Abstract
In this paper, we study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation. We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case. We prove firstly the existence and uniqueness of the weak solution. Then, we discretize the system in two steps: in the first step, a semi-discrete scheme is obtained for discretization in space and, in the second step, a fully-discrete scheme is obtained for discretization in time by the Crank-Nicolson scheme. At each step of the discretization, the a-priori error estimates are obtained.
Highlights
In this work, we study a dissipative numerical property by the finite element method for a flexible Euler-Bernoulli beams with a force control in rotation and velocity rotation
We study a flexible Euler-Bernoulli beam clamped at one end and subjected to a force control in rotation and velocity rotation
We develop a finite element method, stable and convergent which preserves the property of time decay of energy in the continuous case
Summary
We study a dissipative numerical property by the finite element method for a flexible Euler-Bernoulli beams with a force control in rotation and velocity rotation. From Shkalikov’s method (Shkalikov, 1986), a spectral analysis of the operator and the property of the Riesz basis were studied to derive the exponential stability of the system. The time derivative of the energy functional ε(t) along the classical solutions of (1)-(4). The right hand side of (6) serves as a motivation in the design of the control αwxt(1, t) + βwx(1, t), ensures the energy decay of the system in time. Our main contribution is to develop a convergent numerical method which faithfully reproduces some properties of this problem such as stability and energy decay.
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