Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey

Abstract

A survey of several methods under the heading of strong formulation finite element method (SFEM) is presented. These approaches are distinguished from classical one, termed weak formulation finite element method (WFEM). The main advantage of the SFEM is that it uses differential quadrature method (DQM) for the discretization of the equations and the mapping technique for the coordinate transformation from the Cartesian to the computational domain. Moreover, the element connectivity is performed by using kinematic and static conditions, so that displacements and stresses are continuous across the element boundaries. Numerical investigations integrate this survey by giving details on the subject.

Free vibrations of non-uniform CNT/fiber/polymer nanocomposite beams

“…GDQ technique, as an efficient numerical technique, is implemented for solving partial differential equations, especially in the field of solid mechanics [37][38][39][40][41]. Here, this method is utilized to obtain the natural frequencies of non-uniform CNTFPC beam.…”

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part II: Model validation

“…Thus, the unknown functional values f (x i ) at each grid point are found by solving the system of equations with the appropriate boundary conditions. The DQM is a versatile numerical discretisation technique that can be used to develop strong-form finite elements [43]. Using this technique, the relatively simple geometry of a square, flat plate investigated here can be extended to the analysis of more complex geometries.…”

“…Second, the matrix inversion is not computed using the backslash operator in MATLAB, but rather with the Moore-Penrose pseudoinverse as detailed in reference [48]. One possible solution is to discretise the plate into multiple DQM elements using the method proposed by Tornabene et al [43]. Such an approach is equivalent to a strong-form FEM and considerably reduces the bandwidth of the matrices to improve the conditioning of the stiffness matrix.…”

Free Vibration Analysis of Reinforced Composite Functionally Graded Plates with Steady State Thermal Conditions

“…The DQ method approximates a derivative at a point as a linear weighted sum of all the functional values on the domain. How to evaluate the weighting coefficients is one of the key points of this technique [Tornabene et al (2015)]. However, DQM is not always stable increasing the number of collocation points.…”