Parameter Estimation of Bouc-Wen Hysteretic Systems by Sawtooth Genetic Algorithm Aristotelis Charalampakis and Vlasis Koumousis National Technical University of Athens Institute of Structural Analysis and Aseismic Research

National Technical University of Athens Task Identification of a SDOF Bouc-Wen hysteretic system by Genetic Algorithms Genetic Algorithms: Introduced by John Holland in the 1960s Applied in a wide range of problems, especially optimization Very good global optimizer; often coupled with a local optimizer, such as “steepest ascend” hill climbing Suitable for two main reasons: No use of derivative data Massive parallel computing

Bouc – Wen model Concise yet powerful smooth hysteretic model National Technical University of Athens Bouc – Wen model Concise yet powerful smooth hysteretic model Introduced by Bouc in 1967 Extended by Wen in 1976 to produce a variety of hysteretic loops Popular model, used in the fields of magnetism, electricity, materials and elasto‑plasticity of solids. Examples include the response of R/C sections, steel sections, bolted connections, base isolators such as Lead Rubber Bearings (LRB), Friction Pendulum Systems (FPS) etc.

Bouc – Wen model Restoring force: Equation of motion: National Technical University of Athens Bouc – Wen model Restoring force: Equation of motion: Hysteretic parameter:

Bouc – Wen model In state-space form: National Technical University of Athens Bouc – Wen model In state-space form: The above system of three non-linear ODEs is solved following Livermore stiff ODE integrator based on “predictor-corrector” method or 4th order Runge-Kutta

Bouc – Wen model Restoring force: Equation of motion: National Technical University of Athens Bouc – Wen model Restoring force: Equation of motion: Hysteretic parameter: Fy : Yield force Uy : Yield displacement a : ratio of post-yield to pre-yield stiffness c : viscous damping coefficient n : controls the transition from elastic to plastic branch A, beta, gamma : control the shape and size of the hysteretic loop

Bouc – Wen model Simplification #1: Parameter A is redundant (A=1). National Technical University of Athens Bouc – Wen model Simplification #1: Parameter A is redundant (A=1). For a system with m=2.86 and the El Centro earthquake, the following identified system has a very low normalized MSE (0.0076789%). The response of the identified system is almost identical with the one of the true system. Also, the initial stiffness of a system is given by: Parameter True value Estimated value A 1.0000 57.6596 0.1000 0.3125 0.9000 0.9844 n 2.0000 1.9454 a 0.8105 c 5.1507 5.1868 Fy 2.8600 1.9977 Uy 0.1110 0.9020 Therefore, A should be mapped to 1 at all times.

Bouc – Wen model Simple Sinusoidal El Centro (T=25 sec, Amplitude=10) National Technical University of Athens Bouc – Wen model Simple Sinusoidal (T=25 sec, Amplitude=10) El Centro The response of the identified system is almost identical with the true system for the El Centro excitation. This is NOT true for other excitations.

Bouc – Wen model Simplification #2: For strain-softening systems : National Technical University of Athens Bouc – Wen model Simplification #2: For strain-softening systems : For the hysteretic spring:

Sawtooth GA Introduced by V. Koumousis and C. Katsaras National Technical University of Athens Sawtooth GA Introduced by V. Koumousis and C. Katsaras Variable population size and partial reinitialization of the population Population size: mean population amplitude period

GA options Identification of six parameters: National Technical University of Athens GA options Identification of six parameters: Objective function: normalized MSE of the response history Upper bound for the MSE for error-trapping and scaling options Fitness scaling Biased roulette wheel Single point crossover with probability 0.7 Jump mutation with probability Creep mutation with probability Mean population size 25 Amplitude 15 Period 15 Minimum accuracy 1E-06

GA options El Centro accelerogram National Technical University of Athens GA options El Centro accelerogram Three cases of mass: 2.86, 14.3, 28.6 Five cases of viscous damping: 0%, 5%, 10%, 20%, 30% of critical value 100 runs, 3000 generations each “Steepest ascend” hill climbing for the best individual Parameter True Value Lower Bound Upper Bound 0.900 0.000 1.000 2.000 10.000 varies 20.000 0.100 2.860 0.111 0.001

National Technical University of Athens Software

Results Best MSE: Best individual for case I (m = 2.86) 0.014390% National Technical University of Athens Results Best MSE: 0.014390% 0.044956% 0.016365% 0.001775% 0.001615% 0.005248% 0.026854% 0.004543% 0.002519% 0.000335% 0.003343% 0.003093% 0.003270% 0.000871% 0.002052% Best individual for case I (m = 2.86) Parameter c=0.0000 c=0.8585 c=1.7169 c=3.4338 c=5.1507 true value 0.924316 0.968751 0.812500 0.875000 0.937500 0.9000 2.125000 2.037659 1.984375 1.914062 2.054688 2.0000 0.039673 0.781249 1.728515 3.417663 5.161133 - 0.103516 0.101562 0.093749 0.090317 0.109375 0.1000 2.835431 2.797363 2.818928 2.923047 2.840869 2.8600 0.110510 0.108557 0.108375 0.113192 0.110754 0.1110

Results Best individual for case II (m = 14.3) National Technical University of Athens Results Best individual for case II (m = 14.3) Parameter c=0.0000 c=1.9195 c=3.8390 c=7.6780 c=11.5170 True value 0.976563 0.703125 0.898438 0.861328 0.906250 0.9000 2.085655 1.914063 1.958007 1.914062 2.019531 2.0000 0.000000 2.187500 3.750000 7.623291 11.532879 - 0.101563 0.107422 0.097167 0.097655 0.102882 0.1000 2.894043 2.700382 2.884375 2.889209 2.840869 2.8600 0.113496 0.102704 0.111240 0.111640 0.110265 0.1110 Best individual for case III (m = 28.6) Parameter c=0.0000 c=2.7146 c=5.4292 c=10.8584 c=16.2875 True value 0.921875 0.859375 1.000000 0.812501 0.968995 0.9000 2.125000 2.072265 2.177185 1.914062 2.018982 2.0000 0.097656 2.890625 5.625000 10.879517 16.250000 - 0.101563 0.102142 0.097961 0.097655 0.1000 2.836186 2.803783 2.865039 2.826367 2.923047 2.8600 0.111057 0.108740 0.113802 0.108307 0.114168 0.1110

National Technical University of Athens Conclusions Sawtooth GA, coupled with a local optimizer, is applied to the demanding task of the identification of a Bouc-Wen hysteretic system Simplification of the model Very promising results The large chromosome length and the constant ranges of the parameters reduce the ability of the GA to find the exact values Other techniques, such as gradual narrowing of the parameter ranges will lead to better performance

Present work A range reduction scheme was implemented: National Technical University of Athens Present work A range reduction scheme was implemented: The number of generations is very small (typically 3 Sawtooth periods or 60 generations, as opposed to 3000 generations) This GA is applied for a number of repetitions so as to provide an adequate statistical sample of the best parameter values (typically 30 times) The chromosome length is small (typically 10 bits per parameter i.e. 60 as opposed to 132). Much faster execution. The results are analyzed statistically (using weight and truncation) and the new ranges of the parameters are calculated The process is repeated until all parameters are identified

Present work The improvement is phenomenal: National Technical University of Athens Present work The improvement is phenomenal: The new scheme is able to find the exact values of all the parameters (including the insensitive ones) with accuracy of 4 decimal digits by analyzing ~20-40% of the individuals With accuracy of 2 decimal digits by analyzing ~4%-7% of the individuals Robust one-stage identification (all parameters are active and the initial ranges are very wide) Safe narrowing of the ranges because of the large statistical sample The sample can be created very easily by more than one computers and collected by a host computer The scheme reveals the sensitivity of the parameters. The insensitive parameters are late to be identified