减缩积分微极六面体有限单元的开发与应用验证

doi:10.11996/JG.j.2095-302X.2025061153

图学学报 ›› 2025, Vol. 46 ›› Issue (6): 1153-1160.DOI: 10.11996/JG.j.2095-302X.2025061153

• 制造产品核心工业软件 • 上一篇下一篇

减缩积分微极六面体有限单元的开发与应用验证

周天齐¹(), 丁军¹^,²(), 姚宇²

¹ 中国船舶科学研究中心，江苏无锡 214000
² 太湖深海技术科学实验室，江苏无锡 214000

收稿日期:2025-08-30 接受日期:2025-11-05 出版日期:2025-12-30 发布日期:2025-12-27
通讯作者:丁军(1986-)，男，研究员，博士。主要研究方向为船舶与海洋结构物设计制造等。E-mail：dingjun@cssrc.com.cn
第一作者:周天齐(1999-)，男，硕士研究生。主要研究方向为船舶与海洋结构物设计制造。E-mail：836282696@qq.com
基金资助:
国家重点研发计划项目(2022YFB3306200)

Development of reduced integration micropolar hexahedron finite element and application verification

ZHOU Tianqi¹(), DING Jun¹^,²(), YAO Yu²

¹ China Ship Scientific Research Center, Wuxi Jiangsu 214000, China
² Taihu Laboratory of Deepsea Technological Science, Wuxi Jiangsu 214000, China

Received:2025-08-30 Accepted:2025-11-05 Published:2025-12-30 Online:2025-12-27
First author：ZHOU Tianqi (1999-), master student. His main research interests cover design and manufacturing of ships and ocean structures. E-mail：836282696@qq.com
Supported by:
The National Key Research and Development Program of China(2022YFB3306200)

摘要/Abstract

摘要：

微极弹性有限元法在分析具有复杂微结构的材料上有着广泛应用。针对现有实现通常采用完全积分方案而导致的计算效率低以及弯曲工况下出现剪切锁定的问题，提出一种通用高效的缩减积分一阶六面体微极有限单元，并通过海洋结构物通用分析软件SAM进行算例验证。单元算法结合标准拉格朗日插值和均匀应变与曲率公式，能够通过单元分片检验并保证畸形单元计算的准确性。同时，引入一种人工刚度方法，有效抑制缩减积分微极单元的位移和旋转沙漏失稳模式。通过数值验证，包括力和位移分片检验确认单元的收敛性，悬臂梁纯弯曲测试评估单元在剪切锁定问题上的改善效果，以及对船舶常用夹芯结构进行振动响应模态分析，验证了该单元在船舶结构有限元分析中相较于传统实体单元的有效性和优势。

关键词: 微极弹性理论, 有限单元开发, 减缩积分法, 沙漏稳定技术, 数值验证

Abstract:

The micropolar elastic finite element method has found extensive applications in analyzing materials with complex microstructures. To address the problems of low computational efficiency and shear locking under bending conditions caused by the fully integrated scheme commonly adopted in existing implementations, a universal and efficient reduced-integration first-order hexahedral micropolar finite element was proposed and verified using SAM, a general analysis software for marine structures. The element algorithm, combined with the standard Lagrange interpolation and uniform-strain and curvature formulas, passed the element patch test and ensured the calculation accuracy for distorted elements. Additionally, an artificial stiffness method was introduced to effectively suppress displacement and rotational hourglass instability modes in the reduced-integration micropolar elements. Numerical validation, including force and displacement patch tests for convergence, cantilever-beam tests for shear locking, and modal analysis of vibration responses of sandwich structures used in shipbuilding, demonstrated the proposed element’s effectiveness and advantages over traditional solid elements in finite-element analysis of ship structures.

Key words: micropolar elasticity theory, development of finite element, reduced integration, hourglass instabilities, numerical validations

中图分类号:

U661.4
U663

周天齐, 丁军, 姚宇. 减缩积分微极六面体有限单元的开发与应用验证[J]. 图学学报, 2025, 46(6): 1153-1160.

ZHOU Tianqi, DING Jun, YAO Yu. Development of reduced integration micropolar hexahedron finite element and application verification[J]. Journal of Graphics, 2025, 46(6): 1153-1160.

图/表 11

图1 8节点实体单元的沙漏模态((a) Γ1沙漏模态；(b) Γ2沙漏模态；(c) Γ3沙漏模态；(d) Γ4沙漏模态)

Fig. 1 Hourglass mode of 8-node solid element ((a) Hourglass mode Γ1; (b) Hourglass mode Γ2; (c) Hourglass mode Γ3; (d) Hourglass mode Γ4)

图2 受轴向载荷作用的梁示意图

Fig. 2 Schematic of the beam subjected to axial loading

图3 用于位移分片检验的有限单元网格

Fig. 3 The finite element mesh for the displacement patch tests

表1 第1次位移分片检验结果

Table 1 Results for the first displacement patch test

Element	Numerical
Element	σ₁₁	σ₁₂	ε₁₁	ε₁₂
Hex8L^[17]	5.0	1.5	10⁻³	0.75×10⁻³
Hex8R	5.0	1.5	10⁻³	0.75×10⁻³
Hex8EFP^[23]	5.0	1.5	10⁻³	0.75×10⁻³
Hex8IM^[17]	5.0	1.5	10⁻³	0.75×10⁻³

表2 第2次位移分片试验结果

Table 2 Results for the second displacement patch test

Element	Numerical
Element	σ₁₁	σ₁₃	σ₃₁	ε₁₁	ε₁₃	ε₃₁
Hex8L^[17]	5.0	1.0	2.0	10⁻³	0.25×10⁻³	1.25×10⁻³
Hex8R	5.0	1.0	2.0	10⁻³	0.25×10⁻³	1.25×10⁻³
Hex8EFP^[23]	5.0	1.0	2.0	10⁻³	0.25×10⁻³	1.25×10⁻³
Hex8IM^[17]	5.0	1.0	2.0	10⁻³	0.25×10⁻³	1.25×10⁻³

表3 第3次位移分片试验结果

Table 3 Results for the third displacement patch test

Element	σ₁₁	σ₁₃	m₁₁	m₁₂
解析解	5.000	1.483	0.020 0	−0.040 0
Hex8R	5.000	1.483	0.020 0	−0.040 0
Hex8EFP^[23]	4.965	1.483	0.019 9	−0.040 2

图4 承受纯弯曲载荷的悬臂梁位移云图((a) Abaqus位移计算结果；(b) SAM位移计算结果)

Fig. 4 Displacement contour plots of the cantilever beam under pure bending load ((a) Displacement result in Abaqus; (b) Displacement result in SAM)

图5 受纯弯曲载荷的梁模型

Fig. 5 Beam model under pure bending load

图6 承受纯弯曲载荷的悬臂梁应力云图((a) Abaqus应力计算结果；(b) SAM应力计算结果)

Fig. 6 Stress contour plots of the cantilever beam under pure bending load ((a) Stress result in Abaqus; (b) Stress result in SAM)

表4 六面体微极单元结果对比(A为解析解，N为数值解)

Table 4 Comparison of the results of hexahedral micropolar elements (A=analytical, N=numerical)

Element	l_b	u₂		ϕ₃		σ₁₁
Element	l_b	A	N	A	N	A	N
Hex8L^[17]	0	0.941	0.069	0.188	0.012 6	23.660 0	1.968
Hex8IM^[17]			0.941		0.187 5		23.660
Hex8R			0.937		0.186 8		14.947
Hex8EFP^[23]			0.912		0.181 8		23.993
C3D8R^[21]			0.937		0.186 5		14.947
Hex8L^[17]	0.05	0.900	0.069	0.179	0.012 7	22.641 0	1.950
Hex8IM^[17]			0.900		0.179 4		22.641
Hex8R			0.897		0.178 8		14.306
Hex8EFP^[23]			0.874		0.174 3		22.995
C3D8R^[21]			0.895		0.175 6		14.312
Hex8L^[17]	0.15	0.669	0.067	0.133	0.012 9	21.352 0	1.835
Hex8IM^[17]			0.669		0.133 5		21.352
Hex8R			0.668		0.133 1		10.651
Hex8EFP^[23]			0.656		0.130 7		17.245
C3D8R^[21]			0.669		0.133 2		10.861
Hex8L^[17]	0.30	0.359	0.062	0.072	0.012 6	11.450 0	1.599
Hex8IM ^[17]			0.359		0.071 6		11.450
Hex8R			0.359		0.071 5		5.718
Hex8EFP^[23]			0.355		0.070 8		9.341
C3D8R^[21]			0.359		0.071 5		5.785
Hex8L^[17]	0.60	0.126	0.046	0.025	0.009 8	3.163 1	1.144
Hex8IM^[17]			0.060		0.012 0		1.517
Hex8R			0.126		0.025 1		2.004
Hex8EFP^[23]			0.125		0.024 9		3.294
C3D8R^[21]			0.060		0.024 6		2.065

图7 厚度与弯曲特征长度的比率变化对不同网格数夹芯板无量纲固有频率的影响((a) 10×10×2个网格；(b) 66×66×7个网格)

Fig. 7 The influence of the variation in the ratio of thickness to bending characteristic length on the dimensionless natural frequencies of sandwich plates with different numbers of meshes ((a) 10×10×2 grid; (b) 66×66×7 grid)

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减缩积分微极六面体有限单元的开发与应用验证

Development of reduced integration micropolar hexahedron finite element and application verification

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