This thesis describes possible applications of Prony's method in timing analysis of digital circuits. Such applications include predicting the future shape of the waveform in DTA(Dynamic Timing Analysis) and delay look-up table in STA(Static Timing Analysis). Given some equally spaced output values, the traditional Prony's method can be used to extract poles and residues of a linear system, i.e. to characterize a waveform using an exponential function. In this thesis, not only values but also equally spaced derivatives are tested. Still using same idea of the traditional Prony's method, poles and residues can also be extracted with those values and derivatives. The resultant poles and residues will be used to predict the output waveform in DTA analysis. The benefits brought by the using of derivatives include less simulation steps and less CPU time consuming than the regular constant step simulation. As a matter of fact, the Prony's method can precisely approximate a complicated waveform. Such property can be applied for STA analysis. The Prony's approximation can be used to precisely record an output waveform, which is used as an entry of the look-up table of STA. Since the accuracy of STA analysis relies on the accuracy of the input and output waveform in the look-up table, the accuracy of the Prony's approach is promising.

Recent technological developments have made it possible to design various microdevices where fluid flow and heat transfer are involved. For the proper design of such systems, the governing physics needs to be investigated. Due to the difficulty to study complex geometries in micro scales using experimental techniques, computational tools are developed to analyze and simulate flow and heat transfer in microgeometries. However, conventional numerical methods using the Navier-Stokes equations fail to predict some aspects of microflows such as nonlinear pressure distribution, increase mass flow rate, slip flow and temperature jump at the solid boundaries. This necessitates the development of new computational methods which depend on the kinetic theory that are both accurate and computationally efficient. In this study, lattice Boltzmann method (LBM) was used to investigate the flow and heat transfer in micro sized geometries. The LBM depends on the Boltzmann equation which is valid in the whole rarefaction regime that can be observed in micro flows. Results were obtained for isothermal channel flows at Knudsen numbers higher than 0.01 at different pressure ratios. LBM solutions for micro-Couette and micro-Poiseuille flow were found to be in good agreement with the analytical solutions valid in the slip flow regime (0.01 < Kn < 0.1) and direct simulation Monte Carlo solutions that are valid in the transition regime (0.1 < Kn < 10) for pressure distribution and velocity field. The isothermal LBM was further extended to simulate flows including heat transfer. The method was first validated for continuum channel flows with and without constrictions by comparing the thermal LBM results against accurate solutions obtained from analytical equations and finite element method. Finally, the capability of thermal LBM was improved by adding the effect of rarefaction and the method was used to analyze the behavior of gas flow in microchannels. The major finding of this research is that, the newly developed particle-based method described here can be used as an alternative numerical tool in order to study non-continuum effects observed in micro-electro-mechanical-systems (MEMS).

A presença de onda de choque e vórtices de pequena escala exigem métodos numéricos mais sofisticados para simular escoamentos compressíveis em velocidades altas. Alguns desses métodos produzem resultados adequados para regiões com função suave, embora os mesmos não possam ser utilizados diretamente em regiões com função descontínua, resultando em oscilações espúrias. Dessa forma, métodos foram desenvolvidos para solucionar esse problema, apresentando um bom desempenho para regiões com função descontínua; entretanto, estes possuem termos de alta dissipação. Para evitar os problemas encontrados, foram desenvolvidos os métodos híbridos, onde dois métodos com características ideais para cada região são combinados através de uma função detectora que analisa numericamente a variação de uma quantidade em uma região através de fórmulas que envolvem derivadas. Um detector de descontinuidades foi desenvolvido a partir da revisão bibliográfica de diversos métodos numéricos híbridos existentes, sendo avaliadas as principais desvantagens e limitações de cada um. Diversas comparações entre o novo detector e os detectores de descontinuidades já desenvolvidos foram realizadas através da aplicação em funções unidimensionais e bidimensionais. Finalmente, o método híbrido foi aplicado para a solução das equações de Euler unidimensionais e bidimensionais. / The presence of shock and small-scale vortices require more sophisticated numerical methods to simulate compressible flows at high speeds. Some of these methods produce good results for regions with smooth function, altough they cannot be used directly in regions with discontinuous functions, resulting in spurious oscillations. Thus, methods have been developed to solve this problem, showing a good performance for regions with discontinuous functions; however, these methods contain high dissipation terms. To avoid the problems encountered, hybrid methods have been developed, where two methods with ideal characteristics for each region are combined through a function that analyze numerically the variation of a quantity in the region using formulas involving derivatives. A discontinuity detector was developed from the literature review of several existing hybrid methods, evaluating the main disadvantages and limitations of each. The new detector and other developed discontinuity detectors were compared by applying on one and two-dimensional functions. Finally, the hybrid method was applied fo the solution of one and twodimensional Euler equations.

Dans un assemblage combustible de réacteur à neutrons rapides, les aiguilles combustibles sont insérées en faisceau dans un tube hexagonal avec des jeux de montage. Lors de situations transitoires accidentelles de type séisme ou dans des phases de transport et de manutention des assemblages, les aiguilles sont donc susceptibles de se déplacer légèrement dans le faisceau et de s’entrechoquer les unes avec les autres, plusieurs centaines de milliers de contacts auront alors lieu dans le faisceau. L’objectif de cette thèse est de développer une méthode numérique capable simuler le comportement dynamique du faisceau avec jeu, afin d’évaluer les efforts de contact pour pouvoir dimensionner mécaniquement les aiguilles combustibles pour ce type de sollicitation. Le sodium n’est pas représenté, on s’intéresse donc ici au comportement de l’assemblage en air. Le comportement dynamique de l’aiguille a été étudié expérimentalement sur un banc d’essai spécialement conçu dans le cadre de cette thèse et il sera présenté dans un premier temps. L’analyse des essais de vibration libre a permis de mettre en évidence un amortissement non-linéaire dont la phénoménologie sera détaillée. La méthode numérique choisie sera ensuite décrite et comparée à des références analytiques et expérimentales, en particulier afin de justifier des critères de choix des paramètres numériques qui ont été identifiés. Enfin, on s’intéressera à la modélisation d’un assemblage combustible complet, avec différents types de représentation et à la comparaison des résultats à des essais de chocs ce qui conclura la présentation / Understanding of phenomena taking place in a structure with multiple clearances is an industrial challenge. The fuel pellets in the SFR (Sodium-cooled Fast Reactor) prototype ASTRID are enclosed in small and long pins which form a bundle inside a hexagonal assembly. The assessment of stresses in the pins during dynamic loadings is essential for the safety studies of the project. Experimental tests on the test bed CARNAC have been conducted to understand the dynamical behavior of fuel pin with their pellets. Then a numerical model has been chosen to simulate the release of an assembly against a stop. The difficulty is to simulate the dynamical behavior of a structure with a huge number of internal contacts. Numerical method has been validated on a basic problem with a reference semi-analytical method. Simplified models of the pin bundle are created to understand dynamical phenomena of a multicontact system. Then, the whole assembly is modeled. Sub-structuring to accelerate computation and a precise contact law representative of the pin to pin contacts are used. Displacements, energy and contacts force are analyzed with or without clearance. Numerical results are confronted to a previous experiment made in the CEA and provide a very good fit. The average kinetic behavior of assembly is well approximated by a beam structure, if pins are linked. Contact forces are well assess with conservatism using simplified model of a pin row

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<p>In this work, we develop several special computational structures of Neural Networks based on some existing approaches such as Auto-Encoder and DeepONet. Combined with classic numerical methods in scientific computing, finite difference and SAV method, our model is able to solve the operator learning tasks of partial differential equations accurately in both data-driven and non-data-driven settings. The high dimensional problem requires a large number of samples for training in the normal settings of Neural network training. The proposed</p> <p>model equipped with auto-encoder performs the dimension reduction for the input operator, which discovers the intrinsic hidden features, to reduce the number of samples needed for training. In addition, the non-linear basis of the hidden variables are constructed</p> <p>for both the operator variable and the solution of the equation, leading to a concise representation of the solution. For non data-driven setting, our method derives the solution of the equation with only initial and boundary condition, where the normal network can not manage to do it, with the assistance of SAV method. Besides, it preserves the advantages of DeepONet. It performs the operator learning with various initial conditions or parametric equations. The modified energy is defined to estimate the true energy of the system and it has the monotonic decreasing property. It also serves as an indicator of the suitable time step, allowing the model to adjust the time step. Finally, the optimization is a key procedure of network training. We propose a new optimization method based on SAV. It allows a much</p> <p>larger learning rate compared to SGD and ADAM, which are most popular methods used nowadays. Moreover, It also allows the adaptive learning rate to pursue the faster speed converging to the critical point.</p>

A one-dimensional model of a heterogeneous material consisting of a matrix with embedded separated particles is considered, and the melting or solidification of the particles is investigated. The matrix is in imperfect contact with the particles, and the lumped capacity approximation applies to each individual particle. Heat is generated inside the particles or is transferred from the matrix to the particles coupled through a contact conductance. The matrix is not allowed to change phase and energy is either generated inside the matrix or transferred from the boundaries, which is initially conducted through the matrix material. The physical model of this coupled, two-step heat transfer process is solved using the energy method. The investigation is conducted in several phases using a building block approach. First, a lumped capacity system during phase transition is studied, then a one-dimensional homogeneous material during phase change is investigated, and finally the one-dimensional heterogeneous material is analyzed. A numerical solution based on the finite difference method is used to solve the model equations. This method allows for any kind of boundary conditions, any combination of material properties, particle sizes and contact conductance. In addition, computer programs, using Mathematica, are developed for the lumped capacity system, homogeneous material, and heterogeneous material. Results show the effects of control volume thickness, time step, contact conductance, material properties, internal sources, and external sources. / Master of Science

Les structures nanophotoniques sont généralement simulées par des méthodes de volumes, comme la méthode des différences finies dans le domaine temporel (FDTD), ou la méthode des éléments finis (FEM). Toutefois, pour les grandes structures, ou des structures plasmoniques métalliques qui nécessitent, la mémoire et le temps de calcul requis peuvent augmenter de façon spectaculaire.Les méthodes de surface, comme la méthode des éléments de frontière (BEM) ont été développées afin de réduire le nombre d'éléments de maillage. Ces méthodes consistent à exprimer le champ formé dans tout l'espace en fonction des courants électrique et magnétique à la surface de l’objet. Combinées avec la méthode multipôle rapide (FMM) qui permet une accélération du calcul de l'interaction entre les éléments lointain du maillage, de grands systèmes peuvent ainsi être manipulés.Nous avons développé, pour la première fois à notre connaissance, une FMM sur un nouveau formalisme BEM, basé sur les potentiels scalaire et vectoriel au lieu de courants électriques et magnétiques. Cette méthode a été montrée pour permettre une simulation précise des systèmes plasmoniques métalliques, tout en offrant une réduction significative des besoins de calcul. Des systèmes nanophotoniques complexes ont été simulés, comme une lentille plasmonique composé d'un ensemble de nanotubes d'or. / Nanophotonic structures are generally simulated by volume methods, as Finite-difference time-domain (FDTD) method, or Finite element method (FEM). However, for large structures, or metallic plasmonic structures, the memory and time computation required can increase dramatically, and make proper simulation infeasible.Surface methods, like the boundary element method (BEM) have been developed to reduce the number of mesh elements. These methods consist in expressing the electromagnetic filed in whole space as a function of electric and magnetic currents at the surface of scatterers. Combined with the fast multipole method (FMM) that enables a huge acceleration of the calculation of interaction between far mesh elements, very large systems can thus be handled.What we performed is the development of an FMM on a new BEM formalism, based on scalar and vector potentials instead of electric and magnetic currents, for the first time to our knowledge. This method was shown to enable accurate simulation of metallic plasmonic systems, while providing a significant reduction of computation requirements, compared to BEM-alone. Several thousands of unknowns could be handled on a standard computer. More complex nanophotonic systems have been simulated, such as a plasmonic lens consisting of a collection of gold nanorods.

Les vibrations qui sont en général source de dérangement, d’usure et même destruction des machines et structures mécaniques doivent être contrôlées ou éliminées. Pour cette raison, la lutte contre les vibrations est devenue depuis des années un enjeu majeur pour les chercheurs de laboratoire et de développement dans l’industrie afin de développer des solutions efficaces contre ces problèmes. De nombreuses technologies ont donc été développées. Parmi ces technologies, les absorbeurs de vibration non linéaires présentent des performances importantes dans l’atténuation de vibration sur une large bande de fréquences. C’est dans ce contexte que cette thèse se focalise sur l’analyse et l’optimisation des absorbeurs de vibration non linéaires. L’objectif de cette thèse est d’analyser le comportement dynamique non linéaire des systèmes présentant des absorbeurs de vibration non linéaires. Pour cela, un modèle dynamique d’un système à deux degrés de liberté est développé mettant en équations le comportement non linéaire. La résolution des équations de mouvement est faite par la Méthode Asymptotique Numérique (MAN). La performance de cette méthode est montrée via une comparaison avec la méthode de Newton-Raphson. L’analyse des modes non linéaires du système ayant une non-linéarité cubique est faite par une formulation explicite des Fonctions de Réponse en Fréquence non linéaires (FRFs) et les Modes Normaux Non linéaires (MNNs). Un démonstrateur sur la base d’un système simple à deux degré de liberté est mis en place afin de recaler les modèles envisagés sur la base des résultats expérimentaux trouvés. / Vibrations are usually undesired phenomena as they may cause discomfort, disturbance, damage, and sometimes destruction of machines and structures. It must be reduced or controlled or eliminated. For this reason, the vibrations attenuation became a major issue for scientists and researchers in order to develop effective solutions for these problems. Many technologies have been developed. Among these technologies, the nonlinear vibration absorbers have significant performance in the vibration attenuation over a wide frequency band. In this context, this thesis focuses on the analysis and optimization of nonlinear vibration absorbers. The objective of the thesis is to analyze the nonlinear dynamic behavior of systems with nonlinear vibration absorbers. For this, a dynamic model of a two degrees of freedom system is developed. The Asymptotic Numerical Method (ANM) is used to solve the nonlinear equations of motion. The performance of this method is shown via a comparison with the Newton-Raphson method. The nonlinear modal analysis system with cubic nonlinearity is made by an explicit formulation of the nonlinear Frequency Response Functions (FRFs) and Nonlinear Normal Modes (MNNs). An experimental study is performed to validate the numerical results.

The concept of using aluminum as the primary construction material for high speed ships and the hydroelastic behavior of the structure is widely gaining importance as a significant research topic in naval architecture. Aluminum is lighter than steel and hence can be predominantly used in high speed crafts which experiences significant slamming. This thesis work is focused on wedge shaped models. Free fall wedge impact is studied and a FORTRAN 90 computer program is developed to estimate the structural response of the wedge experiencing slamming by the use of matrix methods, finite element techniques and Newmark-Beta numerical time integration methods. The numerical solution is validated by comparison with the static solution. The theoretical hydrodynamic pressures which are used as input for this work was originally developed by using a flat cylinder theory [26]. The wedge drop at 0.6096 m (24 inch) drop height with an impact veloc- ity of v=3.05 m/s is based as the premise and the experimental pressure distributions measured by the pressure-transducers and the theoretical pressure predictions are used as inputs and the structural response is derived. Additionally, the response is compared for three different plate thicknesses and the results are compared against each other. The maximum deflection is comparable to the deflection evaluated from the experiment and tends to attain convergence as well. As the plate thickness reduces there tends to be a significant rise in the deflection values for the wedge plate, in the manner that when the plate thickness is halved there is a deviation of more than 75% in the deflection values as such.