S&T2009-Balili-Oreta-v3

8/6/2019 S&T2009-Balili-Oreta-v3

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Genetic Algorithm Simulations for the Optimum

Design of Reinforced Concrete Space FramesAlden Paul D. Balili, MSCE #1 and Andres Winston C. Oreta, D.Eng. *2

#Parsons Brinckerhoff Philippines, Inc.

Philippines1 [email protected], [email protected]

* Civil Engineering Department, De La Salle UniversityPhilippines

2 [email protected], [email protected]

Abstract This paper discusses the results of simulations

for the Optimization Design of Reinforced Concrete (RC)

Space frames using Genetic Algorithms. A Genetic Algorithm

(GA) intends to mimic the processes of natural selection and

reproduction in nature and apply them in artificial settings.

GA has been applied to the optimization of different

engineering problems and has been proven to be an effectiveoptimization tool. One of the purposes of this study is to

determine the combination of GA constants and improvements

that would give the best performance in the optimization of

reinforced concrete space frames. It was found out that

keeping mutation rates low and having a scaling multiplier of

about 1.5 gives good GA performance. However, it was found

out that using Leader Reproduction (LR) reduces population

diversity that would result in premature convergence of a GA

optimization run. In light of this situation, a new type of LR

was proposed called the Modified Leader Reproduction

(MLR). Comparing the GA runs with LR and MLR, it could

be said that MLR improved the effectiveness and efficiency of

the GA run to acquire the most optimal combination of

sections in an RC space frame.

I. INTRODUCTION

Optimization of concrete is not an easy task, to say the

least. Unlike steel, concrete does not have a database of

sections. Also, the section dimensions in concrete can not be

treated as a continuous variable because of known practical

issues like the size of the formworks and lack of accurate

measurement in the field. In most cases, concrete section

dimensions come in increments of 10mm and some

contractors already have standard formwork dimensions in

their stockpiles. Also, designing concrete is not a simple

case of finding the amount of reinforcement required. There

are constraints like the spacing requirements of the bars and

maximum and minimum steel ratio requirements, etc.

Applying these constraints to the design of multiple sections

for each member would surely tax computer resources

which in turn would lead to longer analysis and design

times. Integrating this with an optimization algorithm would

make things worse as far as computational time is

concerned. Given the construction and computational

constraints, can an effective and efficient optimization

algorithm for concrete be practically achieved?

In this paper, it is proposed that combining an enhanced

Genetic Algorithm (GA) with a database of concrete

sections would achieve a practical and efficient way of

optimizing concrete. The contribution of this study is toshow that Genetic Algorithms can be practically applied to

the design optimization of concrete frames under seismic

loading and constraints specified by the NSCP 2001.

II. GENETIC ALGORITHMS A BRIEF DESCRIPTIONOFTHE

PROCESSES INVOLVED

The GA procedure is shown in Fig. 1. First and foremost,

the GA process starts with a random generation of the

individuals of the population. The main aim of this random

generation is to get all possible sections or traits in the

population. To prevent the possibility that certain sections

are not tested, it is recommended that the population be

large enough.

Fig. 1. A flowchart showing GA procedures.

After the initial population is generated, each individual

would now be tested for fitness. After the fitness of each

individual is determined, the traits of each individual will bepassed on to the next generation through the process of

selection, crossover and mutation.

The selection process is an operator which ensures that

highly fit individuals would have their traits replicated at a

much higher rate than other individuals. However, certain

measures, like fitness scaling, should be implemented to

prevent the early domination of certain traits in the

population and prevent premature convergence.

Once the selection process is finished, crossover and

mutation would now be implemented. Crossover involves

the mixing of traits from two individuals to form two new

children. In theory, the combination of traits from highly

fit individuals is expected to produce individuals withhigher fitness. The mutation operator on the other hand, is a
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random operator which generates new traits that can not be

produced through the crossover process.

The process of fitness evaluation, selection, crossover

and mutation would continue until the convergence criteria

are met. Convergence would occur through the domination

of a certain individual or if a certain number of generations

are reached. For this study, since waiting for the optimal

individual to emerge would take a significant amount of

time, it was decided to set a maximum number of

generations as the condition for convergence.

An additional process recommended by certain

researchers [2] called leader reproduction (LR) was

recommended to prevent the loss of certain highly fit

individuals to the random processes of GA. However, it was

found out that LR led to premature convergence and a new

type of LR was recommended by the author of this paper

called Modified Leader Reproduction (MLR). The result of

the GA runs using MLR and LR will be shown on the next

section.

III. APPLICATIONOF GA TOTHE DESIGNOF CONCRETE FRAMES

In the design of concrete space frames, it is the aim of the

designer to find a balance between economy and structural

soundness. The process of finding this balance usually

involves testing different combinations of sections and

doing multiple re-analysis of the structure until the most

economical and structurally sound solution is found. Due to

the multiple variables involved, it is hard to ascertain if the

end solution to the trial and error process is the true optimal

solution.

Genetic algorithms (GA) is a computer algorithm which

could help aid the designer in finding the most economical

and structurally sound combination of sections. GA

optimizes a problem by mimicking the processes of naturalselection in nature. Through this artificial process of natural

selection done in the confines of the computer, it is aimed

that the best individual or solution could be attained after a

certain number of generations. For more information on GA

basics and terminology, it is recommended to read

Reference [4].

In the present problem, the space frame consisting of

beams and columns shown in Fig. 2 will be optimized. We

can say that the whole space frame is an individual or

solution, and the dimensions of its beams and columns are

the traits or characters which describe that individual. This

individuals fitness will be determined by analysing and

designing the individual using its current traits or sections

and computing the total cost.

The process of GA and its application to the optimization

of space frames is described in Fig. 3. Initially, the sizes of

the beams and columns of the space frame are randomly

selected which become the initial population. These sizes

are then used by a separate Finite Element Analysis

program to determine the member forces which are required

in the design of the members including the amount of steel

reinforcements. A database of the beam and column

sections is used in the design process. The provisions of the

National Structural Code of the Philippines (2001) are

incorporated in the fitness evaluation of the solution or

individual to satisfy the strength and serviceability

requirements. The GA procedures of selection, cross-over,

mutation and leader reproduction are then applied to

generate a new population of solutions.

Population of Chromos

Population of Chromos

Fig. 3. Diagram showing the GA process as applied to space frames

IV. GA SIMULATION RESULTS

In GA, there are a number of constants that must be set

before the run could proceed. Different studies have

recommended different values for these constants and

additional GA operators which they claim improve GA run

performance. To confirm these recommendations, different

GA constants would also e tested for this study. The

following GA variables and GA operators will be tested: (1)

Mutation rates (2) Scaling Multiplier (3) Gray Coding and

(4) Leader Reproduction.

A. Results for Different Mutation Rates

Mutation is a double edged sword as far as GA is

concerned. Its main benefit is that new areas in the solutionspace are explored. While its main drawback is its

tendency to destroy important traits in the individuals in

the population.

Figures 4 shows the average fitness for 5 GA runs for

each tested constant. Figure 5 shows the ratio of the

average maximum fitness for 5 GA runs acquired for each

constant to the true optimal value. It could be observed

that a probability of mutation of 0.001 has a slightly better

performance when it comes to improving the overall

population fitness and acquiring the optimal value. This

confirms Goldbergs[4] statement that it is best to keep

mutation rates low. The good performance of 0.005 could

be attributed to the ability of high mutation rates to exploremore areas in the solution space.

Fig. 2. Model of a two-storey space frame


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Ave. Population Cost of 5 Runs

for Different values of Probability of Mutation without Gray Coding

800000.00

000000.00

200000.00

400000.00

600000.00

800000.00

000000.00

200000.00

400000.00

0 1 2 3 4 5 6 7 8 9 10

Generation (t)

Ave.Cost

PrM 0.001 PrM 0.002 PrM 0.005

Fig. 1. Average fitness for each generation of a GA run for different valuesof Probability of Mutation

Ratio of Optimum Value to Average Minimum Cost Acquired (for 5

runs) for each Probability of Mutation value

without gray coding

0.80

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

0.89

.

Pr 0.001 Pr 0.002 Pr 0.005

OptimalCost/AverageMinimumCost

Fig. 2. Ratio of the true optimal cost to the average optimal cost for 5 GA

runs for different probability of mutation constants.

B. Results for Scaling Multiplier fm

Fig 6. shows the comparison of a GA run using a scaling

multiplier of 1.2 and 1.5.

Population Cost for Probability of Mutation = 0.001

for GA Run using Gray Coding with different values of fm

1500000

2000000

2500000

3000000

3500000

0 5 10 15 20 25

Generation (t)

.

fm 1.5 fm 1.2

Fig. 3. Average fitness of population for scaling multiplier fm 1.5 and 1.2.

It can be seen that a scaling multiplier of 1.5 gives a

slightly better performance in improving the fitness of the

population. This could be attributed to the fact that a higher

scaling multiplier provides sufficient rewarding of the fitter

individual in the population while a lower scaling multiplier

results in almost equal rewarding of highly fit and unfit

individuals.

C. Results for Gray and Binary Coding

Fig. 7 shows the comparison of the average fitness for 5

GA runs using mutation probability of 0.001 for a GA run

with and without gray coding.

Comparison of Average Cost of 5 Runs

for GA run with and without gray coding

1800000.00

2000000.00

2200000.00

2400000.00

2600000.00

2800000.00

3000000.00

3200000.00

0 1 2 3 4 5 6 7 8 9 10Generation (t)

Ave.Cost

PrM 0.001 PrM 0.001 Gray

Fig. 4. Comparison for Probability of Mutation 0.001 for GA run with and

without gray coding.

It can be seen that gray coding gives slightly better

performance for probability of mutation of 0.001 when

based on the average fitness at generation 10. The spikes in

the average fitness are expected due to the stochastic nature

of the basic GA processes involved in a GA run. However,the good genetic material present in the population will

eventually recombine to produce better individuals. This is

the explanation for the spike at generation 9 for probability

of mutation value 0.001 and its improvement to a better

average fitness in generation 10.

D. Results for Leader Reproduction

Leader reproduction is a process wherein the best

individual from the previous generation is reinserted into

the current population in the case that best individual in the

current generation is inferior to the best individual of theprevious generation. The best individual will be reinserted

to the population by replacing an individual with the worst

fitness.

Figure 8 shows the comparison of the GA run with or

without leader reproduction for 25 generations. It is clear

from the graph that leader reproduction significantly

improves the average fitness of the population.Population Cost for P robability of Mutation = 0.001

for GA Run using Gray Coding

1500000

2000000

2500000

3000000

3500000

0 5 10 15 20 25

Generation (t)

Ave.Cost

w LR wo LR

Fig. 5. Comparison of average population cost for run with and without

leader reproduction

GenerationGeneration

Generation

Generation


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E. Evaluation of Initial Results

From the results of our initial GA experiments, it could

be said that having gray coding, leader reproduction and a

mutation probability of 0.001 would give a higher chance

for a FA run to acquire optimal results. To confirm if this is

the case, the best individual from the final run from the

experiment in the previous section was acquired. It was seen

the GA run using the current constants still yieldedunsatisfactory results at the maximum generation.

Since it is the aim of this study to produce a effective and

efficient optimization algorithm, a new type leader

reproduction called Modified Leader Reproduction (MLR)

was conceived.

F. Results of Run with LR and MLR

Figure 9 shows the average fitness for both runs with

leader reproduction and modified leader reproduction.

Figure 10 shows the ratio of the true optimal cost to the

optimal cost attained for each type of leader reproduction.Population Cost for Probability of Mutation = 0.001

for GA Run using LR and MLR

1300000

1800000

2300000

2800000

3300000

0 5 10 15 20 25

Generation (t)

Ave.Cost

LR MLR

Fig. 6. Graph of average fitness of population for GA run with Leader

Reproduction and modified leader reproduction.

Ratio of Optimum Value to Average Minimum Cost Acquired for

different types of leader reproduction

0.90

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

LR MLR

OptimalCost/AverageMinimumCost

Fig. 7. Ratio of true optimal value to minimum cost acquired for GA run

with Leader Reproduction and modified leader reproduction.

As expected, the run with MLR outperformed the one

with LR in terms of getting the average. Also, it is worth

mentioning the MLR did this feat at only generation number

20.

To further test the effectiveness of MLR, another 4-

storey structure (as shown in Fig. 11) was optimized. It was

confirmed that MLR outperformed LR in terms of getting

the optimal solution for the said structure.

Fig. 8. 4-storey building framing 3d view

V. CONCLUSION

From the results of all the experiments in this study, the

following can be concluded:

For any optimization tool that uses GA, it is

recommended to keep mutation rates low and use a

scaling multiplier of 1.5 to give good GA performance.Also, it is confirmed that using gray coding gave better

results for a GA run.

Due to the premature convergence induced by leader

reproduction, a new type of leader reproduction called

modified leader reproduction was proposed. It was

found out that this feature improved the effectiveness

and efficiency of the concrete optimization algorithm to

acquire the optimal values.

Based on the results of the GA optimization run, it could

be said that a concrete optimization algorithm can be

practically integrated to a FEM analysis program. The

only caveat is the analysis time consumed by the FEM

analysis program must be reduced. Using the power of

parallel computing is a possible solution to reduce

analysis times.

ACKNOWLEDGMENT

The first author would like to thank his father, Engr.

Danny Balili for providing data for the unit costs used for

this paper. And finally, the author would like to thank his

wife, Lorie, for being patient and tolerant and for providing

the support and motivation for the author to finish this

study.

REFERENCES

[1] Association of Structural Engineers of the Philippines (ASEP)(2001). National Structural Code of the Philippines 5th Edition,

Volume 1. Quezon City, Philippines.

[2] Balamurugan R., Ramarkrishnan C.V., & Singh N. (2008)

Performance evaluation of a two stage adaptive genetic algorithm(TSAGA) in structural topology optimization. ApplIed Soft

Computing Journal, doi:10.1016/j.asoc.2007.10.022

[3] Camp C.V., Pezeshk S., & Hansson H. (2003) Flexural Design of

Reinforced Concrete Frames Using a Genetic Algorithm, Journal

of Structural Engineering, 129, pp. 105-115.

[4] Goldberg, D.E. (1989). Genetic algorithms in search, optimization

and machine learning. Reading MA: Addisson-Wesley.

[5] Jenkins W.M. (1997) On the application of natural algorithms tostructural design optimization, Engineering Structures, Vol. 19,

No. 4, pp. 302-308

[6] Kassimali, A. (1999) Matrix Analysis of Structures. Pacific Grove,

California: Brooks/Cole Publishing Company.

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