Development of reduced integration micropolar hexahedron finite element and application verification

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

Abstract

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

CLC Number:

U661.4
U663

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.

Figures/Tables 11

References 24

[1]	ERINGEN A C. Theory of micropolar elasticity[M]. New York: Springer, 1999: 101-248.
[2]	AMBARTSUMIAN S A. Micropolar theory of shells and plates[M]. Cham: Springer International Publishing, 2021: 21-24.
[3]	陈东阳. 微极弹性复合材料之理论解析架构与应用[EB/OL]. (2016-05-24) [2025-04-15]. http://140.116.207.99/handle/987654321/163086.
	CHEN D Y. Micropolar elastic composite materials with chirality: theoretical frameworks and applications[EB/OL]. (2016-05-24) [2025-04-15]. http://140.116.207.99/handle/987654321/163086 (in Chinese).
[4]	王洪生. 微极弹性力学的虚功原理[J]. 山东建筑工程学院学报, 1993, 8(4): 1-5.
	WANG H S. Principle of admissible work of microelasticity mechanics[J]. Journal of Shandong Architectural and Civil Engineering Institute, 1993, 8(4): 1-5 (in Chinese).
[5]	YAO Y, ZHOU Y, CHEN L H, et al. A multifunctional three-dimensional lattice material integrating auxeticity, negative compressibility and negative thermal expansion[J]. Composite Structures, 2024, 337: 118032. DOI URL
[6]	YAO Y, HE L H, JIN J H, et al. A novel design of mechanical metamaterial incorporating multiple negative indexes[J]. Materials Research Express, 2023, 10(5): 055801. DOI
[7]	吴浩宇, 杨小超, 王伟, 等. 基于运动学原理的复合材料编织成型工艺仿真技术研究[J]. 图学学报, 2025, 46(5): 1061-1071. DOI
	WU H Y, YANG X C, WANG W, et al. Simulation technology for braiding process of composite materials based on kinematic principles[J]. Journal of Graphics, 2025, 46(5): 1061-1071 (in Chinese). DOI
[8]	HUANG L H, YUAN H, ZHAO H Y. An FEM-based homogenization method for orthogonal lattice metamaterials within micropolar elasticity[J]. International Journal of Mechanical Sciences, 2023, 238: 107836. DOI URL
[9]	MINDLIN R D, TIERSTEN H F. Effects of couple-stresses in linear elasticity[J]. Archive for Rational Mechanics and Analysis, 1962, 11(1): 415-448. DOI URL
[10]	ALBERTINI G, ELBANNA A E, KAMMER D S. A three- dimensional hybrid finite element—Spectral boundary integral method for modeling earthquakes in complex unbounded domains[J]. International Journal for Numerical Methods in Engineering, 2021, 122(23): 6905-6923. DOI URL
[11]	薛雨彤, 王爱增, 岳怡珂, 等. 基于等几何法和模拟退火的复杂壳结构分析及优化[J]. 图学学报, 2024, 45(3): 575-584. DOI
	XUE Y T, WANG A Z, YUE Y K, et al. Complex shell structure analysis and optimization based on isogeometric analysis and simulated annealing algorithm[J]. Journal of Graphics, 2024, 45(3): 575-584 (in Chinese). DOI
[12]	YAO Y, NI Y, HE L H. Rutile-mimic 3D metamaterials with simultaneously negative Poisson's ratio and negative compressibility[J]. Materials & Design, 2021, 200: 109440.
[13]	陈震, 周金宇. 材料二维微结构仿真随机概率圆优化填充算法[J]. 图学学报, 2015, 36(6): 944-949. DOI
	CHEN Z, ZHOU J Y. Random circles optimization filling algorithm of material two dimensional microstructures simulation[J]. Journal of Graphics, 2015, 36(6): 944-949 (in Chinese).
[14]	江见鲸, 何放龙, 何益斌, 等. 有限元法及其应用[M]. 北京: 机械工业出版社, 2006: 75-83.
	JIANG J J, HE F L, HE Y B, et al. Finite element method and application[M]. Beijing: China Machine Press, 2006: 75-83 (in Chinese).
[15]	YAO Y, NI Y, HE L H. Unexpected bending behavior of architected 2D lattice materials[J]. Science Advances, 2023, 9(25): eadg3499.
[16]	许德胜, 凌道盛, 王燕. 八节点非协调实体板单元[J]. 计算力学学报, 2009, 26(2): 264-269.
	XU D S, LING D S WANG Y, et al. Eight node incompatible solid-plate element[J]. Chinese Journal of Computational Mechanics, 2009, 26(2): 264-269 (in Chinese).
[17]	GRBČIĆ S, IBRAHIMBEGOVIĆ A, JELENIĆ G. Variational formulation of micropolar elasticity using 3D hexahedral finite-element interpolation with incompatible modes[J]. Computers & Structures, 2018, 205: 1-14. DOI URL
[18]	FLANAGAN D P, BELYTSCHKO T. A uniform strain hexahedron and quadrilateral with orthogonal hourglass control[J]. International Journal for Numerical Methods in Engineering, 1981, 17(5): 679-706. DOI URL
[19]	KREJA I, CYWIŃSKI Z. Is reduced integration just a numerical trick[J]. Computers & Structures, 1988, 29(3): 491-496. DOI URL
[20]	KOHANSAL-VAJARGAH M, ANSARI R. Quadratic tetrahedral micropolar element for the vibration analysis of three-dimensional micro-structures[J]. Thin-Walled Structures, 2021, 167: 108152. DOI URL
[21]	YAO Y, ZHAO X, CHEN L H, et al. Reduced-integration hexahedral finite element for static and vibration analysis of micropolar continuum[J]. Finite Elements in Analysis and Design, 2025, 251: 104412. DOI URL
[22]	AMEZCUA H R, AYALA A G. A computationally efficient numerical integration scheme for non-linear planestress/strain FEM applications using one-point constitutive model evaluation[J]. Structural Engineering and Mechanics, 2023, 85(1): 89-104.
[23]	GRBAC L, JELENIC G, RIBARIĆ D, et al. Hexahedral finite elements with enhanced fixed-pole interpolation for linear static and vibration analysis of 3D micropolar continuum[J]. International Journal for Numerical Methods in Engineering, 2024, 125(8): e7440. DOI URL
[24]	Taihu Laboratory of Deepsea Technological Science. SAM[EB/OL]. [2025-04-23]. https://www.taihulab.com.cn/info/167.html.

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

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

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

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

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