Validity Domains of Beams Behavioural Models: Efficiency and Reduction with Artificial Neural Networks
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.
[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.
[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.
@article{"International Journal of Information, Control and Computer Sciences:53979", author = "Keny Ordaz-Hernandez and Xavier Fischer and Fouad Bennis", title = "Validity Domains of Beams Behavioural Models: Efficiency and Reduction with Artificial Neural Networks", 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.", volume = "2", number = "6", pages = "1925-8", }
