A Topology Optimization Method Based on End-to-end Deep Learning Framework TOPO-U-Net

doi:10.3969/j.issn.1004-132X.2026.01.018

China Mechanical Engineering ›› 2026, Vol. 37 ›› Issue (1): 174-183.DOI: 10.3969/j.issn.1004-132X.2026.01.018

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A Topology Optimization Method Based on End-to-end Deep Learning Framework TOPO-U-Net

WANG Hao¹^,²(), LUO Haodong³, SHI Yazhong⁴, WANG Liwen⁵(), ZHANG Wei⁵, WANG Zhong⁶

^1.College of Safety Science and Engineering，Civil Aviation University of China，Tianjin，300300
^2.ETTC，Civil Aviation University of China，Tianjin，300300
^3.College of Aeronautical Engineering，Civil Aviation University of China，Tianjin，300300
^4.Southwest China Line Center，Aircraft Maintenance and Engineering Corporation，Chengdu，610000
^5.Institute of Technology and Innovation，Civil Aviation University of China，Tianjin，300300
^6.AECC Chengdu Engine Co. ，Ltd. ，Chengdu，610500

Received:2024-11-20 Online:2026-01-25 Published:2026-02-05
Contact: WANG Liwen

基于端到端深度学习模型TOPO-U型网的结构拓扑优化方法

王浩¹^,²(), 罗浩东³, 施亚中⁴, 王立文⁵(), 张威⁵, 王忠⁶

^1.中国民航大学安全科学与工程学院, 天津, 300300
^2.中国民航大学工程技术训练中心, 天津, 300300
^3.中国民航大学航空工程学院, 天津, 300300
^4.北京飞机维修工程有限公司西南航线中心, 成都, 610000
^5.中国民航大学科技创新研究院, 天津, 300300
^6.中国航发成都发动机有限公司, 成都, 610500

通讯作者: 王立文
作者简介:王浩，男，1985年生，硕士研究生。研究方向为航空器智慧维修、增材制件结构拓扑优化。发表论文5篇。E-mail： wanghao@cauc.edu.cn
王立文^*（通信作者），男，1963年生，研究员、博士研究生导师。研究方向为机场工程自动化、机场地面特种设备。发表论文80余篇。E-mail： hbgdwh@126.com.
基金资助:
国家自然科学基金(U2133202);四川省重大科技专项(2021ZDZX0001);中央高校基本科研业务费专项资金(XJ2024002201)

Abstract

Abstract:

To address the problems such as “grey elements” and “computational cost” in structural topology optimization， an end-to-end deep learning model TOPO-U-Net was proposed. The model integrated high and low feature extraction modules， depth separable convolution， and group normalization. In addition， an evaluation method was designed based on intermediate density element deviation function. Experiments show that the intermediate density deviation rate of the proposed model reaches 85.42%. And， the average optimization computation time is as 1% of that required by the SIMP method， which may significantly reduce the number of “grey elements”， improve the manufacturability of complex structures and computational efficiency of topology optimization.

Key words: solid isotropic material with penalization（SIMP）, deep learning, topology optimization, U-Net

摘要：

针对结构拓扑优化中的“灰度单元”问题和“计算成本”挑战，提出一种基于端到端深度学习模型TOPO-U-Net的结构拓扑优化方法，该模型包括高低阶特征提取模块、深度可分卷积、组归一化，并设计了一种基于中间密度单元偏移函数的评价方法。实验结果表明，所提模型的中间密度偏移率达到85.42%，平均优化计算时间仅为固体各向同性材料惩罚模型方法的1%，显著减少了“灰度单元”数量，提高了设计的可制造性和结构拓扑优化的效率。

关键词: 固体各向同性材料惩罚模型, 深度学习, 拓扑优化, U型网络

CLC Number:

O241.82

WANG Hao, LUO Haodong, SHI Yazhong, WANG Liwen, ZHANG Wei, WANG Zhong. A Topology Optimization Method Based on End-to-end Deep Learning Framework TOPO-U-Net[J]. China Mechanical Engineering, 2026, 37(1): 174-183.

王浩, 罗浩东, 施亚中, 王立文, 张威, 王忠. 基于端到端深度学习模型TOPO-U型网的结构拓扑优化方法[J]. 中国机械工程, 2026, 37(1): 174-183.

Figures/Tables 18

Fig.1 Architecture of TOPO-U-Net

Fig.2 The architecture of high and low feature extraction module

Fig.3 The design domain of sample

Fig.4 The input and output of the neural network

Tab.1 Hyperparameters of the model training process

训练参数	设置	说明
优化器	Adam	梯度下降算法
学习率	0.001	初始学习率
批大小	8	每次训练的样本数
轮次	200	总训练轮次
CUDNN	enable	GPU加速库

Fig.5 Training and validation loss values

Fig.6 Internal details of model prediction results

Tab.2 Comparison of HLFE module ablation experiment

	I_ID	R_IDD/%	R_VFE/%	E_MA
未引入HLFE	0.6735	83.68	2.97	0.0663
引入HLFE	0.7084	85.42	2.96	0.0651

Fig.7 Comparison of topology optimization results

Tab.3 Comparison of three deep learning models

不同模型	I_ID	R_IDD/%	R_VFE/%	E_MA
U-Net	0.6798	83.99	3.76	0.0669
Resnet	0.6467	82.34	2.94	0.0690
TOPO-U-Net	0.7084	85.42	2.96	0.0651

Fig.8 Validation loss variation during training for the 3 neural networks

Figure. 9 Comparison of topology optimization results of SIMP and three loss functions

Tab.4 Average distribution within 6 different density ranges for 4 methods

密度范围	方法	平均分布	密度范围	方法	平均分布
［0， 0.1］	SIMP	1011.1290	（0.5， 0.7］	SIMP	134.3720
	MSE	994.8630		MSE	127.5620
	BCE	976.6470		BCE	129.0490
	DICE	1254.0620		DICE	2.4320
（0.1， 0.3］	SIMP	155.9580	（0.7， 0.9］	SIMP	178.0110
	MSE	175.7750		MSE	204.0700
	BCE	170.0760		BCE	200.2030
	DICE	3.8410		DICE	3.8170
（0.3， 0.5］	SIMP	122.0810	（0.9， 1.0］	SIMP	1598.4490
	MSE	123.3500		MSE	1574.3800
	BCE	125.0660		BCE	1598.9590
	DICE	2.4820		DICE	1933.3660

Tab.5 Average distribution of IDDR for 4 methods

密度范围	方法	平均分布	密度范围	方法	平均分布
［0， 0.1］	SIMP	1011.1290	（0.5， 0.7］	SIMP	134.3720
	MSE	81.1290		MSE	67.7520
	BCE	68.5380		BCE	82.0020
	DICE	398.9640		DICE	110.5270
（0.1， 0.3］	SIMP	155.9580	（0.7， 0.9］	SIMP	178.0110
	MSE	87.7060		MSE	94.1190
	BCE	66.3910		BCE	108.0310
	DICE	143.3360		DICE	171.4260
（0.3， 0.5］	SIMP	122.0810	（0.9， 1］	SIMP	1598.4490
	MSE	67.3130		MSE	40.9640
	BCE	50.3680		BCE	45.2150
	DICE	77.6050		DICE	116.5720

Tab.6 Computational time of topology optimizations

工况
SIMP	2.04	2.03	1.02	0.82	0.18
本文方法	0.01	0.01	0.0099	0.0099	0.01
工况
SIMP	1.46	0.6	1.15	0.94	1.21
本文方法	0.009	0.0099	0.01	0.01	0.01

Tab.7 Distribution of test-dataset samples in different compliance error rates

柔度误差率	样本分布数	柔度误差率	样本分布数
（ $-$ 1， $-$ 0.4）	0	［0，0.1）	47
［ $-$ 0.4， $-$ 0.3）	19	［0.1，0.2）	4
［ $-$ 0.3， $-$ 0.2）	34	［0.2，0.6）	2
［ $-$ 0.2， $-$ 0.1）	122	［0.6，1）	1
［ $-$ 0.1， $-$ 0）	766	≥1	5

Tab.7 Distribution of test-dataset samples in different compliance error rates

柔度误差率	样本分布数	柔度误差率	样本分布数
（ $-$ 1， $-$ 0.4）	0	［0，0.1）	47
［ $-$ 0.4， $-$ 0.3）	19	［0.1，0.2）	4
［ $-$ 0.3， $-$ 0.2）	34	［0.2，0.6）	2
［ $-$ 0.2， $-$ 0.1）	122	［0.6，1）	1
［ $-$ 0.1， $-$ 0）	766	≥1	5

Fig.10 The example of isolated region

Tab.8 Comparison of the number of isolated regions

	孤立区域数
	1	2	3	大于3
SIMP	998	2	0	0
TOPO-U-Net模型	981	17	2	0

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A Topology Optimization Method Based on End-to-end Deep Learning Framework TOPO-U-Net

基于端到端深度学习模型TOPO-U型网的结构拓扑优化方法

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