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AI - Artificial Intellignece - AGA - CEI40


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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

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AI - Artificial Intellignece - AGA - CEI40

  1. 1. 9-Month Program, Intake40, CEI Track ARTIFICIAL INTELLIGENCE FUNDAMENTALS Structural Optimization using Genetic Algorithms Ahmed Gamal Abdel Gawad
  3. 3. CONTENTS GA – Introduction GA – Fundamentals GA – Genotype Representation GA – Population GA – Fitness Function GA – Parent Selection GA – Crossover GA – Mutation Research Paper
  4. 4. GA – Introduction
  5. 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. 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. 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. 8. Applications for GAs  Operational Research  Sociology  Game Theory  Economics  Financial Trading  Biology  Engineering  AI Design
  9. 9. GA – Fundamentals
  10. 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. 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. 12. Terminology
  13. 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. 14. GA Lifecycle
  15. 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. 16. GA – Genotype Representation
  17. 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. 18. Genotype Representation Patterns  Binary Representation  Real Valued Representation  Integer Representation  Permutation Representation
  19. 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. 20. GA – Population
  21. 21. Population  Population is a subset of solutions in the current generation. It can also be defined as a set of chromosomes.
  22. 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. 23. GA – Fitness Function
  24. 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. 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. 26. GA – Parent Selection
  27. 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. 28. Parent Selection Methods  Fitness Proportionate Selection  Tournament Selection  Rank Selection  Random Selection
  29. 29. GA – Crossover
  30. 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. 31. Crossover Operators  One Point Crossover  Multi Point Crossover  Uniform Crossover  Whole Arithmetic Recombination  Davis’ Order Crossover (OX1)
  32. 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. 33. GA – Mutation
  34. 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. 35. Mutation Operators  Bit Flip Mutation  Random Resetting  Swap Mutation  Scramble Mutation  Inversion Mutation
  36. 36. Bit Flip Mutation  We select one or more random bits and flip them. This is used for binary encoded GAs.
  37. 37. Research Paper
  38. 38. About Paper  © 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University.
  39. 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. 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. 41.  The parabolic geometry for the applied arch is presented by the equation: 2. The Numerical Model 2.1 Geometry
  42. 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. 43. 2. The Numerical Model 2.1 Geometry  Fig. 1 shows the preliminary dimensions for a typical concrete arch.
  44. 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. 45. 2. The Numerical Model 2.2 Structural system and applied loads
  46. 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. 47. 2. The Numerical Model 2.2 Structural system and applied loads
  48. 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. 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. 50. 2. The Numerical Model 2.3 The structural analysis
  51. 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. 52. 2. The Numerical Model 2.3 The structural analysis
  53. 53.  The stress-strain relation is given by: 2. The Numerical Model 2.3 The structural analysis
  54. 54. 2. The Numerical Model 2.3 The structural analysis
  55. 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. 56. 2. The Numerical Model 2.3 The structural analysis
  57. 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. 58. 2. The Numerical Model 2.3 The structural analysis
  59. 59. 2. The Numerical Model 2.3 The structural analysis
  60. 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. 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. 62. 2. The Numerical Model 2.4 A verification example
  63. 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. 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. 65. 2. The Numerical Model 2.4 A verification example
  66. 66. 2. The Numerical Model 2.4 A verification example
  67. 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. 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. 69.  So, the objective function can be formulated as following: 3. The Optimization Technique
  70. 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. 71.  The constraints in this model can be formed as follow: 3. The Optimization Technique
  72. 72. 3. The Optimization Technique  The allowable stresses and deformation can be calculated as follow:
  73. 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. 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. 75. 3. The Optimization Technique 3.1 The design variables
  76. 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. 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. 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. 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. 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. 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. 82. 4. The Results
  83. 83. 4. The Results
  84. 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. 85. 4. The Results
  86. 86. 4. The Results
  87. 87.  This rough geometry can be smoothed using curve fitting (for practical considerations) as shown in Figs. 18–20. 4. The Results
  88. 88. 4. The Results
  89. 89. 4. The Results
  90. 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. 91. 4. The Results
  92. 92. 4. The Results
  93. 93. 4. The Results
  94. 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. 95. 4. The Results
  96. 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. 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. 98. ▸ Genetic Algorithms Tutorial, tutorialspoint website ▸ Genetic Algorithms, GeeksforGeeks website ▸ Genetic Algorithms - Jeremy Fisher, Youtube video ▸ Introduction to Genetic Algorithms — Including Example Code, towards data science website code-e396e98d8bf3 ▸ Structural optimization of concrete arch bridges using Genetic Algorithms, Mostafa Z. Abd Elrehim, Mohamed A. Eid, Mostafa G. Sayed 📖References
  99. 99. Any Questions?
  100. 100. THANK YOU 