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A linearly conforming radial point interpolation method (LC-RPIM) for shells. (English) Zbl 1162.74512
Summary: In this paper, a linearly conforming radial point interpolation method (LC-RPIM) is presented for the linear analysis of shells. The first order shear deformation shell theory is adopted, and the radial and polynomial basis functions are employed to construct the shape functions. A strain smoothing stabilization technique for nodal integration is used to restore the conformability and to improve the accuracy. Convergence studies are performed in terms of the number of nodes and the nodal distribution patterns, including the regular distribution and the irregular distribution. Comparisons are made with the existing results available in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach provides very stable and accurate results and effectively eliminates the membrane locking and shear locking in shell problems.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74K25 Shells
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##### References:
 [1] Hughes TJR, Liu WK (1981) Nonlinear finite element analysis of shells: Part I. Three-dimensional shells. Comput Methods Appl Mech Eng 26: 331–362 · Zbl 0461.73061 · doi:10.1016/0045-7825(81)90121-3 [2] Hughes TJR, Liu WK (1981) Nonlinear finite element analysis of shells: Part II. Two-dimensional shells. Comput Methods Appl Mech Eng 27: 167–181 · Zbl 0474.73093 · doi:10.1016/0045-7825(81)90148-1 [3] Belytschko T, Stolarski H (1985) Stress projection for membrane and shear locking in shell elements. Comput Methods Appl Mech Eng 51: 221–258 · Zbl 0581.73091 · doi:10.1016/0045-7825(85)90035-0 [4] Liu WK, Law ES, Lam D, Belytschko T (1986) Resultant-stress degenerated-shell element. Comput Methods Appl Mech Eng 55: 259–300 · Zbl 0587.73113 · doi:10.1016/0045-7825(86)90056-3 [5] Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrical exact shell model. Part II: The linear theory. Comput Methods Appl Mech Eng 73: 53–92 · Zbl 0724.73138 · doi:10.1016/0045-7825(89)90098-4 [6] Crisfield M (1986) Finite elements on solution procedure s for structural analysis, I. Linear analysis. Pineridge Press, Swansea · Zbl 0605.73062 [7] Reddy JN, Liu FL (1985) A higher-order shear deformation theory of laminated elastic shells. Int J Eng Sci 23: 319–330 · Zbl 0559.73072 · doi:10.1016/0020-7225(85)90051-5 [8] Krysl P, Belytschko T (1996) Analysis of thin shells by the element-free Galerkin method. Int J Solids Struct 33: 3057–3080 · Zbl 0929.74126 · doi:10.1016/0020-7683(95)00265-0 [9] Noguchi H, Kawashima T, Miyamura T (2000) Element free analyses of shell and spatial structures. Int J Numer Methods Eng 47: 1215–1240 · Zbl 0970.74079 · doi:10.1002/(SICI)1097-0207(20000228)47:6<1215::AID-NME834>3.0.CO;2-M [10] Li S, Hao W, Liu WK (2000) Numerical simulations of large deformation of thin shell structures using meshfree methods. Comput Mech 25: 102–116 · Zbl 0978.74087 · doi:10.1007/s004660050463 [11] Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method. Part I: Theoretical formulation. Comput Methods Appl Mech Eng 193: 933–951 · Zbl 1060.74670 · doi:10.1016/j.cma.2003.12.001 [12] Li S, Lu H, Han W, Liu WK, Simkins DC Jr. (2004) Reproducing kernel element method. Part II: Globally conforming I m /C n hierarchies. Comput Methods Appl Mech Eng 193: 953–987 · Zbl 1093.74062 · doi:10.1016/j.cma.2003.12.002 [13] Simkins DC Jr., Li S, Lu H, Liu WK (2004) Reproducing kernel element method. Part IV: Globally compatible C n (n triangular hierarchy. Comput Methods Appl Mech Eng 193: 1013–1034 · Zbl 1093.74064 · doi:10.1016/j.cma.2003.12.004 [14] Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139: 49–74 · Zbl 0918.73329 · doi:10.1016/S0045-7825(96)01079-1 [15] Bonet J, Kulasegaram S (1999) Correction and stabilization of smoothed particle hydrodynamics methods with applications in metal forming simulations. Int J Numer Methods Eng 47: 1189–1214 · Zbl 0964.76071 · doi:10.1002/(SICI)1097-0207(20000228)47:6<1189::AID-NME830>3.0.CO;2-I [16] Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50: 435–466 · Zbl 1011.74081 · doi:10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A [17] Wang DD, Chen JS (2004) Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput Methods Appl Mech Eng 193: 1065–1083 · Zbl 1060.74675 · doi:10.1016/j.cma.2003.12.006 [18] Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 0: 937–951 · Zbl 1050.74057 · doi:10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X [19] Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54: 1623–1648 · Zbl 1098.74741 · doi:10.1002/nme.489 [20] Liu GR, Dai KY, Lim KM, Gu YT (2003) A radial point interpolation method for simulation of two-dimensional piezoelectric structures. Smart Mater Struct 12: 171–180 · doi:10.1088/0964-1726/12/2/303 [21] Zhang GY, Liu GR, Wang YY, Huang HT, Zhong ZH, Li GY, Han X (2005) A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. Int J Numer Methods Eng (revised) [22] Liu GR, Li Y, Dai KY, Luan MT, Xue W (2006) A linearly conforming RPIM for solids mechanics problems. Int J Comput Methods (in press) [23] Reddy JN, Arciniega RA (2004) Shear deformation plate and shell theories: from Stavsky to present. Mech Adv Mater Struct 11: 535–582 · doi:10.1080/15376490490452777 [24] Reddy JN, Miravete A (1995) Practical analysis of composite laminates. CRC Press, Boca Raton [25] Koziey BL, Mirza FA (1997) Consistent thick shell element. Comput Struct 65: 531–549 · Zbl 0936.74519 · doi:10.1016/S0045-7949(96)00414-2 [26] Liu WK, Hu Y, Belytschko T (1994) Multiple quadrature underintegrated finite elements. Int J Numer Methods Eng 37: 3263–3289 · Zbl 0811.73063 · doi:10.1002/nme.1620371905 [27] Liu WK, Guo Y, Tang S, Belytschko T (1998) A multiple-quadrature eight-node hexahedral finite element for large deformation elastic analysis. Comput Methods Appl Mech Eng 154: 69–132 · Zbl 0935.74068 · doi:10.1016/S0045-7825(97)00106-0 [28] Brebbia, C, Connor J (1969) Geometrically nonlinear finite element analysis. J Eng Mech 463–483 [29] Palazotto AN, Dennis ST (1992) Nonlinear analysis of shell structures. AIAA, Washington DC · Zbl 0798.73003 [30] Reddy JN (2004) Mechanics of laminated composite plates and shells: theory and analysis. 2nd edn. CRC press, West Palm Beach [31] Varadan TK, Bhaskar K (1991) Bending of laminated orthotropic cylindrical shells-An elastic approach. Compos Struct 17: 141–156 · doi:10.1016/0263-8223(91)90067-9 [32] Vlasov VZ (1964) General theory of shells and its applications in engineering (translation of Obshchaya teoriya obolocheck I yeye prilozheniya v tekhnike) NASA TT F-99, National Aeronautics and Space Administration, Washington, DC
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