KSUPG framework. The accuracy and robustness of the proposed scheme is demonstrated by solving various test cases for hyperbolic partial differential equations like Euler equations and inviscid Burgers equation. In the KSUPG scheme, diffusion terms involve computationally expensive error and exponential functions. To decrease the computational cost, two variants of KSUPG scheme, namely, Peculiar Velocity based KSUPG (PV-KSUPG) scheme and Circular distribution based KSUPG (C-KSUPG) scheme are formulated. The PV-KSUPG scheme is based on peculiar velocity based splitting which, upon taking moments, recovers a convection-pressure splitting type algorithm at the macroscopic level. Both explicit and implicit versions of PV-KSUPG scheme are presented. Unlike KSUPG and PV-KUPG schemes where Maxwellian distribution function is used, the C-KUSPG scheme uses a simpler circular distribution function instead of a Maxwellian distribution function. Apart from being computationally less expensive it is less diffusive than KSUPG scheme.

Orientadores: Maicon Ribeiro Correa, Eduardo Cardoso de Abreu / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T11:41:21Z (GMT). No. of bitstreams: 1 Silva_FelipeAugustoGuedesda_M.pdf: 1119470 bytes, checksum: eeabeb98750e53492e778b99174c0887 (MD5) Previous issue date: 2015 / Resumo: O foco do presente trabalho consiste no estudo computacional de métodos de Galerkin Descontínuo para aproximação numérica de problemas diferenciais de natureza hiperbólica, com enfoque em esquemas explícitos e no uso de aproximações do tipo Runge-Kutta no tempo para aproximação de problemas lineares e não-lineares. Especificamente, serão exploradas as boas propriedades de estabilidade local, no tempo, dos métodos da classe Runge-Kutta em conjunto com funções de fluxo numérico estáveis e com o uso de limitadores de inclinação, com o objetivo de desenvolver métodos Galerkin Descontínuo de alta ordem capazes de obter uma boa resolução de gradientes abruptos e de soluções descontínuas, sem oscilações espúrias, em problemas hiperbólicos. Uma breve discussão sobre esquemas de volumes finitos centrais de alta ordem é apresentada, onde são introduzidos importantes conceitos a serem utilizados na construção dos métodos de Galerkin Descontínuo. Um conjunto representativo de simulações numéricas de modelos hiperbólicos lineares e não-lineares é apresentado e discutido para avaliar a qualidade das aproximações obtidas em uma comparação direta com outras aproximações precisas de volumes finitos ou com soluções exatas, sempre que possível / Abstract: The focus of this work is the computational study of some Discontinuous Galerkin methods for the numerical approximation of first order hyperbolic differential problems, focusing on explicit schemes with discretization based on Runge-Kutta type methods in time, in problems with linear and nonlinear fluxes. Specifically, the good local stability properties of Runge-Kutta methods are combined with stable numerical flux functions and slope limiters in order to propose new higher-order Discontinuous Galerkin methods that achieve high resolution of abrupt gradients and of discontinuous solutions, without spurious oscillations in numerical solutions. Furthermore, a brief discussion about higher-order finite volume central schemes is presented in order to introduce some important concepts to be used in the construction of the DG methods. A representative set of numerical simulations for linear and nonlinear hyperbolic models is presented and discussed, in order to check the accuracy of the obtained Discontinuous Galerkin solutions by comparing their results with those of existing well-established finite volume numerical methods and exact solutions / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

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A presente dissertação propõe a resolução do problema de flexão estocástica em uma viga Euler-Bernoulli, sobre uma fundação do tipo Pasternak, através de um método computacional baseado na simulação de Monte Carlo. A incerteza está presente nos coeficientes elásticos da viga e da fundação. Primeiramente, é estabelecida a formulação matemática do problema que é oriunda, de um modelo físico de deslocamento da viga, que leva em consideração a influência da fundação sobre a resposta do problema. Portanto foi realizado um estudo a cerca dos modelos mais usuais de fundação, que são: o modelo do tipo Winkler, e modelo de Pasternak. Logo a seguir foi provado que o problema variacional abstrato, derivado da formulação forte do problema, apresenta solução e esta é única. Para a obtenção da solução do problema, foi realizada uma fundamentação matemática, dos seguintes assuntos: representação da incerteza, método de Galerkin, série de Neumann, e por fim das cotas inferiores e superiores. Finalmente, o desempenho das cotas inferiores e superiores, em relação à simulação de Monte Carlo direto, foram avaliadas através de vários casos, nos quais a incerteza repousa sobre os diversos coeficientes que compõe a equação de flexão na forma de um problema variacional. A metodologia mostrou-se eficiente, tanto no aspecto da convergência da resposta quanto no que se refere ao custo computacional. / This work proposes the resolution of stochastic bending problem in a Euler- Bernoulli beam, on a foundation type Pasternak, through a computational method based on Monte Carlo simulation. Uncertainty is present in the elastic coefficients of the beam and foundation. First, it is established the mathematical formulation of the problem which is derived from a physical model displacement of the beam, that takes into account the influence of the foundation on the problem of response. This requires an approach that is made up on the most common models of foundation, which are: the model Winkler type and model of Pasternak.In sequence we study the existence and uniqueness of the variational problem. To obtain the solution of the problem, a mathematical reasoning is carried out, to the following matters: representation of uncertainty, Galerkin method, serial Neumann, and finally the lower and upper bounds. Finally, the performance of lower and upper bounds, derived from direct simulation of Monte Carlo were evaluated through various cases where the uncertainty lies in the different coefficients composing the equation bending as a variational problem. The method proved to be efficient, both in the response of the convergence point as regards the computational cost.

A presente dissertação propõe a resolução do problema de flexão estocástica em uma viga Euler-Bernoulli, sobre uma fundação do tipo Pasternak, através de um método computacional baseado na simulação de Monte Carlo. A incerteza está presente nos coeficientes elásticos da viga e da fundação. Primeiramente, é estabelecida a formulação matemática do problema que é oriunda, de um modelo físico de deslocamento da viga, que leva em consideração a influência da fundação sobre a resposta do problema. Portanto foi realizado um estudo a cerca dos modelos mais usuais de fundação, que são: o modelo do tipo Winkler, e modelo de Pasternak. Logo a seguir foi provado que o problema variacional abstrato, derivado da formulação forte do problema, apresenta solução e esta é única. Para a obtenção da solução do problema, foi realizada uma fundamentação matemática, dos seguintes assuntos: representação da incerteza, método de Galerkin, série de Neumann, e por fim das cotas inferiores e superiores. Finalmente, o desempenho das cotas inferiores e superiores, em relação à simulação de Monte Carlo direto, foram avaliadas através de vários casos, nos quais a incerteza repousa sobre os diversos coeficientes que compõe a equação de flexão na forma de um problema variacional. A metodologia mostrou-se eficiente, tanto no aspecto da convergência da resposta quanto no que se refere ao custo computacional. / This work proposes the resolution of stochastic bending problem in a Euler- Bernoulli beam, on a foundation type Pasternak, through a computational method based on Monte Carlo simulation. Uncertainty is present in the elastic coefficients of the beam and foundation. First, it is established the mathematical formulation of the problem which is derived from a physical model displacement of the beam, that takes into account the influence of the foundation on the problem of response. This requires an approach that is made up on the most common models of foundation, which are: the model Winkler type and model of Pasternak.In sequence we study the existence and uniqueness of the variational problem. To obtain the solution of the problem, a mathematical reasoning is carried out, to the following matters: representation of uncertainty, Galerkin method, serial Neumann, and finally the lower and upper bounds. Finally, the performance of lower and upper bounds, derived from direct simulation of Monte Carlo were evaluated through various cases where the uncertainty lies in the different coefficients composing the equation bending as a variational problem. The method proved to be efficient, both in the response of the convergence point as regards the computational cost.

This work is dedicated to the control of the accuracy of computational simulations of sound propagation and scattering. Assuming time-harmonic behaviour, the mathematical models are given as boundary value problems for the Helmholtz equation <i>Delta u+k2u=0 </i> in <i>Oméga</i>. A distinction is made between interior, exterior and coupled problems and this work focuses mainly on interior uncoupled problems for which the Helmholtz equation becomes singular at eigenfrequencies. <p><p>As in other application fields, error control is an important issue in acoustic computations. It is clear that the numerical parameters (mesh size h and degree of approximation p) must be adapted to the physical parameter k. The well known ‘rule of the thumb’ for the h version with linear elements is to resolve the wavelength <i>lambda=2 pi k-1</i> by six elements characterising the approximability of the finite element mesh. If the numerical model is stable, the quality of the numerical solution is entirely controlled by the approximability of the finite element mesh. The situation is quite different in the presence of singularities. In that case, <i>stability</i> (or the lack thereof) is equally (sometimes more) important. In our application, the solutions are ‘rough’, i.e. highly oscillatory if the wavenumber is large. This is a singularity inherent to the differential operator rather than to the domain or the boundary conditions. This effect is called the <i>k-singularity</i>. Similarly, the discrete operator (“stiffness” matrix) becomes singular at eigenvalues of the discretised interior problem (or nearly singular at damped eigenvalues in solid-fluid interaction). This type of singularities is called the <i>lambda-singularities</i>. Both singularities are of global character. Without adaptive correction, their destabilizing effect generally leads to large error of the finite element results, even if the finite element mesh satisfies the ‘rule of the thumb’. <p><p>The k- and lambda-singularities are first extensively demonstrated by numerical examples. Then, two <i>a posteriori</i> error estimators are developed and the numerical tests show that, due to these specific phenomena of dynamo-acoustic computations, <i>error control cannot, in general, be accomplished by just ‘transplanting’ methods that worked well in static computations</i>. However, for low wavenumbers, it is necessary to also control the influence of the geometric (reentrants corners) or physical (discontinuities of the boundary conditions) singularities. An <i>h</i>-adaptive version with refinements has been implemented. These tools have been applied to two industrial examples :the GLT, a bi-mode bus from Bombardier Eurorail, and the Vertigo, a sport car from Gillet Automobiles.<p><p>As a conclusion, it is recommanded to replace the rule of the thumb by a criterion based on the control of the influence of the specific singularities of the Helmholtz operator. As this aim cannot be achieved by the <i>a posteriori</i> error estimators, it is suggested to minimize the influence of the singularities by modifying the formulation of the finite element method or by formulating a “meshless” method.<p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished