L’approximation par harmoniques sphériques (SPN) simplifiées de l’équation de transfert radiatif a été proposée comme un modèle fiable de propagation de la lumière dans les tissus biologiques. Cependant, peu de solutions analytiques ont été trouvées pour ce modèle. De telles solutions analytiques sont d’une grande valeur pour valider les solutions numériques des équations SPN, auxquelles il faut recourir dans le cas de tissus avec des géométries courbes complexes. Dans la première partie de cette thèse, des solutions analytiques pour deux géométries courbes sont présentées pour la première fois, à savoir pour la sphère et pour le cylindre. Pour les deux solutions, les conditions aux frontières générales tenant compte du saut d’indice de réfraction à l’interface du tissus et de son milieu environnant, telles qu’applicables à l’optique biomédicale, sont utilisées. Ces solutions sont validées à l’aide de simulations Monte Carlo basées sur un maillage de discrétisation du milieu. Ainsi, ces solutions permettent de valider rapidement un code numérique, par exemple utilisant les différences finies ou les éléments finis, sans nécessiter de longues simulations Monte Carlo. Dans la deuxième partie de cette thèse, la reconstruction itérative pour l’imagerie par tomographie optique diffuse par fluorescence est proposée sur la base d’une fonction objective et de son terme de régularisation de type Lq-Lp. Pour résoudre le problème inverse d’imagerie, la discrétisation du modèle de propagation de la lumière est effectuée en utilisant la méthode des différences finies. La reconstruction est effectuée sur un modèle de souris numérique en utilisant un maillage multi-échelle. Le problème inverse est résolu itérativement en utilisant une méthode d’optimisation. Pour cela, le gradient de la fonction de coût par rapport à la carte de concentration de l’agent fluorescent est nécessaire. Ce gradient est calculé à l’aide d’une méthode adjointe. Des mesures quantitatives utilisées en l’imagerie médicale sont utilisées pour évaluer la performance de l’approche de reconstruction dans différentes conditions. L’approche Lq-Lp montre des performances quantifiées élevées par rapport aux algorithmes traditionnels basés sur des fonction coût de type somme de carrés de différences. / Abstract : The simplified spherical harmonics (SPN) approximation to the radiative transfer equation has been proposed as a reliable model of light propagation in biological tissues. However, few analytical solutions have been found for this model. Such analytical solutions are of great value to validate numerical solutions of the SPN equations, which must be resorted to when dealing with media with complex curved geometries. In the first part of this thesis, analytical solutions for two curved geometries are presented for the first time, namely for the sphere and for the cylinder. For both solutions, the general refractiveindex mismatch boundary conditions, as applicable in biomedical optics, are resorted to. These solutions are validated using mesh-based Monte Carlo simulations. So validated, these solutions allow in turn to rapidly validate numerical code, based for example on finite differences or on finite elements, without requiring lengthy Monte Carlo simulations. provide reliable tool for validating numerical simulations. In the second part, iterative reconstruction for fluorescence diffuse optical tomography imaging is proposed based on an Lq-Lp framework for formulating an objective function and its regularization term. To solve the imaging inverse problem, the discretization of the light propagation model is performed using the finite difference method. The framework is used along with a multigrid mesh on a digital mouse model. The inverse problem is solved iteratively using an optimization method. For this, the gradient of the cost function with respect to the fluorescent agent’s concentration map is necessary. This is calculated using an adjoint method. Quantitative metrics resorted to in medical imaging are used to evaluate the performance of the framework under different conditions. The results obtained support this new approach based on an Lq-Lp formulation of cost functions in order to solve the inverse fluorescence problem with high quantified performance.

No description available.

Tato práce zkoumá možností řešení kritické síly vzpěrné stability laminátových kompozitních válcových skořepin jak analytickým tak MKP přístupem. Analytické řešení je prezentováno detailně a je zpracován Matlab program pro jeho výpočet. MKP řešení je prováděno ve dvou řešičích softwaru MSC.Nastran, jsou zjišťována vhodná nastavení těchto řešičů a jejich chování. Všechny způsoby řešení jsou na závěr porovnány a je navrženo několik závěrů.

Main topic of this thesis are magnetic fluids. These are specific type of fluids which can simplistically be considered as Bingham fluids. Main issues regarding the magnetic fluids mentioned in this thesis are: rheological properties of the magnetic fluids, behaviour of the magnetic fluids and the use of the magnetic fluids in industrial applications. Main goal is to apply Binghams model on the the magnetic fluids assuming that this model can be applied with a good accuracy. Equations which describe behaviour of the Bingham fluids are constructed. Some assumptions which have to be respected to use this analytical equations for magnetic fluids are formulated. Flow of bingham fluid is analytically solved in some simplificated cases with consideration of laminar flow. Analytical results are confronted with numerical ones obtained from CFD software Fluent for the purpose of verification.

The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary conditions.

Los suelos blandos originan diversos problemas en el momento de las construcciones, como en las cimentaciones de distintas edificaciones u otras obras civiles. Estos tipos de suelos necesariamente tienen que ser analizados y tratados correctamente, pues, estos originan problemas a corto y largo plazo. En las décadas posteriores se han desarrollado métodos para mejorar el terreno, uno de estos métodos consiste en la construcción, sobre el suelo blando, de elementos verticales rígidos que se llaman inclusiones rígidas. Cuyo principal meta, es la mejora sobre el suelo blando, bajo bases o cimentaciones de estructuras y malos rellenos, permitiendo el desarrollo de las edificaciones con mayor velocidad y disminuyendo los asentamientos de forma sustancial. Al aplicarse esta técnica sobre el terreno blando, se disminuye el asentamiento vertical, ya que al colocarse una capa de reparto que distribuye toda la carga de la estructura se transfiere todo el peso a las inclusiones, minimizando la carga sobre el terreno blando y por consecuente el asentamiento. Como parte de esta tesis presentaremos dos métodos para estudiar el asentamiento de edificaciones que descansan sobre un suelo blando mejorado con la técnica de inclusiones rígidas. El desarrollo de la solución es completamente analítico, pero los análisis de elementos finitos se utilizan para verificar la validez de algunas suposiciones, como un modelo geométrico simplificado, la distribución de la carga con la profundidad y las condiciones de contorno. El perfil del suelo se divide en secciones horizontales independientes, y el equilibrio de las tensiones y la compatibilidad de las deformaciones se presentan en las direcciones vertical y horizontal. La solución se presenta desarrollando un análisis numérico (Plaxis) comparándolo con el método analítico (Combarte) y se puede implementar fácilmente en una hoja de cálculo. / The soft soils cause various problems at the time of construction, as in the foundations of different buildings or other civil works. These types of soils necessarily have to be analyzed and treated correctly, as these cause short and long term problems. In the last decades methods have been developed for the improvement of the land, one of these methods consists of the introduction or construction in the soft soil of rigid vertical elements that are called inclusions. Its main application is in soils under large landfills or foundations of structures, allowing the construction of buildings with greater speed and decreasing settlements considerably. By applying this technique on the soft ground, vertical settlement is reduced, since by placing a distribution layer that distributes the entire load of the structure, all the weight is transferred to the inclusions, minimizing the load on the soft ground and consequently settlement. As part of this thesis we will present two methods to study the settlement of buildings that rest on a soft soil improved with the technique of rigid inclusions. The development of the solution is completely analytical, but the finite element analyzes are used to verify the validity of some assumptions, such as a simplified geometric model, the distribution of the load with the depth and the boundary conditions. The floor profile is divided into independent horizontal sections, and the balance of the stresses and the compatibility of the deformations are presented in the vertical and horizontal directions. The solution is presented by developing a numerical analysis (Plaxis) comparing it with the analytical method (Combarieu) and can be easily implemented in a spreadsheet. / Tesis

Rocket combustors and other high-performance chemical propulsion systems are prone to combustion instability. Recent simulations of rocket combustors using detailed chemical kinetics show that the constant pressure assumption used in classical treatments may be suspect due to high rates of heat release. This study is a exploration on the effects of these extraordinary rates of heat addition on the local pressure field, and interactions between the heat release and an acoustic field. <br> <br>The full problem is decomposed into simpler unit problems focused on the particular interactions of physical phenomena involved in combustion instability. The overall strategy consists of analyzing fundamental problems with simplified scenarios and then build up the complexity by adding more phenomena to the analysis. Seven unit problems are proposed in this study. <br> <br>The first unit problem consists of the pressure response to an unsteady heat release source in an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, one a Gaussian spatial distribution with a step temporal profile, and the other a Gaussian spatial distribution with a Gaussian temporal distribution, are used to model the heat source. The analytical solutions predict two different regimes in the pressure response depending on the Helmholtz number, which is defined as the ratio of the acoustic time over the duration of the heat release pulse. A critical Helmholtz number is found to dictate the pressure response regime. For compact cases, in the subcritical regime, the amplitude of the pressure pulse remains constant in space. For noncompact cases, above the critical Helmholtz number, the pressure pulse reaches a maximum at the center of the heat source, and then decays in space converging to a lower far field amplitude. At the limits of very small and very large Helmholtz numbers, the heat release response tends to be a constant pressure process and a constant volume process, respectively. The parameters of the study are chosen to be representative of the extreme conditions in a rocket combustor. The analytical models for both heat source profiles closely match the simulations with a slight overprediction. The differences observed in the analytical solutions are due to neglecting mean flow property variations and the absence of loss mechanisms. The numerical simulations also reveal the presence of nonlinear effects such as weak shocks that cannot be captured by the linear acoustic wave equation. <br> <br>The second unit problem extends the analysis of the pressure response of an unsteady heat release source to an unconfined three-dimensional domain. An analytical model based on the spherical acoustic wave equation with an unsteady heat source is derived and then compared against results from highly-resolved three-dimensional numerical simulations. Two different heat release profiles, a three-dimensional Gaussian spherical distribution with either a step or a Gaussian temporal distribution, are used to model the heat source. Two different regimes in the pressure response depending on the Helmholtz number are found. This analysis also reveals that whereas for the one-dimensional case the pressure amplitude is constant over the distance, for the three-dimensional case it decays with the radial distance from the heat source. In addition, although for moderate heat release values the analytical solution is able to capture the dynamics of the fluid response, for large heat release values the nonlinear effects deviate the highly-resolved numerical solution from the analytical model. <br> <br>The third unit problem studies the pressure response of a fluctuating unsteady heat release source to an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, a flat spatial distribution with sinusoidal temporal profile and a Gaussian spatial distribution and sinusoidal temporal profile, are used to model the heat source. For both cases, the acoustically compact and noncompact regimes depending on the Helmholtz number are analyzed. While in the compact regime the amplitude of the pressure is constant over the distance, in the noncompact regime the amplitude of the pressure fluctuation is larger within the heat source area of application, and once outside the heat source decays to a far field pressure value. In addition, the analytical model does not capture the nonlinear effects present in the highly-resolved numerical simulations for large rates of heat release such as the ones present in rocket combustors.<br> <br>Finally, the last four unit problems focus on the interaction between unsteady heat release and the longitudinal acoustic modes of a combustor. The goal is to assess and quantify how pressure fluctuations due to unsteady heat release amplify a longitudinal acoustic mode. To investigate the nonlinear effects and the limitations based on the acoustic wave equation, the analytical models are compared against highly-resolved numerical simulations. The fourth unit problem consists of the pressure response to a moving rigid surface that generates a forced sinusoidal velocity fluctuation in a one-dimensional open-ended cavity. The fifth unit problem combines an analytical solution from the velocity harmonic fluctuation with an unsteady heat pulse with Gaussian spatial and temporal distribution developed in the first unit problem. The choice of an open-ended cavity simplifies the analysis and serves as a stepping stone to the sixth unit problem, which also includes the pressure reflections provoked by the acoustic boundaries of the duct. This sixth unit problem describes the establishment of a 1L acoustic longitudinal mode inside a closed duct using the harmonic velocity fluctuations from the fourth unit problem. A wall on the left end of the duct is only moved for one cycle at the 1L mode frequency to establish a 1L mode in the initially quiescent fluid. The last unit problem combines the analytical solution of the 1L mode acoustic field developed in the sixth unit problem with an unsteady heat pulse with Gaussian spatial and temporal distribution, and also accounts for pressure reflections. The derivation of the present analytical models includes the identification of relevant length and time scales that are condensed into the Helmholtz number, the phase shift between the longitudinal fluctuating pressure field and the heat source, and ratio of the fluctuating periods. The analytical solution is able to capture with an acceptable degree of accuracy the pressure trace of the numerical solution during the fist few cycles of the 1L mode, but it quickly deviates very significantly from the numerical solution due to wave steepening and the formation of weak shocks. Therefore, models based on the acoustic wave equation can provide a good understanding of the combustion instability behavior, but not accurately predict the evolution of the pressure fluctuations as the nonlinear effects play a major role in the combustion dynamics of liquid rocket engines.

This report details the development and validation of a model for the simulation of supported liquid membrane processes, as applied to the extraction of lanthanides. Supported liquid membranes are systems where two phases, usually aqueous, are separated by a third phase, typically organic, which acts as a membrane, in order to separate solutes from one phase to the other. The model employs an analytical solution to the diffusion equation for the organic phase and linear approximations of the resistances to mass transfer in the aqueous phase boundary layers. The goal of this model is to underline the importance of taking these boundary layer resistances into account. / Detta arbete introducerar, deriverar och evaluerar en matematisk modell för simulering av vätskemembranprocesser, tillämpat på vätskeextraktion av lantanider. Immobiliserade vätskemembran betecknar system där två faser oftast vatten, separeras av en tredje organisk fas som agerar membran för att separera och transportera lösta komponenter från den ena vattenfasen till den andra. Modellen utnyttjar sig av en analytisk lösning till diffusionsekvationen för den organiska fasen och linjära approximationer för motstånden mot masstransport i de båda vattenfasernas gränsskikt. Målet med modellen är att understryka vikten av att inkludera dessa gränsskikt i beräkningarna.

In addition to previous publications, the paper presents the analytical solution of a special boundary value problem which arises in the context of elasticity theory for an extended constitutive law and a non-conservative symmetric ansatz. Besides deriving the general analytical solution, a specific form for linear boundary conditions is given for user convenience.

This doctoral thesis provides the first comprehensive study on the physical processes controlling hydrodynamics and transport in Lake Garda. The investigation is carried out in parallel on three different levels: data collection and analysis, three-dimensional numerical modeling and theoretical study. On the first level, data are collected by building up a network of research institutes and local administrations in the lake area. New data are acquired through traditional field campaigns (CTD, thermistor chains, satellite imagery), while a citizen-science approach, based on local knowledge harvesting, is successfully tested to gather qualitative data on surface circulation. On the second level, a three-dimensional modeling chain is set up, by coupling one-way a mesoscale atmospheric model to a hydrodynamic model. Both models are validated on multiple temporal and spatial scales, allowing to identify the main interactions between the weather forcing and the hydrodynamic response of the lake. Circulations in Lake Garda are found to be very sensitive to the thermal stratification, to the spatial distribution of the wind forcing and to the Earth’s rotation. Surface cyclonic gyre patterns develop in the lake as a residual outcome of alternating wind forcing of local breezes and differential acceleration induced by Earth’s rotation, whereas unidirectional currents flow under a nearly uniform and constant wind. Both model and observations evidences show that, under weak thermal stratification, Ekman transport activates a secondary circulations in the northern part of the lake, driving surface water to the deep layers and possibly preconditioning the lake for subsequent buoyancy-driven deep mixing events. On the third level, the relevance of the Coriolis term in the equations of motion for relatively narrow closed basins is analytically addressed. The classical Ekman problem is solved by including the presence of lateral boundaries and a new analytical solution is formulated. The validity of the new solution is proved by numerical tests of idealized domains of different size, geographical location and turbulent regime, and on Lake Garda as a real test case. The meaningful length scales are discussed, and the significance of Rossby radious as a reference horizontal scale is disproved for steady-state circulations driven by wind and planetary rotation.