Moving morphable components topology optimization approach: Extensions, applications, and prospects
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摘要: 借助显式几何参数描述结构拓扑构型, 移动可变形组件 (Moving Morphable Components, MMC) 拓扑优化方法具备设计变量少、CAD (Computer Aided Design) 系统的无缝兼容性、拓扑描述与分析模型解耦、易于表示制造相关约束等优势, 可以显著提升结构设计效率, 因而自提出以来便受到广泛关注. 本文系统阐述了MMC方法的核心理论框架及扩展体系: 从初始技术方案中超椭圆方程的结构描述、结构柔顺性目标函数、含体积约束的优化列式、灵敏度分析以及核心技术问题, 到涵盖其几何描述、数值技术以及优化求解的多维度扩展. 在分析理论演进脉络的基础之上, 评述MMC方法在非线性、多材料和应力约束等特定优化技术问题的解决方案, 总结MMC方法在轻量化构件设计、动态性能设计、超材料/超结构设计等领域的多学科交叉应用. 最后, 基于现有研究瓶颈讨论若干重点发展方向, 展望MMC方法在智能计算时代的广阔前景.Abstract: By explicitly parameterizing structural topologies with geometric variables, the Moving Morphable Components (MMC) topology optimization method offers several notable advantages, including a reduced number of design variables, seamless compatibility with Computer-Aided Design (CAD) systems, decoupling between topology description and analysis models, and convenient incorporation of manufacturing-related constraints. These features can significantly enhance the efficiency of structural design, which has attracted widespread attention since the method was first proposed. This review systematically presents the core theoretical framework and extension system of the MMC method. Starting from the initial technical formulation—including structural representation based on super-elliptic equations, compliance-based objective functions, optimization formulations with volume constraints, sensitivity analysis, and key technical issues—it further reviews multidimensional extensions encompassing geometric description, numerical techniques, and optimization solvers. Based on an analysis of the theoretical evolution of the method, this paper also reviews MMC-based solution strategies for specific optimization challenges such as nonlinear behavior, multi-material design, and stress-constrained optimization. In addition, interdisciplinary applications of the MMC method are summarized in areas including lightweight structural design, dynamic performance optimization, and metamaterial/metastructure design. Finally, in view of current research bottlenecks, several important future research directions are discussed, and the broad prospects of the MMC method in the era of intelligent computation are envisioned.
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图 2 MMC方法中结构拓扑的几何描述 (Li et al. 2023)
图 3 二维场景中MMC方法的直组件结构形貌 (Zhang et al. 2016b)
图 4 MMC方法中与组件排布相关的技术进展. (a) 基于有效组件的概念控制结构复杂度 (Zhang et al. 2017c), (b) 基于拔靴法思路的双阶段计算模型 (Weiss et al. 2021), (c) 具有生长机制的MMC方法 (Cui et al. 2022), (d) 基于布局优化设置初始组件排布的双阶段计算模型 (Lotfalian et al. 2025), (e) 基于外在几何约束的MMC方法 (Shannon et al. 2025)
图 5 MMC方法中与尺寸控制相关的技术进展. (a) MMC框架下的最小尺寸控制方法 (Zhang et al. 2016a), (b) 基于半圆形端部条状组件的显式宽度调控策略 (Hoang and Jang, 2017), (c) 引入罚函数的等宽控制 (Niu and Wadbro, 2019), (d) 基于虚拟组件骨架的有效连接状态技术 (Wang et al. 2019), (e) 基于移动可变形片的厚度可控方法 (Nguyen et al. 2020)
图 6 MMC方法中与计算策略相关的技术进展. (a) 多分辨率求解框架 (Liu C et al. 2018), (b) 多层级网格细分方法 (Lian et al. 2020), (c) 基于投影变换的MMC方法 (Wang et al. 2021)
图 7 MMC方法中与组件骨架曲线化相关的几何描述扩展. (a) 组件骨架曲线化扩展方案 (Guo et al. 2016), (b) 移动宽度贝齐尔组件方法 (Zhu et al. 2021a), (c) 广义贝齐尔组件方法 (Shannon et al. 2022), (d) 采用时间序列视角的MMC方法 (Li et al. 2023)
图 8 MMC方法中与组件端部相关的几何描述扩展. (a) 连接变形组件方法 (Deng and Chen, 2016), (b) 绝对节点坐标公式组件方法 (Otsuka et al. 2023), (c) 节点驱动MMC方法 (Xu J et al. 2025), (d) 电阻抗断层成像中的半圆形端部组件 (Liu and Du, 2021), (e) 基于面积公式的组件描述 (Yang and Huang, 2020)
图 9 MMC方法中的结构分析扩展. (a) 自由度移除技术 (Zhang et al. 2018c), (b) 单元自适应修剪技术 (Zhou et al. 2024), (c) 组件隐藏机制 (Zhang et al. 2016a), (d) IGA扩展方案 (Hou et al. 2017), (e) 采用R函数均匀化组件的$ {C}^{1} $不连续区域 (Xie et al. 2020a)
图 10 MMC方法中与实时优化设计相关的机器学习扩展. (a) 基于卷积神经网络的实时拓扑优化模型 (Zheng et al. 2021), (b) 引入金字塔注意力机制的实时拓扑优化模型 (Wang L et al. 2022), (c) 采用Wasserstein生成对抗网络的实时拓扑优化模型 (Du et al. 2025)
图 11 MMC方法中与结构分析加速相关的机器学习扩展. (a) 问题无关机器学习系列研究 (Huang et al. 2022, 2023; Zhang L et al. 2024), (b) 将问题无关机器学习应用于MMC方法设计三维晶格结构 (Xu W et al. 2025), (c) 基于问题无关机器学习的大规模、高性能并行拓扑优化算法 (Ma et al. 2025)
图 12 MMC方法在非线性拓扑优化问题中的应用. (a) 具有约束端的移动可变形贝齐尔组件 (Zhu et al. 2022), (b) 基于投影变换的混合拓扑优化方法 (Wang R et al. 2022), (c) 可变换三角网格方法的分层计算框架 (Ding et al. 2021)
图 13 MMC方法在多材料拓扑优化问题中的应用. (a) 引入多种材料属性组件的策略 (Zhang et al. 2018c), (b) 加入纤维取向设计纤维增强复合材料 (Sun et al. 2022), (c) 将杆件与嵌入组件分别映射的策略 (Wang et al. 2018), (d) 基于三周期极小曲面的多材料重叠 (Zhang et al. 2022)
图 14 MMC方法在薄壁结构设计中的应用. (a) 约束材料体积和屈曲荷载因子的设计方案 (Zhang et al. 2018b), (b) 在桁架芯夹层板中的应用 (Chu et al. 2019), (c) 组合内外部TDF描述空心结构组件 (Bai and Zuo, 2020), (d) 基于截面几何性质视角的新优化列式 (Guo et al. 2021), (e) 移动可变形筋肋方法 (Zhang et al. 2025b)
图 15 MMC方法在超材料/超结构设计中的应用及结构基因库相关扩展研究. (a) 深海海绵启发的力学超材料单胞性能提升设计 (Du et al. 2023a), (b) 结合图神经网络的智能逆向设计框架 (Hao et al. 2025), (c) 结合图神经网络的智能逆向设计框架在多尺度优化问题中的应用 (Hao et al. 2026)
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