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On the homogenization of microstructured surfaces

Semin, Adrien; Schmidt, Kersten

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

The direct numerical simulation of microstructured interfaces like multiperforated absorber in acoustics with hundreds or thousands of tiny openings would result in a huge number of basis functions to resolve the microstructure. One is, however, primarily interested in the effective and so homogenized transmission and absorption properties. We introduce the surface homogenization that asymptotically decomposes the solution in a macroscopic part, a boundary layer corrector close to the interface and a near field part close to its ends. The introduction is for a general framework of models of elliptic partial differential equations incorporating the influence of end-points of the microstructured interfaces to the macroscopic part of the solution. The effective transmission and absorption properties are expressed by transmission conditions on an infinitely thin interface and corner conditions at its end-points to ensure the correct singular behaviour, intrinsic to the microstructure. We give details on the computation of the effective parameters and show their dependence on geometrical properties of the microstructure on the example of the wave propagation described by the Helmholtz equation. Numerical experiments indicate with the obtained macroscopic solution representation one can reach very high accuracies with a small number of basis functions.