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

TL;DR

This paper develops a compact difference numerical scheme for Euler-Bernoulli beams and plates with nonlinear nonlocal strong damping, ensuring stability, convergence, and energy dissipation, validated through theoretical analysis and numerical experiments.

Contribution

It introduces a novel fully discrete scheme using Simpson's rule for nonlinear nonlocal damping in beams and plates, with rigorous error estimates and stability analysis.

Findings

Scheme is stable and convergent.

Energy dissipation is preserved in the numerical method.

Numerical experiments confirm theoretical results.

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.