Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the "pure thermodynamic phases", which correspond to the extreme points in the convex set of equilibrium measures, must be constructive equilibria. Finally, for an even smoother class of functions a method is presented to construct a compatible interaction potential and it is checked that the two different structures generate the same sets of Gibbs and equilibrium measures, respectively.

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

In this thesis I present a new numerical scheme for 2.5-D/3-D direct current resistivity modelling in heterogeneous, anisotropic media. This method, named the ‘Gaussian quadrature grid’ (GQG) method, co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it has the main advantages of the spectral element method. The formulation shows that the GQG method is a modification of the spectral element method and does not employ the constant elements and require the mesh generator to match the earth’s surface. This makes it much easier to deal with geological models having a 2-D/3-D complex topography than using traditional numerical methods. The GQG technique can achieve a similar convergence rate to the spectral element method. It is shown that it transforms the 2.5-D/3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method. Comparison with analytic solutions for homogeneous isotropic and anisotropic models shows that the error depends on the Gaussian quadrature order (abscissae number) and the sub-domain size. The higher order or smaller the subdomain size employed, the more accurate the solution. Several other synthetic examples, both homogeneous and inhomogeneous, incorporating sloping, undulating and severe topography are presented and found to yield results comparable to finite element solutions involving a dense mesh. The thesis also presents for the first time explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green’s functions and their gradients, and is developed both from a formal perturbation analysis and by means of a numerical (finite element) method. A critical factor in the equations is the derivative of the electrical conductivity tensor with respect to the principal conductivity values and the angles defining the axes of symmetry; these are given analytically. The Fréchet derivative expressions are given for both the 2.5-D and the 3-D problem using both constant point and constant block model parameterisations. Special cases like the isotropic earth and tilted transversely isotropic (TTI) media are shown to emerge from the general solutions. Numerical examples are presented for the various sensitivities as functions of the dip angle and strike of the plane of stratification in uniform TTI media. In addition, analytic solutions are derived for the electric potential, current density and Fréchet derivatives at any interior point within a 3-D transversely isotropic homogeneous medium having a tilted axis of symmetry. The current electrode is assumed to be on the surface of the Earth and the plane of stratification given arbitrary strike and dip. Profiles can be computed for any azimuth. The equipotentials exhibit an elliptical pattern and are not orthogonal to the current density vectors, which are strongly angle dependent. Current density reaches its maximum value in a direction parallel to the longitudinal conductivity direction. Illustrative examples of the Fréchet derivatives are given for the 2.5-D problem, in which the profile is taken perpendicular to strike. All three derivatives of the Green’s function with respect to longitudinal conductivity, transverse resistivity and dip angle of the symmetry axis (dG/dσ₁,dG/dσ₁,dG/dθ₀ ) show a strongly asymmetric pattern compared to the isotropic case. The patterns are aligned in the direction of the tilt angle. Such sensitivity patterns are useful in real time experimental design as well as in the fast inversion of resistivity data collected over an anisotropic earth. / Thesis (Ph.D.) -- University of Adelaide, School of Chemistry and Physics, 2009

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

Results from models that describe frequency chirps of toroidal Alfvén eigenmode excited by energetic particles are presented here. This structure forms in TAE gap and may or may not chirp into the continuum. Initial work described the particle wave interaction in terms of a generic Hamiltonian for the particle wave interaction, whose spatial dependence was xed in time. In addition, we have developed an improved adiabatic TAE model that takes into account the spatial prole variation of the mode and the nite orbit excursion from the resonant ux surfaces, for a wide range of toroidal mode numbers. We have shown for the generic xed prole model that the results from the adiabatic model agree very well with simulation result except when the adiabatic condition breaks down due to the rapid variations of the wave amplitude and chirping frequency. We have been able to solve the adiabatic problem in the case when the spatial prole is allowed to vary in time, in accord with the structure of the response functions, as a function of frequency. All the models predict that up-chirping holes do not penetrate into the continuum. On the other hand clump structures, which down chirp in frequency may, depending on detailed parameters, penetrate the continuum. The systematic theory is more restrictive than the generic theory, for the conditions that enable clump to penetrate into the continuum. In addition, the systematic theory predicts an important nite drift orbit width eect, which eventually limits and suppresses a down-chirping response in the lower continuum. This interruption of the chirping occurs when the trapped particles make a transition from intersecting both resonant points of the continuum to just one resonant point. / text

We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.

A Natural nEighbours Method (NEM) based on the FRAEIJS de VEUBEKE (FdV) variational principle is developed in the domain of 2D infinitesimal transformations. This method is firstly applied to linear elastic problems and then is extended to materially nonlinear problems and problems of linear elastic fracture mechanics (LEFM). In all these developments, thanks to the FdV variational principle, the displacement field, the stress field, the strain field and the support reaction field are discretized independently. In the spirit of the NEM, nodes are distributed in the domain and on its contour and the corresponding Voronoi cells are constructed. In linear elastic problems the following discretization hypotheses are used: 1. The assumed displacements are interpolated between the nodes with Laplace functions. 2. The assumed support reactions are constant over each edge of Voronoi cells on which displacements are imposed. 3. The assumed stresses are constant over each Voronoi cell. 4. The assumed strains are constant over each Voronoi cell. The degrees of freedom linked with the assumed stresses and strains can be eliminated at the level of the Voronoi cells so that the final equation system only involves the nodal displacements and the assumed support reactions. The support reactions can be further eliminated from the equation system if the imposed support conditions only involve constant imposed displacements (in particular displacements imposed to zero) on a part of the solid contour, finally leading to a system of equations of the same size as in a classical displacement-based method. For the extension to materially non linear problems, similar hypotheses are used. In particular, the velocities are interpolated by Laplace functions and the strain rates are assumed to be constant in each Voronoi cell. The final equations system only involves the nodal velocities. It can be solved step by step by time integration and Newton-Raphson iterations at the level of the different time steps. In the extension of this method for LEFM, a node is located on each crack tip. In the Voronoi cells containing the crack tip, the stress and the strain discretization includes not only a constant term but also additional terms corresponding to the solutions of LEFM for modes 1 and 2. In this approach, the stress intensity coefficients are obtained as primary variables of the solution. The final equations system only involves the nodal displacements and the stress intensity coefficients. Finally, an eXtended Natural nEighbours Method (XNEM) is proposed in which the crack is represented by a line that does not conform to the nodes or the edges of the cells. Based on the hypotheses used in linear elastic domain, the discretization of the displacement field is enriched with Heaviside functions allowing a displacement discontinuity at the level of the crack. In the cells containing a crack tip, the stress and strain fields are also enriched with additional terms corresponding to the solutions of LEFM for modes 1 and 2. The stress intensity coefficients are also obtained as primary variables of the solution. A set of applications are performed to evaluate these developments. The following conclusions can be drawn for all cases (linear elastic, nonlinear, fracture mechanics). In the absence of body forces, the numerical calculation of integrals over the area of the domain is avoided: only integrations on the edges of the Voronoi cells are required, for which classical Gauss numerical integration with 2 integration points is sufficient to pass the patch test. The derivatives of the nodal shape functions are not required in the resulting formulation. The patch test can be successfully passed. Problems involving nearly incompressible materials can be solved without incompressibility locking in all cases. The numerical applications show that the solutions provided by the present approach converge to the exact solutions and compare favourably with the classical finite element method. / Une méthode des éléments naturels (NEM) basée sur le principe variationnel de FRAEIJS de VEUBEKE (FdV) est développée dans le domaine des transformations infinitésimales 2D. Cette méthode est dabord appliquée aux problèmes élastiques linéaires puis est étendue aux problèmes matériellement non linéaires ainsi quà ceux de la mécanique de la rupture élastique linéaire (LEFM). Dans tous ces développements, grâce au principe variationnel de FdV, les champs de déplacements, contraintes, réformations et réactions dappui sont discrétisés de façon indépendante. Dans lesprit de la NEM, des noeuds sont distribués dans le domaine et sur son contour et les cellules de Voronoi associées sont construites. En domaine élastique linéaire, les hypothèses de discrétisation sont les suivantes : 1. Les déplacements sont interpolés entre les noeuds par des fonctions de Laplace. 2. Les réactions dappui sont supposées constantes sur chaque côté des polygones de Voronoi le long desquels des déplacements sont imposés. 3. Les contraintes sont supposées constantes sur chaque cellule de Voronoi. 4. Les déformations sont supposées constantes sur chaque cellule de Voronoi. Les degrés de liberté associés aux hypothèses sur les contraintes et les déformations peuvent être éliminées au niveau des cellules de Voronoi de sorte que le système déquations final nimplique que les déplacement nodaux et les réactions dappui supposées. Ces dernières peuvent également être éliminées de ce système déquations si les conditions dappui nimposent que des déplacements constants (en particulier égaux à zéro) sur une partie du contour du domaine étudié, ce qui conduit à un système déquations de même taille que dans une approche basée sur la discrétisation des seuls déplacements. Pour lextension aux problèmes matériellement non linéaires, des hypothèses similaires sont utilisées. En particulier, les vitesses sont interpolées par des fonctions de Laplace et déformations sont supposées constantes sur chaque cellule de Voronoi. Le système déquations final nimplique que les vitesses nodales. Il peut être résolu pas à pas par intégration temporelle et itérations de Newton-Raphson à chaque pas de temps. Pour lextension de cette méthode aux problèmes de LEFM, un noeud est localisé à chaque pointe de fissure. Dans les cellules de Voronoi correspondantes, la discrétisation des contraintes et des déformations contient non seulement un terme constant mais aussi des termes additionnels correspondant aux solutions de la LEFM pour les modes 1 et 2. Avec cette approche, les coefficients dintensité de contraintes constituent des variables primaires de la solution. Le système déquations final ne contient que les déplacements nodaux et les coefficients dintensité de contraintes. Finalement, une méthode des éléments naturels étendue (XNEM) est proposée dans laquelle la fissure est représentée par une ligne indépendante des noeuds ou des côtés des cellules de Voronoi. La discrétisation utilisée en domaine élastique linéaire est enrichie par des fonctions de Heaviside qui autorisent une discontinuité des déplacements au niveau de la fissure. Dans les cellules contenant une pointe de fissure, les contraintes et les déformations sont aussi enrichies par des termes additionnels correspondant aux solutions de la LEFM pour les modes 1 et 2. Ici aussi, les coefficients dintensité de contraintes constituent des variables primaires de la solution. Une série dapplications numériques sont réalisées afin dévaluer ces développements. Les conclusions suivantes peuvent être tirées. Elles sappliquent à tous les cas (élastique linéaire, non linéaire, mécanique de la rupture) : En labsence de force volumique, le calcul numérique dintégrales sur laire du domaine est évité : seules sont nécessaires des intégrales numériques sur les côtés des cellules de Voronoi. Lutilisation de 2 points de Gauss suffit pour passer le patch test. Les dérivées des fonctions dinterpolation nodales ne sont pas nécessaires dans cette formulation. La formulation passe le patch test. Les problèmes impliquant des matériaux quasi incompressibles sont résolus sans verrouillage. Les applications numériques montrent que les solutions fournies par lapproche développée convergent vers les solutions exactes et se comparent favorablement avec celles de la méthode des éléments finis.

The present thesis is devoted to the study of path probability of random motion on the basis of an extension of Hamiltonian/Lagrangian mechanics to stochastic dynamics. The path probability is first investigated by numerical simulation for Gaussian stochastic motion of non dissipative systems. This ideal dynamical model implies that, apart from the Gaussian random forces, the system is only subject to conservative forces. This model can be applied to underdamped real random motion in the presence of friction force when the dissipated energy is negligible with respect to the variation of the potential energy. We find that the path probability decreases exponentially with increasing action, i.e., P(A) ~ eˉγA, where γ is a constant characterizing the sensitivity of the action dependence of the path probability, the action is given by A = ∫T0 Ldt, a time integral of the Lagrangian L = K-V over a fixed time period T, K is the kinetic energy and V is the potential energy. This result is a confirmation of the existence of a classical analogue of the Feynman factor eiA/ħ for the path integral formalism of quantum mechanics of Hamiltonian systems. The above result is then extended to real random motion with dissipation. For this purpose, the least action principle has to be generalized to damped motion of mechanical systems with a unique well defined Lagrangian function which must have the usual simple connection to Hamiltonian. This has been done with the help of the following Lagrangian L = K - V - Ed, where Ed is the dissipated energy. By variational calculus and numerical simulation, we proved that the action A = ∫T0 Ldt is stationary for the optimal paths determined by Newtonian equation. More precisely, the stationarity is a minimum for underdamped motion, a maximum for overdamped motion and an inflexion for the intermediate case. On this basis, we studied the path probability of Gaussian stochastic motion of dissipative systems. It is found that the path probability still depends exponentially on Lagrangian action for the underdamped motion, but depends exponentially on kinetic action A = ∫T0 Kdt for the overdamped motion.

La présente thèse est consacrée à l’étude de la probabilité du chemin d’un mouvement aléatoire sur la base d’une extension de la mécanique Hamiltonienne/Lagrangienne à la dynamique stochastique. La probabilité d’un chemin est d’abord étudiée par simulation numérique dans le cas du mouvement stochastique Gaussien des systèmes non dissipatifs. Ce modèle dynamique idéal implique que, outre les forces aléatoires Gaussiennes, le système est seulement soumis à des forces conservatrices. Ce modèle peut être appliqué à un mouvement aléatoire réel de régime pseudo-périodique en présence d’une force de frottement lorsque l’énergie dissipée est négligeable par rapport à la variation de l’énergie potentielle. Nous constatons que la probabilité de chemin décroît exponentiellement lorsque le son action augmente, c’est à dire, P(A) ~ eˉγA, où γ est une constante caractérisant la sensibilité de la dépendance de l’action à la probabilité de chemin, l’action est calculée par la formule A = ∫T0 Ldt, intégrale temporelle du Lagrangien. L = K–V sur une période de temps fixe T, K est l’énergie cinétique et V est l’énergie potentielle. Ce résultat est une confirmation de l’existence d’un analogue classique du facteur de Feynman eiA/ħ pour le formalisme intégral de chemin de la mécanique quantique des systèmes Hamiltoniens. Le résultat ci-dessus est ensuite étendu au mouvement aléatoire réel avec dissipation. A cet effet, le principe de moindre action doit être généralisé au mouvement amorti de systèmes mécaniques ayant une fonction unique de Lagrange bien définie qui doit avoir la simple connexion habituelle au Hamiltonien. Cela a été fait avec l’aide du Lagrangien suivant L = K − V − Ed, où Ed est l’énergie dissipée. Par le calcul variationnel et la simulation numérique, nous avons prouvé que l’action A = ∫T0 Ldt est stationnaire pour les chemins optimaux déterminés par l’équation newtonienne. Plus précisément, la stationnarité est un minimum pour les mouvements de régime pseudo-périodique, un maximum pour les mouvements d’amortissement apériodique et une inflexion dans le cas intermédiaire. Sur cette base, nous avons étudié la probabilité du chemin du mouvement stochastique Gaussien des systèmes dissipatifs. On constate que la probabilité du chemin dépend toujours de façon exponentielle de l’action Lagrangien pour les mouvements de régime pseudo-périodique, mais dépend toujours de façon exponentielle de l’action cinétique A = ∫T0 Kdt pour régime apériodique. / The present thesis is devoted to the study of path probability of random motion on the basis of an extension of Hamiltonian/Lagrangian mechanics to stochastic dynamics. The path probability is first investigated by numerical simulation for Gaussian stochastic motion of non dissipative systems. This ideal dynamical model implies that, apart from the Gaussian random forces, the system is only subject to conservative forces. This model can be applied to underdamped real random motion in the presence of friction force when the dissipated energy is negligible with respect to the variation of the potential energy. We find that the path probability decreases exponentially with increasing action, i.e., P(A) ~ eˉγA, where γ is a constant characterizing the sensitivity of the action dependence of the path probability, the action is given by A = ∫T0 Ldt, a time integral of the Lagrangian L = K–V over a fixed time period T, K is the kinetic energy and V is the potential energy. This result is a confirmation of the existence of a classical analogue of the Feynman factor eiA/ħ for the path integral formalism of quantum mechanics of Hamiltonian systems. The above result is then extended to real random motion with dissipation. For this purpose, the least action principle has to be generalized to damped motion of mechanical systems with a unique well defined Lagrangian function which must have the usual simple connection to Hamiltonian. This has been done with the help of the following Lagrangian L = K – V – Ed, where Ed is the dissipated energy. By variational calculus and numerical simulation, we proved that the action A = ∫T0 Ldt is stationary for the optimal paths determined by Newtonian equation. More precisely, the stationarity is a minimum for underdamped motion, a maximum for overdamped motion and an inflexion for the intermediate case. On this basis, we studied the path probability of Gaussian stochastic motion of dissipative systems. It is found that the path probability still depends exponentially on Lagrangian action for the underdamped motion, but depends exponentially on kinetic action A = ∫T0 Kdt for the overdamped motion.

This work aims to investigate some applications of the conjugate duality for scalar and vector optimization problems to the construction of gap functions for variational inequalities and equilibrium problems. The basic idea of the approach is to reformulate variational inequalities and equilibrium problems into optimization problems depending on a fixed variable, which allows us to apply duality results from optimization problems. Based on some perturbations, first we consider the conjugate duality for scalar optimization. As applications, duality investigations for the convex partially separable optimization problem are discussed. Afterwards, we concentrate our attention on some applications of conjugate duality for convex optimization problems in finite and infinite-dimensional spaces to the construction of a gap function for variational inequalities and equilibrium problems. To verify the properties in the definition of a gap function weak and strong duality are used. The remainder of this thesis deals with the extension of this approach to vector variational inequalities and vector equilibrium problems. By using the perturbation functions in analogy to the scalar case, different dual problems for vector optimization and duality assertions for these problems are derived. This study allows us to propose some set-valued gap functions for the vector variational inequality. Finally, by applying the Fenchel duality on the basis of weak orderings, some variational principles for vector equilibrium problems are investigated.