The goal of this thesis is to determine the relationship between the macroscopic stress and the macroscopic strain for a variety of complex multiphase materials exhibiting rate-independent non-linear response at the micro-scale, based on experimental data obtained both at the local and macroscopic scales. A micro-macro secant mean field model (SMF model) based on the result of Eshelby and the approach of Mori and Tanaka is developed to model the behaviour of three particular systems which we have worked out by ourselves: 1) a ferrite-martensite steel produced by rolling in which we quantify the plastic anisotropy due to the morphological texture in terms of the Lankford's coefficient and pseudo yield surface; 2) a composite made of two continuous and interpenetrating phases: an aluminium matrix reinforced by a preform of sintered Inconel601 fibres. We quantify the coupled effects of temperature and phases co-continuity on the phases and overall stresses; 3) a TRIP-aided multiphase steel, in which the dispersed metastable austenite phase transforms to martensite. We derive the relationship between the overall uniaxial elastoplastic response and the progress of phase transformation, itself influenced by the thermodynamical, microstructural and mechanical properties. The stress-state dependence of the martensitic transformation is enlightened and explained. We demonstrate the existence of thermomechanical treatments leading to optima of ductility and strength-ductility balance. Finally, we show that the formability of TRIP-aided multiphase steels depends on the stability criterion.

In this thesis, we propose some innovative developments for the implementation of mean-field homogenization schemes adapted to the prediction of the behavior of elasto-plastic and elasto-viscoplastic composites. For elasto-plastic materials, the local constitutive laws written in a rate form are linearized incrementally over several time-steps so that homogenization schemes developed in the context of linear elasticity can apply over each time interval. Since the original implementation gave too stiff predictions, we propose different stiffness reductions for the matrix tangent operator and study theoretically and numerically the influence on the final macroscopic prediction. Definition of the per phase reference state in also studied and linked to the fields heterogeneity effect. Predictions thus obtained are confronted with those of a secant (or total) formulation of the constitutive laws. For elasto-viscoplastic composites, we use the affine formulation which reduces the constitutive laws to fictitious linear thermo-elastic relations in the Laplace domain where the homogenization can apply. Our main contribution is a full treatment of internal variables in the linearization procedure. This enables to deal with realistic constitutive behaviors and general loading histories. We illustrate the influence of viscous effects under various loading conditions and study the accuracy of the method with respect to the loading rate. For both classes of composites, numerous predictions obtained by mean-field homogenization schemes are confronted against those of three-dimensional finite element simulations and experimental results. For a wide range of materials and loading conditions, a good agreement at the macroscopic level between our predictions and the reference results is observed.

University, Diss., 2006--Augsburg. / Erscheinungsjahr an der Haupttitelstelle: 2005.

Techn. University, Diss., 2001--Berlin.

Universiẗat, Diss., 2003--Dortmund.

No description available.

In this thesis, we consider stationary one-dimensional mean-field games (MFGs) with or without congestion. Our aim is to understand the qualitative features of these games through the analysis of explicit solutions. We are particularly interested in MFGs with a nonmonotonic behavior, which corresponds to situations where agents tend to aggregate. First, we derive the MFG equations from control theory. Then, we compute explicit solutions using the current formulation and examine their behavior. Finally, we represent the solutions and analyze the results. This thesis main contributions are the following: First, we develop the current method to solve MFG explicitly. Second, we analyze in detail non-monotonic MFGs and discover new phenomena: non-uniqueness, discontinuous solutions, empty regions and unhappiness traps. Finally, we address several regularization procedures and examine the stability of MFGs.

Abstract Nuclear Medicine (NM) images inherently suffer from large amounts of noise and blur. The purpose of this research is to reduce the noise and blur while maintaining image integrity for improved diagnosis. The proposal is to further improve image quality after the standard pre- and post-processing undertaken by a gamma camera system. Mean Field Annealing (MFA), the image processing technique used in this research is a well known image processing approach. The MFA algorithm uses two techniques to achieve image restoration. Gradient descent is used as the minimisation technique, while a deterministic approximation to Simulated Annealing (SA) is used for optimisation. The algorithm anisotropically diffuses an image, iteratively smoothing regions that are considered non-edges and still preserving edge integrity until a global minimum is obtained. A known advantage of MFA is that it is able to minimise to this global minimum, skipping over local minima while still providing comparable results to SA with significantly less computational effort. Image blur is measured using either a point or line source. Both allow for the derivation of a Point Spread Function (PSF) that is used to de-blur the image. The noise variance can be measured using a flood source. The noise is due to the random fluctuations in the environment as well as other contributors. Noisy blurred NM images can be difficult to diagnose particularly at regions with steep intensity gradients and for this reason MFA is considered suitable for image restoration. From the literature it is evident that MFA can be applied successfully to digital phantom images providing improved performance over Wiener filters. In this paper MFA is shown to yield image enhancement of planar NM images by implementing a sharpening filter as a post MFA processing technique.

In this thesis, we present three projects. First, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite-difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation. Also, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem. Finally, we study a particle approximation for one-dimensional first-order Mean-Field-Games with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we are dealing with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of the semi-discrete variational problem. Next, we show that our discretization preserves some conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. All results for the discrete problem are illustrated with numerical examples.

The mean field analysis of stochastic dynamical system allows us to gain insight into the qualitative features of their complex behavior, as well as quantitative estimates of certain aspects of their coarse-grained properties. As such, it usually furnishes a first front in approaching new dynamical systems and informs us about their stability landscape in the absence of fluctuations among other things. A knowledge of this landscape can be a valuable tool in model building for describing real world systems and provides a guiding principle for a justifiable choice of form and model parameters. In this work, we contribute to this analysis for two generic classes of high-dimensional models that possess a cyclic symmetry in the network that specifies their stochastic dynamics at the microscopic level. Our analysis is carried out in a manner that can be readily adapted for the mean field analysis of further generalized models that possess this symmetry. Moreover, in the second class of these models, we propose a new basic process that can change the stability landscape of an existing model and, as such, endow us with potential alternatives to model systems with robust biodiverse regimes. / Ph. D.