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Numerical study of a Transmission Problem in Elasticity with kind damping

Kais Ammari, Vilmos Komornik, Mauricio Sepúlveda, Octavio Vera

TL;DR

This work analyzes a one-dimensional elasticity transmission problem with fractional (Caputo-like) damping, identifying the asymptotic behavior of the associated semigroup. By introducing an augmented diffusion-based model and employing semigroup theory, it proves strong stability and, depending on the damping parameter $\eta$, either polynomial decay or, when applicable, exponential-type behavior for the energy; the decay rates are shown to be optimal. A detailed numerical study using a finite-volume spatial discretization and a $\beta$-Newmark time-stepping scheme with an Crank–Nicolson treatment of the fractional term corroborates the theoretical rates, including a demonstrative wavefront reflection/refraction example and energy decay curves that align with the predicted $t^{-1}$ (for $\eta=0$) and $t^{-4}$ (for $\eta>0$, with $\alpha=0.5$) behaviors. The combination of analytical semigroup results and robust numerical validation provides a rigorous understanding of how fractional damping affects energy dissipation in transmission problems and offers a practical framework for simulating such systems. The findings have potential implications for designing damping mechanisms in composite elastic media where localized and fractional damping effects are relevant.

Abstract

We investigate a transmission problem featuring a specific type of damping. Our primary focus is on analyzing the asymptotic behavior of the associated semigroup, $({\mathcal S}_{\mathcal A}(t))_{t\geq 0}$. We demonstrate that this semigroup exhibits a polynomial rate of decay towards zero when the initial data is taken over the domain ${\mathcal D}({\mathcal A})$. Furthermore, we establish that this decay rate is optimal. To support our theoretical findings, we present a comprehensive numerical study that validates and illustrates the sharpness of the obtained decay rates.

Numerical study of a Transmission Problem in Elasticity with kind damping

TL;DR

This work analyzes a one-dimensional elasticity transmission problem with fractional (Caputo-like) damping, identifying the asymptotic behavior of the associated semigroup. By introducing an augmented diffusion-based model and employing semigroup theory, it proves strong stability and, depending on the damping parameter , either polynomial decay or, when applicable, exponential-type behavior for the energy; the decay rates are shown to be optimal. A detailed numerical study using a finite-volume spatial discretization and a -Newmark time-stepping scheme with an Crank–Nicolson treatment of the fractional term corroborates the theoretical rates, including a demonstrative wavefront reflection/refraction example and energy decay curves that align with the predicted (for ) and (for , with ) behaviors. The combination of analytical semigroup results and robust numerical validation provides a rigorous understanding of how fractional damping affects energy dissipation in transmission problems and offers a practical framework for simulating such systems. The findings have potential implications for designing damping mechanisms in composite elastic media where localized and fractional damping effects are relevant.

Abstract

We investigate a transmission problem featuring a specific type of damping. Our primary focus is on analyzing the asymptotic behavior of the associated semigroup, . We demonstrate that this semigroup exhibits a polynomial rate of decay towards zero when the initial data is taken over the domain . Furthermore, we establish that this decay rate is optimal. To support our theoretical findings, we present a comprehensive numerical study that validates and illustrates the sharpness of the obtained decay rates.
Paper Structure (13 sections, 14 theorems, 87 equations, 2 figures)

This paper contains 13 sections, 14 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Augmented Model
  3. Setting of the Semigroup
  4. Strong stability
  5. Polynomial stability for ${\pmb{\color{black}{\eta\geq 0}}}$
  6. Numerical Approximation
  7. Finite Volume Approximation
  8. Linear equations of Motion
  9. Time discretization
  10. Approximation of fractional derivatives
  11. Decay of the discrete energy
  12. A first example: Reflection and refraction of a wavefront
  13. Polynomial decay of energy

Key Result

Theorem 2.1

15 Let $\mu$ be the function Then the relation between the Input$U$ and the Output${\it O}$ is given by the following system which implies that

Figures (2)

  • Figure 1: Simulation of the wavefront for $0<t<1$, with and without dissipation terms.
  • Figure 2: Polynomial decay of the energy for different values of $\eta$ and comparison with the polynomial bounds $y=C_0t^{-1}$ and $y=C_4t^{-4}$; $C_0=2.62$, $C_4=1.75\times 10^{10}$.

Theorems & Definitions (18)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • Proposition 4.4
  • ...and 8 more