4,066 research outputs found
Asymptotics of Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with Oscillating Boundary. Multiple Eigenvalue Case
We study the asymptotic behavior of the solutions of a spectral problem for
the Laplacian in a domain with rapidly oscillating boundary. We consider the
case where the eigenvalue of the limit problem is multiple. We construct the
leading terms of the asymptotic expansions for the eigenelements and verify the
asymptotics
Boundary homogenization for target search problems
In this review, we describe several approximations in the theory of Laplacian
transport near complex or heterogeneously reactive boundaries. This phenomenon,
governed by the Laplace operator, is ubiquitous in fields as diverse as
chemical physics, hydrodynamics, electrochemistry, heat transfer, wave
propagation, self-organization, biophysics, and target search. We overview the
mathematical basis and various applications of the effective medium
approximation and the related boundary homogenization when a complex
heterogeneous boundary is replaced by an effective much simpler boundary. We
also discuss the constant-flux approximation, the Fick-Jacobs equation, and
other mathematical tools for studying the statistics of first-passage times to
a target. Numerous examples and illustrations are provided to highlight the
advantages and limitations of these approaches
A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions
Stents are medical devices designed to modify blood flow in aneurysm sacs, in
order to prevent their rupture. Some of them can be considered as a locally
periodic rough boundary. In order to approximate blood flow in arteries and
vessels of the cardio-vascular system containing stents, we use multi-scale
techniques to construct boundary layers and wall laws. Simplifying the flow we
turn to consider a 2-dimensional Poisson problem that conserves essential
features related to the rough boundary. Then, we investigate convergence of
boundary layer approximations and the corresponding wall laws in the case of
Neumann type boundary conditions at the inlet and outlet parts of the domain.
The difficulty comes from the fact that correctors, for the boundary layers
near the rough surface, may introduce error terms on the other portions of the
boundary. In order to correct these spurious oscillations, we introduce a
vertical boundary layer. Trough a careful study of its behavior, we prove
rigorously decay estimates. We then construct complete boundary layers that
respect the macroscopic boundary conditions. We also derive error estimates in
terms of the roughness size epsilon either for the full boundary layer
approximation and for the corresponding averaged wall law.Comment: Dedicated to Professor Giovanni Paolo Galdi 60' Birthda
Effective toughness of heterogeneous brittle materials
A heterogeneous brittle material characterized by a random field of local
toughness Kc(x) can be represented by an equivalent homogeneous medium of
toughness, Keff. Homogenization refers to a process of estimating Keff from the
local field Kc(x). An approach based on a perturbative expansion of the stress
intensity factor along a rough crack front shows the occurrence of different
regimes depending on the correlation length of the local toughness field in the
direction of crack propagation. A `"weak pinning" regime takes place for long
correlation lengths, where the effective toughness is the average of the local
toughness. For shorter correlation lengths, a transition to "strong pinning"
occurs leading to a much higher effective toughness, and characterized by a
propagation regime consisting in jumps between pinning configurations
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