Stabilization of the Bernoulli-Euler Plate Equation: Numerical Analysis
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32912
Stabilization of the Bernoulli-Euler Plate Equation: Numerical Analysis

Authors: Carla E. O. de Moraes, Gladson O. Antunes, Mauro A. Rincon

Abstract:

The aim of this paper is to study the internal stabilization of the Bernoulli-Euler equation numerically. For this, we consider a square plate subjected to a feedback/damping force distributed only in a subdomain. An algorithm for obtaining an approximate solution to this problem was proposed and implemented. The numerical method used was the Finite Difference Method. Numerical simulations were performed and showed the behavior of the solution, confirming the theoretical results that have already been proved in the literature. In addition, we studied the validation of the numerical scheme proposed, followed by an analysis of the numerical error; and we conducted a study on the decay of the energy associated.

Keywords: Bernoulli-Euler Plate Equation, Numerical Simulations, Stability, Energy Decay, Finite Difference Method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1094347

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2014

References:


[1] N. Burq and G. Lebeau, Micro-Local Approach to the Control for the Plates Equation, Optimization, Optimal Control and Partial Differential Equations, International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / S˜A©rie Internationale dˆaAnalyse Num˜A©rique, BirkhA˜ user Basel, Vol. 107, 111-122, 1992.
[2] J. Lagnese and J.L. Lions, Modelling Analysis and Control of Thin Plates, volume 6 of Recherches en Math´ematiques Appliqu´ees
[Research in Applied Mathematics], Masson, Paris, 1988.
[3] K. Ramdani,M. Takahashi and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations, ESAIM: Control, Optimisation and Calculus of Variation, Vol. 13, 503-527, 2007.
[4] R. Glowinski, C.H. Li and J.L. Lions, A Numerical Approach to the Exact Boundary Controllability of the Wave Equation (I) Dirichlet Controls: Description of the Numerical Methods, Japan Journal of Applied Mathematics, Springer, Volume 7, 1, 1-76, 1990.
[5] J.A. Infante and E. Zuazua, Boundary Observability for the Space Semi-Discretizations of the 1–d Wave Equation, ESAIM: Mathematical Modelling and Numerical Analysis, Cambridge Univ Press, Vol. 33, 02, 407-438, 1999.
[6] K. Ramdani, M. Takahashi and M. Tucsnak, Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems, J. Math. Pures Appl. 85, 17-37, 2006.
[7] W.F. Ames, Numerical Methods For Partial Differential Equations, Computer Science and Scientific Computing, 3rd edition, Academic Press Inc, 1977.