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. 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

3. 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.

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

5. Bouc – Wen model In state-space form:
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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. 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

7. 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 ( %).
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
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.

8. Bouc – Wen model Simple Sinusoidal El Centro (T=25 sec, Amplitude=10)
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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. Bouc – Wen model Simplification #2: For strain-softening systems :
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Bouc – Wen model
Simplification #2: For strain-softening systems :
For the hysteretic spring:

10. 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

11. 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

12. 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

13. National Technical University of Athens
Software

14. Results Best MSE: Best individual for case I (m = 2.86) 0.014390%
National Technical University of Athens
Results
Best MSE: % % % % % % % % % % % % % % %
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
0.9000
2.0000 -
0.1000
2.8600
0.1110

15. 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=
True value
0.9000
2.0000 -
0.1000
2.8600
0.1110
Best individual for case III
(m = 28.6)
Parameter
c=0.0000
c=2.7146
c=5.4292 c= c=
True value
0.9000
2.0000 -
0.1000
2.8600
0.1110

16. 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

17. 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

18. 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