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Numerical stabilization for a mixture system with kind damping

Kais Ammari, Vilmos Komornik, Mauricio Sepúlveda, Octavio Vera

TL;DR

This work analyzes a poroelastic mixture model of two rigid solids with porosity, incorporating a fractional damping mechanism $\partial_t^{\alpha,\eta}$. By introducing an augmented diffusion–based model with auxiliary fields, the authors establish well-posedness via a contraction semigroup and prove strong stability through the Arendt–Batty framework, with polynomial decay rates that depend on whether the diffusion parameter $\eta$ vanishes. They then develop a 2D finite-volume numerical scheme coupled with a $\beta$-Newmark time integrator and Crank–Nicolson treatment of the fractional term to verify the decay rates and examine the spectrum of the discretized operator, revealing stability and non-exponential decay behavior. Numerical experiments show energy dissipation only when fractional damping is present, demonstrate spectrum behavior near the imaginary axis, and illustrate polynomial decay of the energy for long-time simulations, thereby connecting rigorous stability results with computational evidence. The methodology provides a rigorous and practical framework for analyzing stabilization and decay in coupled solid–porosity systems with fractional damping, with potential implications for material design and pavement modeling.

Abstract

In this paper, we conduct a numerical analysis of the strong stabilization and polynomial decay of solutions for the initial boundary value problem associated with a system that models the dynamics of a mixture of two rigid solids with porosity. This mathematical model accounts for the complex interactions between the rigid components and their porous structure, providing valuable information on the mechanical behavior of such systems. Our primary objective is to establish conditions under which stabilization is ensured and to rigorously quantify the rate of decay of the solutions. Using numerical simulations, we assess the effectiveness of different stabilization mechanisms and analyze the influence of key system parameters on the overall dynamics.

Numerical stabilization for a mixture system with kind damping

TL;DR

This work analyzes a poroelastic mixture model of two rigid solids with porosity, incorporating a fractional damping mechanism . By introducing an augmented diffusion–based model with auxiliary fields, the authors establish well-posedness via a contraction semigroup and prove strong stability through the Arendt–Batty framework, with polynomial decay rates that depend on whether the diffusion parameter vanishes. They then develop a 2D finite-volume numerical scheme coupled with a -Newmark time integrator and Crank–Nicolson treatment of the fractional term to verify the decay rates and examine the spectrum of the discretized operator, revealing stability and non-exponential decay behavior. Numerical experiments show energy dissipation only when fractional damping is present, demonstrate spectrum behavior near the imaginary axis, and illustrate polynomial decay of the energy for long-time simulations, thereby connecting rigorous stability results with computational evidence. The methodology provides a rigorous and practical framework for analyzing stabilization and decay in coupled solid–porosity systems with fractional damping, with potential implications for material design and pavement modeling.

Abstract

In this paper, we conduct a numerical analysis of the strong stabilization and polynomial decay of solutions for the initial boundary value problem associated with a system that models the dynamics of a mixture of two rigid solids with porosity. This mathematical model accounts for the complex interactions between the rigid components and their porous structure, providing valuable information on the mechanical behavior of such systems. Our primary objective is to establish conditions under which stabilization is ensured and to rigorously quantify the rate of decay of the solutions. Using numerical simulations, we assess the effectiveness of different stabilization mechanisms and analyze the influence of key system parameters on the overall dynamics.
Paper Structure (14 sections, 17 theorems, 102 equations, 5 figures, 1 table)

This paper contains 14 sections, 17 theorems, 102 equations, 5 figures, 1 table.

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. Example 1: Interaction of two waves with and without the dispative fractional derivative term.
  13. Example 2: Numerical spectrum of operator $A$.
  14. Example 3: Numerical Polynomial decay of the 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 (5)

  • Figure 1: Numerical comparison of energy with and without dissipative term of fractional derivative.
  • Figure 2: Behavior of the sum of both compounds u+v (mixture), for different times without dissipation (graphs on the left), and with dissipation (graphs on the right).
  • Figure 3: Numerical spectrum of the operator $A$, using FVM approximation.
  • Figure 4: Evolution of the solution for long times with not very smooth initial conditions in $\mathcal{D}(A)$.
  • Figure 5: Energy polynomial decay in scale log-log and comparison with curves $\frac{C_1}{t}$ and $\frac{C_2}{t^2}$

Theorems & Definitions (24)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Proposition 3.1
  • Theorem 3.2
  • ...and 14 more