Validity Domains of Beams Behavioural Models: Efficiency and Reduction with Artificial Neural Networks

Authors: Keny Ordaz-Hernandez, Xavier Fischer, Fouad Bennis

Abstract:

In a particular case of behavioural model reduction by ANNs, a validity domain shortening has been found. In mechanics, as in other domains, the notion of validity domain allows the engineer to choose a valid model for a particular analysis or simulation. In the study of mechanical behaviour for a cantilever beam (using linear and non-linear models), Multi-Layer Perceptron (MLP) Backpropagation (BP) networks have been applied as model reduction technique. This reduced model is constructed to be more efficient than the non-reduced model. Within a less extended domain, the ANN reduced model estimates correctly the non-linear response, with a lower computational cost. It has been found that the neural network model is not able to approximate the linear behaviour while it does approximate the non-linear behaviour very well. The details of the case are provided with an example of the cantilever beam behaviour modelling.

Keywords: artificial neural network, validity domain, cantileverbeam, non-linear behaviour, model reduction.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1060008

References:

[1] M. Ang, Jr., W. Wei, and L. Teck-Seng, "On the estimation of the large deflection of a cantilever beam," in Procs. International Conference on Industrial Electronics, Control, and Instrumentation (IECON), vol. 3. Maui, HI, USA: IEEE, Nov. 15-19 1993, pp. 1604-1609.
[2] M. A. Arbib, Ed., The Handbook of Brain Theory and Neural Networks, 2nd ed. Cambridge, Massachusetts: MIT Press, 2003.
[3] J. H. Argyris, O. Hilpert, G. A. Malejannakis, and D. W. Scharpf, "On the geometrical stiffness of a beam in space-a consistent v.w. approach," Computer Methods in Applied Mechanics and Engineering, vol. 20, no. 1, pp. 105-131, Oct. 1979.
[4] M. J. Atalla, "Model updating using neural networks," Ph.D. dissertation, Virginia Polytechnic Institute and State University, Apr. 1 1996.
[5] J. S. Bao, Y. Jin, M. Q. Gu, J. Q. Yan, and D. Z. Ma, "Immersive virtual product development," Journal of Materials Processing Technology, vol. 129, no. 1-3, pp. 592-596, Oct. 2002.
[6] T. Beléndez, C. Neipp, and A. Beléndez, "Large and small deflections of a cantilever beam," Eur. J. Phys., vol. 23, no. 3, pp. 371-379, May 2002.
[7] F. Boyer and D. Primault, "Finite element of slender beams in finite transformations: a geometrically exact approach," International Journal for Numerical Methods in Engineering, vol. 59, no. 5, pp. 669-702, Feb. 7 2004.
[8] G. Capizzi, S. Coco, A. Laudani, and R. Pulvirenti, "A multilayer perceptron neural model for the differentiation of laplacian 3-d finiteelement solutions," IEEE Transactions on Magnetics, vol. 39, no. 3, pp. 1277-1280, May 2003, iSSN: 0018-9464.
[9] A. J. Cartwright, "Interactive prototyping ÔÇö a challenge for computer based design," Research in Engineering Design, vol. 9, no. 1, pp. 10-19, Mar. 1997.
[10] C. L. P. Chen, "A rapid supervised learning neural network for function interpolation and approximation," Neural Networks, IEEE Transactions on, vol. 7, no. 5, pp. 1220-1230, 1996.
[11] G. R. Consolazio, "Iterative equation solver for bridge analysis using neural networks," Computer-Aided Civil and Infrastructure Engineering, vol. 15, no. 2, pp. 107-119, 2000.
[12] C. A. Felippa, "Nonlinear finite element methods," Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Tech. Rep. ASEN 5107, 2004.
[13] S. Fok, W. Xiang, and F. Yap, "Feature-based component models for virtual prototyping of hydraulic systems," The International Journal of Advanced Manufacturing Technology, vol. 18, no. 9, pp. 665-672, Oct. 2001.
[14] R. Grzeszczuk, D. Terzopoulos, and G. Hinton, "Neuroanimator: fast neural network emulation and control of physics-based models," in SIGGRAPH -98: Proceedings of the 25th annual conference on Computer graphics and interactive techniques. New York, NY, USA: ACM Press, 1998, pp. 9-20.
[15] F. Guo, P. Zhang, F. Wang, X. Ma, and G. Qiu, "Finite element analysis based hopfield neural network model for solving nonlinear electromagnetic field problems," in International Joint Conference on Neural Networks IJCNN -99, vol. 6, Washington, DC USA, Jul. 10-16 1999, pp. 4399-4403.
[16] G. A. Hazelrigg, "On the role and use of mathematical models in engineering design," Journal of Mechanical Design, vol. 121, no. 3, pp. 336-341, Sep. 1999.
[17] A. M. Horr and L. C. Schmidt, "Closed-form solution for the timoshenko beam theory using a computer-based mathematical package," Computers & Structures, vol. 55, no. 3, pp. 405-412, May 3 1995.
[18] K. M. Hsiao and F. Y. Hou, "Nonlinear finite element analysis of elastic frames," Computers & Structures, vol. 26, no. 4, pp. 693-701, 1987.
[19] Q. Jing, T. Mukherjee, and G. K. Fedder, "Large-deflection beam model for schematic-based behavioral simulation in NODAS," in Nanotech: Technical Proceedings of the Fifth International Conference on Modeling and Simulation of Microsystems (MSM -02), vol. 1. San Juan, Puerto Rico: NSTI, Apr. 22-25 2002, pp. 136-139.
[20] A. J. M. Jr. and A. A. Fernandez, "The numerical solution of linear ordinary differential equations by feedforward neural networks," Math. Comput. Modeling, vol. 19, no. 12, pp. 1-25, 1994.
[21] ÔÇöÔÇö, "Solution of nonlinear ordinary differential equations by feedforward neural networks," Math. Comput. Modeling, vol. 20, no. 9, pp. 19-44, 1994.
[22] J. Kalkkuhl, K. Hunt, and H. Fritz, "Fem-based neural-network approach to nonlinear modeling with application to longitudinal vehicle dynamics control," IEEE Transactions on Neural Networks, vol. 10, no. 4, pp. 885-897, Jul. 1999, iSSN: 1045-9227.
[23] S. Klinkel and S. Govindjee, "Using finite strain 3d-material models in beam and shell elements," Engineering Computations: Int J for Computer-Aided Engineering, vol. 19, no. 3, pp. 254-271, 2002.
[24] I. Lagaris, A. Likas, and D. Fotiadis, "Artificial neural networks for solving ordinary and partial differential equations," IEEE Transactions on Neural Networks, vol. 9, no. 5, pp. 987-1000, Sep. 1998, iSSN: 1045-9227.
[25] E. N. Lages, G. H. Paulino, I. F. M. Menezes, and R. R. Silva, "Nonlinear finite element analysis using an object-oriented philosophy ÔÇö application to beam elements and to the cosserat continuum," Engineering with Computers, vol. 15, no. 1, pp. 73-89, Apr. 1999, publisher: Springer- Verlag London Ltd, ISSN: 0177-0667 (Paper) 1435-5663 (Online).
[26] Y. C. Liang, W. Z. Lin, H. P. Lee, S. P. Lim, K. H. Lee, and D. P. Feng, "A neural-network-based method of model reduction for the dynamic simulation of mems," Journal of Micromechanics and Microengineering, vol. 11, no. 3, pp. 226-233, May 2001.
[27] Z. Lin, K. Khorasani, and R. V. Patel, "A counter-propagation neural network for function approximation," in Procs. Int. Conf. Systems, Man and Cybernetics. IEEE, 1990, pp. 382-384.
[28] M. Liu, X. Meng, D. Liu, and P. Zhong, "Virtual prototype based architecture of cooperative design and simulation for complex products," in Computer Supported Cooperative Work in Design, 2004. Proceedings. The 8th International Conference on, vol. 2, 26-28 May 2004, pp. 546- 551.
[29] J.-C. Léon, "Visualisation of virtual environments and their applications in the design process," in Virtual Concept. Biarritz, FR: ESTIA, Nov. 5-7 2003, pp. 294-295.
[30] T. S. Low and B. Chao, "The use of finite elements and neural networks for the solution of inverse electromagnetic problems," IEEE Transactions on Magnetics, vol. 28, no. 5, pp. 2811-2813, Sep. 1992, iSSN: 0018- 9464.
[31] K. Martini, "A particle-system approach to real-time non-linear," in Proceedings of the 7th National Conference on Earthquake Engineering. Earthquake Engineering Research Institute, 2002, publication en CDROM.
[32] J. N. Reddy, C. M. Wang, and K. Y. Lam, "Unified finite elements based on the classical and shear deformation theories of beams and axisymmetric circular plates," Communications in Numerical Methods in Engineering, vol. 13, no. 6, pp. 495-510, 1997.
[33] T. D. Sanger, "A tree-structured adaptive network for function approximation in high-dimensional spaces," Neural Networks, IEEE Transactions on, vol. 2, no. 2, pp. 285-293, 1991.
[34] M. Schulz and F. C. Filippou, "Non-linear spatial timoshenko beam element with curvature interpolation," International Journal for Numerical Methods in Engineering, vol. 50, no. 4, pp. 761-785, Feb. 2001.
[35] E. Solano Carrillo, "The cantilevered beam: an analytical solution for general deflections of linear-elastic materials," European Journal of Physics, vol. 27, no. 6, pp. 1437-1445, 2006.
[36] P. Song, V. Krovi, V. Kumar, and R. Mahoney, "Design and virtual prototyping of human-worn manipulation devices," in Procs. ASME DETC, 1999, pp. 11-15.
[37] E. Spacone, V. Ciampi, and F. C. Filippou, "Mixed formulation of nonlinear beam finite element," Computers & Structures, vol. 58, no. 1, pp. 71-83, Jan. 3 1996.
[38] M. Sun, X. Yan, and R. Sclabassi, "Solving partial differential equations in real-time using artificial neural network signal processing as an alternative to finite-element analysis," in Proceedings of the 2003 International Conference on Neural Networks and Signal Processing, 2003., vol. 1, Dec. 14-17 2003, pp. 381-384.
[39] M. R. Thompson, J. H. Maxfield, and P. M. Dew, "Interactive virtual prototyping," in Proc of Eurographics UK -98, Mar. 1998, pp. 107-120.
[40] N. Troussier, F. Pourroy, and M. Tollenaere, "Information structuring for use and reuse of mechanical analysis models in engineering design," Journal of Intelligent Manufacturing, vol. 10, no. 1, pp. 61-71, Mar. 1999.
[41] L. H. Tsoukalas and R. E. Uhrig, Fuzzy and Neural Approaches in Engineering. John Wiley & Sons, 1997.
[42] Z. Waszczyszyn and L. Ziemianski, "Neural networks in mechanics of structures and materials - new results and prospects of applications," Computers & Structures, vol. 79, no. 22-25, pp. 2261-2276, Sep. 2001.
[43] H. Yamashita, N. Kowata, V. Cingoski, and K. Kaneda, "Direct solution method for finite element analysis using hopfield neural network," IEEE Transactions on Magnetics, vol. 31, no. 3, pp. 1964 - 1967, May 1995, iSSN: 0018-9464.
[44] X. Yang, Y. Wang, F. Liu, Q. Yang, and W. Yan, "The use of neural networks combined with fem to optimize the coil geometry and structure of transverse flux induction equipments," IEEE Transactions on Applied Superconductivity, vol. 14, no. 2, pp. 1854-1857, Jun. 2004, iSSN: 1051- 8223.
[45] G. Zachmann, "VR-techniques for industrial applications," in Virtual Reality for Industrial Applications, F. Dai, Ed. Springer, 1998, ch. 1, pp. 13-38.
[46] L. Ziemianski, "Neural networks for dynamic analysis of structures," in Procs. Fifth World Congress on Computational Mechanics (WCCM V), H. A. Mang, F. G. Rammerstorfer, and J. Eberhardsteiner, Eds. Vienna, Austria: Vienna University of Technology, Austria, Jul. 7-12 2002, abstract.
[47] O. Zienkiewicz and R. Taylor, "The finite element method," New York, 1989.