The Strong Formulation Finite Element Method Applied to Structural Mechanics Problems

The Strong Formulation Finite Element Method (SFEM) [1] is a numerical approach that can be used for solving civil, environmental, mechanical, aerospace and naval engineering problems. Generally, practical engineering problems are complex due to geometry, material and load discontinuities. For solving them, it is necessary to divide the whole domain into nite elements of arbitrary shape. The mapping technique is introduced at this level to transform an arbitrarily shaped element to a parent element (computational element). The classic Finite Element Method (FEM) uses the above procedure and the problem at the parent element level is solved by means of weak (variational) formulation. On the contrary, the SFEM summarises a class of methods that is able to approximate total and partial derivatives at discrete points, thus the solution is found in its strong form. The SFEM has its own roots in the Dierential Quadrature Method (DQM), which was introduced in the early 1970s. Nevertheless, DQM does not allow to treat arbitrarily shaped domains and problems where discontinuities are present. These features are proper of nite element approaches, in which the global domain is divided into several smaller elements, and after the assembly procedure they solve the complete system. Therefore the SFEM is an hybrid scheme given by the DQM and the FEM. The most signicant dierence between these two methodologies lays on the formulation used for solving the parent element. In order to clarify the idea about the fact that the SFEM comprehends several numerical techniques, the reader can review a class of methods in the article [2], where it has been claried that the most important and wide-spread numerical approaches are a sub-class of the method of weighted residuals. Moreover a former review article about DQM can be found in [3] where a state of the art of that time was given. Unfortunately the authors limited their analysis to DQM and they did not focus their attention on the generalization of the DQM concepts to a wider prospective. As far as the authors are concerned, the rst paper regarding the present topic was presented in [4]. The discussion was extended in a survey paper published recently [5], where a signicant historical review about strong and weak numerical tools was carried out. The authors provided stability and accuracy of one-dimensional and two-dimensional problems when compared to classic exact solutions related to structural problems, such as rods, beams, membranes and plates. The rst application of the SFEM regarding one-dimensional in-plane multi-stepped and multi-damaged arches was published in [6]. The authors investigated the vibration of thin membranes in a review paper [7], where several well-known numerical applications were compared to the literature. Some other applications were presented concerning the behavior of elastostatic and elastodynamic plane structures in [8, 9, 10, 11]. Later the authors presented the SFEM applied to the modal analysis of Reissner-Mindlin plates [12, 13]. The SFEM based on DQM and Radial Basis Function (RBF) method has been presented in the work [14]. Finally in the works [15, 16] a particular emphasis has been given to the stress recovery procedure for the evaluation of the three dimensional strain and stresses at all the physical points of the problem. As a denition the SFEM is a numerical procedure that decomposes the physical domain or problem in nite elements and used the strong formulation inside each element mapped on the parent (or computational) element. When in the above procedure the weak formulation is used (instead of the strong form), the WFEM is dened. The latter is well-known in literature as FEM. This paper aims to investigate the application of the SFEM to structural mechanics problems. Since its numerical solutions depend on the number of collocation points, the basis functions used, the location of the points and the number of domain divisions, the authors report in graphical form the stability, accuracy and reliability of the present technique. In this way several aspects are raised and remarks are given as closure.