Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 260 and m n indicates the mode number which is plotted in the discrete spectra. It is remarked that   2 d e N n N   for the Rod,   2 2 d e N n N   for the Tim beam and   4 d e N n N   for the EB beam. Moreover   2 2 d e N n N   for the membrane,   2 3 2 d e N n N   for the RM plate and   2 4 d e N n N   for the KL plate. Fig. 2b and 3a illustrate the dynamic discrete spectra by following the same analyses of Fig. 1a and 1b. Firstly a single element of high degree ( 151 N  ) is investigated. Secondly the same structural elements are divided into 100 e n  of low degree ( 7 N  ). The dynamic discrete spectra give a very close picture of the complete dynamic behavior given the total number of degrees of freedom. The percentage error with respect to the exact solution is plotted against the mode number. Globally in Fig. 2b, the accuracy is around 60% (for the Rod and the EB beam) and 50% (for Tim beam), when a single element is taken into account. However, it is uncommon to use a single element with many points in it. Thus, the SFEM approach is investigated in Fig. 3a where 100 e n  and 7 N  . The  -points for the EB beam are taken as (1/2) and the global accuracy results to be good until 30% of the modes, whereas the other Rod and Tim beam models reach the 60% and 40% respectively. These plots report some discrete jumps through the whole spectrum that were not investigated in the present paper. Hence, these aspects will be deeply exhibited in a following work. Finally the dynamic discrete spectra of the WFEM are represented in Fig. 3b for the Rod and the Tim beam. It can be noted that the curves start to detach from the horizontal line with respect to Fig. 3a where SFEM approach is presented. In fact, Rod and Tim beam are accurate until the 20% of the modes. Fig. 4a reports the convergence rate of the SFEM in the double logarithmic plot of the relative error of the first frequency versus the number of elements. In the study the degree of the approximating polynomials is 6 ( 7 N  ) and the number of element varies from 1 to 100. PDQ basis functions and Leg-Gau grid distribution are taken into account. It is remarked that the Rod and the Tim beam have a rate of convergence higher than the EB beam due to the fact that fourth order differential equations need extra points for the imposition of the boundary conditions, hence 2 grid points (per boundary) are lost for the imposition of the boundary conditions. Furthermore, the maximum accuracy that can be reached by the Rod is around 15 10  and the Tim beam is around 12 10  , on the contrary the EB beam reaches 9 10  . For the sake of comparison, the interested reader can refer to the works [36, 37] where the free vibrations of Rods and EB beams are investigated within the Isogeometric Analysis (IGA). For instance Fig. 10-12 of reference [37] show the convergence rates of a Rod with different polynomial orders. As expected, the convergence rate changes from 1/4 to 1/8 increasing the polynomial order. Nevertheless for all three cases the author shows only four points for the Rod convergence. In fact Fig. 10 and 11 consider 10 e n  , 20 e n  , 30 e n  , 40 e n  , whereas Fig. 12 5 e n  , 10 e n  , 15 e n  , 20 e n  . The author is assuming that the minimum convergence is reached, hence no more elements are needed. Nevertheless, the machine error noise is never shown throughout the paper. Ultimately, it is noted that the results proposed by [37] are in perfect agreement with the ones presented by the authors in Fig. 4a. The only exception is given by EB beams with  -points, because a lower trend 1/6 occurs together with a lower accuracy 9 10  . The interested reader can find more details about the so-called round-off error in the book by Boyd [15]. In conclusion, Fig. 4b, 5a, 5b exhibit a summary of the Rod structure behavior using four different numerical approaches: SFEM, WFEM, SEM (Spectral Element Method) and FEM. The first solves the strong form and uses 1 C compatibility conditions, the second solves the weak form and uses 1 C compatibility conditions, the third and the fourth solve the weak form with 0 C continuity, but the former with a general polynomial degree (Lagrange polynomials) and the latter with well-known linear and quadratic functions. Fig. 4b shows that increasing the polynomial order inside each element the error increases. In fact the SFEM error with 7 N  is higher than the FEM one with 2 N  . Nevertheless, the maximum error is always below the machine working precision (black dashed line). A comparison in terms of dynamic discrete spectra in shown in Fig. 5a. It can be noticed that SFEM offers the best accuracy in terms of percentage of accurate modes and the maximum errors are always more limited than all the other techniques. Fig. 5b shows several convergence rates with respect to the first natural frequency. Obviously, increasing the polynomial degree the convergence rate passes from 1/2 to 1/10. Linear FEM ( 2 N  ) has the same trend (1/2) of the ones obtained by SFEM and WFEM with 3 N  and 1 C continuity. The strong formulation requires a higher derivation order with respect to a variational or weak formulation. Considering the second order Rod problem, for a given basis function of order n the strong form derives the approximate solution two times, whereas the weak form only one. Therefore, the strong form solution becomes of order 2 n  and the weak form is 1 n  . Moreover, since the 1 C continuity is considered in the strong form the boundary points are excluded from the computation due to the kinematic condensation.