Full Text:   <1455>

Summary:  <1012>

Suppl. Mater.: 

CLC number: TU31; TP183

On-line Access: 2021-08-20

Received: 2020-08-23

Revision Accepted: 2021-01-04

Crosschecked: 2021-07-20

Cited: 0

Clicked: 2617

Citations:  Bibtex RefMan EndNote GB/T7714


Dung Nguyen Kien


Xiaoying Zhuang


-   Go to

Article info.
Open peer comments

Journal of Zhejiang University SCIENCE A 2021 Vol.22 No.8 P.609-620


A deep neural network-based algorithm for solving structural optimization

Author(s):  Dung Nguyen Kien, Xiaoying Zhuang

Affiliation(s):  Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China; more

Corresponding email(s):   xiaoying.zhuang@gmail.com, zhuang@iop.uni-hannover.de

Key Words:  Structural optimization, Deep learning, Artificial neural networks, Sensitivity analysis

Dung Nguyen Kien, Xiaoying Zhuang. A deep neural network-based algorithm for solving structural optimization[J]. Journal of Zhejiang University Science A, 2021, 22(8): 609-620.

@article{title="A deep neural network-based algorithm for solving structural optimization",
author="Dung Nguyen Kien, Xiaoying Zhuang",
journal="Journal of Zhejiang University Science A",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T A deep neural network-based algorithm for solving structural optimization
%A Dung Nguyen Kien
%A Xiaoying Zhuang
%J Journal of Zhejiang University SCIENCE A
%V 22
%N 8
%P 609-620
%@ 1673-565X
%D 2021
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A2000380

T1 - A deep neural network-based algorithm for solving structural optimization
A1 - Dung Nguyen Kien
A1 - Xiaoying Zhuang
J0 - Journal of Zhejiang University Science A
VL - 22
IS - 8
SP - 609
EP - 620
%@ 1673-565X
Y1 - 2021
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.A2000380

We propose the deep Lagrange method (DLM), which is a new optimization method, in this study. It is based on a deep neural network to solve optimization problems. The method takes the advantage of deep learning artificial neural networks to find the optimal values of the optimization function instead of solving optimization problems by calculating sensitivity analysis. The DLM method is non-linear and could potentially deal with nonlinear optimization problems. Several test cases on sizing optimization and shape optimization are performed, and their results are then compared with analytical and numerical solutions.


方法:1. 采用基于拉格朗日对偶和深度神经网络的方法.2. 将输入数据用于训练神经网络,直到输出值与预测值非常接近为止.3. 通过深度学习插值求解拉格朗日min-max对偶问题,从而找到最小输入值.
结论:1. 该方法可以解决结构优化问题,但它限制了设计变量输入的数量.2. 该方法的准确性取决于输入的区间大小;因此,下一步工作是发展新方法以减少输入数据集的数量.


Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Anitescu C, Atroshchenko E, Alajlan N, et al., 2019. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 59(1):345-359.

[2]Bengio Y, 2012. Practical recommendations for gradient-based training of deep architectures. arXiv:1206.5533. https://arxiv.org/abs/1206.5533

[3]Boyd S, Vandenberghe L, 2004. Convex Optimization. Cambridge University Press, Cambridge, UK.

[4]Braibant V, Fleury C, 1984. Shape optimal design using B-splines. Computer Methods in Applied Mechanics and Engineering, 44(3):247-267.

[5]Canfield RA, 2018. Quadratic multipoint exponential approximation: surrogate model for large-scale optimization. Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimization, p.648-661.

[6]Christensen PW, Klarbring A, 2009. An Introduction to Structural Optimization. Springer, Dordrecht, the Netherlands.

[7]Fletcher R, de la Maza ES, 1989. Nonlinear programming and nonsmooth optimization by successive linear programming. Mathematical Programming, 43(1):235-256.

[8]Fleury C, Braibant V, 1986. Structural optimization: a new dual method using mixed variables. International Journal for Numerical Methods in Engineering, 23(3):409-428.

[9]Ghasemi H, Park HS, Rabczuk T, 2017. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 313:239-258.

[10]Ghasemi H, Park HS, Rabczuk T, 2018. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 332:47-62.

[11]Goodfellow I, Bengio Y, Courville A, 2016. Deep Learning. The MIT Press, Massachusetts, USA.

[12]Goswami S, Anitescu C, Rabczuk T, 2020a. Adaptive fourth-order phase field analysis for brittle fracture. Computer Methods in Applied Mechanics and Engineering, 361:112808.

[13]Goswami S, Anitescu C, Rabczuk T, 2020b. Adaptive fourth-order phase field analysis using deep energy minimization. Theoretical and Applied Fracture Mechanics, 107:102527.

[14]Hajela P, Berke L, 1991. Neurobiological computational models in structural analysis and design. Computers & Structures, 41(4):657-667.

[15]Haslinger J, Mäkinen RAE, 2003. Introduction to Shape Optimization: Theory, Approximation, and Computation. Society for Industrial and Applied Mathematics, Philadelphia, USA.

[16]Hu XH, Eberhart R, 2002. Solving constrained nonlinear optimization problems with particle swarm optimization. Proceedings of the 6th World Multiconference on Systemics, Cybernetics and Informatics, p.203-206.

[17]Jain P, Kar P, 2017. Non-convex optimization for machine learning. Foundations and Trends® in Machine Learning, 10(3-4):142-336.

[18]Kaveh A, 2017. Advances in Metaheuristic Algorithms for Optimal Design of Structures. Springer, Cham, Germany.

[19]Kingma DP, Ba J, 2014. Adam: a method for stochastic optimization. arXiv:1412.6980. https://arxiv.org/abs/1412.6980

[20]Kirsch U, 1993. Structural Optimization: Fundamentals and Applications. Springer, Heidelberg, Germany.

[21]Nguyen-Thanh VM, Zhuang XY, Rabczuk T, 2020. A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics–A/Solids, 80:103874.

[22]Nocedal J, Wright SJ, 2006. Numerical Optimization. Springer, New York, USA.

[23]Papadrakakis M, Lagaros ND, 2002. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer Methods in Applied Mechanics and Engineering, 191(32):3491-3507.

[24]Papadrakakis M, Lagaros ND, Tsompanakis Y, 1998. Structural optimization using evolution strategies and neural networks. Computer Methods in Applied Mechanics and Engineering, 156(1-4):309-333.

[25]Piegl L, Tiller W, 1997. The NURBS Book. Springer, Heidelberg, Germany.

[26]Rogers DF, 2001. An Introduction to NURBS, with Historical Perspective. Morgan Kaufmann Publishers Inc., San Francisco, USA.

[27]Samaniego E, Anitescu C, Goswami S, et al., 2020. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362:112790.

[28]Schittkowski K, Zillober C, Zotemantel R, 1994. Numerical comparison of nonlinear programming algorithms for structural optimization. Structural Optimization, 7(1-2):1-19.

[29]Storn R, Price K, 1997. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 11(4):341-359.

[30]Svanberg K, 1987. The method of moving asymptotes—a new method for structural optimization. International Journal for Numerical Methods in Engineering, 24(2):359-373.

[31]Svanberg K, 2002. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization, 12(2):555-573.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - Journal of Zhejiang University-SCIENCE