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Multiscale modelling of sound propagation through the lung parenchyma. (English) Zbl 1285.93014

Summary: In this paper we develop and study numerically a model to describe some aspects of sound propagation in the human lung, considered as a deformable and viscoelastic porous medium (the parenchyma) with millions of alveoli filled with air. Transmission of sound through the lung above 1 kHz is known to be highly frequency-dependent. We pursue the key idea that the viscoelastic parenchyma structure is highly heterogeneous on the small scale \(\epsilon\) and use two-scale homogenization techniques to derive effective acoustic equations for asymptotically small \(\epsilon\). This process turns out to introduce new memory effects. The effective material parameters are determined from the solution of frequency-dependent micro-structure cell problems. We propose a numerical approach to investigate the sound propagation in the homogenized parenchyma using a discontinuous Galerkin formulation. Numerical examples are presented.

MSC:

93A30 Mathematical modelling of systems (MSC2010)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B40 Asymptotic behavior of solutions to PDEs
74D05 Linear constitutive equations for materials with memory
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
92C50 Medical applications (general)
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