Investigamos analiticamente as propriedades estatísticas dos mínimos locais (estados metaestáveis) de vidros de spin de Ising com interações de p-spins na presença de um campo magnético h. O número médio de mínimos, assim como a sobreposição típica entre pares de mínimos idênticos são calculados para qualquer valor de p. Para p &#62 2 e h pequeno mostramos que a sobreposição típica qt é uma função descontínua da energia. O tamanho na descontinuidade em qt cresce com p e decresce com h, indo a zero para valores finitos do campo magnético [1]. Investigamos as correções ao alcance infinito para o caso em que h = 0 e encontramos que o número de estados metaestáveis aumenta quando o efeito de conectividade finita é considerado e esse aumento torna-se mais pronunciado à medida que p aumenta [2]. Ainda, estudamos a termodinâmica deste modelo utilizando o método das réplicas. Demos ênfase à análise da transição entre os regimes de simetria de réplicas e o primeiro passo de quebra de simetria de réplicas. Em particular, derivamos condições analíticas para o início da transição contínua, assim como para a localização do ponto tricrítico onde a transição entre os dois regimes torna-se descontínua [3]. Como aplicação de interações de ordem alta em sistemas de spins contínuos, estudamos analiticamente as propriedades estatísticas de um ecossistema composto de N espécies interagindo através de interações Gaussianas aleatórias de ordem p &#8805 2 e auto-interações determinísticas u &#8805 0. Para o caso u &#8800 0, o aumento na ordem das interações faz com que o sistema se torne mais cooperativo. Para p &#62 2 há um limite inferior para a concentração de espécies sobreviventes, prevenindo a existência de espécies raras e, conseqüentemente, aumentando a robustez do ecossistema contra perturbações externas [4]. / The statistical properties of the local optima (metastable states) of the infinite range Ising spin glass with p-spin interactions in the presence of an external magnetic field h are investigated analytically. The average number of optima as well as the typical overlap between pairs of identical optima are calculated for general p. For p &#62 2 and small h we show that the typical overlap qt is a discontinuous function of the energy. The size of the jump in qt increases with p and decreases with h, vanishing at finite values of the magnetic field [1]. We study the corrections to the infinite range model for h = 0 and find that the number of local optima increases as the effect of the finite connectivity is considered, and that this increase becomes more pronounced for large p [2]. Furthermore, we study analytically the thermodynamics of this model using the replica method, giving emphasis to the analysis of the transition between the replica symmetric and the one-step of replica symmetry breaking regimes. In particular, we derive analytical conditions for the onset of the continuous transition, as well as for the location of the tricritical point at which the transition between those two regimes becomes discontinuous [3]. As an application of high-order interactions in systems of continuous spins, we study the statistical properties of an ecosystem composed of N species interacting via random Gaussian interactions of order p &#8805 2, and deterministic self-interactions u &#8805 0. For nonzero u the increase of the order of the interactions makes the system more cooperative. We find that for p &#62 2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations [4].

This research deals with determining the acoustic attenuation of heating, ventilation, and air conditioning (HVAC) ductwork. A finite element approach was developed for calculating insertion loss and breakout transmission loss. Procedures for simulating the source and receiving rooms were developed and the effect of structureborne flanking was included. Simulation results have been compared with measurements from the literature and the agreement is very good. With a good model in place, the work was extended in three ways. 1) Since measurements on full-scale equipment are difficult, scale modeling rules were developed and validated. 2) Two different numerical approaches were developed for evaluating the transmission loss of silencers taking into account the effect of higher order modes. 3) A power transfer matrix approach was developed to assess the acoustic performance of several duct components connected in series.

Cette thèse présente le développement d’une méthode ALE pour la modélisation del’interaction laser–plasma. La particularité de cette méthode est l’utilisation d’une étape de projectiond’ordre élevé. Cette étape de projection consiste en une interpolation conservative des quantitésconservatives du maillage Lagrangien sur un maillage régularisé. Afin d’éviter les oscillationsnumériques non-physiques, les flux numériques d’ordre élevé sont combinés avec des fluxnumériques d’ordre moins élevé. Ces flux numériques sont obtenu en considérant les quantitésconservatives constantes par morceaux. Cette méthode pour la discrétisation cellule–centrée consisteà préserver les maximums locaux pour la densité, la vitesse et l’énergie interne. Aspects particuliersde la méthode sont appliquées pour la projection la quantité de mouvement pour la discrétisation’staggered’. Nous l’utilisons ici dans le cadre de la projection sous la forme de la méthode FluxCorrection Remapping (FCR). Dans cette thèse le volet applicatif concerne la modélisation del’interaction d’un laser énergétique avec de plasma et des matériaux microstructures. Un intérêtparticulier est porté à la modélisation de l’absorption du laser par une mousse de faible densité.L’absorption se fait à deux échelles spatiales simultanément. Ce modèle d’absorption laser à deuxéchelles est mis en oeuvre dans le code PALE hydrodynamique. Les simulations numériques de lavitesse de pénétration du laser dans une mousse à faible densité sont en bon accord avec lesdonnées expérimentales. / This thesis presents the overview and the original contributions to a high–orderArbitrary Lagrangian–Eulerian (ALE) method applicable for the laser–generated plasma modeling withthe focus to a remapping step of the ALE method. The remap is the conservative interpolation of theconservative quantities from a low–quality Lagrangian grid onto a better, smoothed one. To avoidnon–physical numerical oscillations, the high–order numerical fluxes of the reconstruction arecombined with the low–order (first–order) numerical fluxes produced by a standard donor remappingmethod. The proposed method for a cell–centered discretization preserves bounds for the density,velocity and specific internal energy by its construction. Particular symmetry–preserving aspects of themethod are applied for a staggered momentum remap. The application part of the thesis is devoted tothe laser radiation absorption modeling in plasmas and microstructures materials with the particularinterest in the laser absorption in low–density foams. The absorption is modeled on two spatial scalessimultaneously. This two–scale laser absorption model is implemented in the hydrodynamic codePALE. The numerical simulations of the velocity of laser penetration in a low–density foam are in agood agreement with the experimental data.

The objective of this research is to develop an efficient and accurate methodology to resolve flow non-linearity of fluid-structural interaction. To achieve this purpose, a numerical strategy to apply the detached-eddy simulation (DES) with a fully coupled fluid-structural interaction model is established for the first time. The following novel numerical algorithms are also created: a general sub-domain boundary mapping procedure for parallel computation to reduce wall clock simulation time, an efficient and low diffusion E-CUSP (LDE) scheme used as a Riemann solver to resolve discontinuities with minimal numerical dissipation, and an implicit high order accuracy weighted essentially non-oscillatory (WENO) scheme to capture shock waves. The Detached-Eddy Simulation is based on the model proposed by Spalart in 1997. Near solid walls within wall boundary layers, the Reynolds averaged Navier-Stokes (RANS) equations are solved. Outside of the wall boundary layers, the 3D filtered compressible Navier-Stokes equations are solved based on large eddy simulation(LES). The Spalart-Allmaras one equation turbulence model is solved to provide the Reynolds stresses in the RANS region and the subgrid scale stresses in the LES region. An improved 5th order finite differencing weighted essentially non-oscillatory (WENO) scheme with an optimized epsilon value is employed for the inviscid fluxes. The new LDE scheme used with the WENO scheme is able to capture crisp shock profiles and exact contact surfaces. A set of fully conservative 4th order finite central differencing schemes are used for the viscous terms. The 3D Navier-Stokes equations are discretized based on a conservative finite differencing scheme, which is implemented by shifting the solution points half grid interval in each direction on the computational domain. The solution points are hence located in the center of the grid cells in the computational domain (not physical domain). This makes it possible to use the same code structure as a 2nd order finite volume method. A finite differencing high order WENO scheme is used since a finite differencing WENO scheme is much more efficient than a finite volume WENO scheme. The unfactored line Gauss-Seidel relaxation iteration is employed for time marching. For the time accurate unsteady simulation, the temporal terms are discretized using the 2nd order accuracy backward differencing. A pseudo temporal term is introduced for the unsteady calculation following Jameson's method. Within each physical time step, the solution is iterated until converged based on pseudo time step. A general sub-domain boundary mapping procedure is developed for arbitrary topology multi-block structured grids with grid points matched on sub-domain boundaries. The interface of two adjacent blocks is uniquely defined according to each local mesh index system (MIS) which is specified independently. A pack/unpack procedure based on the definition of the interface is developed to exchange the data in a 1D array to minimize data communication. A secure send/receive procedure is employed to remove the possibility of blocked communication and achieve optimum parallel computation efficiency. Two terms, "Order" and "Orientation", are introduced as the logics defining the relationship of adjacent blocks. The domain partitioning treatment of the implicit matrices is to simply discard the corner matrices so that the implicit Gauss-Seidel iteration can be implemented within each subdomain. This general sub-domain boundary mapping procedure is demonstrated to have high scalability. Extensive numerical experiments are conducted to test the performance of the numerical algorithms. The LDE scheme is compared with the Roe scheme for their behavior with RANS simulation. Both the LDE and the Roe scheme can use high CFL numbers and achieve high convergence rates for the algebraic Baldwin-Lomax turbulence model. For the Spalart-Allmaras one equation turbulence model, the extra equation changes the Jacobian of the Roe scheme and weakens the diagonal dominance. It reduces the maximum CFL number permitted by the Roe scheme and hence decreases the convergence rate. The LDE scheme is only slightly affected by the extra equation and maintains high CFL number and convergence rate. The high stability and convergence rate using the Spalart-Allmaras one equation turbulence model is important since the DES uses the same transport equation for the turbulence stresses closure. The RANS simulation with the Spalart-Allmaras one equation turbulence model is the foundation for DES and is hence validated with other transonic flows including a 2D subsonic flat plate turbulent boundary layer, 2D transonic inlet-diffuser, 2D RAE2822 airfoil, 3D ONERA M6 wing, and a 3D transonic duct with shock boundary layer interaction. The predicted results agree very well with the experiments. The RANS code is then further used to study the slot size effect of a co-flow jet (CFJ) airfoil. The DES solver with fully coupled fluid-structural interaction methodology is validated with vortex induced vibration of a cylinder and a transonic forced pitching airfoil. For the cylinder, the laminar Navier-Stokes equations are solved due to the low Reynolds number. The 3D effects are observed in both stationary and oscillating cylinder simulation because of the flow separations behind the cylinder. For the transonic forced pitching airfoil DES computation, there is no flow separation in the flow field. The DES results agree well with the RANS results. These two cases indicate that the DES is more effective on predicting flow separation. The DES code is used to simulate the limited cycle oscillation of NLR7301 airfoil. For the cases computed in this research, the predicted LCO frequency, amplitudes, averaged lift and moment, all agree excellently with the experiment. The solutions appear to have bifurcation and are dependent on the initial perturbation. The developed methodology is able to capture the LCO with very small amplitudes measured in the experiment. This is attributed to the high order low diffusion schemes, fully coupled FSI model, and the turbulence model used. This research appears to be the first time that a numerical simulation of LCO matches the experiment. The DES code is also used to simulate the CFJ airfoil jet mixing at high angle of attack. In conclusion, the numerical strategy of the high order DES with fully coupled FSI model and parallel computing developed in this research is demonstrated to have high accuracy, robustness, and efficiency. Future work to further maturate the methodology is suggested.

The degradation of astronomical images caused by atmospheric turbulence will be much more severe in the next generation of terrestrial telescopes and its compensation will require deformable mirrors with up to tens-of-thousands of actuators. Current designs for these correctors consist of scaling up the proven technologies of flexible optical plates deformed under the out-of-plane action of linear actuators. This approach will lead to an exponential growth of cost with the number of actuators, and in very complex mechanisms. This thesis proposes a new concept of optical correction which is modular, robust, lightweight and low-cost and is based on the bimorph in-plane actuation. The adaptive mirror consists of segmented identical hexagonal bimorph mirrors allowing to indefinitely increase the degree of correction while maintaining the first mechanical resonance at the level of a single segment and showing an increase in price only proportional to the number of segments. Each bimorph segment can be mass-produced by simply screen-printing an array of thin piezoelectric patches onto a silicon wafer resulting in very compact and lightweight modules and at a price essentially independent from the number of actuators. The controlled deformation of a screen-printed bimorph mirror was experimentally achieved with meaningful optical shapes and appropriate amplitudes; its capability for compensating turbulence was evaluated numerically. The generation of continuous surfaces by an assembly of these mirrors was numerically simulated and a demonstrator of concept consisting of 3 segments was constructed.

Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software. We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models. The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.

The continued research and development of high-order methods in Computational Fluid Dynamics (CFD) is primarily motivated by their potential to significantly reduce the computational cost and memory usage required to obtain a solution to a desired level of accuracy. In this work, a high-order Central Essentially Non-Oscillatory (CENO) finite-volume scheme is developed for the Euler and Navier-Stokes equations in three dimensions. The proposed CENO scheme is based on a hybrid solution reconstruction procedure using a fixed central stencil. A solution smoothness indicator facilitates the hybrid switching between a high-order k-exact reconstruction technique, and a monotonicity preserving limited piecewise linear reconstruction algorithm. The resulting scheme is applied to the compressible forms of the Euler and Navier-Stokes equations in three dimensions. The latter of which includes the application of this high-order work to the Large Eddy Simulation (LES) of turbulent non-reacting flows.

The development of high-order solution methods remain a very active field of research in Computational Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has been to develop a high-order Central Essentially Non Oscillatory (CENO) finite volume scheme for multi-block unstructured meshes. In particular, solutions to the compressible, inviscid Euler equations are considered. The CENO method achieves a high-order spatial reconstruction based on the k-exact method, combined with hybrid switching to limited piecewise linear reconstruction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-Kutta time marching is applied. The solver described has been validated through a combination of high-order function reconstructions, and solutions to the Euler equations. Cases have been selected to demonstrate high-orders of convergence, the application of the hybrid switching method, and the multi-block techniques which has been implemented.

Large-eddy simulation (LES) is emerging as a promising computational tool for reacting flows. High-order schemes for LES are desirable to achieve improved solution accuracy with reduced computational cost. In this study, a parallel, block-based, three-dimensional high-order central essentially non-oscillatory (CENO) finite-volume scheme for LES of premixed turbulent combustion is developed for Cartesian mesh. This LES formulation makes use of the flame surface density (FSD) for subfilter-scale reaction rate modelling. An algebraic model is used to approximate the FSD. A detailed explanation of the governing equations for LES and the mathematical framework for CENO schemes are presented. The CENO reconstruction is validated and is also applied to three-dimensional Euler equations prior to its application to the equations governing LES of reacting flows.

The development of high-order solution methods remain a very active field of research in Computational Fluid Dynamics (CFD). These types of schemes have the potential to reduce the computational cost necessary to compute solutions to a desired level of accuracy. The goal of this thesis has been to develop a high-order Central Essentially Non Oscillatory (CENO) finite volume scheme for multi-block unstructured meshes. In particular, solutions to the compressible, inviscid Euler equations are considered. The CENO method achieves a high-order spatial reconstruction based on the k-exact method, combined with hybrid switching to limited piecewise linear reconstruction in non-smooth regions to maintain monotonicity. Additionally, fourth-order Runge-Kutta time marching is applied. The solver described has been validated through a combination of high-order function reconstructions, and solutions to the Euler equations. Cases have been selected to demonstrate high-orders of convergence, the application of the hybrid switching method, and the multi-block techniques which has been implemented.