743 research outputs found
A 2-adic approach of the human respiratory tree
We propose here a general framework to address the question of trace
operators on a dyadic tree. This work is motivated by the modeling of the human
bronchial tree which, thanks to its regularity, can be extrapolated in a
natural way to an infinite resistive tree. The space of pressure fields at
bifurcation nodes of this infinite tree can be endowed with a Sobolev space
structure, with a semi-norm which measures the instantaneous rate of dissipated
energy. We aim at describing the behaviour of finite energy pressure fields
near the end. The core of the present approach is an identification of the set
of ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can be
defined in a very natural way by means of Fourier transform, which allows us to
establish precised trace theorems which are formally quite similar to those in
standard Sobolev spaces, with a Sobolev regularity which depends on the growth
rate of resistances, i.e. on geometrical properties of the tree. Furthermore,
we exhibit an explicit expression of the "ventilation operator", which maps
pressure fields at the end of the tree onto fluxes, in the form of a
convolution by a Riesz kernel based on the 2-adic distance.Comment: 22 page
Transparent Boundary Conditions for Wave Propagation in Fractal Trees: Approximation by Local Operators
This work is dedicated to the construction and analysis of high-order transparent boundary conditions for the weighted wave equation on a fractal tree, which models sound propagation inside human lungs. This article follows the works [10, 9], aimed at the analysis and numerical treatment of the model, as well as the construction of low-order and exact discrete boundary conditions. The method suggested in this article is based on the truncation of the meromorphic series that approximate the symbol of the Dirichlet-to-Neumann operator, similarly to the absorbing boundary conditions of B. En-gquist and A. Majda. We analyze its stability, convergence and complexity. The error analysis is largely based on spectral estimates of the underlying weighted Laplacian. Numerical results confirm the efficiency of the method
Homogenization of a Multiscale Viscoelastic Model with Nonlocal Damping, Application to the Human Lungs
International audienceWe 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 foamâlike elastic material containing millions of airâfilled alveoli connected by a treeâ 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 Δ > 0. We use the twoâscale convergence theory to study the asymptotic behavior as Δ 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 Δ 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
Local transparent boundary conditions for wave propagation in fractal trees (I). Method and numerical implementation
This work is dedicated to the construction and analysis of high-order transparentboundary conditions for the weighted wave equation on a fractal tree, which models sound propaga-tion inside human lungs. This article follows the works [9, 6], aimed at the analysis and numerical treatment of the model, as well as the construction of low-order and exact discrete boundary conditions. The method suggested in the present work is based on the truncation of the meromorphicseries that represents the symbol of the Dirichlet-to-Neumann operator, in the spirit of the absorbingboundary conditions of B. Engquist and A. Majda. We analyze its stability and convergence, as wellas present computational aspects of the method. Numerical results confirm theoretical finding
Embedded trace operator for infinite metric trees
We consider a class of infinite weighted metric trees obtained as
perturbations of self-similar regular trees. Possible definitions of the
boundary traces of functions in the Sobolev space on such a structure are
discussed by using identifications of the tree boundary with a surface. Our
approach unifies some constructions proposed by Maury, Salort, Vannier (2009)
for dyadic discrete weighted trees (expansion in orthogonal bases of harmonic
functions on the graph and using Haar-type bases on the domain representing the
boundary), and by Nicaise, Semin (2018) and Joly, Kachanovska, Semin (2019) for
fractal metric trees (approximation by finite sections and identification of
the boundary with a interval): we show that both machineries give the same
trace map, and for a range of parameters we establish the precise Sobolev
regularity of the traces. In addition, we introduce new geometric ingredients
by proposing an identification with arbitrary Riemannian manifolds. It is shown
that any compact manifold admits a suitable multiscale decomposition and,
therefore, can be identified with a metric tree boundary in the context of
trace theorems.Comment: 67 pages. Several wrong citations were correcte
Marriages of Mathematics and Physics: A Challenge for Biology
The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of âgeometric judgmentsâ from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) âspaceâ should be revisited for the purposes of life sciences
Proceedings of the 96th Annual Virginia Academy of Science Meeting, 2018
Proceedings of the 96th Annual Virginia Academy of Science Meeting, May 23-25, 2018, at Longwood University, Farmville, Virginia
Local transparent boundary conditions for wave propagation in fractal trees (ii): error and complexity analysis
International audienceThis work is dedicated to a refined error analysis of the high-order transparent boundary conditions introduced in the companion work [8] for the weighted wave equation on a fractal tree. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the Dirichlet-to-Neumann operator. The error induced by the truncation depends on the behaviour of the eigenvalues and the eigenfunctions of the weighted Laplacian on a self-similar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of self-similar trees
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