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Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media.

Shane Cooper
Published in: Calculus of variations and partial differential equations (2018)
The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasi-momentum dependence are found as resolvent limits of the original operator family. This results in asymptotic spectral behaviour with a rich dependence on quasimomenta.
Keyphrases
  • magnetic resonance
  • optical coherence tomography
  • magnetic resonance imaging
  • contrast enhanced
  • computed tomography
  • molecular dynamics