Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 257 a) b) Figure 4 : a) Relative error of the first frequency for various structural components with 7 N  for a CC Rod, a SS EB beam with  - points (1/2) and a CC Tim beam using PDQ basis functions and Leg-Gau grid distribution. b) Effect of the grid point number inside each element for the static analysis of a CC Rod using SFEM with PDQ basis functions and Leg-Gau grid distribution, WFEM and SEM with PDQ basis functions and Leg-Gau-Lob grid distribution and FEM with linear and quadratic shape functions. However, when a domain decomposition method is taken into account, it could not be necessary to use a polynomial of high degree inside each element, because a good approximation might be captured with a number of points less than 13 N  . It has been demonstrated in [8, 9] that the Vandermonde matrix becomes ill-conditioned when 13 N  . Thus, if a limited number of points is considered the matrix inversion (12) can be used for the evaluation of the weighting coefficients. It is also noted that considering a limited number of points any polynomial basis (Tab. 1) within any distribution (Tab. 2) can be taken into account for the evaluations of the weighting coefficients using the presented general scheme (12). On the contrary, when a single element is taken into account the weighting coefficients in exact form must be used to avoid ill-conditioning problems. The present work also illustrates some applications related to the Weak Formulation Finite Element Method (WFEM), where an integration procedure based on the weighting coefficients of the GDQ method is developed. The present technique has been described in the works [7, 17]. This approach has been termed the Generalized Integral Quadrature (GIQ) method. a) b) Figure 5 : a) Dynamic discrete spectra with 100 e n  for a CC Rod using SFEM with PDQ basis functions and Leg-Gau grid distribution, WFEM and SEM with PDQ basis functions and Leg-Gau-Lob grid distribution and FEM with linear and quadratic shape functions. b) Relative error of the first frequency varying the number of elements e n for a CC Rod using SFEM with PDQ basis functions and Leg-Gau grid distribution, WFEM and SEM with PDQ basis functions and Leg-Gau-Lob grid distribution and FEM with linear and quadratic shape functions. In general, the numerical integration of a function   f x over a domain   , a b can be defined as     1 b N k k k a f x dx w f x     (17) 1 2 1 4 6 8 10 1 1 1 1 6 1 10 1 10