**Auteur :** Alain Bensoussan

**la langue :** en

**Éditeur:** American Mathematical Soc.

**Date de sortie :** 2011-10-26

This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.

**Auteur :** G. Papanicolau

**la langue :** en

**Éditeur:** Elsevier

**Date de sortie :** 1978-01-01

Asymptotic Analysis for Periodic Structures

## Alain Bensoussan Jacques Louis Lions George Papanicolaou Asymptotic Analysis For Periodic Structures

**Auteur :** C W Cai

**la langue :** en

**Éditeur:** World Scientific

**Date de sortie :** 2002-03-25

By using the U-transformation method, it is possible to uncouple linear simultaneous equations, either algebraic or differential, with cyclic periodicity. This book presents a procedure for applying the U-transformation technique twice to uncouple the two sets of unknown variables in a doubly periodic structure to achieve an analytical exact solution. Explicit exact solutions for the static and dynamic analyses for certain engineering structures with doubly periodic properties — such as a continuous truss with any number of spans, cable network and grillwork on supports with periodicity, and grillwork with periodic stiffening members or equidistant line supports — can be found in the book. The availability of these exact solutions not only helps with the checking of the convergence and accuracy of numerical solutions, but also provides a basis for optimization design for these types of structures. The study of the force vibration and mode shape of periodic systems with nonlinear disorder is yet another research area which has attained considerable success by the U-transformation method. This book illustrates the analytical approach and procedure for the problems of localization of the mode shape of nearly periodic systems together with the results. Contents:U-Transformation and Uncoupling of Governing Equations for Systems with Cyclic Bi-PeriodicityBi-Periodic Mass-Spring SystemsBi-Periodic StructuresStructures with Bi-Periodicity in Two DirectionsNearly Periodic Systems with Nonlinear Disorders Readership: Practitioners, researchers and graduate students in mechanical engineering, engineering mechanics, civil engineering, mechanics, and numerical & computational mathematics. Keywords:Double U-Transformation;Cyclic Bi-Periodic Equations;Bi-Periodic Mass-Spring Systems;Bi-Periodic Structures;Localized Modes;Mode Subspace;Cable Network;Grillwork;Grid;Pass Band;Nonlinear Disorder;Stop BandReviews:“This book may be useful to students, postgraduate students, engineers and scientists who analyse periodic structures in different areas of technology and science.”Zentralblatt MATH

**Auteur :** T. Lewi?ski

**la langue :** en

**Éditeur:** World Scientific

**Date de sortie :** 2000

This book gives a systematic and comprehensive presentation of the results concerning effective behavior of elastic and plastic plates with periodic or quasiperiodic structure. One of the chapters covers the hitherto available results concerning the averaging problems in the linear and nonlinear shell models.A unified approach to the problems studied is based on modern variational and asymptotic methods, including the methods of variational inequalities as well as homogenization techniques. Duality arguments are also exploited. A significant part of the book deals with problems important for engineering practice, such as: statical analysis of highly nonhomogeneous plates and shells for which common discretization techniques fail to be efficient, assessing stiffness reduction of cracked 0n/900m]s laminates, and assessing ultimate loads for perfectly plastic plates and shells composed of repeated segments. When possible, the homogenization formulas are cast in closed form expressions. The formulas presented in this manner are then used in constructing regularized formulations of the fundamental optimization problems for plates and shells, since the regularization concepts are based on introducing the composite regions for which microstructural properties play the role of new design variables.