THE OPTIMIZATION OF RIBS POSITION BASED ON STIFFENED PLATES MESHLESS MODEL WITH NONLINEARITY
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摘要: 在加肋板无网格模型中, 肋条的位置对各种工况下加肋板受力性能的影响至关重要. 文章基于一阶剪切变形和移动最小二乘法理论提出一种考虑非线性影响的加肋板无网格模型, 并利用遗传算法优化肋条位置. 首先, 采用离散节点分别对平板和肋条进行离散, 得到加肋板的无网格离散模型; 其次, 通过冯·卡门大挠度理论得到非矩形板几何非线性问题的弯曲控制方程; 再次, 通过哈密顿原理得到加肋非矩形板自由振动问题的控制方程; 最后引入遗传算法, 以肋条的位置为设计变量、非矩形加肋板中心点挠度最小或自振频率最大为目标函数, 对肋条位置进行优化. 在考虑了几何非线性影响的肋条位置优化过程中, 肋条位置改变时只需重新计算位移转换矩阵, 避免了网格重构. 本文以全局荷载下单肋条菱形板为例与理论解进行对比, 进行有效性验证. 再以板的中点挠度最小和自振频率最大为优化目标, 对局部荷载作用下不同形状、不同肋条布置方式的加肋板进行优化, 分析方法的收敛性及稳定性.Abstract: In the stiffened plate's meshless model, the ribs' position is critical to the mechanical performance of the stiffened plate under various working conditions. Based on the first-order shear deformation theory and the moving-least square approximation, a meshless model of the stiffened non-rectangular plate considering nonlinearity is proposed and the position of the ribs is optimized based on the genetic algorithm. Firstly, the meshless model of the stiffened plate is obtained by discretizing the plate and ribs with discrete nodes. Secondly, the bending governing equation for the geometrically nonlinear problem of the stiffened non-rectangular plate is derived from the Von Karman large deflection theory. Then, the governing equation for the free vibration problem of the stiffened non-rectangular plate is derived from the Hamilton principle. Finally, the genetic algorithm is introduced with the position of the ribs as the design variable and the minimal deflection or the maximal natural frequency of the center point of the non-rectangular stiffened plate as the objective function to optimize the position of ribs. In the process of ribs' position optimization considering the influence of geometric nonlinearity, only the displacement transformation matrix needs to be recalculated when the ribs' position changed, and the mesh reconstruction is totally avoided.In this paper, first taking the single-rib rhombus plate under global load as an example, the comparison with the theoretical results is carried out and the validity of the method is verified. Then, taking the minimum center point deflection and the maximum natural frequency of the stiffened plate as the optimization objective, the stiffened plates with different shapes and different rib' arrangements under local load were optimized, and then the convergence and stability of the proposed method were studied.
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图 6 受面外均布力和集中力P作用的加肋圆板
Figure 6. A circular stiffened plate subject to an out-plane force and concentrated force P
图 7 受均布荷载作用的单肋菱形板
Figure 7. Single stiffener stiffened plate subjected to uniformly distributed load
图 8 均布荷载作用下本文解与理论解的对比
Figure 8. Comparison of the presented solution and the theoretical solution
图 9 均布荷载作用下单肋条优化迭代过程的种群分布(第9次计算结果)
Figure 9. Population distribution of single rib optimization iterative process under uniform load (the 9th result)
图 10 均布荷载作用下单肋条优化迭代过程的种群分布(第6次计算结果)
Figure 10. Population distribution of single rib optimization iterative process under uniform load (the 6th result)
图 12 局部荷载作用下单肋条优化迭代过程的种群分布
Figure 12. Population distribution of single rib optimization iterative process under local load
图 14 局部荷载作用下双肋条优化迭代过程的种群分布
Figure 14. Population distribution of double-rib optimization iterative process under local load
图 16 加肋圆板单肋条优化迭代过程的种群分布
Figure 16. Population distribution of single rib optimization iterative process of circular stiffened plate
图 18 加肋半圆板肋条优化迭代过程的种群分布
Figure 18. Population distribution of double-rib optimization iterative process of semicircular stiffened plate
表 1 均布荷载下加肋菱形板肋条位置10次优化结果
Table 1. Results of rib position optimization of the skew stiffened plate under uniformly distributed load
No. Present
results x/mDeflection/
mmTheoretical
results x/mDeflection/
mmRelative errors/
%1 0.757877 9.55306 0.75 9.35654 1.0502 2 0.746311 9.44857 0.75 9.35654 −0.4918 3 0.752845 9.42753 0.75 9.35654 0.3794 4 0.767312 9.82657 0.75 9.35654 2.3083 5 0.747522 9.41837 0.75 9.35654 −0.3304 6 0.524013 12.17842 0.75 9.35654 30.1333 7 0.751636 9.47291 0.75 9.35654 0.6219 8 0.764767 9.72500 0.75 9.35654 1.9690 9 0.754664 9.39735 0.75 9.35654 0.2181 10 0.744043 9.50517 0.75 9.35654 −0.7943 
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表 2 局部布荷载下双肋条位置优化结果
Table 2. Results of rib position optimization under local load
No. rib position x modpoint deflection/mm 1 0.631570 5.11805 2 0.670169 5.51728 3 0.669569 5.51653 4 0.646980 5.52643 5 0.752411 5.53346 6 0.665160 5.51104 7 0.694006 5.12778 8 0.650185 5.52243 9 0.667421 5.51386 10 0.662906 5.05741
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表 3 局部布荷载下双肋条位置优化结果
Table 3. Results of double ribs position optimization under local load
No. x1/m x2/m Midpoint deflection/mm 1 0.66152 0.65521 5.99517 2 0.75392 0.74956 6.06845 3 0.65161 0.60261 5.99052 4 0.66089 0.44267 6.45433 5 0.69129 0.75687 6.10183 6 0.65012 0.62133 5.97900 7 0.75985 0.69062 6.10205 8 0.75541 0.55256 7.05198 9 0.65195 0.68862 6.10291 10 0.70822 0.74912 6.11253
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表 4 加肋圆板单肋条位置优化结果
Table 4. Results of rib position optimization of circular stiffened plate
No. Rib position θ Base frequency/Hz 1 1.52898 67.30954 2 1.54031 67.43255 3 1.54906 67.52759 4 1.55704 67.61421 5 1.25632 65.27400 6 1.41489 66.07094 7 0.93944 61.49203 8 1.52631 67.28055 9 1.43149 66.25119 10 1.41976 66.12386
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表 5 加肋半圆板条位置优化结果
Table 5. Results of double ribs position optimization of semicircular stiffened plate
No. θ1/rad θ2/rad Base frequency/Hz 1 0.45601 0.44118 46.0592 2 0.57187 0.56237 42.4454 3 0.43837 0.43800 46.5886 4 0.38377 0.37132 48.1074 5 0.34707 0.33374 48.1441 6 0.31832 0.31030 47.9313 7 0.27696 0.26809 46.6987 8 0.36818 0.37342 48.1778 9 0.39179 0.40400 48.0054 10 0.33078 0.31961 48.0986
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[1] Yi K, Choi KK, Kim NH, et al. Continuum-based design sensitivity analysis and optimization of nonlinear shell structures using meshfree method. International Journal for Numerical Methods in Engineering, 2010, 68(2): 231-266 [2] 彭细荣, 隋允康. 考虑破损-安全的连续体结构拓扑优化ICM方法. 力学学报, 2018, 50(3): 11 (Peng Xirong, Sui Yunkang. A damage-safe continuum structure topology optimization ICM method. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(3): 11 (in Chinese) doi: 10.6052/0459-1879-17-366 [3] 陈炉云, 张裕芳. 基于遗传算法的复合材料结构-声辐射优化研究. 复合材料学报, 2012, 29(3): 203-207 (Chen Luyun, Zhang Yufang. Research on structure-acoustic radiation optimization of composite materials based on genetic algorithm. Acta Materiae Compositae Sinica, 2012, 29(3): 203-207 (in Chinese) doi: 10.13801/j.cnki.fhclxb.2012.03.033 [4] 李林远, 彭林欣. 基于无网格及混合遗传算法的矩形加肋板肋条布置优化. 应用力学学报, 2018, 35(6): 7 (Li Linyuan, Peng Linxin. Optimization of rib layout of rectangular ribbed slab based on meshless and hybrid genetic algorithm. Chinese Journal of Applied Mechanics, 2018, 35(6): 7 (in Chinese) [5] Wodesenbet E, Kidane S, Pang SS. Optimization for buckling loads of grid stiffened composite panels. Composite Structures, 2003, 60(2): 159-169 doi: 10.1016/S0263-8223(02)00315-X [6] 李琳, 石峰, 项松等. 基于遗传算法和复合二次径向基函数的复合材料层合板自由振动分析. 工程数学学报, 2022, 39(2): 319-329Li Lin, Shi Feng, Xiang Song, et al. Free vibration analysis of composite laminates based on genetic algorithm and compound quadratic radial basis function. Chinese Journal of Engineering Mathematics, 2022, 39(2): 319-329 (in Chinese)) [7] 王选, 刘宏亮, 龙凯等. 基于改进的双向渐进结构优化法的应力约束拓扑优化. 力学学报, 2018, 50(2): 385-397Wang Xuan, Liu Hongliang, Long Kai, et al. Stress constrained topology optimization based on improved bidirectional progressive structural optimization method. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 385-397 (in Chinese)) [8] 米大海, 杨睿, 周亮等. 以频率为目标的加筋平板结构优化设计研究. 机械强度, 2013, 35(2): 4 (Mi Dahai, Yang Rui, Zhou Liang, et al. Research on optimization design of stiffened plate structure with frequency as target. Journal of Mechanical Strength, 2013, 35(2): 4 (in Chinese) doi: 10.16579/j.issn.1001.9669.2013.02.002 [9] Afonso S, Sienz J, Belblidia F. Structural optimization strategies for simple and integrally stiffened plates and shells. Engineering Computations, 2005, 22(4): 429-452 doi: 10.1108/02644400510598769 [10] 王博, 郝鹏, 田阔. 加筋薄壳结构分析与优化设计研究进展. 计算力学学报, 2019, 36(1): 12 (Wang Bo, Hao Peng, Tian Kuo. Research progress on the analysis and optimization design of stiffened thin shell structures. Chinese Journal of Computational Mechanics, 2019, 36(1): 12 (in Chinese) [11] 文立中. 偏压作用下钢木组合柱非线性分析及截面优化设计. [博士论文] 中南林业科技大学, 2021Wen Lizhong. Nonlinear analysis and section optimization design of steel-wood composite column under eccentric pressure. [PhD. Thesis] Central South University of Forestry and Technology, 2021 (in Chinese)) [12] 周俊文, 刘界鹏. 基于多种群遗传算法的钢框架结构优化设计. 土木与环境工程学报, 2022, 44(6): 1-11Zhou Junwen, Liu Jiepeng. Optimal design of steel frame structure based on multiple population genetic algorithm. Journal of Civil and Environmental Engineering, 2022, 44(6): 1-11 (in Chinese) [13] 王栋, 李正浩. 薄板结构加筋布局优化设计方法研究. 计算力学学报, 2018, 35(2): 6 (Wang Dong, Li Zhenghao. Research on optimization design method of reinforcement layout of thin plate structure. Chinese Journal of Computational Mechanics, 2018, 35(2): 6 (in Chinese) [14] Csonka B, Kozák I, Soares C, et al. Shape optimization of axisymmetric shells using a higher order shear deformation theory. Structural Optimization, 1995, 9(2): 117-127 doi: 10.1007/BF01758828 [15] T, Belytschko, Y, et al. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256 [16] 张雄, 宋康祖, 陆明万. 无网格法研究进展及其应用. 计算力学学报, 2001, 20(6): 730-742Zhang Xiong, Song Kangzu, Lu Mingwan. Research progress and application of meshless method.Chinese Journal of Computational Mechanics, 2001, 20(6): 730-742 (in Chinese) [17] 邓立克, 王东东, 王家睿等. 薄板分析的线性基梯度光滑伽辽金无网格法. 力学学报, 2019, 51(3): 690-702 (Deng Like, Wang Dongdong, Wang Jiarui, et al. A gradient smoothing Galerkin meshfree method for thin plate analysis with linear basis function. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 690-702 (in Chinese) doi: 10.6052/0459-1879-19-004 [18] 覃霞, 刘珊珊, 吴宇等. 平行四边形加肋板自由振动分析的无网格法. 工程力学, 2019, 36(3): 24-32, 39 (Qin Xia, Liu Shanshan, Wu Yu, et al. The meshless method for free vibration analysis of parallelogram ribbed plates. Engineering Mechanics, 2019, 36(3): 24-32, 39 (in Chinese) [19] 张雄, 刘岩, 马上. 无网格法的理论及应用. 力学进展, 2009, 39(1): 1-36 (Zhang Xiong, Liu Yan, Ma shang. Theory and application of meshless method. Advances in Mechanics, 2009, 39(1): 1-36 (in Chinese) doi: 10.6052/1000-0992-2009-1-J2008-010 [20] 张建平, 刘庭显, 龚曙光等. 基于无网格法的热力耦合周期性结构多目标拓扑优化. 机械工程学报, 2022, 58(14): 223-232 (Zhang Jianping, Liu Tingxian, Gong Shuguang, et al. Multi-objective topology optimization of thermomechanical coupled periodic structures based on meshless method. Journal of Mechanical Engineering, 2022, 58(14): 223-232 (in Chinese) [21] 彭林欣. 矩形加肋板线性弯曲分析的移动最小二乘无网格法. 计算力学学报, 2012, 29(2): 210-216 (Peng Linxin. Moving least squares meshless method for linear bending analysis of rectangular ribbed plates. Chinese Journal of Computational Mechanics, 2012, 29(2): 210-216 (in Chinese) doi: 10.7511/jslx201202011 [22] 宋超, 彭慧凯, 王东东. 薄板壳屈曲分析的埃尔米特无网格法//中国计算力学大会2014暨第三届钱令希计算力学奖颁奖大会论文集, 2014Song Chao, Peng Huikai, Wang Dongdong. Hermitian meshless method for buckling analysis of thin shells//Proceedings of the China Computational Mechanics Conference 2014 and the 3rd Qian Lingxi Computational Mechanics Award Conference, 2014 (in Chinese) [23] 张驰, 校金友, 张硕. 用无网格法分析功能梯度材料圆板的自由振动. 科学技术与工程, 2014, 13: 6 (Zhang Chi, Xiao Jinyou, Zhang Shuo. Analysis of free vibration of circular plates with functionally gradient materials by meshless method. Science Technology and Engineering, 2014, 13: 6 (in Chinese) [24] 杨柳, 彭建设. 解平行四边形板弯曲问题的GD法. 成都大学学报: 自然科学版, 2014, 33(3): 4 (Yang Liu, Peng Jianshe. GD method for solving parallelogram plate bending problem. Journal of Chengdu University (Natural Science Edition) , 2014, 33(3): 4 (in Chinese) [25] 马文涛. 二维弹性力学问题的光滑无网格伽辽金法. 力学学报, 2018, 50(5): 1115-1124 (Ma Wentao. A smoothed meshfree Galerkin method for 2 D elasticity problem. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(5): 1115-1124 (in Chinese) doi: 10.6052/0459-1879-18-135 [26] 方电新, 李卧东, 王元汉等. 用无网格法计算平板弯曲问题. 岩土力学, 2001, 22(3): 3 (Fang Lixin, Li Wodong, Wang Yuanhan, et al. Calculation of plate bending problems using the meshless method. Rock and Soil Mechanics, 2001, 22(3): 3 (in Chinese) doi: 10.3969/j.issn.1000-7598.2001.03.026 [27] 曾军才, 王久法, 姚望等. 正交各向异性矩形板的自由振动特性分析. 振动与冲击, 2015, 34(24): 123-127 (Zeng Juncai, Wang Jiufa, Yao Wang, et al. Analysis of free vibration characteristics of orthotropic rectangular plates. Journal of Vibration and Shock, 2015, 34(24): 123-127 (in Chinese) doi: 10.13465/j.cnki.jvs.2015.24.021 [28] Zhou L, Zheng WX. Vibration of skew plates by the MLS-Ritz method. International Journal of Mechanical Sciences, 2008, 50(7): 1133-1141 doi: 10.1016/j.ijmecsci.2008.05.002 [29] Reddy JN. Theory and Analysis of Elastic Plates. London: Taylor and Francis, 1999 [30] 彭林欣. 加肋板自由振动的移动最小二乘无单元分析. 振动与冲击, 2011, 30(6): 67-73 (Peng Linxin. Moving least squares element-free analysis of free vibration of ribbed plates. Journal of Vibration and Shock, 2011, 30(6): 67-73 (in Chinese) doi: 10.3969/j.issn.1000-3835.2011.06.015 [31] Chen JS, Pan CH, Wu CT, et al. Reproducing Kernel particle methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1): 195-227 [32] Peng LX, Kitipornchai S, Liew KM. Free vibration analysis of folded plate structures by the FSDT meshfree method. Computational Mechanics, 2007, 39(6): 799-814 doi: 10.1007/s00466-006-0070-9 [33] 彭林欣. 折板结构非线性弯曲分析的移动最小二乘无网格法. 工程力学, 2011, 28(12): 126-132 (Peng Linxin. Moving least squares meshless method for nonlinear bending analysis of folded plate structures. Engineering Mechanics, 2011, 28(12): 126-132 (in Chinese) [34] Qin X, Shen Y, Chen W, et al. Bending and free vibration analyses of circular stiffened plates using the FSDT mesh-free method. International Journal of Mechanical Sciences, 2021(202-203): 106498 [35] Goldberg DE. Genetic algorithms in search, optimization, and machine learning//Ethnographic Praxis in Industry Conference Proceedings, 1988 -
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