Finite Element Method based on a Strong Formulation Isogeometric Analysis

In computational mechanics Finite Elements (FE) are the most common and versatile tool for studying structural components when discontinuities and general shapes are present. FEs are the main items included in all domain decomposition techniques which can be based on the weak or strong formulations. The most famous approach is termed FE method (FEM) and implements a variational formulation with low order approximating polynomials with mapping technique to transform a general shape element into one in the computational space. Novel techniques based on the strong formulation has been recently published [1-10], where a hybrid scheme given by the Differential Quadrature (DQ) method and the FEM has been presented using classic Lagrangian shape functions for the element mapping. Advanced blending function mapping must be employed for treating any discontinuity and distortion effect using the smallest number of elements, since the present strong form technique can deal with variable order approximating polynomials. The DQ method is utilized for solving the mathematical problem, whereas a nonlinear isogeometric mapping is implemented for mapping complex shapes. Therefore, the present methodology can be part of the Isogeometric Analysis (IGA) framework, because it can treat meshes made of NURBS functions that are taken from CAD software. For this reason, this implementation is named Strong Formulation Isogeometric Analysis (SFIGA). Figure 1a shows a general SFIGA mesh wherein four highly distorted elements are used to model a square simply-supported plate. The four elements are described by four NURBS drawn in a CAD software. Since the free vibrations of simply-supported plates have an exact solution, a comparison can be performed on the natural frequencies of the structure depicted in Figure 1a. Figure 1b depicts the convergence trend of the first natural frequency considering different point collocations and number of grid points per element. The basis function of the presented computation is considered as Lagrange basis due to its extreme versatility and numerical stability when the number of points is large. It can be observed that even for a highly distorted mesh a very good accuracy can be reached for the case under investigation. This work gives particular attention to the accuracy, stability and reliability of the SFIGA when it is applied to different structural problems such as beams, plates and shells. All the numerical applications are compared to well-known analytical, semi-analytical and numerical solutions of one- and two-dimensional systems. Ultimately, this work presents typical aspects of an innovative domain decomposition approach that should be of wide interest to the computational mechanics’ community.