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Compact difference method for Euler-Bernoulli beams and plates with nonlinear nonlocal strong damping

Tao Guo, Yiqun Li, Wenlin Qiu

TL;DR

The paper tackles numerical approximation of Euler-Bernoulli beams and plates with nonlinear-nonlocal damping modeled by $\hat{q}(t)=P\big(\int_\Gamma |\Delta u|^2 dx\big)$. It introduces fully discrete compact finite difference schemes that discretize the damping via Simpson-based quadrature and employs novel discrete norms to manage the nonlocal term, establishing rigorous stability, convergence, and energy-dissipation results for both 1D and 2D problems. The analysis covers spatial semi-discretization, fully discrete schemes, and 2D plate extensions, with error bounds of $\mathcal{O}(\tau^2 + h^4)$ in 1D and $\mathcal{O}(\tau^2 + h_1^4 + h_2^4)$ in 2D, complemented by energy-dissipation guarantees. Numerical experiments validate the predicted rates and illustrate the schemes' capability to capture damping-induced energy decay in these nonlinear-nonlocal systems.

Abstract

We investigate the numerical approximation to the Euler-Bernoulli (E-B) beams and plates with nonlinear nonlocal strong damping, which describes the damped mechanical behavior of beams and plates in real applications. We discretize the damping term by the composite Simpson's rule and the six-point Simpson's formula in the beam and plate problems, respectively, and then construct the fully discrete compact difference scheme for these problems. To account for the nonlinear-nonlocal term, we design several novel discrete norms to facilitate the error estimates of the damping term and the numerical scheme. The stability, convergence, and energy dissipation properties of the proposed scheme are proved, and numerical experiments are carried out to substantiate the theoretical findings.

Compact difference method for Euler-Bernoulli beams and plates with nonlinear nonlocal strong damping

TL;DR

The paper tackles numerical approximation of Euler-Bernoulli beams and plates with nonlinear-nonlocal damping modeled by . It introduces fully discrete compact finite difference schemes that discretize the damping via Simpson-based quadrature and employs novel discrete norms to manage the nonlocal term, establishing rigorous stability, convergence, and energy-dissipation results for both 1D and 2D problems. The analysis covers spatial semi-discretization, fully discrete schemes, and 2D plate extensions, with error bounds of in 1D and in 2D, complemented by energy-dissipation guarantees. Numerical experiments validate the predicted rates and illustrate the schemes' capability to capture damping-induced energy decay in these nonlinear-nonlocal systems.

Abstract

We investigate the numerical approximation to the Euler-Bernoulli (E-B) beams and plates with nonlinear nonlocal strong damping, which describes the damped mechanical behavior of beams and plates in real applications. We discretize the damping term by the composite Simpson's rule and the six-point Simpson's formula in the beam and plate problems, respectively, and then construct the fully discrete compact difference scheme for these problems. To account for the nonlinear-nonlocal term, we design several novel discrete norms to facilitate the error estimates of the damping term and the numerical scheme. The stability, convergence, and energy dissipation properties of the proposed scheme are proved, and numerical experiments are carried out to substantiate the theoretical findings.
Paper Structure (14 sections, 15 theorems, 159 equations, 1 figure, 2 tables)

This paper contains 14 sections, 15 theorems, 159 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Beam problem: Spatial semi-discrete scheme
  3. Construction of spatial semi-discrete scheme
  4. Stability
  5. Convergence analysis
  6. Beam problem: Fully discrete scheme
  7. Construction of fully discrete scheme
  8. Stability and energy dissipation
  9. Convergence analysis
  10. Plate problem: Fully discrete scheme
  11. Construction of fully discrete scheme
  12. Stability and dissipative property
  13. Convergence analysis
  14. Numerical experiment

Key Result

Lemma 2.1

Hu For any $\omega, v\in \mathcal{V}^0$, we have $\langle \mathscr{A}\omega,\delta_{x}\delta_{\Bar{x}}v\rangle=\langle \delta_{x}\delta_{\Bar{x}}\omega,\mathscr{A}v\rangle$.

Figures (1)

  • Figure 1: The plot of the energy $\hat{\mathbf{E}}^n$ in Example 1 (left) and in Example 2 (right).

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 3.1
  • Corollary 3.1.1
  • Theorem 3.2
  • ...and 5 more