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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 91
Edited by: B.H.V. Topping, L.F. Costa Neves and R.C. Barros
Paper 101

The Natural Neighbour Radial Point Interpolation Method: A Non-Linear Analysis Review

L.M.J.S. Dinis1, R.M. Natal Jorge1 and J. Belinha2

1Faculty of Engineering, University of Porto, Portugal
2Instituto de Engenharia Mecânica, Pólo FEUP, Porto, Portugal

Full Bibliographic Reference for this paper
L.M.J.S. Dinis, R.M. Natal Jorge, J. Belinha, "The Natural Neighbour Radial Point Interpolation Method: A Non-Linear Analysis Review", in B.H.V. Topping, L.F. Costa Neves, R.C. Barros, (Editors), "Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 101, 2009. doi:10.4203/ccp.91.101
Keywords: natural neighbour radial point interpolator method, meshless methods, non-linear analysis, large deformations, elastoplastic analysis.

This paper presents the large deformation analysis of non-linear elastic and elasto-plastic structures based on the natural neighbour radial point interpolation method (NNRPIM) [1,2], which is an improved meshless method.

The NNPRIM uses the natural neighbour [3] concept in order to enforce the nodal connectivity. Based on the Voronoï diagram [4] small cells are created from the unstructured set of nodes discretizing the problem domain, the "influence-cells". These cells are in fact influence-domains entirely nodal dependent. The Delaunay triangles [5], which are 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. Unlike the finite element method, where geometrical restrictions on elements are imposed for the convergence of the method, in the NNRPIM there are no such restrictions, which permits a random node distribution for the discretized problem. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed in a similar process to the radial point interpolation method (RPIM) [6,7], with some differences that modify the method performance. In the construction of the NNRPIM interpolation functions no polynomial base is required and the radial basis function (RBF) used is the multiquadric RBF. The NNRPIM interpolation functions posses the delta Kronecker property, which simplify the imposition of the natural and essential boundary conditions.

Several non-linear elasto-plasticity problems are studied to demonstrate the effectiveness of the method. The numerical results indicated that the NNRPIM handles large material distortion effectively and provides an accurate solution under large deformation.

L. Dinis, R.N. Jorge, J. Belinha, "Analysis of 3D solids using the natural neighbour radial point interpolation method", Computer Methods in Applied Mechanics and Engineering, 196, 2009-2028, 2007. doi:10.1016/j.cma.2006.11.002
L. Dinis, R.N. Jorge, J. Belinha, "Analysis of plates and laminates using the natural neighbour radial point interpolation method", Engineering Analysis with Boundary Elements, doi:10.1016/j.enganabound.2007.08.006, 2007. doi:10.1016/j.enganabound.2007.08.006
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G.M. Voronoï, "Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième Mémoire: Recherches sur les parallélloèdres primitifs", J. Reine Angew. Math., 134, 198-287, 1908.
R. Sibson, "A brief description of natural neighbor interpolation", in V. Barnett, (ed.), "Interpreting Multivariate Data", Wiley, Chichester, 21-36, 1981.
J.G. Wang, G.R. Liu, "On the Optimal Shape Parameters of Radial Basis Functions used for 2-D Meshless Methods", Computer Methods in Applied Mechanics and Engineering, 191, 2611-2630, 2002. doi:10.1016/S0045-7825(01)00419-4
J.G. Wang, G.R. Liu, "A Point Interpolation Meshless Method based on Radial Basis Functions", International Journal for Numerical Methods in Engineering, 54, 1623-1648, 2002. doi:10.1002/nme.489

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