The Generalized Finite Element Method Applied to Free Vibration of Framed Structures

Gfem Stabilization Techniques Applied to Dynamic Analysis of Non-Uniform Section Bars

“…For the linear bar element Figure 1 , the partition of unity functions are those given by Equation 7 . In this study, the enrichment functions proposed by Arndt 2009 andArndt et al 2011 are written as a group of functions that consists of building a couple of clouds, a sine and a cosine, subordinated to the cover of the master element. These clouds are written in the element domain as two pairs of sine and cosine functions.…”

“…Using particular information to improve the approximation characteristics is the core of the enriched methods, such as the Generalized Finite Element Method GFEM , where enriching functions are used to improve the approximation space Babuška et al, 1994;Melenk, 1995;Duarte and Oden, 1996a;Duarte and Oden, 1996b;Oden and Duarte, 1997;Babuška and Melenk, 1997;Babuška et al, 2004;Belytschko and Black, 1999;Möes et al, 1999 . In this context, the GFEM has applications in several areas, such as fracture mechanics Yazid et al, 2009;Gupta et al, 2015 , flow of biphasic fluids Sauerland and Fries, 2013, electromagnetism Lu and Shanker, 2007, heat transfer with high gradients O'Hara et al, 2009Aragon et al, 2010 and, as approached in this work, in dynamics of structures Arndt, 2009;Torii, 2012;Torii and Machado 2012;Shang, 2014;Torii et al, 2016;Hsu, 2016;Weinhardt et al, 2016;Piedade Neto and Proença, 2016 . Despite the excellent properties of the GFEM, there are aspects that still limit its practical applicability and its efficiency.…”

“…where L e is the element length and j j    is a hierarchical enrichment parameter proposed by Arndt 2009 with α 1 and with j varying from 1 to the number of enrichment layers n l , such as j 1, 2, … n l .…”

“…Addressing numerical sensitivity problems that arise in the application of enrichment proposals, two alternatives are adopted to stabilize the method: application of the concepts of the Stable Generalized Finite Element Method and application of preconditioning changes in enrichment functions, called Heuristic Modification Stabilization, such as presented in Weinhardt et al 2015 Following the idea that small changes in enrichment functions can change the numerical conditioning of the problem without significant loss of accuracy properties of the GFEM, this article presents the effects of a subtle change in the functions used for one-dimensional dynamic analysis such as presented by Arndt 2009 . The stabilization methods are applied in two examples of bar with cross section area variation, and this proposal has been proved to be very effective in improving the stability for the modal and transient analysis.…”

Dynamic analysis of Euler–Bernoulli beam problems using the Generalized Finite Element Method

“…Additionally, there are several researches concerning the development of error estimation in GFEM [18,19]. It has been shown that the enriched FE shape functions associated to trigonometric functions provide remarkable efficiency in free vibration analysis in bars and beams due to the fact that the trigonometric functions present similar profile to the vibration modes [20,21]. GFEM with trigonometric functions also has been applied by Torii and Machado [22] for free vibration analysis employing quadrilateral plane elements.…”