Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 263 In conclusion, in order to have the same accuracy trend SFEM or WFEM with 1 C compatibility conditions should have a higher number of grid points with respect to SFEM or SEM with 0 C compatibility conditions. Remark that the first point of the curves in Fig. 5b refer to a single element mesh. The comparison between SFEM and the other weak methods is meaningless for this case because, excluding the boundary conditions, the SFEM has an 1 n  order with respect to the others due to the strong formulation (second order derivatives with respect to first order ones). It should be also noted that not only SFEM and WFEM but also SEM has the increasing error machine effect when the maximum accuracy is reached (around 12 10  ). Fig. 6-8 show some results related to the static and dynamic stability and accuracy of Membranes, Kirchhoff-Love (KL) and Reissner-Mindlin (RM) plates. Fig. 6a is analogous to Fig. 1a where the number of grid points N varies from 5 to 31 using PDQ basis functions and Che-Gau-Lob grid distribution. The KL plate model is considered with  -points ( 5 10  ). Note that when 15 N  the error is stable increasing the number of points (round-off plateau). Fig. 6b collects the static convergence behavior of a single domain decomposed into several regular elements (no mapping is involved). The number of grid points is fixed 7 N  and the number of elements is variable 1, 4, 9, 16, 25, 36, 49, 64. For the present cases the PDQ basis functions and Leg-Gau grid distribution are considered. As expected the error decreases increasing the number of elements. The dynamic convergence of several structural components is shown in Fig. 7 and 8. Fig. 7a shows the dynamic discrete spectra of the Membrane, the KL and the RM plates using PDQ basis functions and Che-Gau-Lob grid distribution with 31 N  using a single element. The Membrane and KL plate calculate accurate modes until the 25% of total modes, whereas the RM plate gets 10% of accurate modes due to the comparison with a semi-analytical solution. An analogous plot using PDQ basis functions and Leg-Gau grid distribution with 7 N  and 64 e n  is shown in Fig. 7b for the Membrane and the RM plate. Due to the element decomposition the accuracy decreases both for the Membrane and the RM plate. However, the error is limited within the 5% for the 40% and 50% of the modes of the Membrane and the RM plate, respectively. In conclusion a comparison in terms of dynamic discrete spectra for structured and distorted meshes is presented in Fig. 8 for the Membrane and RM plate, respectively. Different number of grid points is considered, in order to get the same number of degrees of freedom changing the number of elements. As expected the modes are more accurate when the mesh is structured, whereas a lower accuracy is observed when irregular meshes are considered. a) b) Figure 8 : a) Dynamic discrete spectra using different meshes (structured and distorted) of a Membrane using PDQ basis functions and Leg-Gau grid distribution. b) Dynamic discrete spectra using different meshes (structured and distorted) of a RM plate using PDQ basis functions and Leg-Gau grid distribution. C ONCLUSIONS AND REMARKS he present paper references the DQM and other methods that, according to the authors’ knowledge, represent similar techniques. The paper aims to present a general view on strong formulation methods. The main novelty of this manuscript is given by the presentation in several forms of the stability and accuracy of the SFEM technique when applied to simple 1D and 2D models and compared to exact solutions (or semi-analytical ones as far as RM plate is concerned). This work should help researchers, of the computational mechanics community, to understand the advantages and disadvantages of a strong formulation approach and, in particular, to the ones who are keen on weak (or T