Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients

Kirill Cherednichenko, Shane Cooper, Sebastien Guenneau

Research output: Contribution to journalArticle

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Abstract


We study the behavior of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent, e.g., the elastic or electromagnetic response of a two-component composite medium. Compared to the standard operators with moderate contrast, they exhibit a number of new effects due to the underlying nonuniform ellipticity of the family. The effective behavior of such media in the vanishing period limit also differs notably from that of multidimensional models investigated thus far by other authors, due to the fact that neither component of the composite forms a connected set. We then discuss a modified problem, where the equation coefficient is set to a positive constant on an interval that is independent of the period. Formal asymptotic analysis and numerical tests with finite elements suggest the existence of localized eigenfunctions (``defect modes''), whose eigenvalues are situated in the gaps of the limit spectrum for the unperturbed problem.


Read More: http://epubs.siam.org/doi/10.1137/130947106
LanguageEnglish
Pages72-98
Number of pages17
JournalMultiscale Modeling and Simulation
Volume13
Issue number1
Early online date8 Jan 2015
DOIs
StatusPublished - 2015

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Periodic Coefficients
Spectral Analysis
Elliptic Problems
Spectrum analysis
spectral analysis
spectrum analysis
operators
composite materials
Asymptotic analysis
Composite materials
ellipticity
eigenvalue
coefficients
Eigenvalues and eigenfunctions
Composite Media
defect
Multidimensional Model
eigenvectors
Connected Set
eigenvalues

Cite this

Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients. / Cherednichenko, Kirill; Cooper, Shane; Guenneau, Sebastien.

In: Multiscale Modeling and Simulation, Vol. 13, No. 1, 2015, p. 72-98.

Research output: Contribution to journalArticle

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