Optimization Design Method for Graded Hollow Cellular Structures

doi:10.3969/j.issn.1004-132X.2022.23.011

Abstract

Abstract: To realize the parametric modeling and improve load-bearing capacity of hollow cellular structures， an optimization design method for graded hollow cellular structures was proposed. The implicit modeling of hollow cellular unit cell was realized by the Boolean operation of I-Wrapped Package（I-WP）triply periodic minimal surfaces（TPMS）with different level parameters. The numerical homogenization method was applied to evaluate the effective elastic properties of hollow cellular structures. By introducing the affine concept of convex optimization theory， the hybrid level set function（LSF）was constructed to realize the parametric modeling of the graded hollow cellular structures. A graded hollow cellular structures topology optimization model was established with the maximum stiffness based on the HLSM. The optimality criteria（OC）was used to solve the model to obtain a cellular structure with superior continuity. The finite element method was performed to analyze the mechanical performance of the optimized structures. Then selective laser sintering（SLS）technology was used to manufacture specimens and the mechanics experiments were carried out. Numerical examples and experimental results show that this method may effectively realize the parametric design of hollow cellular structures， and significantly improve the load-bearing performance of the structures.

Key words: hollow cellular structure, parametric modeling, homogenization method, hybrid level set method（HLSM）, mechanics experiment

摘要： 针对空心多孔结构参数化建模困难、承载性能不足的问题，提出了一种梯度空心多孔结构优化设计方法。将不同水平参数的I-WP（I-wrapped package）型三周期极小曲面进行布尔运算实现了空心多孔结构单胞的隐式建模，并基于数值均匀化法对该结构的等效弹性属性进行了评估。通过引入凸优化理论的仿射概念，构造混合水平集函数，实现了梯度空心多孔结构的参数化建模；以刚度最大化为目标函数，建立了基于混合水平集方法的梯度空心多孔结构拓扑优化模型，通过优化准则算法进行求解，得到了具有良好连续性的梯度多孔结构。对优化结构进行了有限元分析，并采用选择性激光烧结技术制造了模型样件，开展了力学实验。数值案例和实验结果表明，该方法可有效实现空心多孔结构的参数化设计，显著提高了多孔结构的承载性能。

关键词: 空心多孔结构, 参数化建模, 均匀化法, 混合水平集法, 力学实验

CLC Number:

TH112

TIAN Qihua, SHU Zhengtao, FU Junjian, DU Yixian, ZHOU Xiangman, TIAN Lei. Optimization Design Method for Graded Hollow Cellular Structures[J]. China Mechanical Engineering, 2022, 33(23): 2867-2878.

田启华, 舒正涛, 付君健, 杜义贤, 周祥曼, 田磊. 梯度空心多孔结构优化设计方法[J]. 中国机械工程, 2022, 33(23): 2867-2878.

References

［1］RO C J， ROPER C S. Analytical Models of the Geometric Properties of Solid and Hollow Architected Lattice Cellular Materials［J］. Journal of Materials Research， 2018， 33（3）：264-273.
［2］XU J， WU Y， WANG L， et al. Compressive Properties of Hollow Lattice Truss Reinforced Honeycombs（Honeytubes）by Additive Manufacturing：Patterning and Tube Alignment Effects［J］. Materials & Design. 2018， 156：446-457.
［3］SCHAEDLER T A， JACOBSEN A J， TORRENTS A， et al. Ultralight Metallic Microlattices［J］. Science， 2011， 334（6058）：962-965.
［4］陈婷婷，刘杰，谭栋国，等. 金字塔型空心点阵结构的隔声特性研究［J］. 振动与冲击， 2017， 36（23）：209-215.
CHEN Tingting， LIU Jie， TAN Dongguo， et al. Sound Insulation Performance of Pyramidal Hollow Lattice Structures［J］. Journal of Vibration and Shock， 2017， 36（23）：209-215.
［5］张横，丁晓红，沈磊，等. 考虑连接性的三明治阻尼复合结构拓扑优化设计［J］. 中国机械工程， 2021， 32（20）：2403-2410.
ZHNG Heng， DING Xiaohong， SHEN Lei， et al. Topology Optimization of Sandwich Damping Composite Structure with Connective Stiffness Phase［J］. China Mechanical Engineering， 2021， 32（20）：2403-2410.
［6］QUEHEILLALT D T， WADLEY H N G. Cellular Metal Lattices with Hollow Trusses［J］. Acta Materialia， 2005， 53（2）：303-313.
［7］QUEHEILLALT D T， WADLEY H N G. Pyramidal Lattice Truss Structures with Hollow Trusses［J］. Materials Science and Engineering：A， 2005， 397（1/2）：132-137.
［8］殷莎. 基于Ashby设计思想的新型点阵结构——制备工艺与力学性能表征［D］. 哈尔滨：哈尔滨工业大学， 2013.
YIN Sha. Novel Lattice Structures Based Onashbys Desi Gning Criteria：Fabrication and Mechanical Properties Characterization［D］. Harbin：Harbin Institute of Technology， 2013.
［9］TANCOGNE-DEJEAN T， MOHR D. Elastically-Isotropic Elementary Cubic Lattices Composed of Tailored Hollow Beams［J］. Extreme Mechanics Letters， 2018， 22：13-18.
［10］任会兰，申海艇，栗建桥. 单轴压缩下Ti-6Al-4V空心点阵结构的力学行为［J］. 中国科学：技术科学， 2021， 51（3）：293-304.
REN Huilan， SHEN Haiting， LI Jianqiao. Uniaxial Compression Mechanical Behavior of Ti-6Al-4V Hollow Lattice Structures［J］. Scientia Sinica Technologica， 2021， 51（3）：293-304.
［11］PINGLE S M， FLECK N A， DESHPANDE V S， et al. Collapse Mechanism Maps for a Hollow Pyramidal Lattice［J］. Proceedings of the Royal Society A：Mathematical， Physical and Engineering Sciences， 2011， 467：985-1011.
［12］BAI J， ZUO W. Hollow Structural Design in Topology Optimization via Moving Morphable Component Method［J］. Structural and Multidisciplinary Optimization， 2020， 61（1）：187-205.
［13］杜义贤，孙鹏飞，付君健，等. I-WP型极小曲面空心多孔结构设计与力学性能分析［J］.计算机集成制造系统, 2022， 28(9):2739-2747.
DU Yixian, SUN Pengfei, FU Junjian, et al. Design and Mechanical Properties Analysis of I-WP Triply Periodic Minimal Surfaces Hollow Cellular Structure ［J］. Computer Integrated Manufacturing System, 2022， 28(9):2739-2747.
［14］仲梁维，李小伟，胡寿根. 轻质点阵结构的参数化建模及力学性能研究［J］. 中国机械工程， 2014， 25（16）：2253-2261.
ZHONG Liangwei， LI Xiaowei， HU Shougen. Research on Parametric Modeling and Mechanical Properties of Lightweight Lattice Structures［J］. China Mechanical Engineering， 2014， 25（16）：2253-2261.
［15］付君健. 基于参数化水平集的多孔结构拓扑优化方法研究［D］. 武汉：华中科技大学， 2019.
FU Junjian. Research on Parametric Level Set Based Topology Optimization Method for Cellular Structures［D］. Wuhan：Huazhong University of Science and Technology， 2019.
［16］LI D， DAI N， TANG Y， et al. Design and Optimization of Graded Cellular Structures with Triply Periodic Level Surface-based Topological Shapes［J］. Journal of Mechanical Design， 2019， 141（7）：071402.
［17］徐岗，朱亚光，李鑫，等. 插值边界的四边网格离散极小曲面建模方法［J］. 软件学报， 2016， 27（10）：2499-2508.
XU Gang， ZHU Yaguang， LI Xin， et al. Constructing Discrete Minimal Surfaces with Quadrilateral Meshes from Described Boundary［J］. Journal of Software， 2016， 27（10）：2499-2508.
［18］XU F， ZHANG X， ZHANG H. A Review on Functionally Graded Structures and Materials for Energy Absorption［J］. Engineering Structures， 2018， 171：309-325.
［19］LI H， LUO Z， GAO L， et al. Topology Optimization for Concurrent Design of Structures with Multi-patch Microstructures by Level Sets［J］. Computer Methods in Applied Mechanics and Engineering， 2018， 331：536-561.
［20］TERRIAULT P， BRAILOVSKI V. Modeling and Simulation of Large， Conformal， Porosity-graded and Lightweight Lattice Structures Made by Additive Manufacturing［J］. Finite Elements in Analysis and Design， 2018， 138：1-11.
［21］RADMAN A， HUANG X， XIE Y M. Topology Optimization of Functionally Graded Cellular Materials［J］. Journal of Materials Science， 2013， 48（4）：1503-1510.
［22］赵芳垒，敬石开，刘晨燕. 基于局部相对密度映射的变密度多孔结构设计方法［J］. 机械工程学报， 2018， 54（19）：121-128.
ZHAO Fanglei， JING Shikai， LIU Chenyan. Variable Density Cellular Structure Design Method Base on Local Relative Density Mapping［J］. Journal of Mechanical Engineering， 2018， 54（19）：121-128.
［23］蔡金虎，王春洁. 基于映射的梯度点阵结构设计方法［J］. 振动与冲击. 2020， 39（20）：74-81.
CAI Jinhu， WANG Chunjie. A Graded Lattice Structure Design Method Based on Mapping Process［J］. Journal of Vibration and Shock， 2020， 39（20）：74-81.
［24］WANG C， GU X， ZHU J， et al. Concurrent Design of Hierarchical Structures with Three-dimensional Parameterized Lattice Microstructures for Additive Manufacturing［J］. Structural and Multidisciplinary Optimization， 2020， 61（3）：869-894.
［25］WU J， SIGMUND O， GROEN J P. Topology Optimization of Multi-scale Structures：a Review［J］. Structural and Multidisciplinary Optimization， 2021， 63（3）：1455-1480.
［26］LI D， LIAO W， DAI N， et al. Comparison of Mechanical Properties and Energy Absorption of Sheet-based and Strut-based Gyroid Cellular Structures with Graded Densities［J］. Materials， 2019， 12（13）：2183.
［27］OSHER S， SETHIAN J A. Fronts Propagating with Curvature-dependent Speed：Algorithms Based on Hamilton-Jacobi Formulations［J］. Journal of Computational Physics， 1988， 79（1）：12-49.
［28］VAN DIJK N P， MAUTE K， LANGELAAR M， et al. Level-set Methods for Structural Topology Optimization：a Review［J］. Structural and Multidisciplinary Optimization， 2013， 48（3）：437-472.
［29］LIAO B， XIA R F， LI W， et al. 3D-printed Ti6Al4V Scaffolds with Graded Triply Periodic Minimal Surface Structure for Bone Tissue Engineering［J］. Journal of Materials Engineering and Performance， 2021， 30（7）：4993-5004.
［30］GANDY P J F， BARDHAN S， MACKAY A L， et al. Nodal Surface Approximations to the P， G， D and I-WP Triply Periodic Minimal Surfaces［J］. Chemical Physics Letters， 2001， 336（3）：187-195.
［31］OSHER S， FEDKIW R， PIECHOR K. Level Set Methods and Dynamic Implicit Surfaces［J］. Applied Mechanics Reviews， 2004， 57（3）：B15.
［32］ANDREASSEN E， ANDREASEN C S. How to Determine Composite Material Properties Using Numerical Homogenization［J］. Computational Materials Science， 2014， 83：488-495.
［33］BERGER J B， WADLEY H N G， MCMEEKING R M. Mechanical Metamaterials at the Theoretical Limit of Isotropic Elastic Stiffness［J］. Nature， 2017， 543：533-537.
［34］STEPHEN B. Convex Optimization［M］. New York：Cambridge University Press， 2004.
［35］ZONG H， LIU H， MA Q， et al. VCUT Level Set Method for Topology Optimization of Functionally Graded Cellular Structures［J］. Computer Methods in Applied Mechanics and Engineering， 2019， 354：487-505.
［36］付君健，孙鹏飞，杜义贤，等. 基于子结构法的多层级结构拓扑优化［J］. 中国机械工程， 2021， 32（16）：1937-1944.
FU Junjian， SUN Pengfei， DU Yixian， et al. Hierarchical Structure Topology Optimization Based on the Substructure Method［J］. China Mechanical Engineering， 2021， 32（16）：1937-1944.
［37］GRGOIRE A， FRAN O J， ANCA-MARIA T. Structural Optimization Using Sensitivity Analysis and a Level-set Method［J］. Journal of Computational Physics， 2003， 194（1）：363-393.
［38］ZHOU M， ROZVANY G I N. The COC Algorithm， Part Ⅱ：Topological， Geometrical and Generalized Shape Optimization［J］. Computer Methods in Applied Mechanics and Engineering， 1991， 89（1）：309-336.

[1]	WANG Xin, CHEN Geng, LIU Feng, YU Chenfan, ZHANG Lele, . Shakedown　Analysis and Optimization of Airtight Cabin Used in Spacecrafts [J]. China Mechanical Engineering, 2022, 33(20): 2502-2508.
[2]	Zhong Liangwei, Li Xiaowei. Parametric of Skin-frame Lattice Structures and Mechanics Optimization Analysis [J]. China Mechanical Engineering, 2014, 25(20): 2795-2800.
[3]	Zhong Liangwei, Li Xiaowei, Hu Shougen. Research on Parametric Modeling and Mechanical Properties of Lightweight Lattice Structures [J]. China Mechanical Engineering, 2014, 25(16): 2253-2261.
[4]	Zhang Zhenzhou;Zhong Liangwei. Research on Parametric Modeling and Strength Analysis of Strand Loose Tube Composite Cable [J]. China Mechanical Engineering, 2013, 24(17): 2289-2284.
[5]	LI Wen-Wei, CHEN Na, YAN Guo-Dong. Reuse of Virtual Prototyping for Complex Mechanical Products Based on Parametric Modeling [J]. China Mechanical Engineering, 2011, 22(5): 530-535.
[6]	Shen Jingfeng;Yao Fusheng;Qian Wei. Development and Study of Virtual Design System for Micro Gas Turbine [J]. J4, 2008, 19(16): 0-2015.