Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized impedance boundary conditions with vanishing or sign-changing impedance

Lucas Chesnel, Laurent Bourgeois

Abstract

We consider a Laplace type problem with a generalized impedance boundary condition of the form $\partial_νu=-\partial_x(g\partial_xu)$ on a flat part $Γ$ of the boundary. Here $ν$ is the outward unit normal vector to $\partialΩ$, $g$ is the impedance function and $x$ is the coordinate along $Γ$. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases $g=1$ or $g=-1$ have been investigated. In this work, we address situations where $Γ$ contains the origin and $g(x)=\mathbb{1}_{x>0}(x)x^α$ or $g(x)=-\mbox{sign}(x)|x|^α$ with $α\ge0$. In other words, we study cases where $g$ vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of $α$. For $α\in[0,1)$, we show that the associated operators are Fredholm of index zero while it is not the case when $α=1$. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems.

Generalized impedance boundary conditions with vanishing or sign-changing impedance

Abstract

We consider a Laplace type problem with a generalized impedance boundary condition of the form on a flat part of the boundary. Here is the outward unit normal vector to , is the impedance function and is the coordinate along . Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases or have been investigated. In this work, we address situations where contains the origin and or with . In other words, we study cases where vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of . For , we show that the associated operators are Fredholm of index zero while it is not the case when . The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems.
Paper Structure (13 sections, 22 theorems, 139 equations, 11 figures)

This paper contains 13 sections, 22 theorems, 139 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Definition of the problems
  4. Statement of the results
  5. Weighted Sobolev spaces in dimension 1
  6. The vanishing case: a positive result for $s=-1$ and $\alpha \in [0,1)$
  7. The vanishing case: a negative result for $s=-1$ and $\alpha=1$
  8. Problem with a vanishing and sign-changing impedance
  9. Proof of Theorem \ref{['casgeneral_bis']} -- case $\alpha\in[0,1)$
  10. Proof of Theorem \ref{['cas1_bis']} -- case $\alpha=1$
  11. Relationship between the strong and weak formulations
  12. Numerical illustrations
  13. Concluding remarks

Key Result

Proposition 2.1

For all $\alpha \geq 0$, $\mathrm{V}_\alpha(\Omega)$ is a Hilbert space when equipped with the inner product

Figures (11)

  • Figure 1: Left: domain $\Omega$. Center and right: examples of perturbed domains $\Omega^\varepsilon.$
  • Figure 2: Numerical resolution of (\ref{['faible']}) for ten different meshes with $s=-1$ and $\alpha=0.5$.
  • Figure 3: Meshes used in the experiments.
  • Figure 4: Numerical resolution of (\ref{['faible']}) for ten different meshes with $s=-1$ and $\alpha=0.95$.
  • Figure 5: Numerical resolution of (\ref{['faible']}) for ten different meshes with $s=-1$ and $\alpha=1$.
  • ...and 6 more figures

Theorems & Definitions (46)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 36 more