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Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: The translation method

Title
Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: The translation method
Authors
Kang H.Kim E.Milton G.W.
Ewha Authors
김은주
Issue Date
2012
Journal Title
Calculus of Variations and Partial Differential Equations
ISSN
0944-2669JCR Link
Citation
vol. 45, no. 41337, pp. 367 - 401
Indexed
SCI; SCIE; SCOPUS WOS scopus
Abstract
We deal with the problem of estimating the volume of inclusions using a small number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq-Vogelius in the limit when the volume fraction tends to zero. The bounds of this article, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are. © 2011 Springer-Verlag.
DOI
10.1007/s00526-011-0462-3
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연구기관 > 수리과학연구소 > Journal papers
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