Parameter Estimation of Bouc-Wen Hysteretic Systems by Sawtooth Genetic Algorithm

1. 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

2. NationalTechnicalUniversity ofAthens 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

3. NationalTechnicalUniversity ofAthens 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.

4. NationalTechnicalUniversity ofAthens Bouc – Wen model Restoring force: Equation of motion: Hysteretic parameter:

5. NationalTechnicalUniversity ofAthens 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

6. NationalTechnicalUniversity ofAthens 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

7. NationalTechnicalUniversity ofAthens 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: Therefore, A should be mapped to 1 at all times.

8. NationalTechnicalUniversity ofAthens 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.

9. NationalTechnicalUniversity ofAthens Bouc – Wen model Simplification #2: For strain-softening systems : For the hysteretic spring:

10. NationalTechnicalUniversity ofAthens Sawtooth GA • Introduced by V. Koumousis and C. Katsaras • Variable population size and partial reinitialization of the population Population size: mean population amplitude period

11. NationalTechnicalUniversity ofAthens 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

12. NationalTechnicalUniversity ofAthens 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

13. NationalTechnicalUniversity ofAthens Software

14. NationalTechnicalUniversity ofAthens Results Best MSE: Best individual for case I (m = 2.86)

15. NationalTechnicalUniversity ofAthens Results Best individual for case II (m = 14.3) Best individual for case III (m = 28.6)

16. NationalTechnicalUniversity ofAthens 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

17. NationalTechnicalUniversity ofAthens 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

18. NationalTechnicalUniversity ofAthens 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