Metamodeling of nonlinear structural systems with parametric uncertainty subject to stochastic dynamic excitation

  • Spiridonakos, Minas D. (Institute of Structural Engineering, D-BAUG, ETH Zurich) ;
  • Chatzia, Eleni N. (Institute of Structural Engineering, D-BAUG, ETH Zurich)
  • Received : 2014.04.09
  • Accepted : 2014.11.02
  • Published : 2015.04.25


Within the context of Structural Health Monitoring (SHM), it is often the case that structural systems are described by uncertainty, both with respect to their parameters and the characteristics of the input loads. For the purposes of system identification, efficient modeling procedures are of the essence for a fast and reliable computation of structural response while taking these uncertainties into account. In this work, a reduced order metamodeling framework is introduced for the challenging case of nonlinear structural systems subjected to earthquake excitation. The introduced metamodeling method is based on Nonlinear AutoRegressive models with eXogenous input (NARX), able to describe nonlinear dynamics, which are moreover characterized by random parameters utilized for the description of the uncertainty propagation. These random parameters, which include characteristics of the input excitation, are expanded onto a suitably defined finite-dimensional Polynomial Chaos (PC) basis and thus the resulting representation is fully described through a small number of deterministic coefficients of projection. The effectiveness of the proposed PC-NARX method is illustrated through its implementation on the metamodeling of a five-storey shear frame model paradigm for response in the region of plasticity, i.e., outside the commonly addressed linear elastic region. The added contribution of the introduced scheme is the ability of the proposed methodology to incorporate uncertainty into the simulation. The results demonstrate the efficiency of the proposed methodology for accurate prediction and simulation of the numerical model dynamics with a vast reduction of the required computational toll.


nonlinear dynamics;earthquake excitation;metamodeling;nonlinear ARX models;polynomial chaos expansion;system identification;uncertainty quantification


Supported by : ASTRA


  1. Adams, D. and Farrar, C. (2002), "Classifying linear and nonlinear structural damage using frequency domain arx models", Struct. Hlth. Monit., 1(2), 185-201.
  2. Bathe, K.J. (2009), Finite element method, Wiley Encyclopedia of Computer Science and Engineering, Ed. Wah B, Wiley & Sons, Inc.
  3. Beck, J.L. and Katafygiotis, L.S. (1998), "Updating models and their uncertainties. I: Bayesian statistical framework", J. Eng. Mech., 124(4), 455-461.
  4. Blatman, G. and Sudret, B. (2010), "An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis", Prob. Eng. Mech., 25(2), 183-197.
  5. Caicedo, J., Dyke, S. and Johnson, E. (2004), "Natural excitation technique and eigensystem realization algorithm for phase I of the IASC-ASCE benchmark problem, Simulated data", J. Eng. Mech., 130(1), 49-60.
  6. Chatzi, E. and Smyth, A.W. (2009), "The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing", Struct. Control Hlth. Monit., 16(1), 99-123.
  7. Chen, S. and Billings, S.A. (1989), "Modelling and analysis of non-linear time series", Int. J. Control, 50(6), 2151-2171.
  8. Chen, Q., Worden, K., Peng, P. and Leung, A. (2007), "Genetic algorithm with an improved fitness function for (n)arx modeling", Mech. Syst. Signal Pr., 21(2), 994-1007.
  9. Choudhury, S.M., Shah, S.L. and Thornhill, N.F. (2008), "Linear or nonlinear? A bicoherence-based measure of nonlinearity", Chap. 6, Adv. Indust. Control, Springer-Verlag, 77-91.
  10. Christodoulou, K., Ntotsios, E., Papadimitriou C. and Panetsos, P. (2008), "Structural model updating and prediction variability using pareto optimal models", Comput. Meth. Appl. Mech. Eng., 198(1), 138-149.
  11. Coley, D.A. (1999), An introduction to genetic algorithms for scientists and engineers, World Scientific, Singapore.
  12. Corigliano, A. and Mariani, S. (2004), "Parameter identification in explicit structural dynamics, performance of the extended Kalman filter", Comput. Meth. Appl. Mech. Eng., 193(36-38), 3807-3835.
  13. Faravelli, L., Ubertini, F. and Fuggini, C. (2011), "System identification of a super high-rise building via a stochastic subspace approach", Smart Struct. Syst., 7(2), 133-152.
  14. Farrar, C. and Worden, K. (2007), "Structural health monitoring - preface", Philosoph. Transact. Roy. Soc. A, 365(1851), 299-301.
  15. Fraraccio, G., Brugger, A. and Betti, R. (2008), "Identification and damage detection in structures subjected to base excitation", Exper. Mech., 48(4), 521-528.
  16. Gholizadeh, S. and Salajegheh, E. (2009), "Optimal design of structures subjected to time history loading by swarm intelligence and an advanced metamodel", Comput. Meth. Appl. Mech. Eng., 198(37-40), 2936-2949.
  17. Helton, J.C. and Davis, F.J. (2003), "Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems", Reliab. Eng. Syst. Saf., 81(1), 23-69.
  18. Hernandez, E.M. and Bernal, D. (2008), "State estimation in structural systems with model uncertainties", J. Eng. Mech., 134(3), 252-257.
  19. Kalkan, E. and Chopra, A.K. (2010), "Practical guidelines to select and scale earthquake records for nonlinear response history analysis of structures", Usgs open file report 2010-1068, 126 pgs., U.S. Geological Survey, Menlo Park, CA.
  20. Katkhuda, H., Martinez, R. and Haldar, A. (2005), "Health assessment at local level with unknown input excitation", J. Struct. Eng., 131(6), 956-965.
  21. Kerschen, G., Worden, K., Vakakis, A. and Golinval, J. (2006), "Past, present and future of nonlinear system identification in structural dynamics", Mech. Syst. Signal Pr., 20(3), 505-592.
  22. Kopsaftopoulos, F.P. and Fassois, S.D. (2013), "A functional model based statistical time series method for vibration based damage detection, localization, and magnitude estimation", Mech. Syst. Signal Pr., 39(1-2), 143-161.
  23. Lin, J.W., Betti, R., Smyth, A.W. and Longman, R.W. (2001), "On-line identification of non-linear hysteretic structural systems using a variable trace approach", Earthq. Eng. Struct. Dyn., 30(9), 1279-1303.
  24. Lourens, E., Papadimitriou, C., Gillijns, S., Reynders, E., De Roeck, G. and Lombaert, G. (2012), "Joint input-response estimation for structural systems based on reduced-order models and vibration data from a limited number of sensors", Mech. Syst. Signal Pr., 29, 310-327.
  25. Moaveni, B., He, X., Conte, J.P. and Restrepo, J.I. (2010), "Damage identification study of a seven-story full-scale building slice tested on the UCSD-NEES shake table", Struct. Safety, 32(5), 347-356.
  26. Naets, F., Pastorino, R., Cuadrado, J., and Desmet W. (2013), "Online state and input force estimation for multibody models employing extended Kalman filtering", Multibody Syst. Dyn., July, 1-20.
  27. Piroddi, L. and Spinelli, W. (2003), "An identification algorithm for polynomial NARX models based on simulation error minimization", Int. J. Control, 76(17), 1767-1781.
  28. Poulimenos, A.G. and Fassois, S.D. (2006), "Parametric time-domain methods for non-stationary random vibration modelling and analysis - a critical survey and comparison", Mech. Syst. Signal Pr., 20(4), 763-816.
  29. PEER (2012), Peer ground motion database, ground motion database.
  30. Rezaeian, S. and Der Kiureghian, A. (2010), "Simulation of synthetic ground motions for specified earthquake and site characteristics", Earthq. Eng. Struct. Dyn., 39(10), 1155-1180.
  31. Rutherford, A., Park, G. and Farrar, C. (2007), "Nonlinear feature identification based on self-sensing impedance measurement for structural health assessment", Mech. Syst. Signal Pr., 21(1), 322-333.
  32. Samara, P., Sakellariou, J., Fouskitakis, G., Hios, J. and Fassois, S. (2013), "Aircraft virtual sensor design via a time-dependent functional pooling narx methodology", Aerospace Sci. Technol., 29(1), 114-124.
  33. Sapsis, T.P. and Lermusiaux, P.F.J. (2009), "Dynamically orthogonal field equations for continuous stochastic dynamical systems", Physica D: Nonlin. Phenom., 238(23), 2347-2360.
  34. Smyth, A.W., Masri, S.F., Kosmatopoulos, E.B., Chassiakos, A.G. and Caughey, T.K. (2002), "Development of adaptive modeling techniques for non-linear hysteretic systems", Int. J. Nonlin. Mech., 37(8), 1435-1451.
  35. Soize, C. and Ghanem, R. (2004), "Physical systems with random uncertainties, chaos representations with arbitrary probability measure", J. Scientific Comput. SIAM, 26(2), 395-410.
  36. Spiridonakos, M. and Chatzi, E. (2012), "Metamodeling of structural systems through polynomial chaos arx models", International Conference on Uncertainty in Structural Dynamics (USD2012), Leuven, Belgium.
  37. Wagner, S.M. and Ferris, J.B. (2007), "A polynomial chaos approach to ARMA modeling and terrain characterization", SPIE 6564, Modeling and Simulation for Military Operations II, 65640M.
  38. Wei, H.L, Billings, S.A. and Liu, J. (2004), "Term and variable selection for non-linear system identification", Int. J. Control, 77(1), 86-110.
  39. Worden, K. and Tomlinson, G. (2000), Nonlinearity in Structural Dynamics, Detection, Identification and Modelling, Taylor & Francis.
  40. Vanik, M.W., Beck, J.L. and Au, S.K. (2000), "Bayesian probabilistic approach to structural health monitoring", J. Eng. Mech., 126(7), 738-745.
  41. Yun, C.B. and Shinozuka, M. (1980), "Identification of nonlinear structural dynamics systems", J. Struct. Mech., 8(2), 187-203.

Cited by

  1. A State of the Art Review of Modal-Based Damage Detection in Bridges: Development, Challenges, and Solutions vol.7, pp.5, 2017,
  2. Bridge Damage Detection Based on Vibration Data: Past and New Developments vol.3, 2017,
  3. Surrogate Models for Oscillatory Systems Using Sparse Polynomial Chaos Expansions and Stochastic Time Warping vol.5, pp.1, 2017,