Homogenization of Reticulated Structures

ISBN-13: 9781461274377 / Angielski / Miękka / 2011 / 346 str.

Doina Cioranescu; Jeannine Saint Jean Paulin

Opis

This book presents recent works on lattice type structure. Some of the results discussed here have already been published in mathematical journals, but we give here a comprehensive and unified presentation. We have also added some new topics such as those contained in Chapter 4 treating elastic problems for gridworks. The aim of this book is to give continuous simple models for thin reticulated structures (which may have a very complex pattern). This means that we have to treat partial differential equations depending on several small parameters and give the asymptotic behavior with respect to these parameters (which can be the period, the thickness of the material, or the thickness of a plate or of a beam). This book is written from the point of view of the applied mathematician, atten- tion being paid to the mathematical rigor, convergence results, and error estimates. It consists of six chapters and more than a hundred figures. The basic ideas are presented in the first two chapters, while the four last ones study some particular models, using the ideas of Chapters 1 and 2. Chapter 1 is an introduction to homogenization methods in perforated domains. Here the parameter to be taken into consideration is the period. After describing the multiple-scale method (which consists in asymptotic expansions), we focus our attention on the variational method introduced by Tartar, whose main idea is the construction of rapidly oscillating test functions.

Spis treści

1 Homogenization in Perforated Media.- 1. The General Method of Homogenization.- 1.1 The One-Dimensional Periodic Case.- 1.2 A Model Example: the Thermal Problem.- 1.3 Perforated Domains.- 2. The Homogeneous Neumann Problem.- 2.1 Perforated Domains and Variational Formulation.- 2.2 Multiple-Scale Method.- 2.3 Extension Operators.- 2.4 Convergence Theorems.- 2.5 Domains with Nonisolated Holes.- 2.6 Error Estimates.- 3. Other Boundary Value Problems.- 3.1 The Dirichlet Problem.- 3.2 Fourier Conditions.- 3.3 Eigenvalue Problem.- 2 Lattice-Type Structures.- 1. The Two-Dimensional Case.- 1.1 Statement of the Problem and the Main Theorem.- 1.2 Proof of the Main Theorem: Technique of Dilatations.- 1.3 Superposition Method.- 1.4 Error Estimates.- 2. The Three-Dimensional Case.- 2.1 Honeycomb Structures.- 2.2 Reinforced Structures.- 3. Complex Structures and Loss of Ellipticity.- 3.1 A General Method for Diagonal Bars.- 3.2 Linearized Elasticity and Loss of Ellipticity.- 3.3 Examples.- 4. Other Boundary Conditions.- 4.1 The Dirichlet Problem.- 4.2 Fourier Conditions.- 4.3 Eigenvalue Problem.- 3 Thermal Problems for Gridworks.- 1. Statement of the Problem.- 2. Case e = k?.- 2.1 Change of Scale.- 2.2 Limit for ? ? 0.- 2.3 Limit for ? ? 0.- 3. Case ? ? e.- 3.1 The Multiple-Scale Method.- 3.2 The Variational Method.- 3.3 Limit for e ? 0.- 3.4 Limit for ? ? 0.- 4. Case e ? ?.- 4.1 Limit for e ? 0.- 4.2 Limit for ? ? 0.- 4.3 Limit for ? ? 0.- 4.4 Comparison of the Different Limits.- 4 Elasticity Problems for Gridworks.- 1. Statement of the Problem.- 2. Limit Plate Behavior.- 2.1 Main Result.- 2.2 A Priori Estimates and Limits of Displacements.- 2.3 Limits of Stresses and Moments and Limit Equations.- 3. Homogenization Result.- 4. Final Explicit Result and Loss of Ellipticity.- 5. Case ? ? e.- 5.1 Limit for ? ? 0.- 5.2 Limit for e ? 0.- 5.3 Limit for ? ? 0.- 5.4 Loss of Ellipticity.- 6. Plates Without Loss of Ellipticity.- 7. Time-Dependent Plates Models: An Experimental Result.- 5 Thermal Problems for Thin Tall Structures.- 1. Statement of the Problem.- 2. Case e = ??.- 2.1 Limit for ? ? 0.- 2.2 Limit for ? ? 0.- 3. Case ? ? e.- 3.1 Limit for ? ? 0.- 3.2 Limit for e ? 0, Then for ? ? 0.- 3.3 Limit for ? ? 0.- 3.4 Limit for e ? 0.- 4. Case e ? ?.- 5. Comparison of Limit Systems and Solutions.- 6. Numerical Results for a Two-Dimensional Case.- 6.1 Limit for ? ? 0.- 6.2 Limit for e ? 0.- 6.3 Numerical Computation of the Homogenized Solution.- 6.4 Limit for ? ? 0.- 6.5 Numerical Computation of the Solution V?.- 7. Generalization for a Three-Dimensional Structure.- 7.1 Case e = ??.- 7.2 Case ? ? e.- 7.3 Case e ? ?.- 6 Elasticity Problems for Thin Tall Structures.- 1. Statement of the Problem.- 1.1 Geometric Assumptions.- 1.2 Variational Formulation.- 2. Limit for e ? 0.- 2.1 Change of Scale.- 2.2 Assumptions on the Data.- 2.3 Limit for e ? 0: Beam Behaviour.- 3. Limit for e ? 0: Homogenization.- 4. Limit for ? ? 0.- 5. Applications to Other Structures.- 5.1 Towers.- 5.2 Tall Structures with Oblique Bars.- Final Comments.- References.