Author(s) :
Adrien Semin
,
Kersten Schmidt
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 13-2016
MSC 2000
- 35C20 Asymptotic expansions
-
35S05 General theory of PsDO
Abstract :
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.
Keywords :
Asymptotic analysis, periodic surface homogenization, singular asymptotic expansions, stress intensity factor, numerical methods