Shape Optimization by the Homogenization Method: 146 - Rilegato

Allaire, Gregoire

Rilegato

ISBN 10: 0387952985 ISBN 13: 9780387952987

Casa editrice: Springer-Verlag GmbH, 2001

Riassunto

This book provides an introduction to the theory and numerical developments of the homogenization method. Its 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.

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Recensione

"The book is a research monograph, but the structure and completeness of the presentation means that the book constitutes a good basis for a graduate course in applied mathematics. The rigorous mathematical presentation is supplemented with numerous remarks and comments which discuss the subject in broader terms, greatly simplifying the reading process. ... The book is a welcome and up-to-date addition to the literature in the area and it is necessary reading for any researcher and student ..." --M.P. Bendsøe, Structural Multidisciplinary Optimization, 5, 2002

"The book is very well structured, very clearly written, very well motivated, and complete in its treatment of modelling, analysis and simulation. It will be a basic reference for whoever wants to deeply understand homogenization from the point of view of its application to optimal design. The treatment is right to the point, a quality that is very much appreciated by readers. In summary, I believe this text may become a main source for the subject of optimal design and shape optimization." --Pablo Pedregal, Mathematical Reviews, 2002 h

"The book under review presents a comprehensive introduction to the homogenisation method applied to optimal design, including many proofs which were hitherto only scattered throughout the literature ... this one provides the most complete treatment of numerical methods ... A number of realistic examples, mostly for elasticity, has been developed in detail. ... In summary, we would like to warmly recommend this book to anybody working in optimal shape design, composites and homogenisation, as well to those who wish to enter these fields." --Nenad Antonic and Marko Vrdoljak, Zentralblatt MATH, 990:15, 2002

Contenuti

1 Homogenization.- 1.1 Introduction to Periodic Homogenization.- 1.1.1 A Model Problem in Conductivity.- 1.1.2 Two-scale Asymptotic Expansions.- 1.1.3 Variational Characterizations and Estimates of the Effective Tensor.- 1.1.4 Generalization to the Elasticity System.- 1.2 Definition of H-convergence.- 1.2.1 Some Results on Weak Convergence.- 1.2.2 Problem Statement.- 1.2.3 The One-dimensional Case.- 1.2.4 Main Results.- 1.3 Proofs and Further Results.- 1.3.1 Tartar’s Method.- 1.3.2 G-convergence.- 1.3.3 Homogenization of Eigenvalue Problems.- 1.3.4 A Justification of Periodic Homogenization.- 1.3.5 Homogenization of Laminated Structures.- 1.3.6 Corrector Results.- 1.4 Generalization to the Elasticity System.- 1.4.1 Problem Statement.- 1.4.2 H-convergence.- 1.4.3 Lamination Formulas.- 2 The Mathematical Modeling of Composite Materials.- 2.1 Homogenized Properties of Composite Materials.- 2.1.1 Modeling of Composite Materials.- 2.1.2 The G-closure Problem.- 2.2 Conductivity.- 2.2.1 Laminated Composites.- 2.2.2 Hashin-Shtrikman Bounds.- 2.2.3 G-closure of Two Isotropic Phases.- 2.3 Elasticity.- 2.3.1 Laminated Composites.- 2.3.2 Hashin-Shtrikman Energy Bounds.- 2.3.3 Toward G-closure.- 2.3.4 An Explicit Optimal Bound for Shape Optimization.- 3 Optimal Design in Conductivity.- 3.1 Setting of Optimal Shape Design.- 3.1.1 Definition of a Model Problem.- 3.1.2 A first Mathematical Analysis.- 3.1.3 Multiple State Equations.- 3.1.4 Shape Optimization as a Degeneracy Limit.- 3.1.5 Counterexample to the Existence of Optimal Designs.- 3.2 Relaxation by the Homogenization Method.- 3.2.1 Existence of Generalized Designs.- 3.2.2 Optimality Conditions.- 3.2.3 Multiple State Equations.- 3.2.4 Gradient of the Objective Function.- 3.2.5 Self-adjoint Problems.- 3.2.6 Counterexample to the Uniqueness of.- Optimal Designs.- 4 Optimal Design in Elasticity.- 4.1 Two-phase Optimal Design.- 4.1.1 The Original Problem.- 4.1.2 Counterexample to the Existence of Optimal Designs.- 4.1.3 Relaxed Formulation of the Problem.- 4.1.4 Compliance Optimization.- 4.1.5 Counterexample to the Uniqueness of Optimal Designs.- 4.1.6 Eigenfrequency Optimization.- 4.2 Shape Optimization.- 4.2.1 Compliance Shape Optimization.- 4.2.2 The Relaxation Process.- 4.2.3 Link with the Michell Truss Theory.- 5 Numerical Algorithms.- 5.1 Algorithms for Optimal Design in Conductivity.- 5.1.1 Optimality Criteria Method.- 5.1.2 Gradient Method.- 5.1.3 A Convergence Proof.- 5.1.4 Numerical Examples.- 5.2 Algorithms for Structural Optimization.- 5.2.1 Compliance Optimization.- 5.2.2 Numerical Examples.- 5.2.3 Technical Algorithmic Issues.- 5.2.4 Penalization of Intermediate Densities.- 5.2.5 Quasiconvexification versus Convexification.- 5.2.6 Multiple Loads Optimization.- 5.2.7 Eigenfrequency Optimization.- 5.2.8 Partial Relaxation.

Dettagli dell'edizione

• Editore: Springer-Verlag GmbH
• Data di pubblicazione: 2001
• Lingua: Inglese
• ISBN 10: 0387952985
• ISBN 13: 9780387952987
• Rilegatura: Copertina rigida
• Numero di pagine: 480
• Contatto del produttore: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3; Heidelberg; 69115; Germania