الصفحة 1
الصفحة 1
Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials ; Proceedings of the NATO Advanced Research Workshop, held in Warsaw, Poland, 23-26 June 2003
Although several books and conference proceedings have already appeared dealing with either the mathematical aspects or applications of homogenization theory, there seems to be no comprehensive volume dealing with both aspects. The present volume is meant to fill this gap, at least partially, and deals with recent developments in nonlinear homogenization emphasizing applications of current interest. It contains thirteen key lectures presented at the NATO Advanced Workshop on Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials. The list of thirty one contributed papers is also appended.
Multiscale Methods : Averaging and Homogenization
This introduction to multiscale methods gives readers a broad overview of the many uses and applications of the methods. The book begins by setting the theoretical foundations of the subject area, and moves on to develop a unified approach to the simplification of a wide range of problems which possess multiple scales, via perturbation expansions; differential equations and stochastic processes are studied in one unified framework. The book concludes with an overview of a range of theoretical tools used to justify the simplified models derived via the perturbation expansions.
Morphological Models of Random Structures
This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories.
Modeling Excitable Tissue : The EMI Framework
This volume presents a novel computational framework for understanding how collections of excitable cells work. The key approach in the text is to model excitable tissue by representing the individual cells constituting the tissue. This is in stark contrast to the common approach where homogenization is used to develop models where the cells are not explicitly present. The approach allows for very detailed analysis of small collections of excitable cells, but computational challenges limit the applicability in the presence of large collections of cells.
Micromechanics and Nanosimulation of Metals and Composites : Advanced Methods and Theoretical Concepts
This book provides insight into advanced numerical and experimental investigations of the microstructural aspects of strength and damage of metals and metal-based composites. In particular, atomistic and dislocation models, micromechanical simulations and homogenization techniques, as well as experimental observations in combination with computer simulations, are described. These methods are successfully applied to analyze the mechanical behavior of metals and metal-matrix composites.
Homogenization of Partial Differential Equations
Homogenization is a method for modeling processes in microinhomogeneous media, which are encountered in radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. These processes are described by PDEs with rapidly oscillating coefficients or boundary value problems in domains with complex microstructure. From the technical point of view, given the complexity of these processes, the best techniques to solve a wide variety of problems involve constructing appropriate macroscopic (homogenized) models. The present monograph is a comprehensive study of homogenized problems, based on the asymptotic analysis of boundary value problems as the characteristic scales of the microstructure decrease to zero. The work focuses on the construction of nonstandard models: non-local models, multicomponent models, and models with memory.
From Hyperbolic Systems to Kinetic Theory : A Personalized Quest
Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!
Conception optimale de structures = Optimal structural design
Optimal Structural Design deals with all aspects of shape optimization, parametric, geometric and topological, and gives a large place to numerical algorithms, gradient methods and stochastic methods (with an original contribution by Marc Schoenauer for this last point). In particular, most of the structural optimization algorithms have been implemented in the FreeFem ++ finite element software and the programs are freely available on the web. Optimal structural design is devoted to structural or shape optimization and is intended for a mixed audience of applied mathematicians and mechanicians. It discusses parametric, geometric and topology optimization and gives deterministic and stochastic numerical algorithms (implemented in the FreeFem ++ finite element software).
Computational Contact Mechanics
Topics of this book span the range from spatial and temporal discretization techniques for contact and impact problems with small and finite deformations over investigations on the reliability of micromechanical contact models over emerging techniques for rolling contact mechanics to homogenization methods and multi-scale approaches in contact problems. Furthermore, solution algorithms for single- and multi-processor computing environments, enabling methods that span from multi-contact to multi-scale approaches are discussed together with numerical experiments related to soil mechanics using discontinuous deformation analysis.
Composites with micro- and nano-structure : Computational modeling and experiments
The book contains atomistic and continuum numerical methods and experimental validation for composite materials reinforced with particles or fibres, porous materials, homogenization and other important topics for research in this field.
Calculus of variations and nonlinear partial differential equations : With a historical overview by Elvira Mascolo : Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27 - July 2, 2005
This volume provides the texts of lectures given by L. Ambrosio, L. Caffarelli, M. Crandall, L.C. Evans, N. Fusco at the Summer course held in Cetraro (Italy) in 2005. These are introductory reports on current research by world leaders in the fields of calculus of variations and partial differential equations. The topics discussed are transport equations for nonsmooth vector fields, homogenization, viscosity methods for the infinite Laplacian, weak KAM theory and geometrical aspects of symmetrization. A historical overview of all CIME courses on the calculus of variations and partial differential equations is contributed by Elvira Mascolo.
Analysis, Modeling and Simulation of Multiscale Problems
This book reports recent mathematical developments in the Programme "Analysis, Modeling and Simulation of Multiscale Problems", which started as a German research initiative in 2006. Multiscale problems occur in many fields of science, such as microstructures in materials, sharp-interface models, many-particle systems and motions on different spatial and temporal scales in quantum mechanics or in molecular dynamics. The book presents current mathematical foundations of modeling, and proposes efficient numerical treatment.
An Introduction to Computational Micromechanics
This book presents a comprehensive introduction to computational micromechanics, including basic homogenization theory, microstructural optimization and multifield analysis of heterogeneous materials. "An Introduction to Computational Micromechanics".
