Based on the natural neighbor radial point interpolation method (NNRPIM), a 3D analysis of thick composite laminated plates is presented. The NNRPIM uses the natural neighbour concept in order to enforce nodal connectivity. Based on the Voronoï diagram small cells are created from the unstructured set of nodes discretizing the problem domain, the ‘influence-cells’, which are in fact influence domains entirely nodal dependent. The Delaunay triangles, the dual of the Voronoï cells, are used to create a node-dependent background mesh used in the numerical integration of the NNRPIM interpolation functions. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed in a process similar to that in the radial point interpolation method (RPIM) with some differences that modify the method performance. In the construction of the NNRPIM interpolation functions, no polynomial base is required and the used radial basis function (RBF) is the multiquadric RBF. The NNRPIM interpolation functions possess the delta Kronecker property, which simplifies the imposition of the natural and essential boundary conditions. In this work the 3D NNRPIM analysis is used to solve static and dynamic composite laminated plate problems. Thus, several benchmark examples are studied to demonstrate the effectiveness of the method.
Zienkiewicz, O.C. ( 1989). The Finite Element Method, 4th edn, McGraw-Hill, New York.
2.
Belytschko, T., Krongauz, Y., Organ, D., Fleming, M. and Krysl, P. (1996). Meshless Methods: An Overview and Recent Developments, Computer Methods in Applied Mechanics and Engineering, 139(1): 3-47.
3.
Gu, Y.T. (2005). Meshfree Methods and Their Comparisons, International Journal of Computational Methods , 2(4): 477-515.
4.
Monaghan, J.J. (1977). Smoothed Particle Hydrodynamics: Theory and Applications to Non-spherical Stars, Monthly Notices of the Astronomical Society, 181: 375-389.
5.
Belytschko, T., Lu, Y.Y. and Gu, L. (1994). Element-free Galerkin Method, International Journal for Numerical Methods in Engineering , 37: 229-256.
6.
Liu, W.K. , Jun, S. and Zhang, Y.F. (1995). Reproducing Kernel Particle Methods, International Journal for Numerical Methods in Fluids, 20(6): 1081-1106.
7.
Atluri, S.N. and Zhu, T. ( 1998). A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics, Computational Mechanics, 22(2): 117-127.
8.
Liu, G.R. (2002). A Point Assembly Method for Stress Analysis for Two-dimensional Solids, International Journal of Solid and Structures, 39: 261-276.
9.
Wang, J.G. and Liu, G.R. ( 2002). A Point Interpolation Meshless Method Based on Radial Basis Functions, International Journal for Numerical Methods in Engineering, 54: 1623-1648.
10.
Liu, G.R. and Gu, Y.T. (2001). A Point Interpolation Method for Two-dimensional Solids, International Journal for Numerical Methods in Engineering, 50: 937-951.
11.
Traversoni, L. (1994). Natural Neighbour Finite Elements, In: International Conference on Hydraulic Engineering Software, Hydrosoft Proceedings, Computational Mechanics Publications, Southampton, UK, Vol. 2, pp. 291-297.
12.
Sukumar, N., Moran, B. and Belytschko, T. ( 1998). The Natural Element Method in Solid Mechanics, International Journal for Numerical Methods in Engineering, 43(5): 839-887.
13.
Sukumar, N., Moran, B., Semenov, A.Y. and Belikov, V.V. ( 2001). Natural Neighbour Galerkin Methods, International Journal for Numerical Methods in Engineering, 50(1): 1-27.
14.
Cueto, E., Sukumar, N., Calvo, B., Cegoñino, J. and Doblaré, M. ( 2003). Overview and Recent Advances in the Natural Neighbour Galerkin Method, Archives of Computational Methods in Engineering , 10(4): 307-387.
15.
Sergio, R., Idelsohn, S., Oñate, E., Calvo, N. and Del Pin, F. ( 2003). The Meshless Finite Element Method, International Journal for Numerical Methods in Engineering, 58(6): 893-912.
16.
Pagano, N.J. (1970). Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates, Journal of Composite Materials, 4: 20-34.
17.
Pagano, N.J. and Hatfield, S.J. (1972). Elastic Behavior of Multilayered Composite Bi-directional Composites, American Institute of Aeronautics and Astronautics Journal, 10: 931-933.
18.
Liew, K.M., Zhang, J.Z., Ng, T.Y. and Meguid, S.A. ( 2003). Three-dimensional Modelling of Elastic Bonding in Composite Laminates Using Layerwise Differential Quadrature, International Journal of Solids and Structures, 40(7): 1745-1764.
19.
Liew, K.M., Zhang, J.Z., Li, C. and Meguid, S.A. ( 2005). Three-dimensional Analysis of the Coupled Thermo-piezoelectro-mechanical Behaviour of Multilayered Plates Using the Differential Quadrature Technique , International Journal of Solids and Structures, 42(14): 4239-4257.
20.
Zhang, J.Z. , Ng, T.Y. and Liew, K.M. (2003). Three-dimensional Theory of Elasticity for Free Vibration Analysis of Composite Laminates via Layerwise Differential Quadrature Modelling, International Journal for Numerical Methods in Engineering, 57(13): 1819-1844.
21.
Liew, K.M., Zhang, J.K., Ng, T.Y. and Reddy, J.N. ( 2004). Dynamic Characteristics of Elastic Bonding in Composite Laminates: A Free Vibration Study, ASME Journal of Applied Mechanics, 70(6): 860-870.
22.
Zhang, J.Z. , Feng, C. and Liew, K.M. (2006). Three-dimensional Vibration Analysis of Multilayered Piezoelectric Composite Plates, International Journal of Engineering Science, 44(7): 397-408.
23.
Dinis, L., Jorge, R.N. and Belinha, J. ( 2007). Analysis of 3D solids Using the Natural Neighbour Radial Point Interpolation Method, Computer Methods in Applied Mechanics and Engineering, 196: 2009-2028.
24.
Dinis, L., Jorge, R.N. and Belinha, J. ( 2008). Analysis of Plates and Laminates Using the Natural Neighbour Radial Point Interpolation Method, Engineering Analysis with Boundary Elements, 32(3): 267-279.
25.
Voronoï, G. (1907). Nouvelles applications des paramètres continus à la théorie des formes quadratiques , Journal für die Reine und Angewandte Mathematik, 133: 97-178.
26.
Delaunay, B. ( 1934). Sur la sphére vide. A la memoire de Georges Voronoï, Izv, Akad. Nauk SSSR, Otdelenie Matematicheskih i Estestvennyh Nauk, 7: 793-800.
27.
Liew, K.M. and Chen, X.L. ( 2004). Buckling of Rectangular Mindlin Plates Subjected to Partial In-Plane Edge Loads Using the Radial Point Interpolation Method, International Journal of Solids and Structures, 41: 1677-1695.
28.
Liew, K.M. and Chen, X.L. ( 2004). Mesh-free Radial Point Interpolation Method for the Buckling Analysis of Mindlin Plates Subjected to In-plane Point Loads, International Journal for Numerical Methods in Engineering, 60: 1861-1877.
29.
Liew, K.M., Chen, X.L. and Reddy, J.N. ( 2004). Mesh-free Radial Basis Function Method for Buckling Analysis of Non-uniformly Loaded Arbitrarily, Computer Methods in Applied Mechanics and Engineering, 193: 205-224.
30.
Chen, X.L. and Liew, K.M. ( 2004). Buckling of Rectangular Functionally Graded Material Plates Subjected to Nonlinearly Distributed In-plane Edge Loads, Smart Materials and Structures, 13: 1430-1437.
31.
Chen, X.L., Zhao, Z.Y. and Liew, K.M. ( 2008). Stability of Piezoelectric FGM Rectangular Plates Subjected to Non-uniformly Distributed Load, Heat and Voltage, Advances in Engineering Software, 39(2): 121-131.
32.
Sibson, R. ( 1980). A Vector Identity for the Dirichlet Tesselation, Mathematical Proceedings of the Cambridge Philosophical Society, 87: 151-155.
33.
Liu, G.R. , Gu, Y.T. and Dai, K.Y. (2004). Assessment and Applications of Interpolation Methods for Computational Mechanics, International Journal for Numerical Methods in Engineering, 59: 1373-1379.
Reissner, E. (1945). The Effect of Transverse Shear Deformation on the Bending, Journal of Applied Mechanics , 12: 69-77.
36.
Mindlin, R.D. (1977). Influence of Rotatory Inertia and Shear on Flexural Motion of Isotropic, Elastic Plates, Journal of Applied Mechanics, 18: 31-38.
37.
Whitney, J.M. and Sun, C.T. (1973). A Higher Order Theory for Extensional Motion of Laminated Composites, Journal of Sound and Vibration, 30: 85-97.
38.
Ambartsumyan, S.A. (1970). Theory of Anisotropic Plates, 1stedn, Technomic Publishing Co, Stamford.
39.
Nelson, R.B. and Lorch, D.R. (1974). A Refined Theory of Laminated Orthotropic Plates, Journal of Applied Mechanics , 41: 177-183.
40.
Reddy, J.N. (1984). A Simple Higher-order Theory for Laminated Composite Plates, Journal of Applied Mechanics , 45: 745-752.
41.
Leung, A.Y.T. (1991). An Unconstrained Third-order Plate Theory, Computers and Structures, 40(4): 871-875.
