APPROXIMATE TRANSMISSION CONDITIONS THROUGH A WEAKLY OSCILLATING THIN LAYER

We study the behavior of the electro-quasistatic voltage potentials in a material composed by a bidimensional medium surrounded by a weakly oscillating thin layer and embedded in an ambient medium. We build approximate transmission conditions in order to replace the layer by these conditions on the boundary of the interior material. We deal with a weakly oscillating thin layer: the period of the oscillations is greater than the square root of the thinness. Our approach is essentially geometric and based on a suitable change of variable in the layer. This paper extends previous works of the former author, in which the layer had constant thickness.

[1]  Sergei Petrovich Novikov,et al.  The geometry of surfaces, transformation groups, and fields , 1984 .

[2]  Elena Beretta,et al.  Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis , 2003 .

[3]  C. Poignard.  Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer , 2008 .

[4]  Analysis of curvature influence on effective boundary conditions , 2002 .

[5]  G. Pucihar,et al.  Numerical Determination of Transmembrane Voltage Induced on Irregularly Shaped Cells , 2006, Annals of Biomedical Engineering.

[6]  Julián Fernández Bonder,et al.  The best Sobolev trace constant in a domain with oscillating boundary , 2007 .

[7]  Clair Poignard.  Méthodes asymptotiques pour le calcul des champs électromagnétiques dans des milieux à couches minces : application aux cellules biologiques , 2006 .

[8]  Hyeonbae Kang,et al.  Properties of the Generalized Polarization Tensors , 2003, Multiscale Model. Simul..

[9]  T. Tsong,et al.  Electroporation of cell membranes. , 1991, Biophysical journal.

[10]  J L Sebastián,et al.  Analysis of the influence of the cell geometry, orientation and cell proximity effects on the electric field distribution from direct RF exposure. , 2001, Physics in medicine and biology.

[11]  C. Poignard,et al.  Approximate transmission conditions through a rough thin layer: The case of periodic roughness , 2009, European Journal of Applied Mathematics.

[12]  O. Pironneau,et al.  Domain decomposition and wall laws , 1995 .

[13]  K. Foster,et al.  Dielectric properties of tissues and biological materials: a critical review. , 1989, Critical reviews in biomedical engineering.

[14]  M.A. Stuchly,et al.  Modeling assemblies of biological cells exposed to electric fields , 1997, IEEE Transactions on Biomedical Engineering.

[15]  Habib Ammari,et al.  Diffraction at a Curved Grating: TM and TE Cases, Homogenization , 1996 .

[16]  Tomaz Slivnik,et al.  Sequential finite element model of tissue electropermeabilization , 2005, IEEE Transactions on Biomedical Engineering.

[17]  Yves Capdeboscq,et al.  A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction , 2003 .