Bending and vibration analysis of an arbitrary shell by the moving-least square meshfree method

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Journal of Vibration and Shock ›› 2022, Vol. 41 ›› Issue (16) : 125-134.

CHEN Wei1，YANG Jiansheng1，WEI Dongyan1，SHEN Yajing1，PENG Linxin1,2,3

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Abstract

A moving-least square meshfree method for linear bending and free vibration of arbitrary shell structures is proposed in this paper. By using the mapping technology in conjunction with the Mindlin’s theory of plates and shells, the 3D arbitrary shell parametric surface is converted into a 2D meshfree model. Based on the moving-least square (MLS) approximation and the first-order shear deformation theory, the displacement field of arbitrary shell is obtained, and the bending and free vibration control equations are obtained by using the minimum potential energy principle and Hamilton principle, respectively. Because the essential boundary conditions cannot be imposed directly, the full transformation method is used to introduce the essential boundary conditions. At the end of this paper, several examples of different shape shell structures show that the solutions in this paper are in good agreement with the theoretical solutions or ABAQUS finite element solutions, which validate the effectiveness and accuracy of this method in calculating the linear bending and free vibration of arbitrary shells.
Keywords: mapping technology; arbitrary shells; moving-least squares; meshfree method; full transformation method

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mapping technology / arbitrary shells / moving-least squares / meshfree method / full transformation method

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CHEN Wei1，YANG Jiansheng1，WEI Dongyan1，SHEN Yajing1，PENG Linxin1,2,3. Bending and vibration analysis of an arbitrary shell by the moving-least square meshfree method[J]. Journal of Vibration and Shock, 2022, 41(16): 125-134

References

[1] 孙仁傅，王玳瑜，邹定祺. 局部竖向荷载作用下圆柱形薄壳的解析解[J]. 应用数学和力学, 1990(2): 179-188.
SUN Renbo, WANG Daiyu, ZOU Dingqi. Calculation for cylindrical shell under local vertical loadings[J]. Applied Mathematics and Mechanics, 1990 (2): 179-188.
[2] 赵建波，白象忠，司秀勇. 非轴对称载荷作用下圆柱薄壳的解析解[J]. 机械强度, 2014, 36(2): 290-294.
ZHAO Jianbo, BAI Xiangzhong, SI Xiuyong. An analytic solution of length-limited thin-walled cylindrical shells under non-axisymmetric uniform line load[J]. Journal of Mechanical Strength, 2014, 36(2): 290-294.
[3] 吴国伟，王方. 半圆柱壳挠曲问题解析解精确性的研究[J]. 力学季刊, 2018, 39(4): 804-811.
WU Guowei, WANG Fang. Study on the accuracy of the analytical solution for the bending problem of semi-cylindrical shell [J]. Chinese Quarterly of Mechanics, 2018, 39(4): 804-811.
[4] 骆东平. 两曲边为简支的圆柱壳块振动频率的精确解[J]. 固体力学学报, 1985 (4): 470-478.
LUO Dongping. Exact solution for the vibrations of circular cylindrical shell panels with freely supported curved edges[J]. Chinese Journal of Solid Mechanics, 1985 (4): 470-478.
[5] LU D， YANG T J, DU J T, et al. An exact series solution for the vibration analysis of cylindrical shells with arbitrary boundary conditions[J]. Applied Acoustics, 2013, 74(3): 440-449.
[6] RUI L, ZHENG X R, YANG Y S, et al. Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels[J]. Applied Mathematical Modelling, 2019, 76: 900-917.
[7] 陈家瑾. 四边固支球面扁壳的解析解[J]. 应用力学学报, 1994(4): 133-137.
CHEN Jiajin. The analytic solution of the spherical shallow shell under the four side clamped boundary condition[J]. Chinese Journal of Applied Mechanics,1994(4):133-137.
[8] RUI L, ZHOU C, ZHENG X R. On new analytic free vibration solutions of doubly curved shallow shells by the symplectic superposition method within the hamiltonian-system framework[J]. Journal of Vibration And Acoustics-Transactions of The ASME, 2021, 143:0110021.
[9] AHMAD S，IRONS B M，ZIENKIEWICZ O C. Analysis of thick and thin shell structures by curved finite elements[J]. International Journal for Numerical Methods in Engineering, 1970, 2(3): 419-451.
[10] 姚振汉. 真实梁板壳局部应力分析的高性能边界元法[J]. 工程力学, 2015, 32(8): 8-15.
YAO Zhenhan. High-performance boundary element method for the local stress analysis of real beam, plate and shell[J]. Engineering Mechanics, 2015, 32(8): 8-15.
[11] LI J H, SHI Z Y, LIU L Y. A scaled boundary finite element method for static and dynamic analyses of cylindrical shells[J]. Engineering Analysis With Boundary Elements, 2019, 98: 217-231.
[12] BENSON D J, BAZILEVS Y, HSU M C, et al. Isogeometric shell analysis: the Reissner-Mindlin shell[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5/6/7/8): 276-289.
[13] 张升刚，王彦伟，黄正东. 等几何壳体分析与形状优化[J]. 计算力学学报, 2014, 31(1): 115-119.
ZHANG Shenggang, WANG Yanwei, HUANG Zhengdong. Isogeometric shell analysis and shape optimization[J]. Chinese Journal of Computational Mechanics, 2014, 31(1): 115-119.
[14] TANAKA S, SHOGO S, MICHIYA I, et al. Analysis of dynamic stress concentration problems employing spline-based wavelet Galerkin method[J]. Engineering Analysis With Boundary Elements, 2015, 58129-139.
[15] KRYSL P, TED B. Analysis of thin shells by the Element-Free Galerkin method[J]. International Journal of Solids & Structures, 1996, 33(20/21/22): 3057-3080.
[16] NOGUCHI H, TETSUYA K, TOMOSHI M. Element free analyses of shell and spatial structures[J]. International Journal for Numerical Methods in Engineering, 2000, 47(6): 1215-1240.
[17] LI S, HAO W, LIU W K. Numerical simulations of large deformation of thin shell structures using meshfree methods[J]. Computational Mechanics, 2000, 25(2/3): 102-116.
[18] LIU L, LIU G R, TAN V B C. Element free method for static and free vibration analysis of spatial thin shell structures[J]. Computer Methods in Applied Mechanics & Engineering, 2002, 191(51/52): 5923-5942.
[19] LIU L, CHUA L P, GHISTA D N. Element-free Galerkin method for static and dynamic analysis of spatial shell structures[J]. Journal of Sound & Vibration, 2006, 295(1/2): 388-406.
[20] SIMO J C, FOX D D. On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization[J]. Computer Methods in Applied Mechanics and Engineering, 1989, 72(3): 267-304.
[21] KIM N H, CHOI K K, CHEN J S, et al. Meshfree analysis and design sensitivity analysis for shell structures[J]. International Journal for Numerical Methods In Engineering, 2002, 53(9): 2087-2116.
[22] CHEN J S, WANG D D. A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates[J]. International Journal for Numerical Methods in Engineering, 2006, 68(2): 151-172.
[23] SAYAKOUMMANE V, KANOK-NUKULCHAI W. A meshless analysis of shells based on moving kriging interpolation[J]. International Journal of Computational Methods, 2007, 4(4): 543-565.
[24] JARAK T, SORIC J, HOSTER J. Analysis of shell deformation responses by the meshless local Petrov-Galerkin (MLPG) approach[J]. Cmes-Computer Modeling in Engineering & Sciences, 2007, 18(3): 235-246.
[25] ??SORIC J, JARAK T. Meshless local petrov galerkin (mlpg) formulations for analysis of shell-like structures[J]. Computational Method in Applied Sciences, 2009: 277-289.
[26] COSTA J C, PIMENTA P M, WRIGGERS P. Meshless analysis of shear deformable shells: boundary and interface constraints[J]. Computational Mechanics, 2016, 57(4): 679-700.
[27] 李迪，林忠钦，李淑慧. 无网格局部Petrov-Galerkin法求解板壳弹塑性大变形[J]. 应用力学学报, 2010, 27(1): 39-43.
LI Di, LIN Zhongqin, LI Shuhui. Research on elastic-plastic large deformation of plates and shells by meshless local Petrov-Galerkin method[J]. Chinese Journal of Applied Mechanics, 2010, 27(1): 39-43.
[28] 李迪，林忠钦，李淑惠，等. 壳结构的无网格局部Petrov-Galerkin方法[J]. 计算力学学报, 2009, 26(4): 505-509.
LI Di, LIN Zhongqin, LI Shuhui, et al.Meshless local Petrov-Galerkin analyses of shell structure[J]. Chinese Journal of Applied Mechanics, 2009, 26(4): 505-509.
[29] 叶翔. 无网格法在板壳计算中的应用[D]. 南昌：南昌大学, 2005.
[30] 王砚. 无网格方法在结构振动中的应用[D]. 西安：西安理工大学, 2006.
[31] 陈靠伟. 圆柱壳自由振动的无网格方法[J]. 水利与建筑工程学报, 2011, 9(2): 110-112.
CHEN Kaowei. Free vibration analysis for cylindrical shell structures by element-free method[J]. Journal of Water Resources and Architectural Engineering, 2011, 9(2): 110-112.
[32] 曹阳，陈莹婷，姚林泉. 无单元Galerkin方法施加本质边界条件研究进展[J]. 力学季刊, 2020, 41(4): 591-612.
CAO Yang, CHEN Yingting, YAO Linquan. Advances in implementation of essential boundary conditions for element-free Galerkin method[J]. Chinese Quarterly of Mechanics, 2020, 41(4): 591-612.
[33] BELYTSCHKO T, LU Y Y, GU L. Element‐free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 2010, 37(2): 229-256.
[34] REDDY J N. Theory and analysis of elastic plates and shells[M]. London: Taylor & Francis, 1996.
[35] CHEN J S, PAN C, WU C T, et al. Reproducing kernel particle methods for large deformation analysis of non-linear structures[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1/2/3/4): 195-227.
[36] TIMOSHENKO S .Theory of plates and shells[M]. New York ：McGraw-HillBook Company, 1959.
[37] 彭细荣. 有限单元法及其应用[M]. 北京: 清华大学出版社, 2012.