Contents of the presentation: • GA – Introduction • GA – Fundamentals • GA – Genotype Representation • GA – Population • GA – Fitness Function • GA – Parent Selection • GA – Crossover • GA – Mutation • Research Paper

1. 9-Month Program, Intake40, CEI Track
ARTIFICIAL INTELLIGENCE FUNDAMENTALS
Structural Optimization
using Genetic Algorithms
Ahmed Gamal Abdel Gawad

2.  TEACHING ASSISTANT AT MENOUFIYA UNIVERSITY.
 GRADE: EXCELLENT WITH HONORS.
 BEST MEMBER AT ‘UTW-7 PROGRAM’, ECG.
 ITI 9-MONTH PROGRAM, INT40, CEI TRACK STUDENT.
 AUTODESK REVIT CERTIFIED PROFESSIONAL.
 BACHELOR OF CIVIL ENGINEERING, 2016.
 LECTURER OF ‘DESIGN OF R.C.’ COURSE, YOUTUBE.
ABOUT ME

3. CONTENTS
GA – Introduction
GA – Fundamentals
GA – Genotype Representation
GA – Population
GA – Fitness Function
GA – Parent Selection
GA – Crossover
GA – Mutation
Research Paper

4. GA – Introduction

5. Optimization
 Optimization is the process of making something better.
In any process, we have a set of inputs and a set of
outputs as shown in the following figure.

6. Optimization
 Optimization refers to finding the values of inputs in such a
way that we get the “best” output values. The definition of
“best” varies from problem to problem, but in mathematical
terms, it refers to maximizing or minimizing one or more
objective functions, by varying the input parameters.
 The set of all possible solutions or values which the inputs can
take make up the search space. In this search space, lies a
point or a set of points which gives the optimal solution. The
aim of optimization is to find that point or set of points in the
search space.

7. What are Genetic Algorithms?
 Genetic Algorithms (GAs) are adaptive heuristic search
algorithm based on the evolutionary ideas of natural
selection and genetics.
 GAs represent an intelligent exploitation of a random
search used to solve optimization problems.

8. Applications for GAs
 Operational Research
 Sociology
 Game Theory
 Economics
 Financial Trading
 Biology
 Engineering
 AI Design

9. GA – Fundamentals

10. Terminology
 Population − It is a subset of all the possible (encoded) solutions to
the given problem. The population for a GA is analogous to the
population for human beings except that instead of human beings, we
have Candidate Solutions representing human beings.
 Chromosomes − A chromosome is one such solution to the given
problem.
 Gene − A gene is one element position of a chromosome.
 Allele − It is the value a gene takes for a particular chromosome.

11. Terminology
 Genotype − Genotype is the population in the computation space. In
the computation space, the solutions are represented in a way which
can be easily understood and manipulated using a computing system.
 Phenotype − Phenotype is the population in the actual real world
solution space in which solutions are represented in a way they are
represented in real world situations.
 Decoding and Encoding − For simple problems, the phenotype and
genotype spaces are the same. However, in most of the cases, the
phenotype and genotype spaces are different. Decoding is a process of
transforming a solution from the genotype to the phenotype space,
while encoding is a process of transforming from the phenotype to
genotype space.

12. Terminology

13. GA Lifecycle
1. Create a population of random
chromosomes (solutions).
2. Score each chromosome in the
population for fitness.
3. Create a new generation through
a mutation and crossover.
4. Repeat until done.
.
.
5. Emit the fitness chromosome as
the solution.

14. GA Lifecycle

15. GA pseudo-code
GA()
initialize population
find fitness of population
while (termination criteria is reached) do
parent selection
crossover with probability pc
mutation with probability pm
decode and fitness calculation
survivor selection
find best
return best

16. GA – Genotype Representation

17. Genotype Representation
 Choosing a proper representation, having a proper
definition of the mappings between the phenotype and
genotype spaces is essential for the success of a GA.

18. Genotype Representation Patterns
 Binary Representation
 Real Valued Representation
 Integer Representation
 Permutation Representation

19. Binary Representation
 In this type of representation the genotype consists of bit
strings. For some problems, specifically those dealing with
numbers, we can represent the numbers with their binary
representation.

20. GA – Population

21. Population
 Population is a subset of solutions in the current
generation. It can also be defined as a set of
chromosomes.

22. Population Initialization
 Random Initialization − Populate the initial population
with completely random solutions.
 Heuristic initialization − Populate the initial population
using a known heuristic for the problem.

23. GA – Fitness Function

24. Fitness Function
 The fitness function simply defined is a function which
takes a candidate solution to the problem as input and
produces as output how “fit” our how “good” the solution
is with respect to the problem in consideration.

25. Fitness Function
 A fitness function should possess the following
characteristics −
 The fitness function should be sufficiently fast to compute.
 It must quantitatively measure how fit a given solution is or
how fit individuals can be produced from the given solution.

26. GA – Parent Selection

27. Parent Selection
 Parent Selection is the process of selecting parents which
mate and recombine to create off-springs for the next
generation.
 Parent selection is very crucial to the convergence rate of
the GA as good parents drive individuals to a better and
fitter solutions.

28. Parent Selection Methods
 Fitness Proportionate
Selection
 Tournament Selection
 Rank Selection
 Random Selection

29. GA – Crossover

30. Crossover
 The crossover operator is analogous to reproduction and
biological crossover. In this more than one parent is
selected and one or more off-springs are produced using
the genetic material of the parents. Crossover is usually
applied in a GA with a high probability – pc .

31. Crossover Operators
 One Point Crossover
 Multi Point Crossover
 Uniform Crossover
 Whole Arithmetic
Recombination
 Davis’ Order Crossover (OX1)

32. Uniform Crossover
 We essentially flip a coin for each chromosome to decide
whether or not it’ll be included in the off-spring.

33. GA – Mutation

34. Mutation
 A small random tweak in the chromosome, to get a new
solution. It is used to maintain and introduce diversity in
the genetic population and is usually applied with a low
probability – pm .

35. Mutation Operators
 Bit Flip Mutation
 Random Resetting
 Swap Mutation
 Scramble Mutation
 Inversion Mutation

36. Bit Flip Mutation
 We select one or more random bits and flip them. This is
used for binary encoded GAs.

37. Research Paper

38. About Paper
 © 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of
Engineering, Ain Shams University.

39. 1. Abstract
 Concrete bridges are used for both highways and railways
roads. They are characterized by their durability, rigidity,
economy and beauty. Concrete bridges have many types
such as simply supported girder bridges, arch bridges and
rigid frame bridges. However, for very large spans, arch
bridges are more economic in addition to their beauty
appearance.
 In this research, a geometrical structural optimization
study for a deck concrete arch bridges using Genetic
Algorithms technique is presented.

40. 1. Abstract
 This research aims mainly to demonstrate a methodology
to find the least cost design, in term of material volume,
by finding the optimal profile. A Finite Element numerical
model is used to represent the arch structure.
 The MATLAB programing platform is used to develop codes
for Genetic Algorithms optimization technique and Finite
Element analysis method. The resulted design from the
optimization process is compared to traditional design and
an obvious cost reduction is obtained.

41.  The parabolic geometry for the applied arch is presented
by the equation:
2. The Numerical Model
2.1 Geometry

42.  The rise to span ratio for arches varies widely.
 Most arches would have a rise to span ratio in the range
from 0.16 to 0.20.
 The span to crown thickness (t crown) ratio (from existing
concrete arches) can be taken between 70 and 80.
 The springing thickness (t springing) to crown thickness
ratio is between 1.55 and 1.72.
2. The Numerical Model
2.1 Geometry

43. 2. The Numerical Model
2.1 Geometry
 Fig. 1 shows the preliminary dimensions for a typical
concrete arch.

44.  The considered arch bridge consists of two-way directions
separated by an island with 2.00 m width and sidewalk in
each side with 3.00 m width.
 Each way has 2 lanes with lane width = 3.00 m. So, the
total width of the bridge is 20.00 m as shown in Fig. 2.
2. The Numerical Model
2.2 Structural system and applied loads

45. 2. The Numerical Model
2.2 Structural system and applied loads

46.  The structural system consists of a bridge slab, secondary
beams, main beams, and the arch girder as shown in Fig. 3.
 Secondary beams are distributed every 2.00 m along the
bridge (2 edge and 9 intermediate secondary beams).
 The main beams, with 20.00 m length, distributed at least
every 4.00 m in which the bridge slab is divided into a
number of one way bays.
 So, the number of main beams is adaptable by the
program according to the span which is defined by the user.
The two arch girders support the bridge from both sides.
2. The Numerical Model
2.2 Structural system and applied loads

47. 2. The Numerical Model
2.2 Structural system and applied loads

48.  The dead loads is automatically calculated by the program
considering slab thickness, main beam dimensions,
secondary beam dimensions and material properties which
are given by the user.
 According to the Egyptian Code of Practice for design loads
(ECP), there is a typical live loading case for design,
including the dynamic impact factor, shown in Fig. 2.
 Using influence line, the worst position for vehicle to get
the maximum internal forces on arch girder is the center.
2. The Numerical Model
2.2 Structural system and applied loads

49.  Herein, FEA model is presented. This model is classified as
a two dimensional case (plane stress). The assumption of
plane stress is applicable for bodies whose dimension is
very small in one of the coordinate directions.
 Thus, the analysis of thin plates loaded in the plane of the
plate can be made using the assumption of plane stress. In
plane stress distribution, it is assumed that.
2. The Numerical Model
2.3 The structural analysis

50. 2. The Numerical Model
2.3 The structural analysis

51.  The main concept in FEA is to divide the structural
problem to small interconnected elements. The Constant
Strain Triangular (CST) elements are to be used to form
the arch body.
 The CST element is composed of 3 corner nodes; each
node has 2 Degrees of Freedom (DOF), u and v, as shown
in Fig. 5.
2. The Numerical Model
2.3 The structural analysis

52. 2. The Numerical Model
2.3 The structural analysis

53.  The stress-strain relation is given by:
2. The Numerical Model
2.3 The structural analysis

54. 2. The Numerical Model
2.3 The structural analysis

55.  The arch body is divided into (32) CST elements, as shown
in Fig. 6, connected together in (34) nodes. It should be
noticed that the arch is symmetric in geometry and
loads. Nodes (1, 2, 33, and 34) are fixed support.
2. The Numerical Model
2.3 The structural analysis

56. 2. The Numerical Model
2.3 The structural analysis

57.  So, the stiffness matrix size; taking the end conditions in
consideration; is (30 nodes * 2 DOF) = 60 * 60.
 For a linear displacement field with in-plane loads, the
resulting 6 x 6 in-plane stiffness matrix for each CST
element can be expressed as:
2. The Numerical Model
2.3 The structural analysis

58. 2. The Numerical Model
2.3 The structural analysis

59. 2. The Numerical Model
2.3 The structural analysis

60.  From the 32 stiffness matrices of the CST elements, a
global stiffness matrix [K] can be formed through their
contact points and therefore strain vector for elements
can be expressed as:
2. The Numerical Model
2.3 The structural analysis

61.  In order to check the developed numerical model and FEA
code, a verification example is analyzed with SAP program
with the same conditions and the results are compared as
shown in Table 1.
2. The Numerical Model
2.4 A verification example

62. 2. The Numerical Model
2.4 A verification example

63.  From these assumptions, the coordinates that forms the
arch body are mapped in Fig. 7.
2. The Numerical Model
2.4 A verification example

64.  Figs. 8–10 show the extreme values for nodal deformation,
normal and shear stresses.
2. The Numerical Model
2.4 A verification example

65. 2. The Numerical Model
2.4 A verification example

66. 2. The Numerical Model
2.4 A verification example

67.  The comparison between developed model and plane
stress SAP model is illustrated in Table 2.
 The table shows that difference in results between the
developed model and SAP model does not exceed 3% of
values.
2. The Numerical Model
2.4 A verification example

68.  In this paper, Genetic Algorithms technique is used to get
the optimal nodes coordinates to form the optimal
shape of arch girder which has the minimum weight and
therefore has the minimum cost.
 A steady state Genetic Algorithms technique is applied for
this model. The optimization process aims to consume the
element for maximum stresses and deformations
allowed by the Egyptian specifications.
3. The Optimization Technique

69.  So, the objective function can be formulated as following:
3. The Optimization Technique

70.  Two constraints are applied; induced stresses and
deformations should not exceed the allowable stresses
and allowable deformations defined by the Egyptian Code
of Practice for design and construction of concrete
structures (ECP).
3. The Optimization Technique

71.  The constraints in this model can be formed as follow:
3. The Optimization Technique

72. 3. The Optimization Technique
 The allowable stresses and deformation can be calculated
as follow:

73. 3. The Optimization Technique
3.1 The design variables
 The design variables here are the nodes vertical
coordinates. Thirty four nodes are selected to form the
arch body.
 Sixteen nodes in the right side have equal coordinates to
their counterpart in the left side in addition to two nodes
on the axis of symmetry.
 So, only 18 nodes are applied as design variables for the
optimization process with 16 alternative positions for
each node.

74.  The step between alternative positions for each node
depends on the given span as shown in Fig. 11.
 It means that the program searches in (1618) different
geometric design field to get the optimal safe design.
 Number of nodes and search space can be increased to
give more accurate analysis with higher computational
cost.
3. The Optimization Technique
3.1 The design variables

75. 3. The Optimization Technique
3.1 The design variables

76. 3. The Optimization Technique
3.2 Genetic Algorithms operators
 Genetic Algorithms technique starts with generating a set
of possible solutions (parents) randomly as an initial
population to the problem.
 Eight random solutions, which represent different arch
geometries, with length of 18 gene (nodes coordinates)
are used for this model.
 These design vectors are encoded to binary form to
facilitate the application of mating operators. So, the total
length for each solution, chromosome, is 18 * 4 = 72 bit.

77. 3. The Optimization Technique
3.2 Genetic Algorithms operators
 Genetic Algorithm’s mating operators are crossover and
mutation. Each two solutions (parents) are combined
together to create two children solutions.
 Uniform crossover technique is used for this model. The
mutation technique which is used for this model is flip bit
mutation with mutation probability selected by the user.

78. 3. The Optimization Technique
3.2 Genetic Algorithms operators
 All the sixteen solutions (parents and children) are
collected in one pool.
 All solutions are sent to the structural analysis program to
check the safety constraints, stresses and deformations.
 The program reads the induced stresses and deformation
values for each node and compare them to the allowable
limits, according to the ECP.

79. 3. The Optimization Technique
3.2 Genetic Algorithms operators
 Unsafe solutions gets penalty function by increasing
their target weight with a suitable factor.
 The best 8 solutions which have less weight for the arch
girder are selected to be new parents for the next
generation.
 These operators are repeated and new generations are
created until the global optimum solution is reached.

80. 4. The Results
 By using a laptop CORE I3 processor and 2 GB RAM, the
developed model has been applied for 3 examples with
different spans as shown in Table 3.

81.  After running the model for these examples, with average 4
days running time for each example, the induced optimized
shape for each case has been shown in Figs. 12–14.
 It’s noticed that thickness of arch girder is increased in
the lower part of the first and last third while increasing
in the upper part of the middle third of the arch for all
cases.
4. The Results

82. 4. The Results

83. 4. The Results

84.  A skeletal model is developed for the investigated cases
using SAP program. The induced bending moment
diagrams are shown in Figs. 15–17.
 It is clear that the resulted rough optimal design follow
the moment diagram with reversed direction.
4. The Results

85. 4. The Results

86. 4. The Results

87.  This rough geometry can be smoothed using curve fitting
(for practical considerations) as shown in Figs. 18–20.
4. The Results

88. 4. The Results

89. 4. The Results

90.  Fig. 21 shows the progression of the minimum weight
value through 1,000,000 iterations of the optimization
process.
 There are many clear drops in the weight value in the first
100 iterations then the change in weight value occurs at
long intervals until reaches the optimal weight.
4. The Results

91. 4. The Results

92. 4. The Results

93. 4. The Results

94.  The resulted optimal geometry is compared to concrete
arch traditional design.
 This comparison is illustrated in Table 4. The rough
optimal design has arch weight reduction ranges from
35% to 40% compared to traditional design arch weight.
 After solution enhancement, the reduction percentage
became in range of 30–35%.
4. The Results

95. 4. The Results

96. 5. Conclusions
 The proposed optimization methodology proved to be
successful technique for the investigated structural
optimization process.
 The resulted optimal designs, after geometric
enhancement, have a reduction percentage ranges from
30% to 35% compared to traditional designs.

97. 5. Conclusions
 Considering the induced deformations and stresses in the
optimal arch geometry, the shear stress is the control
parameter as the allowable shear strength is reached
compared to normal stress and deformation.
 The resulted optimal profile reflects the bending
moment profile which matches well with the common
design principles.

98. ▸ Genetic Algorithms Tutorial, tutorialspoint website
https://www.tutorialspoint.com/genetic_algorithms/
▸ Genetic Algorithms, GeeksforGeeks website
https://www.geeksforgeeks.org/genetic-algorithms/
▸ Genetic Algorithms - Jeremy Fisher, Youtube video
https://www.youtube.com/watch?v=7J-DfS52bnI
▸ Introduction to Genetic Algorithms — Including Example Code, towards
data science website
https://towardsdatascience.com/introduction-to-genetic-algorithms-including-example-
code-e396e98d8bf3
▸ Structural optimization of concrete arch bridges using Genetic
Algorithms, Mostafa Z. Abd Elrehim, Mohamed A. Eid, Mostafa G. Sayed
📖References

99. Any Questions?

100. THANK YOU 
linkedin.com/in/aGaabdelgawad/