Finite difference method for nonlinear damped viscoelastic Euler-Bernoulli beam model

TL;DR

The paper tackles the nonlinear damped viscoelastic Euler-Bernoulli beam with memory, formulated as $u_{tt}+q(t)u_t+u_{xxxx}-\int_0^t \beta(t-s)u_{xxxx}(s)\,ds=f$ with $q(t)=G\Big(\int_0^1|u_{xx}|^2dx\Big)$. It develops a finite-difference spatial discretization and a backward-Euler/time-averaged-PI temporal scheme, and proves long-time stability, convergence, and existence/uniqueness for both semi-discrete and fully discrete formulations. It also provides rigorous convergence rates in terms of the spatial grid size $h$ and time step $\Delta t$, and confirms the theoretical results through numerical simulations. The framework delivers provably stable and accurate tools for simulating viscoelastic beam dynamics with nonlinear damping and memory effects, with potential impact on design and analysis of damping materials and structures subject to prolonged excitations.

Abstract

We propose and analyze the numerical approximation for a viscoelastic Euler-Bernoulli beam model containing a nonlinear strong damping coefficient. The finite difference method is used for spatial discretization, while the backward Euler method and the averaged PI rule are applied for temporal discretization. The long-time stability and the finite-time error estimate of the numerical solutions are derived for both the semi-discrete-in-space scheme and the fully-discrete scheme. Furthermore, the Leray-Schauder theorem is used to derive the existence and uniqueness of the fully-discrete numerical solutions. Finally, the numerical results verify the theoretical analysis.

Theorems & Definitions (15)

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