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A. E. Charalampakis and C. K. Dimou National Technical University of Athens, Greece (NTUA)

Comparison of Differential Evolution, Particle Swarm Optimization and Genetic Algorithms for the identification of Bouc-Wen hysteretic systems. A. E. Charalampakis and C. K. Dimou National Technical University of Athens, Greece (NTUA).

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A. E. Charalampakis and C. K. Dimou National Technical University of Athens, Greece (NTUA)

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  1. Comparison of Differential Evolution, Particle Swarm Optimization and Genetic Algorithms for the identification of Bouc-Wen hysteretic systems A. E. Charalampakis and C. K. Dimou National Technical University of Athens, Greece (NTUA) CSC2011 The 2nd International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering – Chania, Crete, Greece

  2. Outline • Introduction : Bouc-Wen model • Parameter constraints for mathematical and physical consistency • Effect of each model parameter on the overall response • Experimental data • Identification methods: • Genetic Algorithms (GAs) • Hybrid method • Particle Swarm Optimization (PSO) • Differential Evolution (DE) • Results • Conclusions

  3. R/C sections High strength concrete sections Saatcioglu, M., Ozcebe, G. (1989) “Response of Reinforced Concrete Columns to Simulated Seismic Loading”, ACI Structural Journal, 86(1):3-12. Xiao, Y., Martirossyan, A.(1998) “Seismic performance of high-strength concrete columns”, Journal of Structural Engineering, 124(3):241-251. Masonry shear walls Concrete walls Madan, A., Reinhorn, A. M., Mander, J. B. (2008) “Fiber-Element Model of Posttensioned Hollow Block Masonry Shear Walls under Reversed Cyclic Lateral Loading”, Journal of Structural Engineering, 134(7):1101-1114. Lefas, I. D., Kotsovos, M. D., (1990) “Strength and Deformation Characteristics of Reinforced Concrete Walls Under Load Reversals”, ACI Struct. J., 87(6):716-726.

  4. Steel connections Seismic isolation systems (FPS, LRB, …) Popov, E.P., Stephen, R.M. (1970) “Cyclic Loading of Full-Size Steel Connections”, Report UCB/EERC-70/03. Kikuchi, M., Aiken, I. D. (1997) “An analytical hysteresis model for elastomeric seismic isolation bearings”, Earthquake Engineering and Structural Dynamics, 26:215-231. Wooden shear walls Wood joints Stewart, W. G. (1987) “The seismic design of plywood-sheathed shear walls”, PhD Thesis, Univ. of Canterbury, Christchurch, New Zealand. Foliente, G. C. (1995), “Hysteresis modeling of wood joints and structural systems”, Journal of Structural Engineering, 121(6):1013-1022.

  5. Dynamic response of foundations Gerolymos, N., Gazetas, G. (2006) “Static and dynamic response of massive caisson foundations with soil and interface nonlinearities—validation and results” Soil Dynamics and Earthquake Engineering 26:377–394. Magnetorheological fluid dampers (MR dampers) Kwok, N.M., Ha, Q.P., Nguyen, T.H., Li, J., Samali, B. (2006)“A novel hysteretic model for magnetorheological fluid dampers andparameter identification using particle swarm optimization” Sensors and Actuators A, 132:441–451.

  6. Stick – slip phenomena in elevator rail systems (rubber-to-metal interface) Hoon ,W., Yoon ,Y. K., Haeil, J., Gwang , N. L., (2001) “Nonlinear rate-dependent stick-slip phenomena: modeling and parameter estimation”,International Journal of Solids and Structures, 38:1415-1431. Seat suspension systems Gunstona, T.P., Rebelleb, J., Griffina, M.J. (2004) “A comparison of two methods of simulating seat suspension dynamic performance” Journal of Sound and Vibration 278:117–134.

  7. Introduction: Bouc – Wen model The Bouc-Wen model is a smooth hysteretic model that can describe virtually any type of hysteretic loop. Basic hysteretic loop Strength degradation Stiffness deterioration Asymmetric yield force Pinching Strain hardening

  8. Introduction: Bouc – Wen model The stiffness of the elastic spring is equal to the post-elastic stiffness of the whole system. The restoring force F(t) is given by: (1) The dimensionless hysteretic parameter z(t) obeys the following differential equation: (2) Where: generalized yield force generalized yield displacement post-elastic to initial (elastic) stiffness ratio model parameters initial stiffness

  9. Bouc – Wen model Parameter A The original model has one redundant parameter. Consequently there is a multitude of parameter vectors that produce an identical response for a given excitation[1]. This problem is critical. For instance, during identification of the unknown model parameters we use a specific excitation and we may derive anyone of this multitude of results. This result may be totally inappropriate when the excitation is changed. Consequently, the model cannot predict the response of the actual system. For many reasons, the best way to treat this redundancy is by fixing parameter A to unity. Thus, for reasons of mathematical consistency: [1] F. Ma, H. Zhang, A. Bockstedte, G. C. Foliente, P. Paevere (2004) “Parameter Analysis of the Differential Model of Hysteresis”, Journal of Applied Mechanics ASME, 71:342–349.

  10. Bouc – Wen model Parameter A Simple Sinusoidal Excitation(T=25 sec, Amplitude=10) El Centro The response of the identified system is almost identical with the one of the true system for the El Centro excitation (which was used for the identification process). This does not necessarily hold for other excitations[1]. [1] A.E. Charalampakis, V.K. Koumousis (2006) “Parameter Estimation of Bouc-Wen Hysteretic Systems using Sawtooth Genetic Algorithm”, Proceedings of the Fifth International Conference on Engineering Computational Technology, Las Palmas de Gran Canaria, Spain.

  11. Bouc – Wen model Parameters β and γ Parameters βand γcontrol the shape and size of the hysteretic loops, as demonstrated by Wen [1]. However, they affect the whole response in an uncontrolled manner while they do not bear any physical meaning. [1] Wen, Y.-K. (1976) “Method for random vibration of hysteretic systems”, J. Engrg. Mech. Div., ASCE, 102(2):249-263.

  12. Bouc – Wen model Parameters β and γ For reasons of physical consistency: The two constraints, i.e. A=β+γ=1, reduce the model to a strain-softening formulation with well-defined properties.

  13. Introduction: Bouc-Wen model sharpness of transition between branches ratio of post-elastic to initial stiffness Mathematical consistency[1]: Physical consistency: Hysteretic parameter: [1] F. Ma, H. Zhang, A. Bockstedte, G. C. Foliente, P. Paevere (2004) “Parameter Analysis of the Differential Model of Hysteresis”, Journal of Applied Mechanics ASME, 71:342–349.

  14. Experiment Experiment No. 5, conducted by Popov and Stephen [1] at Berkeley in 1970. A full-scale steel cantilever beam (WF 24x76), bolted-welded (using seven 7/8” bolts) to a rigid column and subjected to cyclic displacement pattern of increasing amplitude. The response exhibited shows clear non-degrading hysteretic behavior. [1] E.P. Popov, R.M. Stephen, “Cyclic Loading of Full-Size Steel Connections”, Report UCB/EERC-70/03, 1970.

  15. Identification method 1: Standard Genetic Algorithm • Genetic algorithms (GAs) are population-based evolutionary algorithms that have been employed in a plethora of applications. Standard GA (SGA) is implementedas the 1st i.d. method, which can be described by the following pseudo-code: • Initialize the population of individuals (chromosomes). • Calculate the fitness of each individual in the population. • Select individuals to form a new population according to each one’s fitness. • Perform crossover and mutation. • Create new population (elitism). • Repeat steps (2) to (5) until some condition is satisfied. • The parameters of SGA are taken as follows: gene length Lg=10 bits; population size P=25 and P=50; single crossover with probability 0.7 ; jump mutation probability 1/P; creep mutation probability Lc/Np/P (Lc=chromosome length, Np=number of variables); biased roulette wheel selection and elitism with one individual.

  16. Identification method 2: Micro-GA Micro-GA, proposed by Goldberg [1] and first implemented by Krishnakumar [2], uses a very small population. The main operator is crossover and the population is restarted when the genetic diversity falls below a certain threshold. The method substitutes the mutation operator with population restart. The optimization process is similar to the one of the SGA. The parameters of Micro-GA are taken as follows: gene length Lg=10 bits; population size P=5; single crossover, minimum different bits fraction 5%. [1] D.E. Goldberg, “Sizing populations for serial and parallel genetic algorithms. In “Proceedings of the third International Conference on Genetic Algorithms (ICGA 89)”, David J. Schaffer, (Editor), George Mason University, United States, Morgan Kaufmann Publishers Inc., 70 79, 1989.D.L. Carroll, FORTRAN Genetic Algorithm (GA) Driver. [Online]. Available: http://www.cuaerospace.com/carroll/ga.html, 1999. [2] K. Krishnakumar, “Micro-genetic algorithms for stationary and nonstationary function optimization”, in “Proceedings of SPIE. Intelligent Control Adaptive Systems”. Philadelphia, PA, United States, 289 296, 1989.

  17. Identification method 3: Hybrid Algorithm A hybrid algorithm, proposed by Charalampakis and Koumousis [1], is also employed. The hybrid method consists of SawTooth-GA [2], a local optimizer, namely the Greedy Ascend Hill Climber, and a bounding method that gradually decreases the size of the DS. The H2 configuration of the hybrid method is used [1]. [1] A.E. Charalampakis, V.K. Koumousis, “Identification of Bouc-Wen hysteretic systems by a hybrid evolutionary algorithm”, Journal of Sound and Vibration, 314, 571 585, 2008. [2] V.K. Koumousis, C.P. Katsaras, “A Saw-Tooth Genetic Algorithm Combining the Effects of Variable Population Size and Reinitialization to Enhance Performance”, IEEE Transactions on Evolutionary Computation, 10(1), 19 28, 2006.

  18. Identification method 4: Particle Swarm Optimization (PSO) Particle Swarm Optimization (PSO) is a stochastic algorithm suitable for global optimization with no need for direct evaluation of gradients. The method, introduced by Kennedy and Eberhart [1], mimics the social behavior of flocks of birds and swarms of insects. In PSO a swarm of N individuals-particles communicate search directions (gradients) revealing the position of their personal best solution. The new position of a particle is a function of its present position, its current search direction and the attraction-repulsion of its personal best and swarm best position. The driving force of optimization in PSO is the evolution/adaptation of the set of solutions instead of the combination of a current set of solutions to generate a new set as in GA variants. A simple PSO variant is employed herein, which features the following properties: (a) population size p=20, (b) cognitive parameter c1=2, (c) social parameter c2=2, (d) constant inertia factor w=0.8. [1] [J. Kennedy, R.C. Eberhart, “Particle swarm optimization”, In “Proceedings of IEEE International Conference on Neural Networks”, 1942–1948, 1995.

  19. Identification method 5: Enhanced PSO The enhanced PSO variant is based on the work of Fourie and Groenwold [1]. It features the following properties: (a) population size p=20 (b) cognitive parameter c1=0.5, (c) social parameter c2=1.6 (d) maximum velocity coefficient γ=0.4, (e) initial inertia factor w0=1.40, (f) maximum steps without improvement h=3, (g) fraction for the decrease of the inertia factor α=0.99, (h) fraction for the decrease of maximum velocity β=0.95, (i) craziness factor Pcr=0.22, (j) elite velocity factor c3=1.30. [1] P.C. Fourie, A.A. Groenwold, “The particle swarm optimization algorithm in size and shape optimization”, Structural and Multidisciplinary Optimization, 23(4), 259–267, 2002.

  20. Identification method 6: Differential Evolution (DE) • Differential Evolution (DE) is a relative new stochastic method which has attracted the attention of the scientific community. It was introduced by Storn and Price [1] and has approximately the same age as PSO. However, it bears no natural paradigm. • An early version was initially conceived under the term “Genetic Annealing” and published in a programmer’s magazine [2]. The DE algorithm is extremely simple; the uncondensed C-style pseudocode of the algorithm spans less than 25 lines [2]. • Three variants of DE are examined. • DE1 or “rand/1/bin” (Classic DE), with (a) p=50 vectors, (b) F=0.5, (c) Cr=0.9. • DE2 or “best/1/bin”, in which the currently best vector is used as base. • DE3 or “rand-best/1/bin”, proposed herein, a mix of DE1 and DE2, where DE1 (random base vectors) is used with a probability rb=25% (DE2 otherwise). [1] R. Storn, K. Price, “Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces”, Journal of Global Optimization, 11, 341 359, 1997. [2] R. Storn, K. Price, J.A. Lampinen, “Differential Evolution – a practical approach to global optimization”, Springer, 2005.

  21. Results Comparative performance of identification methods:(a) PSO vs DE1 (b) PSO and DE1 vs SGA with P=25 and P=50 (c) PSO and DE1 vs μGA with P=5 (d) PSO and DE1 vs hybrid method.

  22. Results (cont.) Comparative performance of Enhanced PSO versus DE1, DE2, and DE3

  23. Results (cont.) Identification results after 5000 analyses (mean values)

  24. Results (cont.) Coefficients of variation after 5000 analyses

  25. Concluding remarks • A Bouc-Wen model representing a full scale bolted-welded steel connection was identified successfully using several Evolutionary Algorithms: • Genetic Algorithms (GAs): Standard GA, μGA • Hybrid method (Sawtooth GA, Hill Climber, Bounding) • Particle Swarm Optimization (PSO) • Differential Evolution (DE) • The results indicate DE is the best algorithm for this problem. In particular, DE3, proposed herein, combines impressive exploration and exploitation capabilities. On the other hand, the classic DE variant (DE1) exhibited impressive robustness and produced excellent results after a reasonable number of function evaluations. • The Enhanced PSO and hybrid methods also performed satisfactorily. • The Bounding component of the hybrid method, which gradually diminishes the size of the DS, was hardly even used in this problem. Thus, the hybrid method is better suited for more difficult problems. • The worst performing algorithm for this problem is μGA.

  26. Thank you for your attention!

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