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## Study on derivation and optimization algorithm about thin plate bending large deformation higher-order nonlinear partial differential equations

Hou Xiang-Lin, Zheng Xi-Jian, Zhang Liang, Liu Tie-Lin
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• #### 摘要

针对薄板弯曲大变形问题, 运用变分原理, 建立了薄板弯曲大变形问题的高阶非线性偏微分方程. 运用有限差分法和动态设计变量优化算法原理, 以离散坐标点的上未知挠度为设计变量, 以离散坐标点的差分方程组构建目标函数, 提出了薄板弯曲大变形挠度求解的动态设计变量优化算法, 编制了相应的优化求解程序. 分析了具有固定边界、均布载荷下的矩形薄板挠度的典型算例. 通过与有限元的结果对比, 表明了本文求解算法的有效性和精确性, 提供了直接求解实际工程问题的基础.

#### Abstract

For a thin plate bending large deformation problem, variational principle is applied, and higher-order nonlinear partial differential equations about thin plate bending large deformation is established. Based on difference method and dynamic design variable optimization method, making unknown deflection of discrete coordinate points as design variables, differential equations sets of the discrete coordinates points as building objective function, a dynamic design variable optimization algorithm for computing thin plate bending deflection is proposed. Universal computing program is designed. Practical example about rectangular thin plate with fixed boundary under uniform load is analyzed. Comparing the program computing result with finite element solution. Effectiveness and feasibility of the method are verified. This method can be used to solve engineering problem.

#### 作者及机构信息

###### 1. 沈阳建筑大学交通与机械工程学院, 沈阳 110168; 2. 中国科学院力学研究所, 北京 100080; 3. 沈阳建筑大学土木工程学院, 沈阳 110168
• 基金项目: 国家自然科学基金(批准号: 10972144);辽宁省自然科学基金(批准号: 201102181)和辽宁省教育厅科学研究项目(批准号: L2010445)资助的课题.

#### Authors and contacts

###### 1. School of Traffic and Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China; 2. Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China; 3. School of Civil Engineering, Shenyang Jianzhu University, Shenyang 110168, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 10972144), the Natural Science Foundation of Liaoning Province, China (Grant No. 201102181), and Science Research Projects of Education Department of Liaoning Province, China (Grant No. L2010445).

#### 参考文献

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#### 施引文献

•  [1] Timoshenk S, Woinowsky-Krieger S 1959 Theory of Plates and Shells (Now York: McGraw-Hill) [2] Alzheimer W E, Davis R T 1968 J. Eng. Mech. Div. Proc. ASCE 4 905 [3] Xu Z L 2006 Elasticity (4th Ed.) (Beijing: Higher Education Press) pp149-150 (in Chinese) [徐芝纶 2006 弹性力学(第4版) (北京: 高等教育出版社) 第149-150页)] [4] Huang Z Y 1957 Acta Phys. Sin. 13 312 (in Chinese) [黄泽言 1957 物理学报 13 312] [5] Qian W C, Ye K Y 1954 Acta Phys. Sin. 10 209 (in Chinese) [钱伟长, 叶开沅 1954 物理学报 10 209] [6] Hu H C 1955 Acta Phys. Sin. 11 19 (in Chinese) [胡海昌 1955 物理学报 11 19] [7] Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜, 唐驾时 2004 物理学报 53 2828] [8] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573 [9] Yang B, Ding H J, Chen W Q 2008 Appl. Math. Mech. 29 9994 [10] Huang J J 2009 Chin. Phys. 18 B 3616 [11] Eerdun Buhe, Temuer Chaolu 2012 Chin. Phys. B 21 035201 [12] Xu H, Chen L H, Mo J Q 2010 Acta Phys. Sin. 60 100201 (in Chinese) [徐惠, 陈丽华, 莫嘉琪 2011 物理学报 60 100201] [13] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504 [14] Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, New York: Chapman & Hall/CRC) pp1-2 [15] Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663] [16] Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林, 刘铁林, 翟仲海 2011 物理学报 60 090202] [17] Hou X L, Zhai Z H, Zheng L, Liu T L 2012 Acta Phys. Sin. 61 010201 (in Chinese) [侯祥林, 翟中海, 郑莉, 刘铁林 2012 物理学报 61 010201] [18] Lao D Z 2007 Variational Method Basis (2nd Ed.) (National Defence Industry Press) pp74-91 (in Chinese) [老大中 2007 变分法基础(第2版) (北京: 国防工业出版社) 第74-91页]
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##### 出版历程
• 收稿日期:  2012-02-21
• 修回日期:  2012-03-10
• 刊出日期:  2012-09-05

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