动态载荷作用下的双模量材料结构拓扑优化

doi:10.3901/JME.260305

摘要/Abstract

摘要： 实际工程中的很多材料在拉伸和压缩状态下具有不同的力学性能(即双模量特性)。然而，针对这类材料结构的拓扑优化方法研究很少，尤其是极少考虑动态载荷下的双模量材料结构设计。因此，提出了一种动态载荷下的双模量材料结构拓扑优化方法。首先，采用混合应力单元离散设计域，以获得更准确地应力计算结果；其次，建立基于单元主应力符号和拉/压容限的单元拉/压状态判断准则，构建了四相材料插值模型实现双模量材料结构优化设计；接着，发展多工况载荷下基于单元拉/压应变能的单元拉/压状态判断准则，实现了多工况载荷下的材料分布优化；最后，通过典型数值算例验证了方法的有效性，并分析了拉/压弹性模量比值、激励频率等参数对优化构型的影响规律，为双模量材料结构在动态载荷下的轻量化设计提供理论支撑。

关键词: 拓扑优化, 混合应力单元, 双模量材料, 动态载荷, 复合材料

Abstract: Many materials used in practical engineering exhibit different mechanical properties under tension and compression (i.e., Bi-modulus characteristics). However, there is limited research on topology optimization methods for structures made of such materials, especially those considering dynamic loads. Therefore, a topology optimization method for Bi-modulus material structures under dynamic loads is proposed. First, the hybrid stress element is used to discretize the design domain to obtain more accurate stress calculation results. Second, a criterion for determining the tensile/compressive state of elements based on the principal stress sign and tensile/compressive tolerance is established, and a four-phase material interpolation model is constructed to achieve the optimal design of Bi-modulus material structures. Next, a criterion for determining the tensile/compressive state of elements based on the tensile/compressive strain energy under multi-load conditions is developed, enabling material distribution optimization under multiple load cases. Finally, the effectiveness of the method is verified through typical numerical examples, and the influence of parameters such as the tensile/compressive elastic modulus ratio and excitation frequency on the optimized configuration is analyzed, providing theoretical support for the lightweight design of Bi-modulus material structures under dynamic loads.

Key words: topology optimization, hybrid stress element, bi-modulus material, dynamic load, composite materials

中图分类号:

TH122

蔡金虎, 曹豪浩, 荣见华, 赵磊, 刘鑫, 尹来容, 李方义. 动态载荷作用下的双模量材料结构拓扑优化[J]. 机械工程学报, 2026, 62(13): 294-308.

CAI Jinhu, CAO Haohao, RONG Jianhua, ZHAO Lei, LIU Xin, YIN Lairong, LI Fangyi. Structural Topology Optimization of Bi-modulus Materials under Dynamic Loads[J]. Journal of Mechanical Engineering, 2026, 62(13): 294-308.

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链接本文: http://www.cjmenet.com.cn/CN/10.3901/JME.260305

http://www.cjmenet.com.cn/CN/Y2026/V62/I13/294

参考文献

[1] 高强，王健，张严，等. 拓扑优化方法及其在运载工程中的应用与展望[J]. 机械工程学报，2024，60(4)：369-390. GAO Qiang，WANG Jian，ZHANG Yan，et al. Topology optimization method and its application and prospect in vehicle engineering[J]. Journal of Mechanical Engineering，2024，60(4)：369-390.
[2] BENDSØE M P，KIKUCHI N. Generating optimal topologies in structural design using a homogenization method[J]. Computer Methods in Applied Mechanics and Engineering，1988，71(2)：197-224.
[3] 郑静，荣轩霈，姜潮，等. 考虑惯性载荷的结构热力耦合拓扑优化[J]. 机械工程学报，2024，60(13)：130-140. ZHENG Jing，RONG Xuanpei，JIANG Chao，et al. Structural thermodynamic coupling topology optimization considering inertial loads[J]. Journal of Mechanical Engineering，2024，60(13)：130-140.
[4] 林起崟，贾一鸣，周意葱，等. 面向装配结构的拓扑优化参数影响分析[J]. 机械工程学报，2024，60(15)：291-303. LIN Qiyin，JIA Yiming，ZHOU Yicong，et al. Influence analysis of topology optimization parameters for assembly structures[J]. Journal of Mechanical Engineering，2024，60(15)：291-303.
[5] 白影春，刘康，李超，等. 基于大规模并行计算的三维薄壁填充结构特征频率拓扑优化设计[J]. 机械工程学报，2024，60(11)：32-40. BAI Yingchun，LIU Kang，LI Chao，et al. Eigenfrequency topology optimization design of three-dimensional thin-walled filled structure based on large-scale parallel computing[J]. Journal of Mechanical Engineering，2024，60(11)：32-40.
[6] 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.
[7] QIU Z，LI Q H，LIU S T. Topology optimization method for continuous fiber reinforced composites with different moduli in tension and compression[J]. Computer Methods in Applied Mechanics and Engineering，2024，423：116867.
[8] CAI K，CAO J，SHI J，et al. Optimal layout of multiple bi-modulus materials[J]. Structural and Multidisciplinary Optimization，2016，53(4)：801-811.
[9] SHI J，CAO J，CAI K，et al. Layout optimization for multi-bi-modulus materials system under multiple load cases[J]. Engineering with Computers，2016，32(4)：745-753.
[10] DU Z L，ZHANG W S，ZHANG Y P，et al. Structural topology optimization involving bi-modulus materials with asymmetric properties in tension and compression[J]. Computational Mechanics，2018，63(2)：335-363.
[11] LI Y，XIE Y M. Evolutionary topology optimization for structures made of multiple materials with different properties in tension and compression[J]. Composite Structures，2021，259：113497.
[12] LI Y，YUAN P F，XIE Y M. Topology optimization of structures composed of more than two materials with different tensile and compressive properties[J]. Composite Structures，2023，306：116609.
[13] RONG X P，ZHENG J，JIANG C. Topology optimization for structures with bi-modulus material properties considering displacement constraints[J]. Computers & Structures，2023，276：106952.
[14] GUO X Z，ZHOU K M. Topology optimization for variable stiffness design of fiber-reinforced composites with bi-modulus materials[J]. Optimization and Engineering，2023，24(4)：2745-2762.
[15] JANG H H，LEE H A，LEE J Y，et al. Dynamic response topology optimization in the time domain using equivalent static loads[J]. AIAA Journal，2012，50(1)：226-234.
[16] 江旭东，吴昊，滕晓艳，等. 时域动载荷作用下多微结构多尺度并行动力学拓扑优化[J]. 振动与冲击，2024，43(12)：53-64. JIANG Xudong，WU Hao，TENG Xiaoyan，et al. Topology optimization of multi-microstructure and multi-scale parallel dynamics under time-domain dynamic loads[J]. Journal of Vibration and Shock，2024，43(12)：53-64.
[17] YUN K S，YOUN S K. Microstructural topology optimization of viscoelastic materials of damped structures subjected to dynamic loads[J]. International Journal of Solids and Structures，2018，147：67-79.
[18] DU J B，OLHOFF N. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps[J]. Structural and Multidisciplinary Optimization，2007，34：91-110.
[19] SHU L，WANG M Y，FANG Z D，et al. Level set based structural topology optimization for minimizing frequency response[J]. Journal of Sound and Vibration，2011，330(24)：5820-5834.
[20] ZHAO J P，YOON H，YOUN B D. An efficient concurrent topology optimization approach for frequency response problems[J]. Computer Methods in Applied Mechanics and Engineering，2019，347：700-734.
[21] ZHAO J P，YOON H，YOUN B D. An adaptive hybrid expansion method (AHEM) for efficient structural topology optimization under harmonic excitation[J]. Structural and Multidisciplinary Optimization，2020，61：895-921.
[22] ZHOU P Z，PENG Y C，DU J B. Topology optimization of bi-material structures with frequency-domain objectives using time-domain simulation and sensitivity analysis[J]. Structural and Multidisciplinary Optimization，2021，63(2)：575-593.
[23] 蔡金虎，黄静，黄龙，等. 动态载荷不确定条件下的周期性结构稳健拓扑优化[J]. 机械工程学报，2024，60(23)：304-319. CAI Jinhu，HUANG Jing，HUANG Long，et al. Robust topology optimization of periodic structures under dynamic load uncertainty[J]. Journal of Mechanical Engineering，2024，60(23)：304-319.
[24] NIU B，HE X M，SHAN Y，et al. On objective functions of minimizing the vibration response of continuum structures subjected to external harmonic excitation[J]. Structural and Multidisciplinary Optimization，2018，57：2291-2307.
[25] LIU T，ZHU J H，HE F，et al. A MAC based excitation frequency increasing method for structural topology optimization under harmonic excitations[J]. International Journal for Simulation and Multidisciplinary Design Optimization，2017，8：A4.
[26] ZHU J H，HE F，LIU T，et al. Structural topology optimization under harmonic base acceleration excitations[J]. Structural and Multidisciplinary Optimization，2018，57：1061-1078.
[27] LIU T，ZHU J H，ZHANG W H，et al. Integrated layout and topology optimization design of multi-component systems under harmonic base acceleration excitations[J]. Structural and Multidisciplinary Optimization，2019，59：1053-1073.
[28] CLYNE T W，HULL D. An Introduction to Composite Materials[M]. 3rd ed. Cambridge：Cambridge University Press，2019.
[29] 王选，时元昆，杨博，等. 基于响应面方法的破损−安全结构可靠性拓扑优化[J]. 力学学报，2023，55(5)：1206-1216. WANG Xuan，SHI Yuankun，YANG Bo，et al. Topology optimization of damaged-safe structures based on response surface methodology[J]. Chinese Journal of Theoretical and Applied Mechanics，2023，55(5)：1206-1216.
[30] PIAN T H H，SUMIHARA K. Rational approach for assumed stress finite elements[J]. International Journal for Numerical Methods in Engineering，1984，20(9)：1685-1695.
[31] 曾攀. 有限元基础教程[M]. 北京：高等教育出版社，2009. ZENG Pan. Finite element basic tutorial[M]. Beijing：Higher Education Press，2009.