Nonlinear Variational Methods for Estimating Effective Properties of Multiscale Materials

  • Dag Lukkassen
  • Annette Meidell
  • Lars-Erik Persson
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 35)

Abstract

We consider homogenization of sequences of integral functionals with natural growth conditions. Some means are analyzed and used to discuss some fairly new bounds for the homogenized integrand corresponding to integrands which are periodic in the spatial variable. These bounds, which are obtained by variational methods, are compared with the nonlinear bounds of Wiener and Hashin–Shtrikman type. We also point out conditions that make our bounds sharp. Several applications are presented. Moreover, we also discuss bounds for some linear reiterated two-phase problems with m different micro-levels in the spatial variable. In particular, the results imply that the homogenized integrand becomes optimal as m turns to infinity. Both the scalar case (the conductivity problem) and the vector-valued case (the elasticity problem) are considered. In addition, we discuss bounds for estimating the effective behavior described by homogenizing a problem which is a generalization of the Reynold equation.

Keywords

Manifold Total Heat Milton Iterate Cell 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Dag Lukkassen
    • 1
  • Annette Meidell
    • 2
  • Lars-Erik Persson
    • 3
  1. 1.Narvik University College, and Norut NarvikNarvikNorway
  2. 2.Narvik University CollegeNarvikNorway
  3. 3.Department of MathematicsLulea UniversityLuleaSweden

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