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Green's Kernels and Meso-Scale Approximations in Perforated Domains

  • Book
  • © 2013

Overview

Authors:
  1. Vladimir Maz'ya

    1. Department of Mathematics, Linköping University, Linköping, Sweden

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    You can also search for this author in PubMed   Google Scholar

  2. Alexander Movchan

    1. Dept. Mathematical Sciences, University of Liverpool, Liverpool, United Kingdom

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    You can also search for this author in PubMed   Google Scholar

  3. Michael Nieves

    1. School of Engineering, Liverpool John Moores University, Liverpool, United Kingdom

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  • Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale approximations for bodies with non-periodic multiple perforations Contains illustrations and numerical examples for a range of physically realisable configurations
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2077)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xvii
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  2. Green’s Functions in Singularly Perturbed Domains

    1. Front Matter

      Pages 1-1
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  3. Green’s Functions in Singularly Perturbed Domains

    1. Uniform Asymptotic Formulae for Green’s Functions for the Laplacian in Domains with Small Perforations

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 3-19

    2. Mixed and Neumann Boundary Conditions for Domains with Small Holes and Inclusions: Uniform Asymptotics of Green’s Kernels

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 21-57

    3. Green’s Function for the Dirichlet Boundary Value Problem in a Domain with Several Inclusions

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 59-73

    4. Numerical Simulations Based on the Asymptotic Approximations

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 75-81

    5. Other Examples of Asymptotic Approximations of Green’s Functions in Singularly Perturbed Domains

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 83-94

  4. Green’s Tensors for Vector Elasticity in Bodies with Small Defects

    1. Front Matter

      Pages 95-95
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  5. Green’s Tensors for Vector Elasticity in Bodies with Small Defects

    1. Green’s Tensor for the Dirichlet Boundary Value Problem in a Domain with a Single Inclusion

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 97-137

    2. Green’s Tensor in Bodies with Multiple Rigid Inclusions

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 139-167

    3. Green’s Tensor for the Mixed Boundary Value Problem in a Domain with a Small Hole

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 169-188

  6. Meso-scale Approximations: Asymptotic Treatment of Perforated Domains Without Homogenization

    1. Front Matter

      Pages 189-189
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    2. Meso-scale Approximations for Solutions of Dirichlet Problems

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 191-219

    3. Mixed Boundary Value Problems in Multiply-Perforated Domains

      • Vladimir Maz’ya, Alexander Movchan, Michael Nieves
      Pages 221-247

  7. Back Matter

    Pages 249-260
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Keywords

About this book

There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution.
Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions.
The main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables.
This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.

Authors and Affiliations

  • Department of Mathematics, Linköping University, Linköping, Sweden

    Vladimir Maz'ya

  • Dept. Mathematical Sciences, University of Liverpool, Liverpool, United Kingdom

    Alexander Movchan

  • School of Engineering, Liverpool John Moores University, Liverpool, United Kingdom

    Michael Nieves

Bibliographic Information

  • Book Title: Green's Kernels and Meso-Scale Approximations in Perforated Domains

  • Authors: Vladimir Maz'ya, Alexander Movchan, Michael Nieves

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-00357-3

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2013

  • Softcover ISBN: 978-3-319-00356-6Published: 14 June 2013

  • eBook ISBN: 978-3-319-00357-3Published: 07 June 2013

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XVII, 258

  • Number of Illustrations: 7 b/w illustrations, 10 illustrations in colour

  • Topics: Partial Differential Equations, Approximations and Expansions

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