A Small Sample Data Driven and Constraint Fusion Multi-Objective Topology Optimization Method

Ying Jiahuan; Zhou Huanlin; Meng Zeng

doi:10.21656/1000-0887.460112

Volume 47 Issue 9

Sep. 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(9): 1107-1116

Ying Jiahuan, Zhou Huanlin, Meng Zeng. A Small Sample Data Driven and Constraint Fusion Multi-Objective Topology Optimization Method[J]. Applied Mathematics and Mechanics, 2026, 47(9): 1107-1116. doi: 10.21656/1000-0887.460112

Citation:

PDF( 3014 KB)

A Small Sample Data Driven and Constraint Fusion Multi-Objective Topology Optimization Method

doi: 10.21656/1000-0887.460112

College of Civil Engineering, Hefei University of Technology, Hefei 230009, P. R. China

Received Date: 2025-06-04
Rev Recd Date: 2025-09-02
Publish Date: 2026-09-01

Abstract

Abstract

The traditional volume-compliance multi-objective topology optimization methods rely on intensive finite element calculations, often suffering from low computational efficiency and sparse coverage of the Pareto solution set. To address these issues, an efficient multi-objective topology optimization method based on small sample data driving and constraint fusion was proposed. The core of this method is the establishment of a small sample conditional generative adversarial network embedded with physical constraints, which learns from a limited number of finite element calculations to generate a high-precision and comprehensive Pareto optimal solution set for multi-objective topology optimization. By means of the small sample data driven techniques, the relationships between volume fractions and multi-objective topological structures were established. The results show that, the proposed method ensures computational accuracy while minimizing reliance on large-scale finite element calculations. Furthermore, the fusion of multiple physical constraints guarantees high precision mapping between physical fields and topologies, providing an efficient and reliable solution for multi-objective optimization design.
- multi-objective topology optimization,
- small sample data,
- data driven,
- physical constraint,
- generative adversarial network

(Contributed by Zhou Huanlin, Member of the Editorial Board of AMM & Meng Zeng, Member of the Youth Editorial Board of AMM)

FullText(HTML)

References(25)

References

[1]	Sigmund O. A 99 line topology optimization code written in Matlab[J]. Structural and Multidisciplinary Optimization, 2001, 21 (2): 120-127. doi: 10.1007/s001580050176
[2]	Wang D, Xiang C, Pan Y, et al. A deep convolutional neural network for topology optimization with perceptible generalization ability[J]. Engineering Optimization, 2022, 54 (6): 973-988. doi: 10.1080/0305215X.2021.1902998
[3]	Zheng S, Fan H, Zhang Z, et al. Accurate and real-time structural topology prediction driven by deep learning under moving morphable component-based framework[J]. Applied Mathematical Modelling, 2021, 97: 522-535. doi: 10.1016/j.apm.2021.04.009
[4]	Yan J, Geng D, Xu Q, et al. Real-time topology optimization based on convolutional neural network by using retrain skill[J]. Engineering With Computers, 2023, 39 (6): 4045-4059. doi: 10.1007/s00366-023-01846-3
[5]	Li J, Wang L, Liu H, et al. Probabilistic reliability-based topology optimization of multi-scale structure under load uncertainty[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 435: 117656. doi: 10.1016/j.cma.2024.117656
[6]	Wang L, Shi D, Zhang B, et al. Real-time topology optimization based on deep learning for moving morphable components[J]. Automation in Construction, 2022, 142: 104492. doi: 10.1016/j.autcon.2022.104492
[7]	Geng D, Yan J, Xu Q, et al. Real-time structure topology optimization using CNN driven moving morphable component method[J]. Engineering Structures, 2023, 290: 116376. doi: 10.1016/j.engstruct.2023.116376
[8]	Xia Z, Zhang H, Zhuang Z, et al. A machine-learning framework for isogeometric topology optimization[J]. Structural and Multidisciplinary Optimization, 2023, 66 (4): 83. doi: 10.1007/s00158-023-03539-3
[9]	Zhang Z, Yao W, Li Y, et al. Topology optimization via implicit neural representations[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 411: 116052. doi: 10.1016/j.cma.2023.116052
[10]	Chi H, Zhang Y, Tang T L E, et al. Universal machine learning for topology optimization[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 375: 112739. doi: 10.1016/j.cma.2019.112739
[11]	谢川, 徐超, 周丹发, 等. 带襟翼导轨翼肋后缘尺寸-拓扑综合优化的摄动神经网络代理模型法[J]. 应用数学和力学, 2024, 45 (1): 61-71. doi: 10.21656/1000-0887.440033 Xie Chuan, Xu Chao, Zhou Danfa, et al. The perturbation neural network surrogate model method for size-topology synthetical optimization of wing rib trailing edges with flap tracks[J]. Applied Mathematics and Mechanics, 2024, 45 (1): 61-71. (in Chinese) doi: 10.21656/1000-0887.440033
[12]	Tan R K, Qian C, Li K, et al. An adaptive and scalable artificial neural network-based model-order-reduction method for large-scale topology optimization designs[J]. Structural and Multidisciplinary Optimization, 2022, 65 (12): 348. doi: 10.1007/s00158-022-03456-x
[13]	崔富豪. 基于深度学习的拓扑优化算法研究[D]. 烟台: 烟台大学, 2024. Cui Fuhao. Research on topology optimization algorithm based on deep learning[D]. Yantai: Yantai University, 2024. (in Chinese)
[14]	郭旭, 廉艳平. 数据驱动的结构分析与设计专题序[J]. 力学学报, 2024, 56 (7): 1838-1839. Guo Xu, Lian Yanping. Preface of theme articles on data-driven structure analysis and design[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56 (7): 1838-1839. (in Chinese)
[15]	Huang M, Cui T, Liu C, et al. A problem-independent machine learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures[J]. Extreme Mechanics Letters, 2023, 63: 102041. doi: 10.1016/j.eml.2023.102041
[16]	Du Z, Luo J, Xu Z, et al. Higher-order topological insulators by ML-enhanced topology optimization[J]. International Journal of Mechanical Sciences, 2023, 255: 108441. doi: 10.1016/j.ijmecsci.2023.108441
[17]	Du Z, Ma X, Hao W, et al. Real-time generative design of diverse optimized structures with controllable structural complexities and high quality[J]. Extreme Mechanics Letters, 2025, 77: 102321. doi: 10.1016/j.eml.2025.102321
[18]	Ryu N, Seo M, Min S. Multi-objective topology optimization incorporating an adaptive weighed-sum method and a configuration-based clustering scheme[J]. Computer Methods in Applied Mechanics and Engineering, 2021, 385: 114015. doi: 10.1016/j.cma.2021.114015
[19]	Marler R T, Arora J S. Survey of multi-objective optimization methods for engineering[J]. Structural and Multidisciplinary Optimization, 2004, 26 (6): 369-395. doi: 10.1007/s00158-003-0368-6
[20]	Sanchis J, Martínez M, Blasco X, et al. A new perspective on multiobjective optimization by enhanced normalized normal constraint method[J]. Structural and Multidisciplinary Optimization, 2008, 36 (5): 537-546. doi: 10.1007/s00158-007-0185-4
[21]	Deaton J D, Grandhi R V. A survey of structural and multidisciplinary continuum topology optimization: post 2000[J]. Structural and Multidisciplinary Optimization, 2014, 49 (1): 956.
[22]	刘应, 傅小恒, 唐莉萍. 非凸多目标优化问题有效解集的非空性与有界性的渐近刻画[J]. 应用数学和力学, 2025, 46 (4): 519-527. doi: 10.21656/1000-0887.450235 Liu Ying, Fu Xiaoheng, Tang Liping. Asymptotic characterization of non-emptiness and boundedness of efficient solution sets for nonconvex multi-objective optimization problems[J]. Applied Mathematics and Mechanics, 2025, 46 (4): 519-527. (in Chinese) doi: 10.21656/1000-0887.450235
[23]	Suresh K. A 199-line Matlab code for Pareto-optimal tracing in topology optimization[J]. Structural and Multidisciplinary Optimization, 2010, 42 (5): 665-679. doi: 10.1007/s00158-010-0534-6
[24]	Mirza M, Osindero S. Conditional generative adversarial nets[PP/OL]. arXiv[2025-06-04]. https://arxiv.org/abs/1411.1784.
[25]	Zhao Z, Singh S, Lee H, et al. Improved consistency regularization for GANs[J]. Proceedings of the AAAI Conference on Artificial Intelligence, 2021, 35 (12): 11033-11041. doi: 10.1609/aaai.v35i12.17317