An open-boundary integrable model of three coupled XY spin chains

The integrable open-boundary conditions for the model of three coupled one-dimensional XY spin chains are considered in the framework of the quantum inverse scattering method. The diagonal boundary K-matrices are found and a class of integrable boundary terms is determined. The boundary model Hamiltonian is solved by using the coordinate space Bethe ansatz technique and Bethe ansatz equations are derived. (C) 1998 Elsevier Science B.V.

Exact solution for the Bariev model with boundary fields

The Bariev model with open boundary conditions is introduced and analysed in detail in the framework of the Quantum Inverse Scattering Method. Two classes of independent boundary reflecting K-matrices leading to four different types of boundary fields are obtained by solving the reflection equations. The models are exactly solved by means of the algebraic nested Bethe ansatz method and the four sets or Bethe ansatz equations as well as their corresponding energy expressions are derived. (C) 2001 Elsevier Science B.V. All rights reserved.

Three-point boundary value problems for second-order, ordinary, differential equations

We establish existence results for solutions to three-point boundary value problems for nonlinear, second-order, ordinary differential equations with nonlinear boundary conditions. (C) 2001 Elsevier Science Ltd. All rights reserved.

Topological methods for some boundary value problems

We establish existence of solutions for a finite difference approximation to y = f(x, y, y ') on [0, 1], subject to nonlinear two-point Sturm-Liouville boundary conditions of the form g(i)(y(i),y ' (i)) = 0, i = 0, 1, assuming S satisfies one-sided growth bounds with respect to y '. (C) 2001 Elsevier Science Ltd. All rights reserved.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

MASS CONSERVATION ENHANCEMENT OF FREE BOUNDARY FLOWS IN COMPOSITES MANUFACTURING

Undesirable void formation during the injection phase of the liquid composite molding process can be understood as a consequence of the non-uniformity of the flow front progression, caused by the dual porosity of the fiber perform. Therefore the best examination of the void formation physics can be provided by a mesolevel analysis, where the characteristic dimension is given by the fiber tow diameter. In mesolevel analysis, liquid impregnation along two different scales; inside fiber tows and within the spaces between them; must be considered and the coupling between these flow regimes must be addressed. In such case, it is extremely important to account correctly for the surface tension effects, which can be modeled as capillary pressure applied at the flow front. When continues Galerkin method is used, exploiting elements with velocity components and pressure as nodal variables, strong numerical implementation of such boundary conditions leads to ill-posing of the problem, in terms of the weak classical as well as stabilized formulation. As a consequence, there is an error in mass conservation accumulated especially along the free flow front. This article presents a numerical procedure, which was formulated and implemented in the existing Free Boundary Program in order to significantly reduce this error.

Mass Conservation Enhancement of Free Boundary Mesolevel Flows during LCM Processes of Composites Manufacturing

Undesirable void formation during the injection phase of the liquid composite moulding process can be understood as a consequence of the non-uniformity of the flow front progression, caused by the dual porosity of the fibre perform. Therefore the best examination of the void formation physics can be provided by a mesolevel analysis, where the characteristic dimension is given by the fibre tow diameter. In mesolevel analysis, liquid impregnation along two different scales; inside fibre tows and within the open spaces between them; must be considered and the coupling between these flow regimes must be addressed. In such case, it is extremely important to account correctly for the surface tension effects, which can be modelled as capillary pressure applied at the flow front. Numerical implementation of such boundary conditions leads to ill-posing of the problem, in terms of the weak classical as well as stabilized formulation. As a consequence, there is an error in mass conservation accumulated especially along the free flow front. This contribution presents a numerical procedure, which was formulated and implemented in the existing Free Boundary Program in order to significantly reduce this error.

Factorization of elliptic boundary value problems by invariant embedding and application to overdetermined problems

Dissertação para obtenção do Grau de Doutor em Matemática

Structure-property relationships in polymer nanocomposites

Tese de Doutoramento em Ciência e Engenharia de Polímeros e Compósitos

Bifurcations and chaos in single-roll natural convection with low Prandtl number

Convective flows of a small Prandtl number fluid contained in a two-dimensional cavity subject to a lateral thermal gradient are numerically studied by using different techniques. The aspect ratio (length to height) is kept at around 2. This value is found optimal to make the flow most unstable while keeping the basic single-roll structure. Two cases of thermal boundary conditions on the horizontal plates are considered: perfectly conducting and adiabatic. For increasing Rayleigh numbers we find a transition from steady flow to periodic oscillations through a supercritical Hopf bifurcation that maintains the centrosymmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation the system initiates a complex scenario of bifurcations. In the conductive case these include a quasiperiodic route to chaos. In the adiabatic one the dynamics is dominated by the interaction of two Neimark-Sacker bifurcations of the basic periodic solutions, leading to the stable coexistence of three incommensurate frequencies, and finally to chaos. In all cases, the complex time-dependent behavior does not break the basic, single-roll structure.

Low-cost MR-compatible moving heart phantom.

For the development and evaluation of cardiac magnetic resonance (MR) imaging sequences and methodologies, the availability of a periodically moving phantom to model respiratory and cardiac motion would be of substantial benefit. Given the specific physical boundary conditions in an MR environment, the choice of materials and power source of such phantoms is heavily restricted. Sophisticated commercial solutions are available; however, they are often relatively costly and user-specific modifications may not easily be implemented. We therefore sought to construct a low-cost MR-compatible motion phantom that could be easily reproduced and had design flexibility. A commercially available K'NEX construction set (Hyper Space Training Tower, K'NEX Industries, Inc., Hatfield, PA) was used to construct a periodically moving phantom head. The phantom head performs a translation with a superimposed rotation, driven by a motor over a 2-m rigid rod. To synchronize the MR data acquisition with phantom motion (without introducing radiofrequency-related image artifacts), a fiberoptic control unit generates periodic trigger pulses synchronized to the phantom motion. Total material costs of the phantom are US$ < 200.00, and a total of 80 man-hours were required to design and construct the original phantom. With schematics of the present solution, the phantom reproduction may be achieved in approximately 15 man-hours. The presented MR-compatible periodically moving phantom can easily be reproduced, and user-specific modifications may be implemented. Such an approach allows a detailed investigation of motion-related phenomena in MR images.

Seismic characterization of heterogeneous sedimentary and fractured rocks based on Biot's theory of poroelasticity

Understanding and quantifying seismic energy dissipation, which manifests itself in terms of velocity dispersion and attenuation, in fluid-saturated porous rocks is of considerable interest, since it offers the perspective of extracting information with regard to the elastic and hydraulic rock properties. There is increasing evidence to suggest that wave-induced fluid flow, or simply WIFF, is the dominant underlying physical mechanism governing these phenomena throughout the seismic, sonic, and ultrasonic frequency ranges. This mechanism, which can prevail at the microscopic, mesoscopic, and macroscopic scale ranges, operates through viscous energy dissipation in response to fluid pressure gradients and inertial effects induced by the passing wavefield. In the first part of this thesis, we present an analysis of broad-band multi-frequency sonic log data from a borehole penetrating water-saturated unconsolidated glacio-fluvial sediments. An inherent complication arising in the interpretation of the observed P-wave attenuation and velocity dispersion is, however, that the relative importance of WIFF at the various scales is unknown and difficult to unravel. An important generic result of our work is that the levels of attenuation and velocity dispersion due to the presence of mesoscopic heterogeneities in water-saturated unconsolidated clastic sediments are expected to be largely negligible. Conversely, WIFF at the macroscopic scale allows for explaining most of the considered data while refinements provided by including WIFF at the microscopic scale in the analysis are locally meaningful. Using a Monte-Carlo-type inversion approach, we compare the capability of the different models describing WIFF at the macroscopic and microscopic scales with regard to their ability to constrain the dry frame elastic moduli and the permeability as well as their local probability distribution. In the second part of this thesis, we explore the issue of determining the size of a representative elementary volume (REV) arising in the numerical upscaling procedures of effective seismic velocity dispersion and attenuation of heterogeneous media. To this end, we focus on a set of idealized synthetic rock samples characterized by the presence of layers, fractures or patchy saturation in the mesocopic scale range. These scenarios are highly pertinent because they tend to be associated with very high levels of velocity dispersion and attenuation caused by WIFF in the mesoscopic scale range. The problem of determining the REV size for generic heterogeneous rocks is extremely complex and entirely unexplored in the given context. In this pilot study, we have therefore focused on periodic media, which assures the inherent self- similarity of the considered samples regardless of their size and thus simplifies the problem to a systematic analysis of the dependence of the REV size on the applied boundary conditions in the numerical simulations. Our results demonstrate that boundary condition effects are absent for layered media and negligible in the presence of patchy saturation, thus resulting in minimum REV sizes. Conversely, strong boundary condition effects arise in the presence of a periodic distribution of finite-length fractures, thus leading to large REV sizes. In the third part of the thesis, we propose a novel effective poroelastic model for periodic media characterized by mesoscopic layering, which accounts for WIFF at both the macroscopic and mesoscopic scales as well as for the anisotropy associated with the layering. Correspondingly, this model correctly predicts the existence of the fast and slow P-waves as well as quasi and pure S-waves for any direction of wave propagation as long as the corresponding wavelengths are much larger than the layer thicknesses. The primary motivation for this work is that, for formations of intermediate to high permeability, such as, for example, unconsolidated sediments, clean sandstones, or fractured rocks, these two WIFF mechanisms may prevail at similar frequencies. This scenario, which can be expected rather common, cannot be accounted for by existing models for layered porous media. Comparisons of analytical solutions of the P- and S-wave phase velocities and inverse quality factors for wave propagation perpendicular to the layering with those obtained from numerical simulations based on a ID finite-element solution of the poroelastic equations of motion show very good agreement as long as the assumption of long wavelengths remains valid. A limitation of the proposed model is its inability to account for inertial effects in mesoscopic WIFF when both WIFF mechanisms prevail at similar frequencies. Our results do, however, also indicate that the associated error is likely to be relatively small, as, even at frequencies at which both inertial and scattering effects are expected to be at play, the proposed model provides a solution that is remarkably close to its numerical benchmark. -- Comprendre et pouvoir quantifier la dissipation d'énergie sismique qui se traduit par la dispersion et l'atténuation des vitesses dans les roches poreuses et saturées en fluide est un intérêt primordial pour obtenir des informations à propos des propriétés élastique et hydraulique des roches en question. De plus en plus d'études montrent que le déplacement relatif du fluide par rapport au solide induit par le passage de l'onde (wave induced fluid flow en anglais, dont on gardera ici l'abréviation largement utilisée, WIFF), représente le principal mécanisme physique qui régit ces phénomènes, pour la gamme des fréquences sismiques, sonique et jusqu'à l'ultrasonique. Ce mécanisme, qui prédomine aux échelles microscopique, mésoscopique et macroscopique, est lié à la dissipation d'énergie visqueuse résultant des gradients de pression de fluide et des effets inertiels induits par le passage du champ d'onde. Dans la première partie de cette thèse, nous présentons une analyse de données de diagraphie acoustique à large bande et multifréquences, issues d'un forage réalisé dans des sédiments glaciaux-fluviaux, non-consolidés et saturés en eau. La difficulté inhérente à l'interprétation de l'atténuation et de la dispersion des vitesses des ondes P observées, est que l'importance des WIFF aux différentes échelles est inconnue et difficile à quantifier. Notre étude montre que l'on peut négliger le taux d'atténuation et de dispersion des vitesses dû à la présence d'hétérogénéités à l'échelle mésoscopique dans des sédiments clastiques, non- consolidés et saturés en eau. A l'inverse, les WIFF à l'échelle macroscopique expliquent la plupart des données, tandis que les précisions apportées par les WIFF à l'échelle microscopique sont localement significatives. En utilisant une méthode d'inversion du type Monte-Carlo, nous avons comparé, pour les deux modèles WIFF aux échelles macroscopique et microscopique, leur capacité à contraindre les modules élastiques de la matrice sèche et la perméabilité ainsi que leur distribution de probabilité locale. Dans une seconde partie de cette thèse, nous cherchons une solution pour déterminer la dimension d'un volume élémentaire représentatif (noté VER). Cette problématique se pose dans les procédures numériques de changement d'échelle pour déterminer l'atténuation effective et la dispersion effective de la vitesse sismique dans un milieu hétérogène. Pour ce faire, nous nous concentrons sur un ensemble d'échantillons de roches synthétiques idéalisés incluant des strates, des fissures, ou une saturation partielle à l'échelle mésoscopique. Ces scénarios sont hautement pertinents, car ils sont associés à un taux très élevé d'atténuation et de dispersion des vitesses causé par les WIFF à l'échelle mésoscopique. L'enjeu de déterminer la dimension d'un VER pour une roche hétérogène est très complexe et encore inexploré dans le contexte actuel. Dans cette étude-pilote, nous nous focalisons sur des milieux périodiques, qui assurent l'autosimilarité des échantillons considérés indépendamment de leur taille. Ainsi, nous simplifions le problème à une analyse systématique de la dépendance de la dimension des VER aux conditions aux limites appliquées. Nos résultats indiquent que les effets des conditions aux limites sont absents pour un milieu stratifié, et négligeables pour un milieu à saturation partielle : cela résultant à des dimensions petites des VER. Au contraire, de forts effets des conditions aux limites apparaissent dans les milieux présentant une distribution périodique de fissures de taille finie : cela conduisant à de grandes dimensions des VER. Dans la troisième partie de cette thèse, nous proposons un nouveau modèle poro- élastique effectif, pour les milieux périodiques caractérisés par une stratification mésoscopique, qui prendra en compte les WIFF à la fois aux échelles mésoscopique et macroscopique, ainsi que l'anisotropie associée à ces strates. Ce modèle prédit alors avec exactitude l'existence des ondes P rapides et lentes ainsi que les quasis et pures ondes S, pour toutes les directions de propagation de l'onde, tant que la longueur d'onde correspondante est bien plus grande que l'épaisseur de la strate. L'intérêt principal de ce travail est que, pour les formations à perméabilité moyenne à élevée, comme, par exemple, les sédiments non- consolidés, les grès ou encore les roches fissurées, ces deux mécanismes d'WIFF peuvent avoir lieu à des fréquences similaires. Or, ce scénario, qui est assez commun, n'est pas décrit par les modèles existants pour les milieux poreux stratifiés. Les comparaisons des solutions analytiques des vitesses des ondes P et S et de l'atténuation de la propagation des ondes perpendiculaires à la stratification, avec les solutions obtenues à partir de simulations numériques en éléments finis, fondées sur une solution obtenue en 1D des équations poro- élastiques, montrent un très bon accord, tant que l'hypothèse des grandes longueurs d'onde reste valable. Il y a cependant une limitation de ce modèle qui est liée à son incapacité à prendre en compte les effets inertiels dans les WIFF mésoscopiques quand les deux mécanismes d'WIFF prédominent à des fréquences similaires. Néanmoins, nos résultats montrent aussi que l'erreur associée est relativement faible, même à des fréquences à laquelle sont attendus les deux effets d'inertie et de diffusion, indiquant que le modèle proposé fournit une solution qui est remarquablement proche de sa référence numérique.

Monte Carlo thermodynamic and structural properties of the TIP4P water model: dependence on the computational conditions

Classical Monte Carlo simulations were carried out on the NPT ensemble at 25°C and 1 atm, aiming to investigate the ability of the TIP4P water model [Jorgensen, Chandrasekhar, Madura, Impey and Klein; J. Chem. Phys., 79 (1983) 926] to reproduce the newest structural picture of liquid water. The results were compared with recent neutron diffraction data [Soper; Bruni and Ricci; J. Chem. Phys., 106 (1997) 247]. The influence of the computational conditions on the thermodynamic and structural results obtained with this model was also analyzed. The findings were compared with the original ones from Jorgensen et al [above-cited reference plus Mol. Phys., 56 (1985) 1381]. It is notice that the thermodynamic results are dependent on the boundary conditions used, whereas the usual radial distribution functions g(O/O(r)) and g(O/H(r)) do not depend on them.

Fixed point results for multivalued contractions on graphs and their applications

Nous présentons dans cette thèse des théorèmes de point fixe pour des contractions multivoques définies sur des espaces métriques, et, sur des espaces de jauges munis d’un graphe. Nous illustrons également les applications de ces résultats à des inclusions intégrales et à la théorie des fractales. Cette thèse est composée de quatre articles qui sont présentés dans quatre chapitres. Dans le chapitre 1, nous établissons des résultats de point fixe pour des fonctions multivoques, appelées G-contractions faibles. Celles-ci envoient des points connexes dans des points connexes et contractent la longueur des chemins. Les ensembles de points fixes sont étudiés. La propriété d’invariance homotopique d’existence d’un point fixe est également établie pour une famille de Gcontractions multivoques faibles. Dans le chapitre 2, nous établissons l’existence de solutions pour des systèmes d’inclusions intégrales de Hammerstein sous des conditions de type de monotonie mixte. L’existence de solutions pour des systèmes d’inclusions différentielles avec conditions initiales ou conditions aux limites périodiques est également obtenue. Nos résultats s’appuient sur nos théorèmes de point fixe pour des G-contractions multivoques faibles établis au chapitre 1. Dans le chapitre 3, nous appliquons ces mêmes résultats de point fixe aux systèmes de fonctions itérées assujettis à un graphe orienté. Plus précisément, nous construisons un espace métrique muni d’un graphe G et une G-contraction appropriés. En utilisant les points fixes de cette G-contraction, nous obtenons plus d’information sur les attracteurs de ces systèmes de fonctions itérées. Dans le chapitre 4, nous considérons des contractions multivoques définies sur un espace de jauges muni d’un graphe. Nous prouvons un résultat de point fixe pour des fonctions multivoques qui envoient des points connexes dans des points connexes et qui satisfont une condition de contraction généralisée. Ensuite, nous étudions des systèmes infinis de fonctions itérées assujettis à un graphe orienté (H-IIFS). Nous donnons des conditions assurant l’existence d’un attracteur unique à un H-IIFS. Enfin, nous appliquons notre résultat de point fixe pour des contractions multivoques définies sur un espace de jauges muni d’un graphe pour obtenir plus d’information sur l’attracteur d’un H-IIFS. Plus précisément, nous construisons un espace de jauges muni d’un graphe G et une G-contraction appropriés tels que ses points fixes sont des sous-attracteurs du H-IIFS.