Hierarchical Structure Topology Optimization Based on Substructure Method

doi:10.3969/j.issn.1004-132X.2021.16.006

Abstract

Abstract: To avoid the separation of scales in the hierarchical structures topology optimization, and to maintain the connection between different cellular structures, a hierarchical structure topology optimization method was proposed based on the substructure method. The topology optimization of cellular structures was divided into two hierarchies with the parametric level set method. The topology configuration of the cellular structure was optimized in meso-hierarchy. The spatial distribution of the cellular structure was optimized in macro-hierarchy. The relationship between macro-structure and meso-structure was established using the substructure method. The meso-structures was condensed into a super element, which was then used as the basic element of macro-structures for the structural analysis and optimization. Numerical examples show that the proposed method is effective for the 2D and 3D hierarchical structure topology optimization. The proposed method may effectively guarantee the connection between different cellular structures when considering multiple types of cellular structures in the hierarchical structure design.

Key words: topology optimization, hierarchical structure, parametric level set method, substructure method

摘要： 为避免多层级结构拓扑优化中的尺度分离问题，保证不同多孔结构之间的连接性，提出了一种基于子结构法的多层级结构拓扑优化方法。采用参数化水平集方法，将多孔结构拓扑优化分为两个层级，在细观层级优化多孔结构拓扑构型，在宏观层级优化多孔结构最优空间分布。采用子结构法建立宏细观结构之间的联系，将细观结构凝聚为超单元，凝聚的超单元又作为宏观结构分析和优化的基本单元。数值算例表明：所提方法可有效实现二维、三维多层级结构的拓扑优化；在多层级结构设计中考虑多种类型的多孔结构时，所提方法能有效保证不同多孔结构之间的连接性。

关键词: 拓扑优化, 多层级结构, 参数化水平集法, 子结构法

CLC Number:

TH122

FU Junjian, SUN Pengfei, DU Yixian, TIAN Qihua, GAO Liang. Hierarchical Structure Topology Optimization Based on Substructure Method[J]. China Mechanical Engineering, 2021, 32(16): 1937-1944，1951.

付君健, 孙鹏飞, 杜义贤, 田启华, 高亮. 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): 1937-1944，1951.

References

［1］XIA L. Multiscale Structural Topology Optimization［M］.Oxford：Elsevier, 2016.

［2］LIU L, YAN J, CHENG G. Optimum Structure with Homogeneous Optimum Truss-like Material［J］. Computers & Structures, 2008, 86(13/14):1417-1425.

［3］NIU B, YAN J, CHENG G. Optimum Structure with Homogeneous Optimum Cellular Material for Maximum Fundamental Frequency［J］. Structural and Multidisciplinary Optimization, 2009, 39(2):115-132.

［4］YAN J, CHENG G, LIU L. A Uniform Optimum Material Based Model for Concurrent Optimization of Thermoelastic Structures and Materials［J］. International Journal for Simulation and Multidisciplinary Design Optimization, 2008, 2(4):259-266.

［5］VICENTE W, ZUO Z, PAVANELLO R, et al. Concurrent Topology Optimization for Minimizing Frequency Responses of Two-level Hierarchical Structures［J］. Computer Methods in Applied Mechanics and Engineering, 2016, 301:116-136.

［6］ZHANG W, SUN S. Scale-related Topology Optimization of Cellular Materials and Structures［J］. International Journal for Numerical Methods in Engineering, 2006, 68(9):993-1011.

［7］孙士平. 材料和结构的拓扑优化关键理论与方法研究［D］. 西安：西北工业大学, 2006.

SUN Shiping.Investigation on the Key Theory and Methods of Topology Optimization of Materials and Structures［D］. Xi'an：Northwestern Polytechnical University, 2006.

［8］LI H, LUO Z, ZHANG N, et al. Integrated Design of Cellular Composites Using a Level-set Topology Optimization Method［J］. Computer Methods in Applied Mechanics and Engineering, 2016, 309:453-475.

［9］LI H, LUO Z, GAO L, et al. Topology Optimization for Functionally Graded Cellular Composites with Metamaterials by Level Sets［J］. Computer Methods in Applied Mechanics and Engineering, 2018, 328:340-364.

［10］SIVAPURAM R, DUNNING P D, KIM H A. Simultaneous Material and Structural Optimization by Multiscale Topology Optimization［J］. Structural and Multidisciplinary Optimization, 2016, 54(5):1267-1281.

［11］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.

［12］GAO J, LUO Z, LI H, et al. Topology Optimization for Multiscale Design of Porous Composites with Multi-domain Microstructures［J］. Computer Methods in Applied Mechanics and Engineering, 2019, 344:451-476.

［13］SCHURY F, STINGL M, WEIN F. Efficient Two-scale Optimization of Manufacturable Graded Structures［J］. SIAM Journal on Scientific Computing, 2012, 34(6):B711-B733.

［14］XIA L, BREITKOPF P. Multiscale Structural Topology Optimization with an Approximate Constitutive Model for Local Material Microstructure［J］. Computer Methods in Applied Mechanics and Engineering, 2015, 286:147-167.

［15］XIA L, BREITKOPF P. Concurrent Topology Optimization Design of Material and Structure within FE2 Nonlinear Multiscale Analysis Framework［J］. Computer Methods in Applied Mechanics and Engineering, 2014, 278:524-542.

［16］WANG Y, CHEN F, WANG M Y. Concurrent Design with Connectable Graded Microstructures［J］. Computer Methods in Applied Mechanics and Engineering, 2017, 317:84-101.

［17］孙士平, 刘宇锋. 周期性圆环结构的拓扑优化设计［J］. 机械设计与制造, 2014(10):12-14.

SUN Shiping, LIU Yufeng. Topology Optimization Design of Periodical Ring Structure［J］. Machinery Design & Manufacture, 2014(10):12-14.

［18］FU J, XIA L, GAO L, et al. Topology Optimization of Periodic Structures with Substructuring［J］. Journal of Mechanical Design, 2019, 141(7):071403.

［19］WU Z, XIA L, WANG S, et al. Topology Optimization of Hierarchical Lattice Structures with Substructuring［J］. Computer Methods in Applied Mechanics and Engineering, 2019, 345:602-617.

［20］OSHER S, FEDKIW R. Level Set Methods and Dynamic Implicit Surfaces［M］. New York：Springer, 2006.

［21］LUO Z, TONG L, KANG Z. A Level Set Method for Structural Shape and Topology Optimization Using Radial Basis Functions［J］. Computers & Structures, 2009, 87(7):425-434.

［22］WANG S, WANG M Y. Radial Basis Functions and Level Set Method for Structural Topology Optimization［J］. International Journal for Numerical Methods in Engineering, 2010, 65(12):2060-2090.

［23］WEI P, LI Z, LI X, et al. An 88-line MATLAB Code for the Parameterized Level Set Method Based Topology Optimization Using Radial Basis Functions［J］. Structural and Multidisciplinary Optimization, 2018, 58(2):831-849.

［24］GUYAN R J. Reduction of Stiffness and Mass Matrices［J］. AIAA Journal, 1965, 3(2):380-389.

［25］SAAD Y. Iterative Methods for Sparse Linear Systems［M］.Philadelphia：SIAM, 2003.

［26］李昂, 刘初升. 基于超单元技术的大型复杂结构的拓扑优化设计［J］. 中国机械工程, 2017, 28(20):2467-2474.

LI Ang, LIU Chusheng. Topology Optimization Designs of Large and Complex Structures Based on Super Element Technique ［J］. China Mechanical Engineering, 2017, 28(20):2467-2474.

［27］WANG M Y, WANG X, GUO D. A Level Set Method for Structural Topology Optimization［J］. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1/2):227-246.

［28］ALLAIRE G, JOUVE F, TOADER A M. Structural Optimization Using Sensitivity Analysis and a Level-set Method［J］. Journal of Computational Physics, 2004, 194(1):363-393.

［29］ZHOU M, ROZVANY G I N. The COC Algorithm, Part Ⅱ：Topological, Geometry and Generalized Shape Optimization［J］. Computer Methods in Applied Mechanics & Engineering, 2015, 89(1/3):309-336.

[1]	YANG Feng, LUO Shijie, YANG Jianghong, WANG Yingjun, . A GPU-accelerated High-efficient Multi-grid Algorithm for ITO [J]. China Mechanical Engineering, 2024, 35(04): 602-613.
[2]	SU Yonglei, ZHANG Zhifei. Integrated Casting Triangular Beam Lightweight Improving with Multi-performance Constraints of Body Systems [J]. China Mechanical Engineering, 2024, 35(04): 691-699.
[3]	LI Rongqi, YAN Tao, HE Zhicheng, MI Dong, JIANG Chao, ZHENG Jing. A Topology Optimized Design Method for High-performance Structures with Fluid-thermal-mechanics Coupling [J]. China Mechanical Engineering, 2024, 35(03): 487-497.
[4]	WANG Chao, CHENG Aiguo, ZHANG Chenglin, YU Wanyuan, HE Zhicheng. Lightweight Design of Protective Structures of Battery Packs for Bottom-scraping Safety [J]. China Mechanical Engineering, 2023, 34(19): 2343-2352.
[5]	LIU Yingjie, HU Qiang, ZHAO Xinming, ZHANG Shaoming, HUANG Shuai, WANG Yonghui. Research on Topology Optimization and Additive Manufacturing of Automotive Engine Connection Brackets [J]. China Mechanical Engineering, 2023, 34(18): 2238-2267.
[6]	DONG Xiaohu, WANG Shitao, ZHOU Dechun. High Rigidity and Lightweight Design of Purlins in Photovoltaic Tracker Bracket [J]. China Mechanical Engineering, 2023, 34(10): 1207-1213.
[7]	WEN Guilin, CHEN Gaoxi, WANG Hongxin, XUE Liang, WEI Peng, LIU Jie, . Time Domain Dynamics Topology Optimization of Functionally Gradient Material Structures with Self-weight Load [J]. China Mechanical Engineering, 2022, 33(23): 2774-2782.
[8]	DING Yandong, LUO Nianmeng, YANG Aodi, WANG Shuting, ZHU Haoran, XIE Xianda. Efficient Multigrid Isogeometric Topology Optimization under Bézier Element Stiffness Mapping [J]. China Mechanical Engineering, 2022, 33(23): 2801-2810.
[9]	YANG Yuhao, ZHENG Wei, WANG Yingjun, . An Efficient Isogeometric Topology Optimization Method Using DOF Reduction and Convergence Acceleration [J]. China Mechanical Engineering, 2022, 33(23): 2811-2821.
[10]	MENG Liang, ZHONG Mingzhe, LI Wenbiao, XIA Liang, GAO Tong, ZHU Jihong, ZHANG Weihong, . Topology Optimization Design of Aero-engine External System Brackets for Additive Manufacturing [J]. China Mechanical Engineering, 2022, 33(23): 2822-2832.
[11]	JIA Lianhui, LI Xiaoke, YUAN Wenzheng, HE Wenbin, LIAO Zhaojin. Lightweight Design of Front and Middle Shield Structures Based on Topology Optimization and Kriging Model [J]. China Mechanical Engineering, 2022, 33(23): 2888-2897.
[12]	HE Zhicheng, YANG Dingding, JIANG Chao, WU Yi, JIANG Hexin. Strength-constrainted Topology Optimization Based on Additive Manufacturing Anisotropy [J]. China Mechanical Engineering, 2022, 33(19): 2372-2380，2393.
[13]	ZHANG Heng, DING Xiaohong, SHEN Lei, XU Shipeng. Topology Optimization of Sandwich Damping Composite Structure with Connective Constraint [J]. China Mechanical Engineering, 2021, 32(20): 2403-2410.
[14]	LI Xinqing, ZHAO Qinghai, ZHANG Hongxin, ZHANG Tiezhu, CHEN Jianliang. Steady-state Heat Conduction Topology Optimization Design for Periodic Functional Gradient Structures [J]. China Mechanical Engineering, 2021, 32(19): 2348-2356.
[15]	NIU Bin, YAN Jiaming, MAO Yuming, LIU Haiyang. Collaborative Design Optimization of Damping Layers and Stiffener Layout of Thin-walled Stiffened Plate Structures for Dynamics Performances [J]. China Mechanical Engineering, 2021, 32(16): 1912-1920.