Eventos Anais de eventos
COBEM 2023
27th International Congress of Mechanical Engineering
DESIGN OF SHAPE FUNCTIONS FOR ELIMINATING THE GIBBS PHENOMENON
Submission Author:
Francisco Alex Monteiro , SP
Co-Authors:
Eliseu Lucena Neto, Francisco Alex Monteiro, Sérgio Cordeiro
Presenter: Sérgio Cordeiro
doi://10.26678/ABCM.COBEM2023.COB2023-1504
Abstract
One of the major challenges in approximate methods is the mitigation of spurious oscillations from the obtained solution and its derivatives, near the boundary of the problem domain, known as the Gibbs phenomenon. Eventual errors caused by such oscillations may be reduced but not eliminated as the number of degree of freedom is increased. Because of the Gibbs phenomenon, the Fourier series may face difficulties in representing even sufficiently smooth functions defined on a compact interval, when their periodic extension have discontinuities at the interval boundaries. Such difficulties can be properly surmounted expanding the modified function f-φ, instead of merely expanding the original function f. The suitably chosen function φ helps improve the convergence of the approximation by enforcing not only the continuity of the periodic extension of f-φ but also of its derivatives. Once the auxiliary function φ is stated, one says that the series of f has been enriched because of its modification towards a faster convergence. Aiming to design complete sets of shape functions that provide fast convergence according to the theory of Fourier series and eliminate the Gibbs phenomenon up to derivatives of order p, this work proposes a rational approach to construct hierarchic sets of shape functions of any C^q continuity (q=0,1,2,…) derived from proper C^p periodic extension of f-φ.
Keywords
Gibbs phenomenon, Improved convergence, Hierarchic set
