Developing three-dimensional mesh-free Galerkin method for structural analysis of general polygonal geometries

In this paper, triangular prismatic cells for background integration on mesh-free methods are introduced and the Gauss integration scheme is developed in these cells. The cells are used in the mesh-free Galerkin method for free vibration, static and dynamic analysis of general polygonal plates of various thicknesses. The moving least square shape functions are used to construct the approximation of field variables, and the Hamilton principle is used to drive the weak form equations of motion. For the dynamic case, the resulted set of differential equations are solved by the Wilson method. The main objective of this work is developing of a mesh-free method for analysis of triangle and polygonal geometries and more important is to use the triangular prismatic cells and corresponding generalized Gaussian quadrature rules for integration in mesh-free methods. The cells can be used in a compound with cubic cells for discretization of any complicate three-dimensional geometries. To show this capability, free vibration analysis of a general pentagon plate is also performed. For all cases, the results show the flexibility and accuracy of mesh-free methods for irregular and complicate sharp corner geometries.