深圳大学学报理工版

在有限元分析过程中,定位单元刚度矩阵中的元素在整体刚度矩阵中的位置,对结构分析模型的正确建立起着重要作用.为了探究单元刚度矩阵向整体刚度矩阵的集成规律,推导了单元刚度矩阵和整体刚度矩阵之间的对应关系,得到单元刚度矩阵元素在整体刚度矩阵中的位置,依据单元结点号与结构整体坐标系下的结点编号的对应关系建立了统一的关联表,形成基于关联表的整体矩阵集成方法.与目前常用的结点平衡法相比,新建立的整体矩阵集成方法物理意义明确,建模过程简便快捷,且可靠稳定.算例分析结果表明该方法适合用有限元编程实现.

Locating the positions of elements of the unit stiffness matrix in the global stiffness matrix plays an important part in establishment of structural analysis model by finite element method. In order to explore the integration law of the unit stiffness matrix to the global stiffness matrix, we derive the correspondence between the element stiffness matrix and the global stiffness matrix. We identify the positions of elements of the unit stiffness matrix in the global stiffness matrix, and generate the unified link table based on the correspondence between the element node number in the unit stiffness matrix and the node number in the global coordinate system. On the basis of the link table, we establish the integrated global stiffness matrix. We find that the new global matrix ensemble method has clear physical significance and rigorous logic reasoning, as well as with simple, fast, reliable and stable modeling process. The example analysis shows that the proposed method is suitable for finite element programming.

引言
1 利用能量原理建立整体矩阵
2 利用关联表建立整体矩阵
3 算例分析
4 结语

图1 桁架结构平面图<br/>Fig.1 Plane truss structure

图1 桁架结构平面图
Fig.1 Plane truss structure

表1 局部结点与整体结点的关联表<br/>Table 1 Link table between local nodes and global nodes

表1 局部结点与整体结点的关联表
Table 1 Link table between local nodes and global nodes

表2 关联向量计算表<br/>Table 2 Table of link vector

表2 关联向量计算表
Table 2 Table of link vector

图2 单元内力(单位:N)<br/>Fig.2 Diagram of element internal force(unit:N)

图2 单元内力(单位:N)
Fig.2 Diagram of element internal force(unit:N)

[1] 董长军,刘世忠,李爱军.变曲率曲线梁的单元刚度矩阵分析[J]. 西南交通大学学报自然科学版,2017, 52(3): 474- 481.
[2] XUE Huiyu. A combined finite element-stiffness equation transfer method for steady state vibration response analysis of structures[J]. Journal of Sound and Vibration, 2003, 265(4): 783-793.
[3] JONCKHEERE S, DECKERS E, GENECHTEN B V,et al. A direct hybrid finite element-wave based method for the steady-state analysis of acoustic cavities with poro-elastic damping layers using the coupled Helmholtz-Biot equations[J]. Computer Methods in Applied Mechanics & Engineering, 2013, 263(24):144-157.
[4] MENCIK J M. New advances in the forced response computation of periodic structures using the wave finite element(WFE)method[J]. Computational Mechanics, 2014, 54(3): 789-801.
[5] LEE J. A finite element beam analysis by interface matching[J]. Journal of Mechanical Science and Technology, 2016, 30(12): 5587-5594.
[6] MIGNOLET P M, CHRISTIAN S, AVALOS J. Nonparametric stochastic modeling of structures with uncertain boundary conditions/coupling between substructures[J]. American Institute of Aeronautics and Astronautics, 2013, 51(6): 1296-1308.
[7] MACE B R, MANCONI E. Modeling wave propagation in two-dimensional structures using finite element analysis[J]. Journal of Sound and Vibration,2008, 318(4): 884-902.
[8] MAKHIJA D, MAUTE K. Numerical instabilities in level set topology optimization with the extended finite element method[J]. Structural and Multidisciplinary Optimization,2014,49(2):185-197.
[9] LANG C, MAKHIJA D, DOOSTAN A, et al. A simple and efficient preconditioning scheme for heaviside enriched XFEM[J]. Computational Mechanics,2014, 54(5): 1357-1374.
[10] 罗少云, 左岩岩, 路丽芳. 矩阵位移法在连系梁计算中的应用[J]. 数学的实践与认识, 2018, 48(5): 309-316.
[11] LIU Peng, LIN Kun, LIU Hongjun, et al. Free transverse vibration analysis of axially functionally graded tapered euler-bernoulli beams through spline finite point method[J]. Shock and Vibration, 2016(5):1-23.
[12] BORESI A P, CHONG K P, SAIGAL S. Approximate solution methods in engineering mechanics[M]. New York, USA: John Wiley & Sons Inc., 2003: 76-77.
[13] LI Shuangbei, HUANG Lixin, JIANG Linjie, et al. B-spline finite point method for the analysis of piezoelectric laminated composite plates and its application in material parameter identification[J]. Composite Structures, 2014,107(1): 346-362.
[14] 罗帅, 颜全胜. 桁架结构有限元模型的坐标转换方法[J]. 四川建筑科学研究, 2015, 41(3): 10-13.
[15] XIA Zhi, DU Kui. A tensor product finite element method for the diffraction grating problem with transparent boundary conditions[J]. Computers and Mathematics with Applications, 2017, 73(4): 628- 639.

备注

引言

1 利用能量原理建立整体矩阵

2 利用关联表建立整体矩阵

3 算例分析

4 结语

期刊信息

备注

引言

1 利用能量原理建立整体矩阵

2 利用关联表建立整体矩阵

3 算例分析

4 结 语

期刊信息

4 结语