TY - JOUR

T1 - Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs

AU - Cazeaux, Paul

AU - Grandmont, Céline

N1 - Publisher Copyright:
© 2015 World Scientific Publishing Company.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/6/15

Y1 - 2015/6/15

N2 - We are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree-shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model.

AB - We are interested in the mathematical modeling of the deformation of the human lung tissue, called the lung parenchyma, during the respiration process. The parenchyma is a foam-like elastic material containing millions of air-filled alveoli connected by a tree-shaped network of airways. In this study, the parenchyma is governed by the linearized elasticity equations and the air movement in the tree by the Poiseuille law in each airway. The geometric arrangement of the alveoli is assumed to be periodic with a small period ε > 0. We use the two-scale convergence theory to study the asymptotic behavior as ε goes to zero. The effect of the network of airways is described by a nonlocal operator and we propose a simple geometrical setting for which we show that this operator converges as ε goes to zero. We identify in the limit the equations modeling the homogenized behavior under an abstract convergence condition on this nonlocal operator. We derive some mechanical properties of the limit material by studying the homogenized equations: the limit model is nonlocal both in space and time if the parenchyma material is considered compressible, but only in space if it is incompressible. Finally, we propose a numerical method to solve the homogenized equations and we study numerically a few properties of the homogenized parenchyma model.

KW - Mathematical modeling

KW - fluid-structure interaction

KW - periodic homogenization

KW - two-scale convergence method

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U2 - 10.1142/S0218202515500293

DO - 10.1142/S0218202515500293

M3 - Article

AN - SCOPUS:84928558567

VL - 25

SP - 1125

EP - 1177

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 6

ER -