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Fig.2: (a) Voronoi diagram for elliptical domain (b) simplex subcells.
The authors write:
“…the meshless differential and integral operators are used to discretize the matricized form of Hamilton's principle. In this paper, the moving least-squares (MLS) approximation (see Appendix A) is employed to perform the numerical discretization for shape function and its derivatives. Also, for construction of integral operator, the background cells are utilized. Furthermore, Voronoi diagram is employed to create background cells, as illustrated in Fig. 2(a). By the use of main grid points, each cell can be divided into triangular (simplex) subcells, as shown in Fig. 2(b).”
FROM:
Reza Ansari, Jalal Torabi and Amir Hosein Shakouri (Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran),
“Vibration analysis of functionally graded carbon nanotube-reinforced composite elliptical plates using a numerical strategy”, Aerospace Science and Technology, Vol. 60, pp 152-161, January 2017, https://doi.org/10.1016/j.ast.2016.11.004
ABSTRACT: A numerical meshless discretization technique is developed within the framework of variational formulation to present the linear free vibration analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) elliptical plates. The effective material properties of nanocomposite plate are continuously varied across the thickness direction and are evaluated based on the extended rule of mixture. The governing equations are derived on the basis of the first order shear deformation theory. To this end, the matrix form of Hamilton's principle is first presented. Then, based on the moving least-squares (MLS) approximation and background cells approach, the meshless differential and integral operators are constructed to perform the discretization process. After conducting the comparison and convergence study, various numerical results are reported to explore the effects of concerned parameters on the natural frequencies of composite elliptical plates reinforced with carbon nanotubes (CNTs). Results reveal that functionally grading of CNTs through the thickness direction can considerably improve the vibrational characteristics of FG-CNTRC elliptical plates.
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