广义周期点阵结构的高阶渐近展开分析方法

doi:10.7527/S1000-6893.2025.32386

Abstract

Abstract:

This paper proposes a novel high-order asymptotic expansion analysis method for generalized periodic lattice structures， aimed at accurately predicting their physical and mechanical behaviors and equivalent performances. The proposed method converts the homogenization problem of generalized periodic lattice structures into the classical two-scale homogenization problem for a cubic unit cell， thereby elucidating the intrinsic mapping mechanisms of the generalized two-scale homogenization approach. This conversion clarifies the fundamental properties of the equivalent performance of generalized periodic two-scale homogenization， substantially reducing the computational and programming complexities involved. By employing typical numerical examples， this paper compares the outcomes from the proposed mapping method with those derived from classical periodic two-scale homogenization and fine-scale finite element methods. The results affirm the universality and effectiveness of the proposed method， which exhibits high computational precision for addressing both static and dynamic issues， such as natural frequency calculations， thus providing strong theoretical support and practical pathways for designing high-performance， lightweight structures in generalized periodic lattice configurations.

Key words: generalized periodic lattice structures, multi-scale analysis, homogenization, asymptotic expansion analysis, mapping method, high-order approximation

CLC Number:

Shijie XU, Weihong ZHANG. A high-order asymptotic expansion analysis method for generalized periodic lattice structures[J]. Acta Aeronautica et Astronautica Sinica, 2025, 46(21): 532386.

Figures/Tables 18

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig. 6

Table 1

Comparison of result and error among four different computing methods

参数	单胞数目	计算方法	应力场分布	最大Von-Mises 等效应力/MPa	应力误差/%	最大位移/mm	最大位移误差/%
$a = - 0.15$	$8 × 8$	直接离散		4.29		53.83
		经典周期高阶		3.00	30.0	46.55	13.5
		广义周期高阶		3.62	15.6	50.47	6.3
		广义周期零阶		2.16	49.5	49.92	7.8
	$12 × 12$	直接离散	图7 （a）	4.48		52.21
		经典周期高阶	图7 （d）	3.15	29.7	46.55	10.8
		广义周期高阶	图7 （c）	3.98	13.1	50.47	3.6
		广义周期零阶	图7 （b）	2.26	49.5	49.92	4.6
	$16 × 16$	直接离散		4.72		51.52
		经典周期高阶		3.32	29.6	46.55	9.7
		广义周期高阶		4.20	11.1	50.47	2.2
		广义周期零阶		2.51	46.7	49.92	3.1
$a = - 0.3$	$8 × 8$	直接离散		5.61		62.05
		经典周期高阶		2.99	46.6	46.55	25.0
		广义周期高阶		4.86	13.3	60.58	4.2
		广义周期零阶		3.56	36.6	59.85	5.3
	$12 × 12$	直接离散	图8 （a）	6.00		61.80
		经典周期高阶	图8 （d）	3.14	47.5	46.55	24.7
		广义周期高阶	图8 （c）	5.42	9.6	60.58	2.1
		广义周期零阶	图8 （b）	3.88	35.3	59.85	3.7
	$16 × 16$	直接离散		6.30		61.49
		经典周期高阶		3.11	47.2	46.55	24.3
		广义周期高阶		5.86	6.9	60.58	1.5
		广义周期零阶		4.02	36.2	59.85	2.7

Table 1

Fig.7

Fig.8

Fig.9

Fig.10

Table 2

Maximum Von-Mises equivalent stress and error of rotationally symmetrical generalized periodic lattice structure

单胞数目	精细尺度有限元直接离散（直接离散）		广义周期高阶渐近展开（广义周期高阶）			广义周期零阶渐近展开（广义周期零阶）
单胞数目	时间/s	最大Von-Mises 等效应力/MPa	时间/s	最大Von-Mises 等效应力/MPa	误差/%	时间/s	最大Von-Mises 等效应力/MPa	误差/%
$4 × 48$	676	1.636 9	14.4	1.619 2	1.08	9.4	1.398 2	14.6
$6 × 72$	1 287	1.648 0	26.9	1.640 9	0.43	9.4	1.428 4	13.3
$8 × 96$	4 268	1.653 8	46.9	1.647 8	0.30	9.4	1.445 7	12.6

Table 2

Fig.11

Fig.12

Table 3

First five natural frequencies and errors obtained from fine-scale finite element direct discretization and generalized periodic high-order asymptotic expansion method

单胞数目	计算方法	计算时间/s	模态振型	第1阶	第2阶	第3阶	第4阶	第5阶
$2 × 24$	直接离散	199.5	图12 （a）	21.09	42.85	57.41	66.30	92.06
	广义周期零阶广义周期高阶	9.4	图12 （b）图12 （c）	20.53	40.08	58.56	67.63	76.42
	相较直接离散误差/%			2.7	6.5	2.0	2.0	17.0
	同振型误差			2.7	6.5	11.7	17.8	17.0
$4 × 48$	直接离散	792.0	图13 （a）	20.86	41.04	61.05	65.57	81.68
	广义周期零阶广义周期高阶	9.4	图13 （b）图13 （c）	20.53	40.08	58.56	67.63	76.42
	相较直接离散误差/%			1.6	2.3	4.1	3.1	6.4
$8 × 96$	直接离散	2707.1	图14 （a）	20.67	40.37	59.18	67.07	77.68
	广义周期零阶广义周期高阶	9.4	图14 （b）、图14 （c）	20.53	40.08	58.56	67.63	76.42
	相较直接离散误差/%			0.7	0.7	1.0	0.8	1.6

Table 3

Fig.13

Fig.14

Fig.15

References 41

[1]	COMPTON B G， LEWIS J A. 3D printing： 3D-printing of lightweight cellular composites （adv. mater. 34/2014）［J］. Advanced Materials， 2014， 26（34）： 6043.
[2]	Intralattice［EB/OL］. （2023-03-03）［2024-08-12］. .
[3]	LIU Z Q， MEYERS M A， ZHANG Z F， et al. Functional gradients and heterogeneities in biological materials： Design principles， functions， and bioinspired applications［J］. Progress in Materials Science， 2017， 88： 467-498.
[4]	ZHAO Z L， ZHOU S W， FENG X Q， et al. Morphological optimization of scorpion telson［J］. Journal of the Mechanics and Physics of Solids， 2020， 135： 103773.
[5]	OSORIO L， TRUJILLO E， VAN VUURE A W， et al. Morphological aspects and mechanical properties of single bamboo fibers and flexural characterization of bamboo/epoxy composites［J］. Journal of Reinforced Plastics and Composites， 2011， 30（5）： 396-408.
[6]	CHEN W J， ZHENG X N， LIU S T. Finite-element-mesh based method for modeling and optimization of lattice structures for additive manufacturing［J］. Materials， 2018， 11（11）： 2073.
[7]	ZHAO Z L， ZHOU S W， FENG X Q， et al. On the internal architecture of emergent plants［J］. Journal of the Mechanics and Physics of Solids， 2018， 119： 224-239.
[8]	STROMBERG L L， BEGHINI A， BAKER W F， et al. Application of layout and topology optimization using pattern gradation for the conceptual design of buildings［J］. Structural and Multidisciplinary Optimization， 2011， 43（2）： 165-180.
[9]	YE Q， GUO Y， CHEN S K， et al. Topology optimization of conformal structures on manifolds using extended level set methods （X-LSM） and conformal geometry theory［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 344： 164-185.
[10]	TANG Y L， KURTZ A， ZHAO Y F. Bidirectional Evolutionary Structural Optimization （BESO） based design method for lattice structure to be fabricated by additive manufacturing［J］. Computer-Aided Design， 2015， 69： 91-101.
[11]	NGUYEN J， PARK S I， ROSEN D. Heuristic optimization method for cellular structure design of light weight components［J］. International Journal of Precision Engineering and Manufacturing， 2013， 14（6）： 1071-1078.
[12]	TANG Y L， DONG G Y， ZHAO Y F. A hybrid geometric modeling method for lattice structures fabricated by additive manufacturing［J］. The International Journal of Advanced Manufacturing Technology， 2019， 102（9）： 4011-4030.
[13]	DEDÈ L， BORDEN M J， HUGHES T J R. Isogeometric analysis for topology optimization with a phase field model［J］. Archives of Computational Methods in Engineering， 2012， 19（3）： 427-465.
[14]	MOSES E， FUCHS M B， RYVKIN M. Topological design of modular structures under arbitrary loading［J］. Structural and Multidisciplinary Optimization， 2002， 24（6）： 407-417.
[15]	GAO T， ZHANG W H. Topology optimization involving thermo-elastic stress loads［J］. Structural and Multidisciplinary Optimization， 2010， 42（5）： 725-738.
[16]	MAHARAJ Y， JAMES K A. Metamaterial topology optimization of nonpneumatic tires with stress and buckling constraints［J］. International Journal for Numerical Methods in Engineering， 2020， 121（7）： 1410-1439.
[17]	ZUO Z H， XIE Y M， HUANG X D. Optimal topological design of periodic structures for natural frequencies［J］. Journal of Structural Engineering， 2011， 137（10）： 1229-1240.
[18]	XU Z， ZHANG W H， ZHOU Y， et al. Multiscale topology optimization using feature-driven method［J］. Chinese Journal of Aeronautics， 2020， 33（2）： 621-633.
[19]	WANG C， ZHU J H， ZHANG W H， et al. Concurrent topology optimization design of structures and non-uniform parameterized lattice microstructures［J］. Structural and Multidisciplinary Optimization， 2018， 58（1）： 35-50.
[20]	ZHU J H， ZHOU H， WANG C， et al. A review of topology optimization for additive manufacturing： Status and challenges［J］. Chinese Journal of Aeronautics， 2021， 34（1）： 91-110.
[21]	BABUŠKA I. Homogenization approach in engineering［C］∥Computing Methods in Applied Sciences and Engineering. Berlin： Springer， 1976： 137-153.
[22]	KESAVAN S. Homogenization of elliptic eigenvalue problems： Part 1［J］. Applied Mathematics and Optimization， 1979， 5（1）： 153-167.
[23]	KESAVAN S. Homogenization of elliptic eigenvalue problems： Part 2［J］. Applied Mathematics and Optimization， 1979， 5（1）： 197-216.
[24]	BENSOUSSAN A， LIONS J L， PAPANICOLAOU G. Asymptotic analysis for periodic structures［M］. Providence， R.I. American Mathematical Society， 2011
[25]	BRIANE M. Three models of non periodic fibrous materials obtained by homogenization［J］. ESAIM： Mathematical Modelling and Numerical Analysis， 1993， 27（6）： 759-775.
[26]	TSALIS D， BAXEVANIS T， CHATZIGEORGIOU G， et al. Homogenization of elastoplastic composites with generalized periodicity in the microstructure［J］. International Journal of Plasticity， 2013， 51： 161-187.
[27]	TSALIS D， CHATZIGEORGIOU G， CHARALAMBAKIS N. Homogenization of structures with generalized periodicity［J］. Composites Part B： Engineering， 2012， 43（6）： 2495-2512.
[28]	徐仕杰，张卫红. 广义周期点阵结构的等效性能均匀化映射方法研究［J］. 中国科学（技术科学）， 2024， 54（6）： 1036-1056.
	XU S J， ZHANG W H. Research on equivalent performance homogenization mapping method of generalized periodic lattice structure［J］. SCIENTIA SINICA Technologica， 2024， 54（6）： 1036-1056 （in Chinese）.
[29]	XU S J， ZHANG W H. Energy-based homogenization method for lattice structures with generalized periodicity［J］. Computers & Structures， 2024， 302： 107478.
[30]	DONG H， ZHENG X J， CUI J Z， et al. Multi-scale computational method for dynamic thermo-mechanical performance of heterogeneous shell structures with orthogonal periodic configurations［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 354： 143-180.
[31]	YANG Z Q， LIU Y Z， SUN Y， et al. A second-order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations［J］. International Journal for Numerical Methods in Engineering， 2022， 123（1）： 128-157.
[32]	TOMPSON A F B. Numerical simulation of chemical migration in physically and chemically heterogeneous porous media［J］. Water Resources Research， 1993， 29（11）： 3709-3726.
[33]	HOU T Y， WU X H. A multiscale finite element method for elliptic problems in composite materials and porous media［J］. Journal of Computational Physics， 1997， 134（1）： 169-189.
[34]	CAO L Q， CUI J Z. Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains［J］. Numerische Mathematik， 2004， 96（3）： 525-581.
[35]	CAO L Q. Iterated two-scale asymptotic method and numerical algorithm for the elastic structures of composite materials［J］. Computer Methods in Applied Mechanics and Engineering， 2005， 194（27-29）： 2899-2926.
[36]	OLEINIK O A， SHAMAEV A S， YOSIFIAN G A. Mathematical problems in elasticity and homogenization［M］. Amsterdam： Elsevier Science Publishers B.V.， 2012.
[37]	GHOSH S， LEE K， MOORTHY S. Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and Voronoi cell finite element model［J］. Computer Methods in Applied Mechanics and Engineering， 1996， 132（1-2）： 63-116.
[38]	SCHRÖDER J. A numerical two-scale homogenization scheme： The FE2-method［M］∥Plasticity and Beyond. Vienna： Springer Vienna， 2014： 1-64.
[39]	RAO Y P， XIANG M Z， CUI J Z. A strain gradient brittle fracture model based on two-scale asymptotic analysis［J］. Journal of the Mechanics and Physics of Solids， 2022， 159： 104752.
[40]	曹礼群，崔俊芝，王崇愚. 非均质材料多尺度物理问题与数学方法［C］∥中国科学院技术科学论坛学术报告会. 2002.
	CAO L Q， CUI J Z， WANG C Y. Multi-scale physical problems and mathematical methods for heterogeneous materials［C］∥Academic Report Meeting of the Technology Science Forum of China Academy of Sciences. 2002 （in Chinese）.
[41]	张卫红，徐仕杰，朱继宏. 循环对称结构的多尺度拓扑优化方法［J］. 计算力学学报， 2021， 11（4）： 512-522.
	ZHANG W H， XU S J， ZHU J H. Multi-scale topology optimization method for cyclic symmetric structures［J］. Chinese Journal of Computational Mechanics， 2021， 11（4）： 512-522 （in Chinese）.

A high-order asymptotic expansion analysis method for generalized periodic lattice structures

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 18

References 41

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	Wei LI, Xin SONG, Xiangqi LIN, Zhilin SU, Chengbo WANG, Tong LI. Fast optimization method for nail load homogenization of composite joint structures [J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(23): 230345-230345.
[2]	Xiaogang XU, Yang ZHANG, Tianbo WANG, Xudong MA, Gang CHEN. Mixing enhancement effect of multi-size water droplet spray [J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(S2): 729130-729130.
[3]	Xiaoqian CHEN, Yong ZHAO, Senlin HUO, Zeyu ZHANG, Bingxiao DU. A review of topology optimization design methods for multi-scale structures [J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(15): 528863-528863.
[4]	LONG Kai, CHEN Zhuo, GU Chunlu, WANG Xuan. Structural topology optimization method with maximum displacement constraint on load-bearing surface [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2020, 41(7): 223577-223577.
[5]	LI Zengcong, TIAN Kuo, ZHAO Haixin. Efficient variable-fidelity models for hierarchical stiffened shells [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2020, 41(7): 623435-623435.
[6]	LI Peng, ZHOU Wanlin, LI Bei, SHAO Jintao. Pin-load distribution homogenization method of adjusting bolt-hole clearance and bolt torqued in multi-bolt composite joints [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2019, 40(3): 422548-422548.
[7]	GAO Tao, QI Wenkai, SHEN Cheng. Prediction of FRP stiffness based on a new implementation of homogenization method [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(5): 220579-220579.
[8]	WANG Bo, TIAN Kuo, ZHENG Yanbing, HAO Peng, ZHANG Ke. A rapid buckling analysis method for large-scale grid-stiffened cylindrical shells [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(2): 220379-220387.
[9]	LONG Kai, GU Xianguang, WANG Xuan. Lightweight design method for continuum structure under vibration using multiphase materials [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(10): 221022-221022.
[10]	XING Yufeng, CHEN Lei. Accuracy analysis of mathematical homogenization method for several periodical composite structure [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2015, 36(5): 1520-1529.
[11]	Wang Xingming;Xing Yufeng. Developments in Research on 3D Braided Composites [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2010, 31(5): 914-927.
[12]	Sun Shiping;Qin Guohua;Zhang Weihong. Integrated Optimal Design of Multiphase Porous Materials and Structures [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2009, 30(1): 186-192.
[13]	Xu Yingjie;Zhang Weihong;Yang Jungang;Wang Haibin. Prediction of Effective Elastic Modulus of Plain Weave Multi-phase and Multi-layer Silicon Carbide Ceramic Matrix Composite [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2008, 29(5): 1350-1355.
[14]	ZHANG Wei-hong;WANG Lei;SUN Shi-ping. Topology Optimization for Microstructures of Composite Materials Based on Thermal Conductivity [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2006, 27(6): 1229-1233.
[15]	ZHU Ji-hong;ZHANG Wei-hong;QIU Ke-peng. Investigation of Localized Modes in Topology Optimization of Dynamic Structures [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2006, 27(4): 619-623.