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A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells. (English) Zbl 1217.74143
Summary: A formulation for thin plates with only the deflection as nodal variables has been proposed using the generalized gradient smoothing technique and the radial point interpolation method (RPIM). The deflection fields are approximated using the RPIM shape functions which possess the Kronecker delta property for easy impositions of essential boundary conditions. Three types of smoothing domains, which are also serving as the numerical integrations domains, are constructed based on the background three-node triangular cells and the generalized gradient smoothing operation is performed over each of them to obtain the smoothed curvatures. The generalized smoothed Galerkin weak form is then used to create the discretized system equations. The essential boundary conditions of rotations are imposed in the process of constructing the curvature field, and the translation boundary conditions are imposed as in the standard FEM. A number of numerical examples, including both static and free vibration analysis, are studied using the present methods and the numerical results are compared with the analytical ones and those in the open literatures. The results show that the present formulation can obtain very stable and accurate solutions, even for the extremely irregular background cells.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K20 Plates
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[1] Monaghan, An introduction of SPH, Computer Physics Communications 48 pp 89– (1982) · Zbl 0673.76089
[2] Nayroles, Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics 10 (5) pp 307– (1992) · Zbl 0764.65068
[3] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 (2) pp 229– (1994) · Zbl 0796.73077
[4] Liu, Reproducing kernel particle methods, International Journal for Numerical Methods in Engineering 20 pp 1081– (1995) · Zbl 0881.76072
[5] Onate, A finite point method in computational mechanics: applications to convective transport and fluid flow, International Journal for Numerical Methods in Engineering 39 pp 3839– (1996) · Zbl 0884.76068
[6] Duarte, An h-p adaptive methods using clouds, Computer Methods in Applied Mechanics and Engineering 139 pp 237– (1996) · Zbl 0918.73328
[7] Atluri, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics 22 (2) pp 117– (1998) · Zbl 0932.76067
[8] Liu, A point interpolation method for two-dimensional solids, International Journal for Numerical Methods in Engineering 50 pp 937– (2001) · Zbl 1050.74057
[9] Wang, A point interpolation meshless method based on radial basis functions, International Journal for Numerical Methods in Engineering 54 pp 1623– (2002) · Zbl 1098.74741
[10] Liu, Meshfree Methods: Moving Beyond the Finite Element Method (2009) · Zbl 1205.74003
[11] Liu, An Introduction to Meshfree Methods and their Programming (2005)
[12] Krysl, Analysis of thin plates by the element-free Galerkin method, Computational Mechanics 17 (1) pp 26– (1995) · Zbl 0841.73064
[13] Krysl, Analysis of thin shells by the element-free Galerkin method, International Journal of Solids and Structures 33 (20-22) pp 3057– (1996) · Zbl 0929.74126
[14] Liu, A mesh-free Galerkin method for static and free vibration analyses of thin plates of complicated shape, Journal of Sound and Vibration 241 (5) pp 839– (2001)
[15] Liu, Element free method for static and free vibration analysis of spatial thin shell structures, Computer Methods in Applied Mechanics and Engineering 191 pp 5923– (2002) · Zbl 1083.74610
[16] Donning, Meshless methods for shear-deformable beams and plates, Computer Methods in Applied Mechanics and Engineering 152 (1-2) pp 47– (1998) · Zbl 0959.74079
[17] Noguchi, Element free analyses of shell and spatial structures, International Journal for Numerical Methods in Engineering 47 (6) pp 1215– (2000) · Zbl 0970.74079
[18] Kanok-Nukulchai, On elimination of shear locking in the element-free Galerkin method, International Journal for Numerical Methods in Engineering 52 (7) pp 705– (2001) · Zbl 1128.74347
[19] Chen, A stabilized conforming nodal integration for Galerkin meshfree methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081
[20] Wang, Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissener plate formulation, Computer Methods in Applied Mechanics and Engineering 193 (12-14) pp 1065– (2004) · Zbl 1060.74675
[21] Chen, A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates, International Journal for Numerical Methods in Engineering 68 (2) pp 151– (2006) · Zbl 1130.74055
[22] Wang, A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration, International Journal for Numerical Methods in Engineering 74 (3) pp 368– (2008) · Zbl 1159.74460
[23] Zhang, Moving least-squares approximation with discontinuous derivative basis functions for shell structures with slope discontinuities, International Journal for Numerical Methods in Engineering 76 (8) pp 1202– (2008) · Zbl 1195.74304
[24] Gu, A meshless local Petrov-Galerkin (MLPG) formulation for static and free vibration analyses of thin plates, Computer Modeling in Engineering and Sciences 2 (4) pp 463– (2001) · Zbl 1102.74310
[25] Long, A meshless local Petrov-Galerkin (MLPG) method for solving the bending problem of a thin plate, Computer Modeling in Engineering and Sciences 3 pp 53– (2002) · Zbl 1147.74414
[26] Soric, Meshless local Petrov-Galerkin (MLPG) formulation for analysis of thick plates, Computer Modeling in Engineering and Sciences 6 (4) pp 349– (2004)
[27] Sladek, Meshless formulations for simply supported and clamped plate problems, International Journal for Numerical Methods in Engineering 55 (3) pp 359– (2002) · Zbl 1098.74740
[28] Sladek, Meshless local Petrov-Galerkin (MLPG) method for shear deformable shells analysis, Computer Modeling in Engineering and Sciences 13 (2) pp 103– (2006) · Zbl 1232.74073
[29] Liu, Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory, Archive of Applied Mechanics 75 (4-5) pp 248– (2006) · Zbl 1119.74630
[30] Liu, Applications of point interpolation method for spatial general shells structures, Computer Methods in Applied Mechanics and Engineering 196 (9-12) pp 1633– (2007) · Zbl 1173.74476
[31] Liu, Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method, Composites Science and Technology 68 (2) pp 354– (2008)
[32] Liu, A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems, International Journal for Numerical Methods in Engineering 66 (7) pp 1153– (2006) · Zbl 1110.74871
[33] Morley, The constant-moment plate-bending element, Journal of Strain Analysis 6 (1) pp 20– (1971)
[34] Kolahi, A large-strain elasto-plastic shell formulation using the Morley triangle, International Journal for Numerical Methods in Engineering 52 pp 829– (2001) · Zbl 1017.74074
[35] Onate, Derivation of thin plate bending elements with one degree of freedom per node: a simple three node triangle, Engineering Computations 10 pp 543– (1993)
[36] Onate, Ratation-free triangular plate and shell elements, International Journal for Numerical Methods in Engineering 47 pp 557– (2000) · Zbl 0968.74070
[37] Cui, A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique, CMES-Computer Modeling in Engineering and Sciences 38 (3) pp 217– (2008) · Zbl 1357.74059
[38] Liu, A generalized gradient smoothing technique and smoothed bilinear form for Galerkin formulation of a wide class of computational methods, International Journal of Computational Methods 5 (2) pp 199– (2008) · Zbl 1222.74044
[39] Liu, A G space theory and weakened weak (W2) form for a unified formulation of compatible and incompatible methods, Part I: theory and Part II: applications to solid mechanics problems, International Journal for Numerical Methods in Engineering 2009 81 pp 1093– (2010)
[40] Liu, On G space theory, International Journal of Computational Methods 6 (2) pp 257– (2009) · Zbl 1264.74266
[41] Zhang, The upper boundary property for solid mechanics of linearly conforming radial point interpolation method (LC-RPIM), International Journal of Computational Methods 4 (3) pp 521– (2007) · Zbl 1198.74123
[42] Liu, Edge-based smoothed point interpolation methods, International Journal of Computational Methods 5 (4) pp 621– (2008) · Zbl 1264.74284
[43] Liu, A normed G space and weakened weak (W2) formulation of a cell-based smoothed point interpolation method, International Journal of Computational Methods 6 (1) pp 147– (2009) · Zbl 1264.74285
[44] Cui, A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids, Engineering Analysis with Boundary Elements 34 pp 144– (2010) · Zbl 1244.74214
[45] Hardy, Theory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968-1988), Computers and Mathematics with Applications 19 pp 163– (1990) · Zbl 0692.65003
[46] Ugural, Stresses in Plates and Shells (1981)
[47] Zienkiewicz, Solid Mechanics 2 (2000)
[48] Alwood, A polygonal finite element for plate bending problems using the assumed stress approach, International Journal for Numerical Methods in Engineering 1 pp 135– (1969)
[49] Dhatt G Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypothesis 255 278
[50] Jirousek, The hybrid-Trefftz finite element model and its application to plate bending, International Journal for Numerical Methods in Engineering 23 pp 651– (1986) · Zbl 0585.73135
[51] Abbassian, Free Vibration Benchmarks (1987)
[52] Gorman, Free Vibration Analysis of Rectangular Plates (1982) · Zbl 0478.73037
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