A finite element approximation to a viscoelastic Euler–Bernoulli beam with internal damping

Abstract

We analyze a finite element approximation to a viscoelastic Euler–Bernoulli beam with internal damping that undergoes vibrations under external excitation. We prove the wellposedness of the problem and regularity estimates of the exact solution to the model. We then utilize these results to prove an optimal-order error estimate of the numerical approximation assuming only the regularity of the data of the model but not that of the exact solution. Because the model exhibits its salient features that are different from those of conventional elastic Euler–Bernoulli beams, a new estimate technique is used in the analysis. We finally carry out numerical experiments to substantiate the error estimate and to investigate the dynamic response of the viscoelastic Euler–Bernoulli beam, in comparison with the conventional Euler–Bernoulli beam.