×

Analysis of 3D solids using the natural neighbour radial point interpolation method. (English) Zbl 1173.74469

Summary: An improved meshless method is proposed, based on the combination of the natural neighbour finite element method with the radial point interpolation method, the natural neighbour radial point interpolation method - NNRPIM. The nodal connectivity and the node dependent integration background mesh are constructed resorting to the Voronoï tessellation and to the Delaunay triangulation. Within NNRPIM the obtained shape functions pass through all nodes inside the influence-cell providing shape functions with the delta Kronecker property. Optimization tests and examples of well-known, 2D and 3D problems are solved in order to prove the high accuracy and convergence rate of the proposed method.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates

Software:

Mfree2D
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bathe, K.J., Finite element procedures, (1996), Prentice-Hall Englewood Cliffs, NJ · Zbl 0511.73065
[2] Zienkiewicz, O.C., The finite element method, (1989), McGraw-Hill · Zbl 0435.73073
[3] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. engrg., 139, 1, 3-47, (1996) · Zbl 0891.73075
[4] Gu, Y.T., Meshfree methods and their comparisons, Int. J. comput. methods, 2, 4, 477-515, (2005) · Zbl 1137.74302
[5] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput. mech., 10, 307-318, (1992) · Zbl 0764.65068
[6] Lancaster, P.; Salkauskas, K., Surfaces generation by moving least squares methods, Math. comput., 37, 141-158, (1981) · Zbl 0469.41005
[7] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin method, Int. J. numer. methods engrg., 37, 229-256, (1994) · Zbl 0796.73077
[8] Dolbow, J.; Belytschko, T., An introduction to programming the meshless element free Galerkin method, Arch. comput. mech., 5, 3, 207-241, (1998)
[9] Lu, Y.; Belytschko, T.; Gu, L., A new implementation of the element free Galerkin method, Comput. methods appl. mech. engrg., 113, 397-414, (1994) · Zbl 0847.73064
[10] Monaghan, J.J., Smoothed particle hydrodynamics: theory and applications to non-spherical stars, Mon. notices astronom. soc., 181, 375-389, (1977) · Zbl 0421.76032
[11] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int. J. numer. methods fluids, 20, 6, 1081-1106, (1995) · Zbl 0881.76072
[12] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput. mech., 22, 2, 117-127, (1998) · Zbl 0932.76067
[13] Oñate, E.; Idelsohn, S.; Zienkiewicz, O.C.; Taylor, R.L., A finite point method in computational mechanics – applications to convective transport and fluid flow, Int. J. numer. methods engrg., 39, 3839-3866, (1996) · Zbl 0884.76068
[14] Bathe, K.J.; De, S., Towards an efficient meshless computational technique: the method of finite spheres, Engrg. comput., 18, 170-192, (2001) · Zbl 0985.74079
[15] Liu, G.R.; Gu, Y.T., A point interpolation method for two-dimensional solids, Int. J. numer. methods engrg., 50, 937-951, (2001) · Zbl 1050.74057
[16] Wang, J.G.; Liu, G.R.; Wu, Y.G., A point interpolation method for simulating dissipation process of consolidation, Comput. methods appl. mech. engrg., 190, 5907-5922, (2001) · Zbl 1055.74015
[17] Liu, G.R., A point assembly method for stress analysis for two-dimensional solids, Int. J. solid struct., 39, 261-276, (2002) · Zbl 1090.74699
[18] Traversoni, L., Natural neighbour finite elements, Int. conf. hydraulic engrg. software, hydrosoft proc., comput. mech. pub., 2, 291-297, (1994)
[19] Sukumar, N.; Moran, B.; Yu Semenov, A.; Belikov, V.V., Natural neighbour Galerkin methods, Int. J. numer. methods engrg., 50, 1, 1-27, (2001) · Zbl 1082.74554
[20] Braun, J.; Sambridge, M., A numerical method for solving partial differential equations on highly irregular evolving grids, Nature, 376, 655-660, (1995)
[21] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, Int. J. numer. methods engrg., 43, 5, 839-887, (1998) · Zbl 0940.74078
[22] Cueto, E.; Doblaré, M.; Gracia, L., Imposing essential boundary conditions in the natural element method by means of density-scaled-shapes, Int. J. numer. methods engrg., 49, 4, 519-546, (2000) · Zbl 0989.74077
[23] Cueto, E.; Sukumar, N.; Calvo, B.; Cegoñino, J.; Doblaré, M., Overview and recent advances in the natural neighbour Galerkin method, Arch. comput. methods engrg., 10, 4, 307-387, (2003) · Zbl 1050.74001
[24] Sergio, R.; Idelsohn, S.; Oñate, E.; Calvo, N.; Del Pin, F., The meshless finite element method, Int. J. numer. methods engrg., 58, 6, 893-912, (2003) · Zbl 1035.65129
[25] Wang, J.G.; Liu, G.R., A point interpolation meshless method based on radial basis functions, Int. J. numer. methods engrg., 54, 1623-1648, (2002) · Zbl 1098.74741
[26] Wang, J.G.; Liu, G.R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Comput. methods appl. mech. engrg., 191, 2611-2630, (2002) · Zbl 1065.74074
[27] Kansa, E.J., A scattered data approximation scheme with applications to computational fluid-dynamics - I and II, Comput. math. appl., 19, 127-161, (1990) · Zbl 0692.76003
[28] Liu, G.R.; Bernard, B.T.; Chun, L., A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis, Comput. methods appl. mech. engrg., 195, 4843-4861, (2006) · Zbl 1128.74050
[29] Liu, G.R.; Zhang, G.Y.; Gu, Y.T.; Wang, Y.Y., A meshfree radial point interpolation method (RPIM) for three-dimensional solids, Comput. mech., 36, 6, 421-430, (2005) · Zbl 1138.74420
[30] Liu, G.R.; Dai, K.Y.; Lim, K.M.; Gu, Y.T., A point interpolation meshfree method for static and frequency analysis of two-dimensional piezoelectric structures, Comput. mech., 29, 6, 510-519, (2002) · Zbl 1146.74371
[31] Liu, G.R.; Dai, K.Y.; Lim, K.M.; Gu, Y.T., A radial point interpolation method for simulation of two-dimensional piezoelectric structures, Smart mater. struct., 12, 171-180, (2003)
[32] Chen, X.L.; Liew, K.M., Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads, Smart mater. struct., 13, 1430-1437, (2004)
[33] Dai, K.Y.; Liu, G.R.; Lim, K.M., A meshfree method for static and free vibrations analysis of shear deformable laminated composite plates, J. sound vib., 269, 3-5, 633-652, (2004)
[34] Liew, K.M.; Chen, X.L., Mesh-free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in-plane point loads, Int. J. numer. methods engrg., 60, 1861-1877, (2004) · Zbl 1060.74669
[35] Liew, K.M.; Chen, X.L., Buckling of rectangular Mindlin plates subjected to partial in-plane edge loads using the radial point interpolation method, Int. J. solids struct., 41, 1677-1695, (2004) · Zbl 1075.74533
[36] Liew, K.M.; Chen, X.L.; Reddy, J.N., Mesh-free radial basis function method for buckling analysis of non-uniformly loaded arbitrarily, Comput. methods appl. mech. engrg., 193, 205-224, (2004) · Zbl 1075.74700
[37] Liu, Y.; Hon, Y.C.; Liew, K.M., A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems, Int. J. numer. methods engrg., 66, 1153-1178, (2006) · Zbl 1110.74871
[38] Sibson, R., A brief description of natural neighbor interpolation, (), 21-36
[39] Voronoï, G.M., 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) · JFM 39.0274.01
[40] Delaunay, B., Sur la sphére vide. A la memoire de georges voronoï. izv. akad. nauk SSSR, Otdelenie matematicheskih i estestvennyh nauk, 7, 793-800, (1934) · JFM 60.0946.06
[41] Liu, G.R., Mesh free methods, moving beyond the finite element method, (2002), CRC Press
[42] L. Dinis, J. Belinha. Analysis of 2D problems resorting to a new meshless method, in: III European Conference on Computational Mechanics - Solids, Structures and Coupled Problems in Engineering, 2006, Lisbon, LNEC, 5-9 June.
[43] L. Dinis, J. Belinha. A numerical comparison of distinct meshless methods for the analysis of composite laminates, in: III European Conference on Computational Mechanics - Solids, Structures and Coupled Problems in Engineering, 2006, Lisbon, LNEC, 5-9 June.
[44] Sibson, R., A vector identity for the Dirichlet tesselation, Math. proc. Cambridge philos. soc., 87, 151-155, (1980) · Zbl 0466.52010
[45] Lawson, C.L., Software for C1 surface interpolation, () · Zbl 0407.68033
[46] Watson, D.F., Contouring: A guide to the analysis and display of spatial data, (1992), Pergamon Press Oxford
[47] Dai, K.Y.; Liu, G.R.; Han, X.; Li, Y., Inelastic analysis of 2D solids using a weak-form RPIM based on deformation theory, Comput. methods appl. mech. engrg., 195, 4179-4193, (2006) · Zbl 1176.74209
[48] Hardy, R.L., Theory and applications of the multiquadrics – biharmonic method (20 years of discovery 1968-1988), Comput. math. appl., 19, 127-161, (1990) · Zbl 0692.65003
[49] Golberg, M.A.; Chen, C.S.; Bowman, H., Some recent results and proposals for the use of radial basis functions in the BEM, Engrg. anal. boundary elements, 23, 285-296, (1999) · Zbl 0948.65132
[50] Liu, G.R.; Gu, Y.T.; Dai, K.Y., Assessment and applications of interpolation methods for computational mechanics, Int. J. numer. methods engrg., 59, 1373-1379, (2004) · Zbl 1041.74562
[51] Timoshenko, S.; Goodier, J.N., Theory of elasticity, (1970), McGraw Hill · Zbl 0266.73008
[52] Irons, B.M.; Razzaque, A., Experience with the patch test for convergence of nite elements, () · Zbl 0315.73091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.