Approximate transmission conditions through a rough thin layer: The case of periodic roughness

Abstract

We study the behaviour of the steady-state voltage potentials in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be ϵ-periodic, ϵ being the magnitude of the mean thickness of the layer. For ϵ tending to zero, we determine approximate transmission conditions in order to replace the rough thin layer by these conditions on the boundary of the interior material. This paper extends the previous works (Poignard, 2009,Math. Meth. Appl. Sci., vol. 32, pp. 435–453; Poignardet al., 2008,IEEE Trans. Magnet., vol. 44, no. 6, pp. 1154–1157) of the third author, which deal with smooth thin layers.