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On eigenvibrations of branched structures with heterogeneous mass density

Yuriy Golovaty, Delfina Gómez, Maria-Eugenia Pérez-Martínez

TL;DR

This work analyzes the spectra of the Laplace-Beltrami operator on a stratified surface $\Omega$ formed by multiple smooth sheets joined along a junction $\gamma$, with a highly heterogeneous mass density $\rho^\varepsilon$ concentrated in a thin band $\omega^\varepsilon$ of width $O(\varepsilon)$ around $\gamma$ and scaling like $O(\varepsilon^{-m})$, for $m\ge1$. Using matched asymptotics and Dirichlet-to-Neumann maps, the authors derive a limit problem for $m=1$ that couples the bulk and the junction via a transmission condition along $\gamma$, and they prove spectral convergence of $\lambda^\varepsilon$ to a limit spectrum consisting of bulk- and junction-induced modes. For $m>1$, they characterize low-frequency vibrations on the $\varepsilon^{m-1}$ scale, leading to a mass-concentrated limit problem on $\gamma$ with a transmission-Kirchhoff condition; the upper part of the spectrum is shown to converge to a graph-like operator spectrum associated with the interface graph $\mathcal{D}$, with precise quasimode constructions. Overall, the paper provides a rigorous framework for understanding how concentrated masses along junctions in branched stratified structures shape both local and global vibrational modes and establishes robust spectral convergence and asymptotics as the perturbation parameter vanishes.

Abstract

We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set $Ω$ which is composed of smooth surfaces joined along a line $γ$, the junction. Through this junction we impose the Kirchhoff-type vertex conditions, which imply the continuity of the solutions and some balance for normal derivatives, and Neumann conditions on the rest of the boundary of the surfaces. Assuming that the density is $O(\varepsilon^{-m})$ along small bands of width $O(\varepsilon)$, which collapse into the line $γ$ as $\varepsilon$ tends to zero, and it is $O(1)$ outside these bands, we address the asymptotic behavior, as $\varepsilon\to 0$, of the eigenvalues and of the corresponding eigenfunctions for a parameter $m\geq 1$. We also study the asymptotics for high frequencies when $m\in(1,2)$.

On eigenvibrations of branched structures with heterogeneous mass density

TL;DR

This work analyzes the spectra of the Laplace-Beltrami operator on a stratified surface formed by multiple smooth sheets joined along a junction , with a highly heterogeneous mass density concentrated in a thin band of width around and scaling like , for . Using matched asymptotics and Dirichlet-to-Neumann maps, the authors derive a limit problem for that couples the bulk and the junction via a transmission condition along , and they prove spectral convergence of to a limit spectrum consisting of bulk- and junction-induced modes. For , they characterize low-frequency vibrations on the scale, leading to a mass-concentrated limit problem on with a transmission-Kirchhoff condition; the upper part of the spectrum is shown to converge to a graph-like operator spectrum associated with the interface graph , with precise quasimode constructions. Overall, the paper provides a rigorous framework for understanding how concentrated masses along junctions in branched stratified structures shape both local and global vibrational modes and establishes robust spectral convergence and asymptotics as the perturbation parameter vanishes.

Abstract

We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set which is composed of smooth surfaces joined along a line , the junction. Through this junction we impose the Kirchhoff-type vertex conditions, which imply the continuity of the solutions and some balance for normal derivatives, and Neumann conditions on the rest of the boundary of the surfaces. Assuming that the density is along small bands of width , which collapse into the line as tends to zero, and it is outside these bands, we address the asymptotic behavior, as , of the eigenvalues and of the corresponding eigenfunctions for a parameter . We also study the asymptotics for high frequencies when .

Paper Structure

This paper contains 12 sections, 13 theorems, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Vibrating systems with concentrating masses. A historical review
  3. Stratified sets as a generalization of metric graphs. Singularly perturbed problem on graphs
  4. Main results and the structure of paper
  5. Statement of problem
  6. Spectral properties of the perturbed problem
  7. Formal asymptotics and the limit operator. The case $m=1$
  8. Asymptotics of eigenvalues and eigenfunctions
  9. Properties of the limit operator
  10. Convergence of spectra in the case $m=1$
  11. Low frequency eigenvibrations in the case $m>1$
  12. Asymptotics of upper part of $\sigma(\mathcal{A}_\varepsilon)$ in the case $m\in(1,2)$

Key Result

Lemma 1

The operator $\mathcal{B}$ is closed, self-adjoint, bounded from below, and has a compact resolvent.

Figures (4)

  • Figure 1: Stratified set $\Omega^*$.
  • Figure 2: The graph $G$ and set $\omega=G\times \gamma$.
  • Figure 3: Turbines and propellers
  • Figure 4: (a) The set $\Sigma$: domain $A$ is where asymptotics \ref{['convvp']} holds, while domain $B$ is where the asymptotics is not valid, but other approaches of eigenvalues could exist. (b) An illustration of possible values $\lambda_{j}^{\varepsilon}$ for $m=3/2.$

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 13 more