The General Theory Of Homogenization

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The General Theory of Homogenization

Luc Tartar
The General Theory of Homogenization Book PDF

Author : Luc Tartar
Publisher : Springer Science & Business Media
Page : 471 pages
File Size : 45,7 Mb
Release : 2009-12-03
Category : Science
ISBN : 9783642051951

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The General Theory of Homogenization by Luc Tartar Pdf

Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of François Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly. For a better understanding of 20th century science, new mathematical tools must be introduced, like the author’s H-measures, variants by Patrick Gérard, and others yet to be discovered.

Homogenization Methods for Multiscale Mechanics

Chiang C. Mei,Bogdan Vernescu
Homogenization Methods for Multiscale Mechanics Book PDF

Author : Chiang C. Mei,Bogdan Vernescu
Publisher : World Scientific
Page : 349 pages
File Size : 54,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814282444

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Homogenization Methods for Multiscale Mechanics by Chiang C. Mei,Bogdan Vernescu Pdf

In many physical problems several scales present either in space or in time, caused by either inhomogeneity of the medium or complexity of the mechanical process. A fundamental approach is to first construct micro-scale models, and then deduce the macro-scale laws and the constitutive relations by properly averaging over the micro-scale. The perturbation method of multiple scales can be used to derive averaged equations for a much larger scale from considerations of the small scales. In the mechanics of multiscale media, the analytical scheme of upscaling is known as the Theory of Homogenization The authors share the view that the general methods of homogenization should be more widely understood and practiced by applied scientists and engineers. Hence this book is aimed at providing a less abstract treatment of the theory of homogenization for treating inhomogeneous media, and at illustrating its broad range of applications. Each chapter deals with a different class of physical problems. To tackle a new problem, the novel approach of first discussing the physically relevant scales, then identifying the small parameters and their roles in the normalized governing equations is adopted. The details of asymptotic analysis are only explained afterwards.

An Introduction to Homogenization

Doïna Cioranescu,Patrizia Donato
An Introduction to Homogenization Book PDF

Author : Doïna Cioranescu,Patrizia Donato
Publisher : Oxford University Press on Demand
Page : 262 pages
File Size : 46,9 Mb
Release : 1999
Category : Mathematics
ISBN : 0198565542

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An Introduction to Homogenization by Doïna Cioranescu,Patrizia Donato Pdf

Composite materials are widely used in industry: well-known examples of this are the superconducting multi-filamentary composites which are used in the composition of optical fibres. Such materials are complicated to model, as different points in the material will have different properties. The mathematical theory of homogenization is designed to deal with this problem, and hence is used to model the behaviour of these important materials. This book provides a self-contained and authoritative introduction to the subject for graduates and researchers in the field.

Homogenization Theory for Multiscale Problems

Xavier Blanc,Claude Le Bris
Homogenization Theory for Multiscale Problems Book PDF

Author : Xavier Blanc,Claude Le Bris
Publisher : Springer Nature
Page : 469 pages
File Size : 48,7 Mb
Release : 2023-04-29
Category : Mathematics
ISBN : 9783031218330

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Homogenization Theory for Multiscale Problems by Xavier Blanc,Claude Le Bris Pdf

The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.

Homogenization of Multiple Integrals

Andrea Braides,Anneliese Defranceschi
Homogenization of Multiple Integrals Book PDF

Author : Andrea Braides,Anneliese Defranceschi
Publisher : Oxford University Press
Page : 322 pages
File Size : 51,6 Mb
Release : 1998
Category : Mathematics
ISBN : 019850246X

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Homogenization of Multiple Integrals by Andrea Braides,Anneliese Defranceschi Pdf

An introduction to the mathematical theory of the homogenization of multiple integrals, this book describes the overall properties of such functionals with various applications ranging from cellular elastic materials to Riemannian metrics.

Nonlinear Reaction-Diffusion Processes for Nanocomposites

Jesús Ildefonso Díaz,David Gómez-Castro,Tatiana A. Shaposhnikova
Nonlinear Reaction Diffusion Processes for Nanocomposites Book PDF

Author : Jesús Ildefonso Díaz,David Gómez-Castro,Tatiana A. Shaposhnikova
Publisher : Walter de Gruyter GmbH & Co KG
Page : 200 pages
File Size : 45,5 Mb
Release : 2021-06-21
Category : Mathematics
ISBN : 9783110648997

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Nonlinear Reaction-Diffusion Processes for Nanocomposites by Jesús Ildefonso Díaz,David Gómez-Castro,Tatiana A. Shaposhnikova Pdf

The behavior of materials at the nanoscale is a key aspect of modern nanoscience and nanotechnology. This book presents rigorous mathematical techniques showing that some very useful phenomenological properties which can be observed at the nanoscale in many nonlinear reaction-diffusion processes can be simulated and justified mathematically by means of homogenization processes when a certain critical scale is used in the corresponding framework.

Harmonic Analysis and Applications

Carlos E. Kenig
Harmonic Analysis and Applications Book PDF

Author : Carlos E. Kenig
Publisher : American Mathematical Soc.
Page : 345 pages
File Size : 55,9 Mb
Release : 2020-12-14
Category : Education
ISBN : 9781470461270

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Harmonic Analysis and Applications by Carlos E. Kenig Pdf

The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.

G-Convergence and Homogenization of Nonlinear Partial Differential Operators

A.A. Pankov
G Convergence and Homogenization of Nonlinear Partial Differential Operators Book PDF

Author : A.A. Pankov
Publisher : Springer Science & Business Media
Page : 276 pages
File Size : 40,6 Mb
Release : 1997-09-30
Category : Mathematics
ISBN : 079234720X

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G-Convergence and Homogenization of Nonlinear Partial Differential Operators by A.A. Pankov Pdf

Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space.

Plasticity and Beyond

Jörg Schröder,Klaus Hackl
Plasticity and Beyond Book PDF

Author : Jörg Schröder,Klaus Hackl
Publisher : Springer Science & Business Media
Page : 412 pages
File Size : 43,5 Mb
Release : 2013-09-20
Category : Science
ISBN : 9783709116258

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Plasticity and Beyond by Jörg Schröder,Klaus Hackl Pdf

The book presents the latest findings in experimental plasticity, crystal plasticity, phase transitions, advanced mathematical modeling of finite plasticity and multi-scale modeling. The associated algorithmic treatment is mainly based on finite element formulations for standard (local approach) as well as for non-standard (non-local approach) continua and for pure macroscopic as well as for directly coupled two-scale boundary value problems. Applications in the area of material design/processing are covered, ranging from grain boundary effects in polycrystals and phase transitions to deep-drawing of multiphase steels by directly taking into account random microstructures.

Partial Differential Equations arising from Physics and Geometry

Mohamed Ben Ayed,Mohamed Ali Jendoubi,Yomna Rébaï,Hasna Riahi,Hatem Zaag
Partial Differential Equations arising from Physics and Geometry Book PDF

Author : Mohamed Ben Ayed,Mohamed Ali Jendoubi,Yomna Rébaï,Hasna Riahi,Hatem Zaag
Publisher : Cambridge University Press
Page : 471 pages
File Size : 41,8 Mb
Release : 2019-05-02
Category : Mathematics
ISBN : 9781108431637

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Partial Differential Equations arising from Physics and Geometry by Mohamed Ben Ayed,Mohamed Ali Jendoubi,Yomna Rébaï,Hasna Riahi,Hatem Zaag Pdf

Presents the state of the art in PDEs, including the latest research and short courses accessible to graduate students.

Shape Optimization by the Homogenization Method

Gregoire Allaire
Shape Optimization by the Homogenization Method Book PDF

Author : Gregoire Allaire
Publisher : Springer Science & Business Media
Page : 470 pages
File Size : 45,6 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781468492866

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Shape Optimization by the Homogenization Method by Gregoire Allaire Pdf

This book provides an introduction to the theory and numerical developments of the homogenization method. It's main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials; a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.

Homogenization of Differential Operators and Integral Functionals

V.V. Jikov,S.M. Kozlov,O.A. Oleinik
Homogenization of Differential Operators and Integral Functionals Book PDF

Author : V.V. Jikov,S.M. Kozlov,O.A. Oleinik
Publisher : Springer Science & Business Media
Page : 583 pages
File Size : 43,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642846595

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Homogenization of Differential Operators and Integral Functionals by V.V. Jikov,S.M. Kozlov,O.A. Oleinik Pdf

It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.

Analysis and Stochastics of Growth Processes and Interface Models

Peter Mörters,Roger Moser,Mathew Penrose,Hartmut Schwetlick,Johannes Zimmer
Analysis and Stochastics of Growth Processes and Interface Models Book PDF

Author : Peter Mörters,Roger Moser,Mathew Penrose,Hartmut Schwetlick,Johannes Zimmer
Publisher : Oxford University Press
Page : 347 pages
File Size : 50,9 Mb
Release : 2008-07-24
Category : Mathematics
ISBN : 9780199239252

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Analysis and Stochastics of Growth Processes and Interface Models by Peter Mörters,Roger Moser,Mathew Penrose,Hartmut Schwetlick,Johannes Zimmer Pdf

This is a collection of topical survey articles by researchers in the fields of applied analysis and probability theory, working on the mathematical description of growth phenomena. Particular emphasis is given to the interplay of the usually separate fields of applied analysis and probability theory.

Optimal Design of Multi-Phase Materials

Juan Casado-Díaz
Optimal Design of Multi Phase Materials Book PDF

Author : Juan Casado-Díaz
Publisher : Springer Nature
Page : 119 pages
File Size : 40,8 Mb
Release : 2022-03-31
Category : Mathematics
ISBN : 9783030981914

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Optimal Design of Multi-Phase Materials by Juan Casado-Díaz Pdf

This book aims the optimal design of a material (thermic or electrical) obtained as the mixture of a finite number of original materials, not necessarily isotropic. The problem is to place these materials in such a way that the solution of the corresponding state equation minimizes a certain functional that can depend nonlinearly on the gradient of the state function. This is the main novelty in the book. It is well known that this type of problems has no solution in general and therefore that it is needed to work with a relaxed formulation. The main results in the book refer to how to obtain such formulation, the optimality conditions, and the numerical computation of the solutions. In the case of functionals that do not depend on the gradient of the state equation, it is known that a relaxed formulation consists of replacing the original materials with more general materials obtained via homogenization. This includes materials with different properties of the originals but whose behavior can be approximated by microscopic mixtures of them. In the case of a cost functional depending nonlinearly on the gradient, it is also necessary to extend the cost functional to the set of these more general materials. In general, we do not dispose of an explicit representation, and then, to numerically solve the problem, it is necessary to design strategies that allow the functional to be replaced by upper or lower approximations. The book is divided in four chapters. The first is devoted to recalling some classical results related to the homogenization of a sequence of linear elliptic partial differential problems. In the second one, we define the control problem that we are mainly interested in solving in the book. We obtain a relaxed formulation and their main properties, including an explicit representation of the new cost functional, at least in the boundary of its domain. In the third chapter, we study the optimality conditions of the relaxed problem, and we describe some algorithms to numerically solve the problem. We also provide some numerical experiments carried out using such algorithms. Finally, the fourth chapter is devoted to briefly describe some extensions of the results obtained in Chapters 2 and 3 to the case of dealing with several state equations and the case of evolutive problems. The problems covered in the book are interesting for mathematicians and engineers whose work is related to mathematical modeling and the numerical resolution of optimal design problems in material sciences. The contents extend some previous results obtained by the author in collaboration with other colleagues.

Periodic Homogenization of Elliptic Systems

Zhongwei Shen
Periodic Homogenization of Elliptic Systems Book PDF

Author : Zhongwei Shen
Publisher : Springer
Page : 291 pages
File Size : 41,6 Mb
Release : 2018-09-04
Category : Mathematics
ISBN : 9783319912141

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Periodic Homogenization of Elliptic Systems by Zhongwei Shen Pdf

This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.