变节点余能原理基面力元法单元模型性能研究

王耀; 胥民尧; 纵岗; 侯长超

doi:10.21656/1000-0887.450059

变节点余能原理基面力元法单元模型性能研究

doi: 10.21656/1000-0887.450059

盐城工业职业技术学院建筑工程学院，江苏盐城 224005

基金项目:

2023年度国家外国专家项目(个人类) G2023014042L

江苏省高职院校教师访学研修项目 2024GRFX071

江苏省高等学校自然科学研究A类项目 19KJB560006

详细信息

通讯作者:
王耀(1986—)，男，副教授，博士(通讯作者. E-mail: yaowang@yctei.edu.cn)

中图分类号: O343.1;O343.2
计量
- 文章访问数: 82
- HTML全文浏览量: 29
- PDF下载量: 16
- 被引次数: 0
出版历程
- 收稿日期: 2024-03-05
- 修回日期: 2024-05-04
- 刊出日期: 2025-03-01

Performances of Variable-Node Elements With the Base Force Element Method Under the Complementary Energy Principle

School of Architecture and Engineering, Yancheng Polytechnic College, Yancheng, Jiangsu 224005, P.R.China

摘要

摘要: 为了解决疏密单元网格交界面节点位移不协调、求解方程构造复杂及空间可扩展性能差的问题，基于余能原理基面力元法提出了一种可变节点数量及位置的单元模型，并针对任意单元类型建立了一种具有统一形式的显式求解方法. 首先，建立了一种二维可变边中节点单元模型，介绍了边中节点的柔度贡献矩阵及节点位移显式表达式；随后，将该模型扩展到三维层次，建立了一种可变面中节点单元模型，并将节点柔度贡献矩阵及节点位移表达式扩展到三维单元. 基于上述模型，建立了平面及空间问题的疏密网格悬挂单元模型，并通过端部承受弯矩荷载、集中荷载及拉伸荷载的悬臂梁算例，论证了平面及空间可变节点单元模型、疏密网格悬挂单元模型的数值精度和适用性. 研究表明，基于余能原理基面力元法建立的平面及空间可变节点单元模型具有较高的数值精度；此外，疏密网格之间的交界面无需进行任何处理措施，也无需构造插值函数或约束函数，仅依靠疏密网格交界面共用的边中节点(2D)及面中节点(3D)即可确保疏密网格交界面的节点位移协调；同时，模型和方法独立于单元类型、单元维度、节点数量及分布等因素，具有优异的空间可扩展性能及易于程序化的特点.
- 基面力元法 /
- 变节点单元 /
- 余能原理 /
- 悬挂单元模型
Abstract: To address the problems of uncoordinated nodal displacements at the intersection of the sparse and dense elements, complicated construction of solving equations, and poor spatial scalability, an element model with variable node numbers and positions was proposed with the base force element method (BFEM) under the complementary energy principle, and an explicit solution method in a unified form was established for arbitrary element types. First, a 2D variable mid-edge node element model was established, and the explicit expressions of the contribution of nodes compliance matrix and nodal displacements were introduced. Subsequently, the element model was extended to the 3D form, a variable mid-face node element model was proposed, and the above expressions were also extended to the 3D forms. Hereafter, a sparse and dense mesh hanging element model was established, and the numerical accuracy and applicability of the planar and spatial variable-node element model were demonstrated with the cantilever beam subjected to a bending moment load, a concentrated load, and a tensile load at the end. The numerical results show that, the variable-node element model and the hanging element model in the 2D and 3D forms based on the BFEM have high numerical accuracy. In addition, the nodal displacement coordination at the intersection interface between sparse and dense elements can be ensured only by the shared mid-edge node (2D) and the mid-face node (3D) at the interface without any processing treatment or construction of interpolation functions and constraint functions. Meanwhile, the element models and methods are independent of the element type, element dimensions, nodes' number, and nodes' distribution, etc., and have excellent spatial scalability and programmability.
- base force element method /
- variable-node element /
- complementary energy principle /
- hanging element model

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图 1 位移与变形

Figure 1. The displacement and deformation

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图 2 基面力^[14]

Figure 2. Base forces^[14]

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图 3 四面体及面上的力^[14]

Figure 3. The tetrahedron and the forces^[14]

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图 4 任意形状的基面力元法单元

Figure 4. The element with an arbitrary shape based on the BFEM

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图 5 二维可变节点基面力元法单元

Figure 5. The 2D variable-node element based on the BFEM

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图 6 三维六面体单元

Figure 6. 3D hexahedron element

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图 7 三维可变节点基面力元法单元

Figure 7. The 3D variable-node element based on the BFEM

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图 8 悬臂梁构件

Figure 8. The cantilever beam

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图 9 网格剖分模型

Figure 9. The mesh models

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图 10 数值解与理论解对比(承受弯矩荷载)

Figure 10. The comparison of numerical and theoretical solutions (moment load)

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图 11 弯曲正应力分布云图(承受弯矩荷载)

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 11. The distributions of bending normal stresses (moment load)

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图 12 数值解与理论解对比(承受竖向集中荷载)

Figure 12. The comparison of numerical and theoretical solutions (concentrated force)

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图 13 剪应力分布云图及变化规律

Figure 13. The distribution of shear stresses and variation trends

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图 14 数值解与理论解对比(承受水平拉伸荷载)

Figure 14. The comparison of numerical and theoretical solutions(tension force)

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图 15 悬挂单元网格剖分模型

Figure 15. The mesh models of hanging elements

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图 16 数值解与理论解对比(悬挂单元模型)

Figure 16. The comparison of numerical and theoretical solutions (hanging element model)

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图 17 弯曲正应力分布云图(悬挂单元模型)

Figure 17. The distributions of bending normal stresses (hanging element model)

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图 18 不同网格尺寸的悬挂单元模型

Figure 18. The hanging element model with different mesh sizes

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图 19 网格尺寸对数值分析精度的影响

Figure 19. The effects of the mesh size on the numerical accuracy

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图 20 三维悬臂梁模型

Figure 20. The 3D cantilever beam

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图 21 数值解与理论解对比(三维悬臂梁模型)

Figure 21. The comparison of numerical and theoretical solutions (3D cantilever beam)

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图 22 弯曲正应力分布云图(三维悬臂梁模型)

Figure 22. The distribution of bending normal stresses (3D cantilever beam)

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图 23 柱加载模型及网格模型

Figure 23. The loading model and mesh model for the column

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图 24 三维六面体悬挂网格模型

Figure 24. The 3D hexahedron hanging element method

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图 25 数值解与理论解对比(三维六面体悬挂网格模型)

Figure 25. The comparison of numerical and theoretical solutions (3D hexahedron hanging element method)

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图 26 弯曲正应力分布云图(三维六面体悬挂网格模型)

Figure 26. The distribution of bending normal stresses (3D hexahedron hanging element method)

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表 1 不同位置的节点位移及误差

Table 1. The nodal displacements and errors at different locations

№.	x/m	theoretical solution	mesh Ⅰ (error)	mesh Ⅱ (error)	mesh Ⅲ (error)	mesh Ⅳ (error)
1	1.000 5	-0.006 01	-0.006 10(1.497 5%)*	-0.006 14(2.163 0%)*	-0.006 42(6.821 9%)*	-0.006 57(9.317 8%)*
2	2.001 0	-0.024 02	-0.024 26(0.999 1%)	-0.024 37(1.457 1%)	-0.025 02(4.163 1%)	-0.025 56(6.411 3%)
3	2.501 2	-0.037 54	-0.037 83(0.772 5%)	-0.038 00(1.255 3%)	-0.038 85(3.489 6%)	-0.039 59(5.460 8%)
4	3.001 5	-0.054 05	-0.054 47(0.777 0%)	-0.054 73(1.258 0%)	-0.055 73(3.108 2%)	-0.056 65(4.810 3%)
5	4.002 0	-0.096 1	-0.096 70(0.624 3%)	-0.097 04(0.978 1%)	-0.098 52(2.518 2%)	-0.099 91(3.964 6%)
6	4.502 2	-0.121 62	-0.122 26(0.526 2%)	-0.122 71(0.896 2%)	-0.124 43(2.310 4%)	-0.126 00(3.601 3%)
7	4.989 9	-0.149 4	-0.150 06(0.140 1%)	-0.150 32(0.313 6%)	-0.151 70(1.234 5%)	-0.153 15(2.202 2%)

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表 2 单元应力对比结果(受拉为正)

Table 2. Comparison of element stresses (tension is positive)

element №.	numerical result in this paper	QIANG Tianchi’s numerical result^[24]	theoretical result
1		1.000 0	1.000 0
2		0.999 9	1.000 0
3	0.999 999 978 ~ 1.000 000 023	0.995 4	1.000 0
4		1.003 7	1.000 0
		-	1.000 0

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表 3 节点位移对比结果

Table 3. Comparison of nodal displacements

nodal №.	numerical result in this paper	QIANG Tianchi’s numerical result^[24]	theoretical result
1	2.999 495~3.000 504	2.994 2	3.000 0

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