@misc{DelourmeSchmidtSemin, author = {Delourme, B{\´e}rang{\`e}re and Schmidt, Kersten and Semin, Adrien}, title = {On the homogenization of thin perforated walls of finite length}, series = {Asymptotic Analysis}, volume = {97}, journal = {Asymptotic Analysis}, number = {3-4}, issn = {0921-7134}, doi = {10.3233/ASY-151350}, pages = {211 -- 264}, abstract = {The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization.}, language = {en} } @misc{SchmidtSemin, author = {Schmidt, Kersten and Semin, Adrien}, title = {Absorbing boundary conditions for the viscous acoustic wave equation}, series = {Mathemtatical Methods in the Applied Sciences}, volume = {39}, journal = {Mathemtatical Methods in the Applied Sciences}, number = {17}, issn = {1099-1476}, doi = {10.1002/mma.3755}, pages = {5043 -- 5065}, abstract = {We consider different acoustic models with viscosity in a semi-infinite waveguide with rigid walls, for which we propose and analyse absorbing boundary conditions on a truncated subdomain. The considered models are (i) the viscous acoustic equations in a stagnant mean flow, which exhibit for small viscosities boundary layers on the infinite walls, (ii) the limit equations for vanishing viscosity and (iii) a first-order approximation for low viscosity. The limit model (i) is well known as the Helmholtz equation for the pressure with homogeneous Neumann boundary conditions. For each of these models, the absorbing conditions appear as Dirichlet-to-Neumann (DtN) maps. The DtN boundary conditions for the singularly perturbed model (i) and the approximative model (iii) tend to the DtN boundary conditions of the limit problem (ii) if the viscosity approaches zero, and, hence, provide a uniform accuracy in the viscosity. The convergence of truncated DtN boundary conditions and the behaviour for viscosities tending to zero are shown in numerical experiments.}, language = {en} } @techreport{SchmidtSeminDelourme, author = {Schmidt, Kersten and Semin, Adrien and Delourme, B{\´e}rang{\`e}re}, title = {On the homogenization of the Helmholtz problem with thin perforated walls of finite length}, pages = {12}, abstract = {On the homogenization of the Helmholtz problem with thin perforated walls of finite length}, language = {en} }