Similar articles

Abstract: 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.

一种基于深度神经网络的结构优化求解算法

[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.