Numerical Analysis of Geometrically Nonlinear Problems Based on the S-R Decomposition Theorem
-
摘要: 为了探究几何非线性问题的数值求解方法,采用理论推导、MATLAB编程计算、有限元模拟相结合的方法,基于S-R和分解定理及更新拖带坐标描述法,运用插值型无单元Galerkin方法对几何非线性问题的增量变分方程进行了推导,并通过四点Gauss积分法和不动点迭代法对其进行求解.最后以平面悬臂梁的大变形问题为例进行求解计算,发现与ANSYS的计算结果拟合相似度很高,说明了所采用的几何非线性力学理论及数值计算方法的正确性和合理性,为求解几何非线性问题提供了一种新的依据.
-
关键词:
- 几何非线性问题 /
- S-R和分解定理 /
- 更新拖带坐标法 /
- 插值型无单元Galerkin法
Abstract: To explore the numerical solution method for geometrically nonlinear problems, the theoretical derivation, the MATLAB programming and the finite element simulation were used together. Based on the S-R decomposition theorem, the interpolated element-free Galerkin method was applied to deduce the incremental variational equations through the updated co-moving coordinate formulation, which were solved with the 4-point Gauss integration method and the fixed point iteration method. Finally, the large deformations of exemplary elastic and elastoplastic planar cantilever beams were calculated and the results agreed well with those from the ANSYS simulation. The examples illustrate the correctness and rationality of the proposed geometrically nonlinear mechanics theory and the present numerical calculation method. The work provides a new basis for the solutions to geometrically nonlinear problems. -
[1] 张龙飞, 胡全星. 固体火箭发动机试车架中板簧弹阻力有限元计算分析方法[J]. 火炮发射与控制学报, 2015,36(4): 50-54.(ZHANG Long-fei, HU Quan-xing. The finite element method for calculating elastic resistance of plate spring in SRM test frame[J]. Journal of Gun Launch & Control,2015,36(4): 50-54.(in Chinese)) [2] 李明霞, 董联杰. 层状反倾边坡变形特征及影响因素分析[J]. 计算力学学报, 2015,32(6): 831-837.(LI Ming-xia, DONG Lian-jie. Analysis on influential factors and deformation characteristics of toppling slope[J]. Chinese Journal of Computational Mechanics,2015,32(6): 831-837.(in Chinese)) [3] 秦勇, 邱爱慈, 张永民. 高聚能重复强脉冲波煤储层增渗新技术试验与探索[J]. 煤炭科学与技术,2014,42(6): 1-7, 70.(QIN Yong, QIU Ai-ci, ZHANG Yong-min. Experiment and discovery on permeability improved technology of coal reservoir based on repeated strong pulse waves of high energy accumulation[J]. Coal Science and Technology,2014,42(6): 1-7, 70.(in Chinese)) [4] 肖舒敏, 闫云聚, 姜波澜. 基于小波神经网络方法的桥梁结构损伤识别研究[J]. 应用数学和力学,2016,37(2): 149-159.(XIAO Shu-min, YAN Yun-ju, JIANG Bo-lan. Damage identification for bridge structures based on the wavelet neural network method[J]. Applied Mathematics and Mechanics,2016,37(2): 149-159.(in Chinese)) [5] 许进升, 杨晓红, 赵磊, 等. 聚合物时温等效模型有限元应用研究[J]. 应用数学和力学, 2015,36(5): 539-547.(XU Jin-sheng, YANG Xiao-hong, ZHAO Lei, et al. Finite element application of the time-temperature superposition principle (TTSP) to polymer[J]. Applied Mathematics and Mechanics,2015,36(5): 539-547.(in Chinese)) [6] 宋天霞, 邹时智, 杨文兵. 非线性结构有限元计算[M]. 武汉: 华中理工大学出版社, 1996.(SONG Tian-xia, ZOU Shi-zhi, YANG Wen-bing. The Finite Element Calculation of Nonlinear Structure[M]. Wuhan: Huazhong University of Science and Technology Press, 1996.(in Chinese)) [7] 沈亚鹏, 薛奇. 平面粘弹性大变形问题的研究[J]. 上海交通大学学报, 1990,24(5/6): 7-15.(SHEN Ya-peng, XUE Qi. Research of viscoelastic large deformation plane problems[J]. Journal of Shanghai Jiaotong University,1990,24(5/6): 7-15.(in Chinese)) [8] 李平. 非线性大变形有限元分析的更新拖带坐标法及其应用[D]. 博士学位论文. 北京: 中国矿业学院北京研究生部, 1991.(LI Ping. The updated co-moving coordinate formulation for the nonlinear large deformation finite element analysis and application[D]. PhD Thesis. Beijing: Graduate school of China University of Mining and Technology, 1991.(in Chinese)) [9] 罗丹. 基于S-R和分解定理的几何非线性问题的无网格Galerkin法分析[D]. 硕士学位论文. 长沙: 湖南大学, 2011.(LUO Dan. Based on S-R decomposition theorem analysis element free Galerkin method on geometric nonlinear problems[D]. Master Thesis. Changsha: Hunan University, 2011.(in Chinese)) [10] 陈芳祖, 罗丹. 基于S-R和分解定理的无网格Galerkin法求解几何非线性问题[J]. 湖南大学学报(自然科学版), 2012,39(1): 42-46.(CHEN Fang-zu, LUO Dan. Element free Galerkin method for geometrically nonlinear problems based on the S-R decomposition theorem[J]. Journal of Hunan University(Natural Sciences),2012,39(1): 42-46.(in Chinese)) [11] 陈至达. 理性力学[M]. 重庆: 重庆出版社, 1999.(CHEN Zhi-da. Rational Mechanics [M]. Chongqing: Chongqing Press, 1999.(in Chinese)) [12] 陈至达. 有限变形力学基础[M]. 徐州: 中国矿业大学出版社, 2000.(CHEN Zhi-da. The Foundation of the Finite Deformation Mechanics [M]. Xuzhou: The University of Mining and Technology Press, 2000.(in Chinese)) [13] FENG De-shan, GUO Rong-wen, WANG Hong-hua. An element-free Galerkin method for ground penetrating radar numerical simulation[J]. Journal of Central South University,2015,22: 261-269. [14] JIANG Chen, LIU Gui-rong, HAN Xu, et al. A smoothed finite element method for analysis of anisotropic large deformation of passive rabbit ventricles in diastole[J]. International Journal for Numerical Methods in Biomedical Engineering,2015,31(1): 1-25. [15] Ju S H, Hsu H H. Solving numerical difficulties for element-free Galerkin analyses[J]. Computational Mechanics,2014,53(2): 273-281.
点击查看大图
计量
- 文章访问数: 922
- HTML全文浏览量: 111
- PDF下载量: 492
- 被引次数: 0