Getting Acquainted with Homogenization and Multiscale

✍ Scribed by Leonid Berlyand, Volodymyr Rybalko

Publisher: Springer International Publishing,Birkhäuser
Year: 2018
Tongue: English
Leaves: 187
Series: Compact Textbooks in Mathematics
Edition: 1st ed.
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

The objective of this book is to navigate beginning graduate students in mathematics and engineering through a mature field of multiscale problems in homogenization theory and to provide an idea of its broad scope. An overview of a wide spectrum of homogenization techniques ranging from classical two-scale asymptotic expansions to Gamma convergence and the rapidly developing field of stochastic homogenization is presented. The mathematical proofs and definitions are supplemented with intuitive explanations and figures to make them easier to follow. A blend of mathematics and examples from materials science and engineering is designed to teach a mixed audience of mathematical and non-mathematical students.

✦ Table of Contents

Front Matter ....Pages i-xviii
Preliminaries (Leonid Berlyand, Volodymyr Rybalko)....Pages 1-15
What Is Homogenization and Multiscale? First Examples (Leonid Berlyand, Volodymyr Rybalko)....Pages 17-33
Brief History and Surprising Examples in Homogenization (Leonid Berlyand, Volodymyr Rybalko)....Pages 35-43
Formal Two-Scale Asymptotic Expansions and the Corrector Problem (Leonid Berlyand, Volodymyr Rybalko)....Pages 45-52
Compensated Compactness and Oscillating Test Functions (Leonid Berlyand, Volodymyr Rybalko)....Pages 53-58
Two-Scale Convergence (Leonid Berlyand, Volodymyr Rybalko)....Pages 59-74
Examples of Explicit Effective Coefficients: Laminated Structures and 2D Checkerboards (Leonid Berlyand, Volodymyr Rybalko)....Pages 75-83
Introduction to Stochastic Homogenization (Leonid Berlyand, Volodymyr Rybalko)....Pages 85-101
Γ-Convergence in Nonlinear Homogenization Problems (Leonid Berlyand, Volodymyr Rybalko)....Pages 103-122
An Example of a Nonlinear Problem: Homogenization of Plasticity and Limit Loads (Leonid Berlyand, Volodymyr Rybalko)....Pages 123-138
Continuum Limits for Discrete Problems with Fine Scales (Leonid Berlyand, Volodymyr Rybalko)....Pages 139-168
Back Matter ....Pages 169-178

✦ Subjects

Mathematics; Computational Science and Engineering; Mathematical and Computational Engineering; Partial Differential Equations

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