Numerical study of a Transmission Problem in Elasticity with kind damping

1. Problem Statement

The paper investigates a transmission problem involving two distinct elastic materials joined at an interface ($x=0$) within a bounded domain $(-L, L)$. The system models the displacement $u(x,t)$ in $(-L, 0)$ and $v(x,t)$ in $(0, L)$.

Key Features of the Model:

• Governing Equations: The system consists of wave equations for both materials, coupled with a fractional damping term in the Caputo sense ($\partial^{\alpha, \eta}_t$).
  $$\rho_i w_{tt} - k_i w_{xx} + \partial^{\alpha, \eta}_t w = 0$$
  where $w \in \{u, v\}$, $\rho$ is mass density, $k$ is the elastic constant, and $0 < \alpha < 1$.
• Damping Mechanism: Unlike standard frictional damping (which yields exponential decay if localized) or Kelvin-Voigt damping (which yields analytic semigroups), this study focuses on fractional derivative damping. The term $\partial^{\alpha, \eta}_t$ includes a weight parameter $\eta \geq 0$.
• Boundary and Transmission Conditions:
  • Dirichlet conditions at the outer boundaries: $u(-L, t) = 0, v(L, t) = 0$.
  • Continuity of displacement and stress at the interface: $u(0, t) = v(0, t)$ and $k_1 u_x(0, t) = k_2 v_x(0, t)$.
• Objective: To analyze the asymptotic behavior (stability) of the associated semigroup $(S_A(t))_{t \geq 0}$ and validate the theoretical decay rates numerically.

2. Methodology

The authors employ a dual approach combining rigorous functional analysis with advanced numerical simulation.

A. Theoretical Framework (Augmented Model)

To handle the non-local nature of the fractional derivative, the authors utilize an augmented model strategy:

1. Diffusion Representation: They replace the fractional derivative term $\partial^{\alpha, \eta}_t$ with an auxiliary diffusion equation in an extended phase space (involving a frequency variable $\xi \in \mathbb{R}$). This transforms the integro-differential system into a coupled system of PDEs and ODEs.
2. Semigroup Theory: The system is formulated as an abstract evolution equation $\frac{d}{dt}U(t) = AU(t)$ in a Hilbert space $\mathcal{H}$.
   • Well-posedness: Using the Lax-Milgram theorem and properties of the operator $A$, they prove the existence and uniqueness of solutions (Corollary 3.2).
   • Strong Stability: They apply the Arendt-Batty Theorem to prove strong stability. This involves showing that the operator $A$ has no purely imaginary eigenvalues and that the intersection of its spectrum with the imaginary axis is countable (specifically $\{0\}$ or empty).
   • Polynomial Stability: They derive decay rates based on the behavior of the resolvent operator $(i\lambda I - A)^{-1}$.

B. Numerical Approximation

To validate the theoretical findings, the authors develop a robust numerical scheme:

1. Spatial Discretization: A Finite Volume Method (FVM) is used for the spatial domain, ensuring conservation properties. The transmission interface is handled by integrating the equation over the control volume containing the interface.
2. Fractional Derivative Handling: Instead of direct discretization of the fractional derivative (which can be unstable), they discretize the augmented diffusion system (the auxiliary variable $\phi$).
3. Time Discretization:
   • A $\beta$-Newmark scheme is used for the mechanical wave equations to ensure second-order accuracy.
   • A Crank-Nicolson scheme is applied to the auxiliary diffusion equations.
   • This combination ensures the discrete energy is non-increasing, mimicking the physical dissipation.

3. Key Contributions and Results

Theoretical Results

1. Strong Stability: The system is proven to be strongly stable for all $\eta \geq 0$. The energy decays to zero as $t \to \infty$.
2. Optimal Polynomial Decay Rates:
   • Case $\eta = 0$: The semigroup decays polynomially with a rate of $O(t^{-1/2})$ for the state vector, implying the energy decays as $O(t^{-1})$.
   • Case $\eta > 0$: The presence of the exponential weight $\eta$ alters the decay rate. The energy decays as $O(t^{-\frac{2}{1-\alpha}})$.
   • Optimality: The authors demonstrate that these rates are optimal, meaning the system cannot decay faster than these polynomial rates.
3. Contrast with Other Dampings: The study highlights that localized fractional damping is weaker than localized frictional damping (which is exponential) but stronger than localized Kelvin-Voigt damping (which is not exponentially stable).

Numerical Results

1. Validation of Wave Phenomena: Simulations successfully capture wave reflection at boundaries and reflection/refraction at the transmission interface, validating the FVM spatial discretization.
2. Verification of Decay Rates:
   • For $\eta = 0$, the numerical energy decay follows the curve $y = C_0 t^{-1}$ on a log-log scale.
   • For $\eta > 0$ (with $\alpha=0.5$), the decay follows $y = C_4 t^{-4}$, matching the theoretical prediction of $-\frac{2}{1-\alpha} = -4$.
3. Sensitivity Analysis: The study notes that while $\eta$ regularizes the solution, excessively high values can destabilize the numerical scheme, requiring careful parameter selection.

4. Significance

• Bridging Theory and Computation: The paper provides a rare and rigorous link between the abstract theory of fractional damping in transmission problems and concrete numerical evidence. Many papers prove stability but fail to demonstrate the specific decay rates numerically due to the difficulty of simulating long-time asymptotic behavior.
• Optimality Proof: By establishing that the decay rates are optimal, the paper clarifies the limitations of using fractional damping for control in transmission systems. It shows that while effective, it does not yield the rapid exponential stabilization seen in frictional damping.
• Methodological Innovation: The use of the augmented model (coupling the wave equation with a diffusion equation in frequency space) combined with a conservative Newmark-Crank-Nicolson scheme offers a stable and accurate framework for simulating fractional viscoelastic systems.
• Practical Implications: The results are relevant for engineering applications involving composite materials with viscoelastic interfaces, where understanding the long-term energy dissipation (polynomial vs. exponential) is crucial for structural integrity and vibration control.

In conclusion, this work establishes that a transmission problem with Caputo fractional damping exhibits polynomial stability with rates dependent on the fractional order $\alpha$ and the weight parameter $\eta$, and successfully validates these theoretical predictions through a conservative numerical scheme.