Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 251 Focussed on: Computational Mechanics and Mechanics of Materials in Italy The strong formulation finite element method: stability and accuracy Francesco Tornabene, Nicholas Fantuzzi, Michele Bacciocchi University of Bologna francesco.tornabene@unibo.it, nicholas.fantuzzi@unibo.it , michele.bacciocchi@studio.unibo.it A BSTRACT . The Strong Formulation Finite Element Method (SFEM) is a numerical solution technique for solving arbitrarily shaped structural systems. This method uses a hybrid scheme given by the Differential Quadrature Method (DQM) and the Finite Element Method (FEM). The SFEM takes the best from DQM and FEM giving a highly accurate strong formulation based technique with the adaptability of finite elements. The present study investigates the stability and accuracy of SFEM when applied to 1D and 2D structural components, such as rods, beams, membranes and plates using analytical and semi-analytical well-known solutions. The numerical results show that the present approach can be very accurate using a small number of grid points and elements, when it is compared to standard FEM. K EYWORDS . Strong Formulation Finite Element Method; Differential Quadrature Method; Finite Element Method; Free Vibration Analysis; Static Analysis; Numerical Stability. I NTRODUCTION his manuscript illustrates a general view of a class of methods which approximates functions at points, called collocation or sampling points. The complete review of the present methodology can be found in the previous works by the authors [1-6] and in the book by the authors [7]. The name which comprehends all this approaches is termed Strong Formulation Finite Element Method and it has its basis in the Differential Quadrature (DQ) Method (DQM) [8-12] as far as the numerical solution is concerned and in the Finite Element concept [13] as far as element connection is considered. The DQ method is part of a more general family of methods called Spectral Methods (SMs) [14-16] with the advantage of considering a free collocation once certain basis functions are set. However, DQM is not always stable increasing the number of collocation points. For this reason Generalized Differential Quadrature (GDQ) method was introduced by Shu [17]. Shu followed the former studies of Quan and Chang [10, 11], who developed a mathematical procedure for the definition of the weighting coefficients in exact form, thus the accuracy and stability is not lost increasing the number of grid points. Other papers that used the mapping technique for strong form based formulations can be found in [18, 19]. It should be mentioned that, the authors in their previous papers gave explicit formulae for the corner point conditions [1-6] that are an extension of the conditions given by other researchers. The following sections expose the mathematical basics of the DQ and GDQ methods. Only some details are given for the sake of conciseness, nevertheless for further details the interested reader can read the references [18-34]. In the numerical applications, the stability, reliability and accuracy of the Strong Formulation Finite Element Method (SFEM) and the Weak Formulation Finite Element Method (WFEM) are shown through several figures. T