Multi-Layered Structures of Arbitrary Shape via Generalized Differential Quadrature Finite Element Method

Over the years both the Finite Element Method (FEM) and the Ritz method have been used to study the static and dynamic behaviour of composite structures. However, the Ritz method is suitable for solving problems of simple geometries and it cannot be applied to arbitrarily shaped structures. In order to solve a problem with arbitrary boundary conditions and any kind of shape and discontinuity, the Generalized Differential Quadrature Finite Element Method (GDQFEM) is used in this work. Basically GDQFEM uses the Generalized Differential Quadrature (GDQ) scheme for solving the equations inside each transformed element. As a result, all the well known advantages of GDQ method are valid in GDQFEM applications too. GDQFEM is an advanced numerical tool that consists in dividing the complex given problem geometry into several sub-domains or elements, where the original problem can be solved more simply. The mapping technique is used to transform the fundamental system of equations and the compatibility conditions, between adjacent edges, on the physical domain into the computational domain. In this work, the numerical implementation of the GDQFEM is used to analyse the static and dynamic behavior of multi-layered structures with arbitrary shape. Since GDQFEM is a numerical scheme, it is able to transform a partial differential system of equations into an algebraic one, which can be implemented into a numerical code. Unlike FEM, GDQFEM is based on the strong formulation of the problem and the compatibility conditions among all the elements of the mesh are enforced in a different way. The fundamental system of equations is written in terms of the generalized displacements of the model under consideration. Finally, the assembly procedure allows building up the global algebraic system of the whole problem. The validity of the proposed methodology is verified within analytical results and numerical solutions obtained through FEM. It is important noticing that the convergence of the GDQFEM can be achieved either increasing the number of grid points per element or the number of elements of the mesh, keeping the number of points per element constant.