2006/07/18 by Clair Poignard, Poignard, Clair
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical methods in inverse problems #math.AP
paper · pdf · doi:10.48550/arxiv.math/0607403
26 pages. Submitted
openalex publication_date 2006/07/18 · arxiv created 2007/03/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an ambient medium and a heterogeneous entity composed of a bidimensional material surrounded by a thin membrane. The electromagnetic constants of these materials are different. By analogy with biological cells, we call this entity a cell. We study the asymptotic behavior of the electric field in the transverse magnetic (TM) mode, when the thickness of the membrane tends to zero. The membrane is of thickness of hf(s), with s a curvilinear coordinate. We provide a rigorous derivation of the first two terms of the asymptotic expansion for h tending to zero. In the membrane, these terms are given explicitly in local coordinates in terms of the boundary data and of the function f, while outside the membrane they are the solutions of a scalar Helmholtz equation with appropriate boundary and transmissions conditions given explicitly in terms of the boundary data and of the above function f. We prove that the remainder terms are of order O(h<sup>3/2</sup>). In addition, if the complex dielectric permittivity in the membrane, denoted by zm, tends to zero faster than h, we give the difference between the exact solution and the above asymptotic with zm = 0; it is of order O(h3/2+|zm|).