Advances in Computational Design

  • 제7권1호
  • /
  • Pages.1-17
  • /
  • 2022
  • /
  • 2383-8477(pISSN)
  • /
  • 2466-0523(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Numerical solution of beam equation using neural networks and evolutionary optimization tools

  • Babaei, Mehdi (Department of Civil Engineering, University of Bonab) ;
  • Atasoy, Arman (Department of Civil Engineering, Istanbul Rumeli University) ;
  • Hajirasouliha, Iman (Department of Civil & Structural Engineering, The University of Sheffield) ;
  • Mollaei, Somayeh (Department of Civil Engineering, University of Bonab) ;
  • Jalilkhani, Maysam (Department of Civil Engineering, Urmia University of Technology)
  • 투고 : 2021.04.05
  • 심사 : 2021.09.03
  • 발행 : 2022.01.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this study, a new strategy is presented to transmit the fundamental elastic beam problem into the modern optimization platform and solve it by using artificial intelligence (AI) tools. As a practical example, deflection of Euler-Bernoulli beam is mathematically formulated by 2nd-order ordinary differential equations (ODEs) in accordance to the classical beam theory. This fundamental engineer problem is then transmitted from classic formulation to its artificial-intelligence presentation where the behavior of the beam is simulated by using neural networks (NNs). The supervised training strategy is employed in the developed NNs implemented in the heuristic optimization algorithms as the fitness function. Different evolutionary optimization tools such as genetic algorithm (GA) and particle swarm optimization (PSO) are used to solve this non-linear optimization problem. The step-by-step procedure of the proposed method is presented in the form of a practical flowchart. The results indicate that the proposed method of using AI toolsin solving beam ODEs can efficiently lead to accurate solutions with low computational costs, and should prove useful to solve more complex practical applications.

키워드

참고문헌

  1. Babaei, M. (2013), "A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization", Appl. Soft Comput., 13(7), 3354-3365. https://doi.org/10.1016/j.asoc.2013.02.005.
  2. Babaei, M. and Sheidaii, M.R. (2013), "Optimal design of double layer scallop domes using genetic algorithm", Appl. Mathem. Modelling, 37(4), 2127-2138. https://doi.org/10.1016/j.apm.2012.04.053.
  3. Babaei, M. and Sheidaii, M.R. (2014), "Automated optimal design of double-layer latticed domes using particle swarm optimization", Struct. Multidiscip. O., 50(2), 221-235. https://doi.org/10.1007/s00158-013-1042-2.
  4. Berg, J. and Nystrom, K. (2018), "A unified deep artificial neural network approach to partial differential equations in complex geometries", Neurocomput., 317, 28-41. https://doi.org/10.1016/j.neucom.2018.06.056.
  5. Dissanayake, M. and Phan-Thien, N. (1994), "Neural-network-based approximations for solving partial differential equations", Commun. Numer. Meth. Eng., 10(3), 195-201. https://doi.org/10.1002/cnm.1640100303.
  6. Eberhart, R. and Kennedy, J. (1995), "Particle swarm optimization", Proceedings of the IEEE International Conference on Neural Networks, Perth, WA, Australia, November.
  7. Fang, Z., Roy, K., Chen, B., Sham, C.W., Hajirasouliha, I. and Lim, J.B. (2021), "Deep learning-based procedure for structural design of cold-formed steel channel sections with edge-stiffened and un-stiffened holes under axial compression", Thin Wall. Struct., 166, 108076. https://doi.org/10.1016/j.tws.2021.108076.
  8. Holland, J.H. (1992), Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, The MIT press, U.S.
  9. Jafarian, A., Mokhtarpour, M. and Baleanu, D. (2017). "Artificial neural network approach for a class of fractional ordinary differential equation", Neural Comput. Appl., 28(4), 765-773. https://doi.org/10.1007/s00521-015-2104-8.
  10. Koza, J.R. (1992), Genetic Programming: On the Programming of Computers by Means of Natural Selection, MIT press, Cambridge, Massachusetts, London, England.
  11. Lagaris, I. E., Likas, A. and Fotiadis, D. (1998), "Artificial neural networks for solving ordinary and partial differential equations", IEEE T. Neural. Networ., 9(5), 987-1000. https://doi.org/10.1109/72.712178.
  12. Lee, H. and Kang I.S. (1990), "Neural algorithm for solving differential equations", J. Comput. Phys., 91(1), 110-131. https://doi.org/10.1016/0021-9991(90)90007-N.
  13. Luh, G.C., Lin, C.Y. and Lin, Y.S. (2011), "A binary particle swarm optimization for continuum structural topology optimization", Appl. Soft Comput., 11(2), 2833-2844. https://doi.org/10.1016/j.asoc.2010.11.013.
  14. Magill, M., Qureshi, F., and de Haan, H.W. (2018), "Neural networks trained to solve differential equations learn general representations", Adv. Neural Inform. Process. Syst., 1097-1105. https://arxiv.org/abs/1807.00042.
  15. Mojtabaei, S.M., Hajirasouliha, I., Ye, J. (2021), "Optimization of cold-formed steel beams for best seismic performance in bolted moment connections", J. Constr. Steel Res., 181, 106621. https://doi.org/10.1016/j.jcsr.2021.106621.
  16. Nabian, M.A. and Meidani, H. (2019). "A deep learning solution approach for high-dimensional random differential equations", Probabilist. Eng. Mech., 57, 14-25. https://doi.org/10.1016/j.probengmech.2019.05.001.
  17. Omondi, A.R. and Rajapakse J.C. (2006), FPGA Implementations of Neural Networks, Springer, Boston, MA. https://doi.org/10.1007/0-387-28487-7.
  18. Parastesh, H., Mojtabaei, S.M., Taji, H., Hajirasouliha, I. and Sabbagh, A.B. (2021), "Constrained optimization of anti-symmetric cold-formed steel beam-column sections", Eng. Struct., 228, 111452. https://doi.org/10.1016/j.engstruct.2020.111452.
  19. Phan, D.T., Mojtabaei, S.M., Hajirasouliha, I., Ye, J. and Lim, J.B.P (2020), "Coupled element and structural level optimisation framework for cold-formed steel frames", J. Constr. Steel Res., 168, 105867. https://doi.org/10.1016/j.jcsr.2019.105867.
  20. Schmidhuber, J. (2015), "Deep learning in neural networks: An overview", Neural Networks, 61, 85-117. https://doi.org/10.1016/j.neunet.2014.09.003.
  21. Sirignano, J. and Spiliopoulos, K. (2018), "DGM: A deep learning algorithm for solving partial differential equations", J. Comput. Phys., 375, 1339-1364. https://doi.org/10.1016/j.jcp.2018.08.029.
  22. Sivanandam, S.N. and Deepa, S.N. (2008), Introduction to genetic algorithms, Springer, Berlin, Heidelberg, Germany.
  23. Sun, H., Hou, M., Yang, Y., Zhang, T., Weng, F. and Han, F. (2019). "Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm", Neural Process. Lett., 50(2), 1153-1172. https://doi.org/10.1007/s11063-018-9911-8.
  24. Van Milligen, B.P., Tribaldos, V. and Jimenez, J.A. (1995), "Neural network differential equation and plasma equilibrium solver", Phys. Rev. Lett., 75(20), 3594. https://doi.org/10.1103/PhysRevLett.75.3594.
  25. Yadav, N., Yadav, A. and Kumar, M. (2015), An Introduction to Neural Network Methods for Differential Equations, Springer, Netherlands.
  26. Ye, J., Becque, J., Hajirasouliha, I., Mojtabaei, S.M. and Lim, J.B.P. (2018). "Development of optimum cold-formed steel sections for maximum energy dissipation in uniaxial bending". Eng. Struct., 161, 55-67. https://doi.org/10.1016/j.engstruct.2018.01.070.
  27. Ye, J., Hajirasouliha, I., Becque, J. and Eslami, A. (2016). "Optimum design of cold-formed steel beams using particle swarm optimisation method", J. Constr. Steel Res., 122, 80-93. https://doi.org/10.1016/j.jcsr.2016.02.014.
  28. Yentis, R. and Zaghloul, M. (1996), "VLSI implementation of locally connected neural network for solving partial differential equations", IEEE T. Circ. Syst. I, 43(8), 687-690. https://doi.org/10.1109/81.526685.