Genetic Algorithms

1. USING MICRO GENETIC ALGORITHM TO
OPTIMIZE CABLE NET STRUCTURES
Son Thaia, Min Joo Choa, Nam-Il Kima, Jaehong Leea∗
a Department of Architectural Engineering, Sejong University, Seoul, Korea
11th Asian Pacific Conference on Shell and Spatial Structures - APCS
SPATIAL STRUCTURES REFLECTING ORIENTAL ANTIQUITY
AND MODERN TECHNIQUE
Xi’an - China, May 14-16 2015
Son Thai et al. (Sejong University) Xi’an - China, May 2015 1 / 27

2. Table Of Contents
1 Introduction
Optimization problems of Cable Structures
Micro Genetic Algorithm
2 Design Optimization
3 Numerical examples
Plane Cable Net
Hyperbolic Cable Net
4 Conclusion
Son Thai et al. (Sejong University) Xi’an - China, May 2015 2 / 27

3. Introduction: Optimization problems of Cable Structures
Cable structures:
Widely utilized in civil engineering, aerospace structures (large spaces,
long span structure: cable-supported bridge, cable-stayed bridge, roof
system.)
Distinctive characteristics: light weight, high degree of flexibility, high
strength of axial stiffness.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 3 / 27

4. Introduction: Optimization problems of Cable Structures
Drawback of Cable structures:
Having a very weak initial stiffness and exhibiting highly geometrical
non-linearity.
Difficulty in determination of the initial equilibrium configuration.
Nonlineariry of a complex system causes a lot of computational cost.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 4 / 27

5. Introduction: Optimization problems of Cable Structures
Previous Cable Structures Optimization Works:
Optimal results were obtained by using analytical approach (e.i.
calculus of variations method), need a lot of effort to handle with the
nonlinear problems.
Consider cross-sectional areas or pretension forces as the optimal
objective parameters.
Applied to quite simple and specific model of cable structures.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 5 / 27

6. Introduction: Optimization problems of Cable Structures
Therefore
It is necessary to find a generalized procedure and reduce the
calculation effort.
The searching technique is combined with a Nonlinear Finite Element
Method (using Isoparametric cable element) to develop the proposed
optimization method.
Proposed searching technique: Micro - genetic Algorithm (µGA).
Son Thai et al. (Sejong University) Xi’an - China, May 2015 6 / 27

7. Introduction: Micro Genetic Algorithm
Genetic Algorithm (GA)
Developed by John Holland in the 1960s and 1970s.
Model of biological evolution based on Charles Darwin’s Theory of
natural selection.
A search heuristic among popular evolutionary algorithm.
Used for discrete problem by encoding possibilities into array of bits.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 7 / 27

8. Introduction: Micro Genetic Algorithm
0 0 1 0 1 00 1 0 0 0 1 1 1 00 1 0 1
1 0 1 0 1 00 1 0 1 0 0 1 0 00 0 0 1
1 0 0 0 1 00 1 0 0 0 0 1 0 10 1 0 0
1 0 1 1 1 00 1 0 1 0 1 1 1 00 1 0 1
Genes of var. 1
Individual 1
Individual 2
Individual n
Individual 3
Population
1 0 1 1 1 00 1 0 1 0 1 1 1 00 1 0 1
0 0 1 0 1 00 1 0 0 0 1 1 1 00 1 0 1
{
1 0 1 1 1 10 1 1 1 0 1 1 1 00 1 0 1
1 0 1 0 1 10 1 1 1 1 1 1 1 10 1 0 1
Crossover
Selection
Mutation
Genes of var. 2 Genes of var. 3
Son Thai et al. (Sejong University) Xi’an - China, May 2015 8 / 27

9. Introduction: Micro Genetic Algorithm
Micro-Genetic Algorithm (µGA) is a modified version of GA, first proposed
by Krishnakumar.
µGA refer to a small-population (5 individuals) GA with
reinitialization.
The fittest individual is carried to the next generation.
Having faster convergence and avoiding the fluctuation.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 9 / 27

10. Table of Contents
1 Introduction
Optimization problems of Cable Structures
Micro Genetic Algorithm
2 Design Optimization
3 Numerical examples
Plane Cable Net
Hyperbolic Cable Net
4 Conclusion
Son Thai et al. (Sejong University) Xi’an - China, May 2015 10 / 27

11. Design optimization
The optimization problem can be expressed as:
Min V (Ai ) =
n
1
Ai li
where:
V : total volume of Cable net structure.
Ai and li : cross-sectional area and length of element ith
Son Thai et al. (Sejong University) Xi’an - China, May 2015 11 / 27

12. Design optimization
subjected to two conditional functions
f 1
i (Ai , Fi ) = σi
σa
i
− 1 ≤ 0; i = 1, ..., ne
f 2
j (Ai , Fi ) =
∆j
∆limit
− 1 ≤ 0; i = 1, ..., ne ; j = 1, ..., nj
and two side constraints of design variable
Amin ≤ Ai ≤ Amax
Fmin ≤ Fi ≤ Fmax
i = 1, ..., ne
where:
Fi : pretension force of element ith.
ne, nj: total number of cable elements and internal nodes.
σi and σa: the tensile stress and the allowable stress.
∆j and ∆limit: dis-placement of node jth and limited displacement
value.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 12 / 27

13. Design optimization
Fitness function:
F (Ai , Fi ) = N0 − {V (Ai , Fi ) + η [g1 (Ai , Fi ) + g2 (Ai , Fi )]}
in which
N0, η: relative magnitude parameters
g1(Ai , Fi ) and g2(Ai , Fi ) :penalty functions:
g1 (Ai , Fi ) =
ne
i=1
(f 1
i )
2
; i = 1, ..., ne
f 1
i =
0, if f 1
i ≤ 0
f 1
i , if f 1
i > 0
g2 (Ai , Fi ) =
nj
j=1
(f 2
j )
2
; i = 1, ..., ne ; j = 1, ..., nj
f 2
j =
0, if f 2
j ≤ 0
f 2
j , if f 2
j > 0
Son Thai et al. (Sejong University) Xi’an - China, May 2015 13 / 27

14. Design optimization
Nonlinear Cable Analysis
Fitness Initial Evaluation
Create new generation
End
Converged ?
µGA operators:
Selection, Crossover, mutation
Save the elitist individual
Initialize the First
Generation
Start
No
Yes
Son Thai et al. (Sejong University) Xi’an - China, May 2015 14 / 27

15. Table of Contents
1 Introduction
Optimization problems of Cable Structures
Micro Genetic Algorithm
2 Design Optimization
3 Numerical examples
Plane Cable Net
Hyperbolic Cable Net
4 Conclusion
Son Thai et al. (Sejong University) Xi’an - China, May 2015 15 / 27

16. Plane Cable Net
Table: Properties of Plane Cable Net.
Symbol Definition Data
A Cross-sectional area 146.45 mm2
E Elastic modulus 82,737 MPa
σy Yeild stress 420 MPa
Fi Pretensioned force of inclined element 23.687 kN
Fh Pretensioned force of horizontal element 24.283 kN
P Vertical load at internal nodes 35.586 kN
Son Thai et al. (Sejong University) Xi’an - China, May 2015 16 / 27

17. Plane Cable Net
Table: Comparison of vertical displacement of plane cable net
Resercher Displacement of node 4 (mm)
x-direction y-direction z-direction
Jayaraman and Knudson -36.92 -40.2 -446.32
Densai et al -40.17 -40.17 -446.11
Thai and Kim -40.13 -40.43 -446.5
Present work (Isoparametric elemet) -40.16 -40.16 -445.95
Son Thai et al. (Sejong University) Xi’an - China, May 2015 17 / 27

18. Plane Cable Net
Optimization Cases
A A + F A + Fi + Fh
Table: Conditional parameters for Optimization problems
Constant Condition
Allowable stress σa = 200, 410, 600 MPa
Limited displacement ∆limit = 0.2, 0.45, 0.7 m
Cross-sectional Area 1 mm2 ≤ A ≤ 500 mm2, ∆A = 1 mm2
Pretension force 1 kN ≤ F ≤ 100 kN , ∆F = 1 kN
Son Thai et al. (Sejong University) Xi’an - China, May 2015 18 / 27

19. Plane Cable Net
Table: The optimal cross-sectional areas A (mm2
) and pretensioned forces F (kN)
σa (MPa) ∆limit (m) A A + F A + Fi + Fh
200 0.45 A = 303 A = 299 A = 299
F = 4.88 Fi = 5.26
Fh = 4.13
410 0.2 A = 347 A = 149 A = 148
F = 49.99 Fi = 55.24
Fh = 22.89
410 0.45 A = 145 A = 145 A = 145
F = 24.68 Fi = 34.63
Fh = 13.65
410 0.7 A = 145 A = 142 A = 142
F = 10.35 Fi = 12.51
Fh = 6.05
600 0.45 A = 145 A = 99 A = 99
F = 36.87 Fi = 50.59
Fh = 9.36
Son Thai et al. (Sejong University) Xi’an - China, May 2015 19 / 27

20. Plane Cable Net
A
A + F
A + F i + F h
0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
4 0 0
Cross-sectionalArea(mm2
)
L i m i t e d d i s p l a c e m e n t δl i m i t
( m )
Figure: optimal cross-sectional areas
when σa = 410MPa and ∆limit = 0.2m,
0.45m and 0.7m
A
A + F
A + F i + F h
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
4 0 0
Cross-sectionalArea(mm2
)
A l l o w a b l e S t r e s s σa
( M P a )
Figure: optimal cross-sectional areas
when ∆limit = 0.45MPa and σa =
200MPa, 410Mpa and 600MPa
Son Thai et al. (Sejong University) Xi’an - China, May 2015 20 / 27

21. Hyperbolic Cable Net
Table: Properties of Hyperbolic parabolic Cable Net
Symbol Definition Data
A Cross-sectional area 0.785 mm2
E Elastic modulus 128,300 MPa
F Pretensioned force 0.2 kN
P Vertical load at internal nodes 0.0157 kN
Son Thai et al. (Sejong University) Xi’an - China, May 2015 21 / 27

22. Table: Comparison of vertical displacements for hyperbolic net (mm)
Node Experiment Dynamic Thai and Kim Abad et al. Present
relaxation study
5 19.50 19.30 19.56 19.51 19.38
6 25.30 25.30 25.70 25.57 25.32
7 22.80 23.00 23.37 23.27 22.95
10 25.40 25.90 25.91 25.82 25.58
11 33.60 33.80 34.16 33.94 33.77
12 28.80 29.40 29.60 29.42 29.33
15 25.20 26.40 25.86 25.61 25.45
16 30.60 31.70 31.43 31.01 31.09
17 21.00 21.90 21.56 21.24 21.29
20 21.00 21.90 21.57 20.84 21.19
21 19.80 20.50 20.14 19.20 19.77
22 14.20 14.80 14.55 13.83 14.30
Son Thai et al. (Sejong University) Xi’an - China, May 2015 22 / 27

23. Hyperbolic Cable Net
Optimization Cases
A,F Ax + Ay , Fx + Fy A1-A7, F1-F7
Table: Conditional parameters for hyperbolic paraboloid net
Constraint Condition
Allowable stress σa = 360 MPa
Limited displacement δlimit = 0.034 m
Cross-sectional area 0.1 m2 ≤ A ≤ 1 m2; ∆A = 0.001 m2
Pretension force 0 kN ≤ F ≤ 0.5 kN; ∆F = 0.001 kN
Son Thai et al. (Sejong University) Xi’an - China, May 2015 23 / 27

24. Hyperbolic Cable Net
Table: Optimal volumes (10−5
m3
)for
hyperbolic cable net
A A + F A + 2F A+7F
1.2444 1.2392 1.2343 1.2072
2A 2A + F 2A + 2F 2A+7F
1.1850 1.1757 1.1710 1.1513
7A 7A + F 7A + 2F 7A + 7F
1.1111 1.1094 1.1051 1.0888
0 . 8
0 . 9
1 . 0
1 . 1
1 . 2
1 . 3
1 . 4
N u m b e r o f v a r i a b l e f o r c r o s s - s e c t i o n a l a r e a
72
Optimalvolume(10-5
m3
)
A r e a s + 7 P r e t e n s i o n e d f o r c e s
A r e a s + 2 P r e t e n s i o n e d f o r c e s
A r e a s + 1 P r e t e n s i o n e d f o r c e
O n l y a r e a s
1
Son Thai et al. (Sejong University) Xi’an - China, May 2015 24 / 27

25. Table of Contents
1 Introduction
Optimization problems of Cable Structures
Micro Genetic Algorithm
2 Design Optimization
3 Numerical examples
Plane Cable Net
Hyperbolic Cable Net
4 Conclusion
Son Thai et al. (Sejong University) Xi’an - China, May 2015 25 / 27

26. Conclusion
A generalized method for optimizing the total volume of cable net
structures has been developed by employed Micro - Genetic Algorithm.
The proposed method can be extended for other optimization
problems of the cable net structure (i.e. inelastic problems, dynamic
problems, ...)
The proposed procedure can be used as an aid tooth in preliminary
design of the cable net structures.
Son Thai et al. (Sejong University) Xi’an - China, May 2015 26 / 27

27. The End
Thank You For Your Attention
Son Thai et al. (Sejong University) Xi’an - China, May 2015 27 / 27