Strong and Weak Formulations for the Analysis of Arbitrarily Shaped Laminated Composite Structures

A numerical approach is developed to deal with arbitrarily shaped structures. Two different methodologies are used to this aim, which are based on the Differential Quadrature and Integral Quadrature methods, respectively. These numerical methods are able to approximate both derivatives and integrals [1]. Therefore, the strong and weak formulations of the governing equations can be solved. As shown in the paper [2], these approaches are accurate, reliable and stable, when employed to obtain the mechanical response of various kinds of structures, such as plates, shells and membranes. In particular, their effectiveness is proven by means of the comparison with the analytical solutions available in the literature, both for isotropic and composite structures. With respect to other approaches such as the Finite Element Method (FEM), the proposed methodologies are able to get the solution with few degrees of freedom. In addition, the convergence behavior is faster than the FEM. A domain decomposition based on Isogeometric analysis is developed to analyze the mechanical behavior of arbitrarily shaped structures. The so-called blending functions are used to deal with discontinuities and distortions by means of a reduced number of elements [3, 4]. Thus, a nonlinear mapping is achieved by employing NURBS curves. According to the numerical method used in the computation, the strong and weak formulations are solved within each element. The effect of distorted meshes on the solution is investigated, as well. The numerical methods at issue are named Strong Formulation Finite Element Method (SFEM) and Weak Formulation Finite Element Method (WFEM). References [1] Tornabene, F., Fantuzzi, N., Ubertini, F., Viola, E., "Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey", Applied Mechanics Reviews, 67, 02081-1-55 (2015). [2] Tornabene, F., Fantuzzi, Bacciocchi, M., "Strong and weak formulations based on differential and integral quadrature methods for the free vibration analysis of composite plates and shells: Convergence and accuracy", Engineering Analysis with Boundary Elements. In press. DOI: 10.1016/j.enganabound.2017.08.020. [3] Fantuzzi, N., Tornabene, F., "Strong Formulation Isogeometric Analysis (SFIGA) for Laminated Composite Arbitrarily Shaped Plates", Composites Part B - Engineering, 96, 173-203 (2016). [4] Tornabene, F., Fantuzzi, Bacciocchi, M., "The GDQ Method for the Free Vibration Analysis of Arbitrarily Shaped Laminated Composite Shells Using a NURBS-Based Isogeometric Approach", Composite Structures, 154, 190-218 (2016).