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Bloch Waves in Bubbly Crystal Near the First Band Gap: A High-Frequency Homogenization Approach
Ammari, Habib,Lee, Hyundae,Zhang, Hai Society for Industrial and Applied Mathematics 2019 SIAM journal on mathematical analysis Vol.51 No.1
<P>This paper is concerned with the high-frequency homogenization of bubbly phononic crystals. It is a follow-up of the work [H. Ammari et al., <italic toggle='yes'>J. Differential Equations</I>, 263 (2017), pp. 5610--5629], which shows the existence of a subwavelength band gap. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. In this paper, we prove that the first Bloch eigenvalue achieves its maximum at the corner of the Brillouin zone. Moreover, by computing the asymptotic of the Bloch eigenfunctions in the periodic structure near that critical frequency, we demonstrate that these eigenfunctions can be decomposed into two parts: one part is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is varying. They rigorously justify, in the nondilute case, the observed superfocusing of acoustic waves in bubbly crystals near and below the maximum of the first Bloch eigenvalue and confirm the band gap opening near and above this critical frequency.</P>
Ammari, H.,Kang, H.,Kim, K.,Lee, H. Academic Press 2013 Journal of differential equations Vol.254 No.12
We consider the Lame system of linear elasticity when the inclusion has the extreme elastic constants. We show that the solutions to the Lame system converge in appropriate H<SUP>1</SUP>-norms when the shear modulus tends to infinity (the other modulus, the compressional modulus is fixed), and when the bulk modulus and the shear modulus tend to zero. Using this result, we show that the asymptotic expansion of the displacement vector in the presence of small inclusion is uniform with respect to Lame parameters.
Admittivity imaging from multi-frequency micro-electrical impedance tomography
Ammari, Habib,Giovangigli, Laure,Nguyen, Loc Hoang,Seo, Jin-Keun Academic Press 2017 Journal of mathematical analysis and applications Vol.449 No.2
<P><B>Abstract</B></P> <P>The aim of this paper is to propose an optimal control optimization algorithm for reconstructing admittivity distributions (i.e., both conductivity and permittivity) from multi-frequency micro-electrical impedance tomography. A convergent and stable optimal control scheme is shown to be obtainable from multi-frequency data. This opens a door for convergence analysis of optimal control type approaches in imaging from internal data. The results of this paper have potential applicability in cancer imaging, cell culturing and differentiation, food sciences, and biotechnology.</P>
Transient anomaly imaging by the acoustic radiation force
Ammari, H.,Garapon, P.,Guadarrama Bustos, L.,Kang, H. Academic Press 2010 Journal of differential equations Vol.249 No.7
This paper is devoted to provide a solid mathematical foundation for a promising imaging technique based on the acoustic radiation force, which acts as a dipolar source. From the rigorously established asymptotic expansions of near- and far-field measurements of the transient wave induced by the anomaly, we design asymptotic imaging methods leading to a quantitative estimation of physical and geometrical parameters of the anomaly.
Ammari, H.,Kang, H.,Lee, H.,Lim, M.,Zribi, H. Academic Press 2009 Journal of differential equations Vol.247 No.11
When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution to the conductivity equation blows up in general. In this paper, we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form: one of them has a bounded gradient and the gradient of the other part blows up. Using the decomposition, we derive the best possible estimates for the blow-up of the gradient. We then consider the case when the inclusions have positive permittivities. We show quantitatively that in this case the size of the blow-up is reduced.
Mathematical models and reconstruction methods in magneto-acoustic imaging
AMMARI, HABIB,CAPDEBOSCQ, YVES,KANG, HYEONBAE,KOZHEMYAK, ANASTASIA Cambridge University Press 2009 European journal of applied mathematics Vol.20 No.3
<P>In this paper, we provide the mathematical basis for three different magneto-acoustic imaging approaches (vibration potential tomography, magneto-acoustic tomography with magnetic induction and magneto-acoustic current imaging) and propose new algorithms for solving the inverse problem for each of them.</P>