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IGA Laplace Eigenfrequencies Distributions and Estimations: Impact of Reparametrization on Eigenfrequency Behavior

Lamsahel Noureddine, Abdeladim El Akri, Ahmed Ratnani

Abstract

This work addresses the Galerkin isogeometric discretization of the one-dimensional Laplace eigenvalue problem subject to homogeneous Dirichlet boundary conditions on a bounded interval. We employ GLT theory to analyze the behavior of the eigenfrequencies when a reparametrization is applied to the computational domain. Under suitable assumptions on the reparametrization transformation, we prove that a structured pattern emerges in the distribution of eigenfrequencies when the problem is reframed through GLT-symbol analysis. Additionally, we establish results that refine and extend those of [3], including a uniform discrete Weyl's law. Furthermore, we derive several eigenfrequency estimates by establishing that the symbol exhibits asymptotically linear behavior near zero.

IGA Laplace Eigenfrequencies Distributions and Estimations: Impact of Reparametrization on Eigenfrequency Behavior

Abstract

This work addresses the Galerkin isogeometric discretization of the one-dimensional Laplace eigenvalue problem subject to homogeneous Dirichlet boundary conditions on a bounded interval. We employ GLT theory to analyze the behavior of the eigenfrequencies when a reparametrization is applied to the computational domain. Under suitable assumptions on the reparametrization transformation, we prove that a structured pattern emerges in the distribution of eigenfrequencies when the problem is reframed through GLT-symbol analysis. Additionally, we establish results that refine and extend those of [3], including a uniform discrete Weyl's law. Furthermore, we derive several eigenfrequency estimates by establishing that the symbol exhibits asymptotically linear behavior near zero.
Paper Structure (11 sections, 20 theorems, 157 equations)

This paper contains 11 sections, 20 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the Paper
  4. Preliminaries
  5. Galerkin B-spline IGA Discretization
  6. Preliminaries on GLT Sequences
  7. The IGA GLT-Symbol
  8. Symbol analysis
  9. Impact of reparametrization on the behavior of eigenfrequencies
  10. Improved eigenfrequency estimations
  11. Conclusions, further comments, and future work

Key Result

Proposition 2.1

Let $(L_n)_{n\in\mathbb{N}^*} \sim_{\lambda} \omega$ with $\omega: [0,1] \times [0,\pi] \longrightarrow \mathbb{R}$ having a bounded essential range. Let $\xi$ be the monotone rearrangement of $\omega$. Then, we have

Theorems & Definitions (37)

  • Definition 2.1: Spectral symbol
  • Definition 2.2: Outliers
  • Definition 2.3: Monotone rearrangement of the symbol
  • Proposition 2.1
  • Theorem 2.1: Discrete Weyl’s law, bianchi2021analysis
  • Corollary 2.1
  • Corollary 2.2
  • Remark 2.1
  • Theorem 2.2: garoni2017generalized, IGA GLT symbol
  • Corollary 2.3
  • ...and 27 more