Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach
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Numerische Mathematik
Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach Patrick Joly1 · Maryna Kachanovska1 Received: 10 August 2019 / Revised: 3 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work (Joly et al. in Netw Heterog Media 14(2):205–264, 2019), dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (transparent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments. Mathematics Subject Classification 65M12 · 65M60 · 65M06 · 35R02
1 Introduction Sound propagation in a human lung can be used for non-invasive diagnosis of the respiratory diseases, see e.g. [44] for some experimental studies, a PhD thesis [25], and, in particular, the Audible Human Project [47] and references therein. A human lung can be viewed as a network of small tubes (bronchioles), immersed into the lung tissue (parenchyma) and coupled with their ends to microscopic cavities in the parenchyma (alveoli). The physical phenomenon of sound propagation in a lung is highly complex, due to the fractal geometry of lung airways, heterogeneity of parenchyma, interactions/couplings between various types of tissues, and, eventually, multiscale nature of
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Maryna Kachanovska [email protected] Patrick Joly [email protected]
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POEMS (INRIA-CNRS-ENSTA), Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91120 Palaiseau, France
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P. Joly, M. Kachanovska
the problem. Thus, in practice, one uses simplified models. For instance, in the mathematical literature, in [12,13], sound propagation in a highly heterogeneous parenchyma is modelled using the homogenization techniques. In [40] Sobolev spaces associated to the Laplace equation on a fractal tree that models the network of bronchioli are studied, and in [21] the wave equation with a viscous non-local term on a dyadic infinite tree is analyzed, see as well the monograph [39]. This point of view at the bronchioli as a self-similar network (with possibly multiple levels of self-similar structure) seems to be rather classical (though indeed simplified) in the medical and medical engineering literature, see in particular [16,26,42,48] for the related discussion. In this article we adapt this, simplified, approach of studying wave propagation in lungs. In the limit when the thickness of the bronchiolar tubes tends to zero, the problem becomes essentially one-dimensional inside each
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