Calculus Of Variations Homogenization And Continuum Mechanics
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Calculus Of Variations Homogenization And Continuum Mechanics
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Author : Guy Bouchitte
language : en
Publisher: World Scientific
Release Date : 1994-06-28
Calculus Of Variations Homogenization And Continuum Mechanics written by Guy Bouchitte and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-06-28 with categories.
The aim of the workshop was to promote a better understanding of the connections between recent problems in Theoretical or Computational Mechanics (bounds in composites, phase transitions, microstructure of crystals, optimal design, nonlinear elasticity) and new mathematical tools in the Calculus of Variations (relaxation and Γ-convergence theory, Young and H-measures, compensated compactness and quasiconvexity).
Calculus Of Variations Homogenization And Continuum Mechanics
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Author : Guy Bouchitté
language : en
Publisher:
Release Date : 1994
Calculus Of Variations Homogenization And Continuum Mechanics written by Guy Bouchitté and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with SCIENCE categories.
Variational Methods For Discontinuous Structures
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Author : Raul Serapioni
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Variational Methods For Discontinuous Structures written by Raul Serapioni and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
In recent years many researchers in material science have focused their attention on the study of composite materials, equilibrium of crystals and crack distribution in continua subject to loads. At the same time several new issues in computer vision and image processing have been studied in depth. The understanding of many of these problems has made significant progress thanks to new methods developed in calculus of variations, geometric measure theory and partial differential equations. In particular, new technical tools have been introduced and successfully applied. For example, in order to describe the geometrical complexity of unknown patterns, a new class of problems in calculus of variations has been introduced together with a suitable functional setting: the free-discontinuity problems and the special BV and BH functions. The conference held at Villa Olmo on Lake Como in September 1994 spawned successful discussion of these topics among mathematicians, experts in computer science and material scientists.
Variational Principles Of Continuum Mechanics
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Author : Victor Berdichevsky
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-09-18
Variational Principles Of Continuum Mechanics written by Victor Berdichevsky and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-09-18 with Science categories.
The book reviews the two features of the variational approach: its use as a universal tool to describe physical phenomena and as a source for qualitative and quantitative methods of studying particular problems. Berdichevsky’s work differs from other books on the subject in focusing mostly on the physical origin of variational principles as well as establishing their interrelations. For example, the Gibbs principles appear as a consequence of the Einstein formula for thermodynamic fluctuations rather than as the first principles of the theory of thermodynamic equilibrium. Mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for the direct study of variational problems. In addition, a thorough account of variational principles discovered in various branches of continuum mechanics is given. This book, the second volume, describes how the variational approach can be applied to constructing models of continuum media, such as the theory of elastic plates; shells and beams; shallow water theory; heterogeneous mixtures; granular materials; and turbulence. It goes on to apply the variational approach to asymptotical analysis of problems with small parameters, such as the derivation of the theory of elastic plates, shells and beams from three-dimensional elasticity theory; and the basics of homogenization theory. A theory of stochastic variational problems is considered in detail too, along with applications to the homogenization of continua with random microstructures.
Variational Principles Of Continuum Mechanics
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Author : Victor Berdichevsky
language : en
Publisher: Springer
Release Date : 2009-11-09
Variational Principles Of Continuum Mechanics written by Victor Berdichevsky and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-11-09 with Technology & Engineering categories.
I Fundamentals.- Variational Principles.- Thermodynamics.- Continuum Mechanics.- Principle of least action in continuum mechanics.- Direct methods of calculus of variations.- II Variational features of classical continuum models.- Statics of a geometrically linear elastic body.- Statics of a geometrically nonlinear elastic body.- Dynamics of elastic bodies.- Ideal incompressible fluid.- Ideal compressible fluid.- Steady motion of ideal fluid and elastic body.- Principle of least dissipation.- Motion of rigid bodies in fluids.
Discrete Variational Problems With Interfaces
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Author : Roberto Alicandro
language : en
Publisher: Cambridge University Press
Release Date : 2024-01-31
Discrete Variational Problems With Interfaces written by Roberto Alicandro and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-01-31 with Mathematics categories.
A systematic presentation of discrete-to-continuum results and methods, offering new perspectives on intrinsically discrete problems.
Variational Principles Of Continuum Mechanics
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Author : Victor Berdichevsky
language : en
Publisher: Springer
Release Date : 2009-10-01
Variational Principles Of Continuum Mechanics written by Victor Berdichevsky and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10-01 with Technology & Engineering categories.
Thereareabout500booksonvariationalprinciples. Theyareconcernedmostlywith the mathematical aspects of the topic. The major goal of this book is to discuss the physical origin of the variational principles and the intrinsic interrelations between them. For example, the Gibbs principles appear not as the rst principles of the theory of thermodynamic equilibrium but as a consequence of the Einstein formula for thermodynamic uctuations. The mathematical issues are considered as long as they shed light on the physical outcomes and/or provide a useful technique for direct study of variational problems. Thebookisacompletelyrewrittenversionoftheauthor’smonographVariational Principles of Continuum Mechanics which appeared in Russian in 1983. I have been postponing the English translation because I wished to include the variational pr- ciples of irreversible processes in the new edition. Reaching an understanding of this subject took longer than I expected. In its nal form, this book covers all aspects of the story. The part concerned with irreversible processes is tiny, but it determines the accents put on all the results presented. The other new issues included in the book are: entropy of microstructure, variational principles of vortex line dynamics, va- ational principles and integration in functional spaces, some stochastic variational problems, variational principle for probability densities of local elds in composites with random structure, variational theory of turbulence; these topics have not been covered previously in monographic literature.
Variational Methods For Discontinuous Structures
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Author : Gianni Dal Maso
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Variational Methods For Discontinuous Structures written by Gianni Dal Maso and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This volume contains the Proceedings of the International Workshop Variational Methods For Discontinuous Structures, which was jointly organized by the Dipar timento di Matematica Francesco Brioschi of Milano Politecnico and the Interna tional School for Advanced Studies (SISSA) of Trieste. The Conference took place at Villa Erba Antica (Cernobbio) on the Lago di Como on July 4- 6, 2001. In past years the calculus of variations faced mainly the study of continuous structures, say particularly problems with smooth solutions. One of the deepest and more delicate problems was the regularity of weak solutions. More recently, new sophisticated tools have been introduced in order to study discontinuities: in many variational problems solutions develop singularities, and sometimes the most interesting part of a solution is the singularity itself. The conference intended to focus on recent developments in this direction. Some of the talks were devoted to differential or variational modelling of image segmentation, occlusion and textures synthesizing in image analysis, varia tional description of micro-magnetic materials, dimension reduction and structured deformations in elasticity and plasticity, phase transitions, irrigation and drainage, evolution of crystalline shapes; in most cases theoretical and numerical analysis of these models were provided. viii Preface Other talks were dedicated to specific problems of the calculus of variations: variational theory of weak or lower-dimensional structures, optimal transport prob lems with free Dirichlet regions, higher order variational problems, symmetrization in the BV framework.
Optimal Design Of Multi Phase Materials
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Author : Juan Casado-Díaz
language : en
Publisher: Springer Nature
Release Date : 2022-03-31
Optimal Design Of Multi Phase Materials written by Juan Casado-Díaz and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-31 with Mathematics categories.
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.
Modern Methods In The Calculus Of Variations
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Author : Irene Fonseca
language : en
Publisher: Springer
Release Date : 2010-11-29
Modern Methods In The Calculus Of Variations written by Irene Fonseca and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-29 with Science categories.
This is the first of two books on methods and techniques in the calculus of variations. Contemporary arguments are used throughout the text to streamline and present in a unified way classical results, and to provide novel contributions at the forefront of the theory. This book addresses fundamental questions related to lower semicontinuity and relaxation of functionals within the unconstrained setting, mainly in L^p spaces. It prepares the ground for the second volume where the variational treatment of functionals involving fields and their derivatives will be undertaken within the framework of Sobolev spaces. This book is self-contained. All the statements are fully justified and proved, with the exception of basic results in measure theory, which may be found in any good textbook on the subject. It also contains several exercises. Therefore,it may be used both as a graduate textbook as well as a reference text for researchers in the field. Irene Fonseca is the Mellon College of Science Professor of Mathematics and is currently the Director of the Center for Nonlinear Analysis in the Department of Mathematical Sciences at Carnegie Mellon University. Her research interests lie in the areas of continuum mechanics, calculus of variations, geometric measure theory and partial differential equations. Giovanni Leoni is also a professor in the Department of Mathematical Sciences at Carnegie Mellon University. He focuses his research on calculus of variations, partial differential equations and geometric measure theory with special emphasis on applications to problems in continuum mechanics and in materials science.