Topology Optimization of Continuum Structure ...

A Dissertation report on Topology Optimization of Continuum Structure Genetic Algorithm Optimization Tool A THESIS Submitted by BHAUMIKKUMAR M. AMIN In partial fulfillment for the award of the Degree of MASTER OF TECHNOLOGY IN MECHANICAL ENGINEERING (MACHINE DESIGN) Under the guidance of PROF. GHANSHYAM G. TEJANI (Assistant Professor)

DEPARTMENT OF MECHANICAL ENGINEERING SCHOOL OF ENGINEERING, R.K. UNIVERSITY, RAJKOT, GUJARAT-360020 JUNE-2014

CERTIFICATE This is to certify that the thesis entitled Topology Optimization of Continuum

Structure Genetic Algorithm Optimization Tool submitted by Bhaumikkumar K. Amin in fulfillment of the degree of Mechanical Engineering (Machine Design) in Mechanical Engineering Department, R.K. University, RAJKOT under my guidance and supervision. In my opinion the standards fulfilling the requirements of the M.tech Degree as prescribed in the regulations of the University has been attained. Date:

Place: Rajkot

Signature and Name of Guide:

PROF. Ghanshyam G. Tejani

Signature and Name of Head:

PROF. Kartik D. Kothari

Signature and Name of Director:

Dr. C. D. Sankhavara

Seal of Institute

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DECLARATION

I hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which has been accepted for the award of any other degree or diploma of the university or other institute of higher learning, except where due acknowledgment has been made in the text.

Furthermore, to the extent that I have included copyrighted material that surpasses the bounds of fair dealing within the meaning of the Indian Copyright Act, I certify that I have obtained a written permission from the copyright owner(s) to include such material(s) in my thesis and have included copies of such copyright clearances to my appendix.

I declare that this is a true copy of my thesis, including any final revisions, as approved by my thesis review committee.

Date:

Signature:

Place: Rajkot

Name: Bhaumikkumar K. Amin

Enrollment No.: 12SOEMD21001

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ACKNOWLEDGEMENTS

I take pride to express thanks from bottom of my heart to all those respected people who have directly and indirectly helped me. I heartily thank my guide Prof. Ghanshyam.G. Tejani who greatly helped me in my thesis. He is constant source of guidance and inspiration to me. He has been very patient and extended excellent support never accepting less than my best efforts. I heartily thank my head of department Prof. Kartik. D. Kothari who greatly helped me in my Dissertation work. I also thank dean of R.K. University Dr. Sankhavara without whose guideline and motivation. I would have not completed my Dissertation work. I am also thankful to my all colleagues for building good team work to make the success of my Dissertation work. Most importantly I would like to dedicate this work to my parents, who have been there for me always and are responsible for what I am today and without whom, the joy of completing the educational phase of my life would have been incomplete.

Date:

Signature:

Place: Rajkot

Bhaumikkumar K.Amin

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THESIS APPROVAL

This is to certify that research work embodied in this entitled: Topology Optimization of Continuum Structure Genetic Algorithm Optimization Tool was carried out by Bhaumikkumar K. Amin (12SOEMD-21001) at Department of Mechanical Engineering, School of Engineering, R.K. University ,In partial fulfillment for the award of the Degree of M.Tech in Mechanical Engineering (Machine Design) by R.K. University.

Date:

Place: Rajkot

Examinar(s) Name and Signature:

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2.

3.

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Contents CERTIFICATE

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DECLARATION

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ACKNOWLEDGEMENTS

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THESIS APPROVAL

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TABLE OF CONTENTS

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LIST OF FIGURES

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LIST OF SYMBOLS, ABBREVIATIONS AND NOMENCLATURE

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ABSTRACT

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INTRODUCTION 1.1 Size optimization . . . . . . . . . . . 1.2 Shape optimization . . . . . . . . . . 1.3 Topology optimization . . . . . . . . 1.3.1 Homogenization . . . . . . . 1.3.2 SIMP . . . . . . . . . . . . . 1.3.3 ESO . . . . . . . . . . . . . . 1.3.4 Level set . . . . . . . . . . . 1.4 Application of topology optimization . 1.5 Objective of Research . . . . . . . . . 1.6 Thesis Outline . . . . . . . . . . . . . LITERATURE REVIEW 2.1 Literature review of homogenization 2.2 Literature review of SIMP . . . . . 2.3 Literature review of ESO/BESO . . 2.4 Literature review of level-set . . . . 2.5 Literature review of GA . . . . . . .

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GENETIC ALGORITHMS 3.1 Overview of genetic algorithms . . . . . . . . . . 3.2 History of GA . . . . . . . . . . . . . . . . . . . 3.3 Main ingredients of GA . . . . . . . . . . . . . . 3.3.1 Chromosomes . . . . . . . . . . . . . . 3.3.2 Selection . . . . . . . . . . . . . . . . . 3.3.3 Crossover . . . . . . . . . . . . . . . . . 3.3.4 Mutation . . . . . . . . . . . . . . . . . 3.4 Flow chart . . . . . . . . . . . . . . . . . . . . . 3.4.1 The basic genetic algorithm is as follows: 3.5 Advantages and limitations of GA . . . . . . . . 3.6 Application of GA . . . . . . . . . . . . . . . .

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METHODOLOGY 4.1 The general scheme of topology optimization using SIMP method 4.2 MATLAB implementation with GA tool . . . . . . . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Implementation of GA SIMP code in MATLAB . . . . . . 4.2.3 Finite Element Analysis . . . . . . . . . . . . . . . . . . 4.2.4 Element Stiffness Matrix . . . . . . . . . . . . . . . . . . 4.2.5 Global Stiffness Matrix . . . . . . . . . . . . . . . . . . . 4.2.6 Optimality Criteria . . . . . . . . . . . . . . . . . . . . . 4.2.7 Mesh-Independency Filtering . . . . . . . . . . . . . . . 4.3 Research field of Topology Optimization . . . . . . . . . . . . . . 4.3.1 Checkerboard patterns . . . . . . . . . . . . . . . . . . . 4.3.2 Mesh dependency . . . . . . . . . . . . . . . . . . . . . .

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RESULT AND DISCUSSION

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CONCLUSION AND FUTURE SCOPE 35 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Future scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

REFERENCES / BIBLIOGRAPHY

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APPENDICES

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List of Figures 1.1 1.2

1.3

a) Sizing b) Shape and c) Topology optimization [16] . . . . . . . . . . . . Microstructures for two-dimensional continuum topology optimization problems: a) Perforated microstructure with rectangular holes in square unit cells, and b) Layered microstructure constructed from two different isotropic material. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Relative stiffness vs volume density for the SIMP material model for different values of the penalization power. [1] . . . . . . . . . . . . . . . .

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2.1

BESO optimal designs for a two material structure . . . . . . . . . . . . . 13

3.1 3.2

Bit string cross over of parents. [18] . . . . . . . . . . . . . . . . . . . . . 17 Flowchart of genetic algorithm. [18] . . . . . . . . . . . . . . . . . . . . . 20

4.1 4.2 4.3 4.4 4.5

General scheme of topology optimization using SIMP Optimization Tool . . . . . . . . . . . . . . . . . . . Typical Element in FEA Model . . . . . . . . . . . . DOF in Global FEM Mesh . . . . . . . . . . . . . . Checkerboard Pattern . . . . . . . . . . . . . . . . .

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5.1 5.2 5.3 5.4

Optimal Topology Structure with Objective Value: 413.3108 Optimal Topology Structure with Objective Value: 229.5530 Optimal Topology Structure with Objective Value: 204.43 . Plot between Fitness Function and Generation . . . . . . . .

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6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Baroda bus station.[19] . . . . . . . . . . . . . . . . . . . . . . Baroda bus station.[19] . . . . . . . . . . . . . . . . . . . . . . Baroda bus station.[19] . . . . . . . . . . . . . . . . . . . . . . Baroda bus station.[19] . . . . . . . . . . . . . . . . . . . . . . Mumbai airport.[20] . . . . . . . . . . . . . . . . . . . . . . . Mumbai airport.[20] . . . . . . . . . . . . . . . . . . . . . . . Mumbai airport.[20] . . . . . . . . . . . . . . . . . . . . . . . Mumbai airport.[20] . . . . . . . . . . . . . . . . . . . . . . . South-west view of an office building designed using BESO.[21]

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6.10 (a) second floor inside view; (b) another inside view of second floor; (c) first floor inside view; (d) ground floor outside view; (e) west side view; (f) south-west view BESO.[21] . . . . . . . . . . . . . . . . . . . . . . . . . 44

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List of Symbols, Abbreviation and Nomenclature SIMP................................................... Solid Isotropic Micro-structure with Penalization ESO..................................................... Evolutionary Structural Optimization BESO.................................................. Bi-directional evolutionary structural optimization GA...................................................... Genetic algorithms STOP................................................... Structure topology optimization problems EFG.................................................... Element free Galerkin method ERR.................................................... Element removal ratio C......................................................... Constraint φ......................................................... The range of allowed values for the design variable 0 Eijkl .................................................... A material property of the base material

p.......................................................... Penalization exponent w......................................................... Weight of function ρ.......................................................... Density distribution OC...................................................... Optimality criteria ui ........................................................ Element displacement vector ki ........................................................ Element stiffness matrix

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ABSTRACT This research investigates the use of advance engineering optimization techniques GA tool for topology optimization of continuum structures. Topology optimization is a process in which materials are distributed in a way that the structure achieves its best performance. Topology optimization of continuum structures is one of the most challenging in research field of structural optimization.The topology optimization of continuum structure most important and advanced way of optimizing various mechanical & civil structure. Several types of structural optimization are investigated, including size, shape and topology optimization methods. Most focus is put on the topology optimization, where methods as SIMP, ESO, homogenization and level set. The four optimization types have applied on civil and mechanical engineering structures, to investigate the usefulness of structural optimization. Genetic algorithms (GA) for structure topology optimization problems (STOPs) developed due to GA is flexible and effective to be applied to various complicated engineering design problems.

This research in based on investigation of advance optimization teaches such as genetic algorithm for topology optimization of continuum structure and provide alternate new method.

Key Words: SIMP, BESO, Homogenization, Level set, Genetic Algorithm.

Department of Mechanical Engineering, R.K. University-Rajkot

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Chapter 1 INTRODUCTION Topology optimization is an incredibly powerful tool in many areas of design such as optics, electronics and structural mechanics. The field emerged from structural design and so topology optimization applied in this context is also known as structural optimization. Applying topology optimization to structural design typically involves considering quantities such as weight, stresses, stiffness, displacements, buckling loads and resonant frequencies, with some measure of these defining the objective function and others constraining the system. For other applications aerodynamic performance, optical performance or conductance may be of interest, in which case the underlying state. Equations are very different to those considered in the structural case. The topology optimization can be done by using the two main techniques, first the Microstructure Material Approaches and second the Macrostructure Geometrical approaches. Simply topology means study of geometric properties that are not change by stretching. In Greek toposmeans location, place, space or domain. Mathematically we can say, topology is concerned with objects that are deformable. E.g. rubber.[16] The main objective is of this research to find the optimum distribution of a specified volume fraction of material in a selected design domain or simply to find the optimal lay-out of a structure within a specified region. The optimum distribution is measured in terms of the overall stiffness of the structure such that the higher the stiffness the more optimal the distribution of the allotted material in the domain. Topology optimization can be applied to both continuum and discrete structures depending on the application.[1] Topology optimization is rapidly expanding research field, which has interesting theoretical implications in mathematics, mechanics, multiphasic and computer science, but also important practical applications by the manufacturing, industries, and is likely to have a significant role in micro- and nano technologies. In topology optimization, both the shape of the exterior boundary and configuration of interior boundaries can be optimized all at once.[16] Figure 1.1 [16] shows the difference between sizing, shape and topology optimization prob1

lems.The differences between these three structural optimization categories mainly consist of the definition of design variables.

Figure 1.1: a) Sizing b) Shape and c) Topology optimization [16] In the sizing optimization problem, the layout of the structure is prescribed, whereas in shape optimization problem, the exterior and interior boundaries can be treated as design variables. Additionally, the design variables can be either continuous or discrete. Depending on the chosen variables and constraints the problem will be one of the three types: size, shape or topology optimization. Which is discussed in below section.

1.1

Size optimization

In size optimization, structural members are varied to find the optimal design. This implies that before such an optimization can be done, the structure to be analyzed must be defined. The geometry change is very small when varying these design variables, so it does not involve the redefinition of shapes of outer boundaries and inner holes of a structure. One of the primary disadvantages of sizing optimization is that the topology of the structure remains fixed throughout the optimization procedure. Therefore, if a sub-optimal topology is chosen when formulating the optimization problem, the resulting structure will also be

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sub-optimal. The optimal design of a sizing optimization is only best design that can come out of the predetermined structural geometric definition.During the size optimization, no elements are added or removed. This gives that the topology of the design is kept throughout the optimization process. There is however the possibility, that the diameter, or area, of a bar gets so small, that it from an engineering point of view hardly exists.[4]

1.2

Shape optimization

Shape optimization means the geometric definitions of the boundaries of outer circumference and inner holes of structure. Shape optimization requires the finite element model to change during the optimization procedure. The design variables used for shape optimization because boundary variation and these are referred to as the reshape design variables. Compared to sizing optimization, the computational cost of shape optimization is higher due to the need to constantly update the finite element model. Shape optimization is divided into two categories; parametric variable variation and boundary variation. In parametric variable variation, the design variables are parameters defining certain features of the shape or important dimensions. For example, the side of a square hole or the radius of a circular hole. Many examples of shape variation using parametric variables. In boundary variation, parts of the boundary of the solid are treated as the design variables. For example, the nodal coordinates of the nodes on the boundary of the shape. The geometric configuration of a structure is required before the shape optimization can be performed. If the shape optimization is based on a specific geometric configuration, any design that is not included in the set of the predetermined geometric modeling will not be created during the optimization. Therefore, the shape optimization converges to different optimal shapes for different starting topologies. Another drawback to shape optimization is the finite element model must be continually updated to ensure that the mesh does not become highly distorted. This task is difficult because mesh refinement and recreation must be performed automatically during the optimization procedure.[5]

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1.3

Topology optimization

Topology optimization is the mostly used structural optimization, to find the problem of the structures topology, i.e. not only to the size and shape of the members but also how the members are connected to each other. This method can be divided into those who operate on the state of each element in the finite element mesh, and those who use lines or surfaces to define the material boundary. When optimizing the topology of a structure it is natural to demand the solution to consist of clearly separated material and void, preferably with a material distribution possible to manufacture. This implies the use of a discrete variable for each element state, either material or void. On the other hand, using continuous variables open up for the use of efficient mathematical programming schemes.[16] The issue with the topology optimization problem formulation is its lack of a true solution. This is due to the fact that refining the mesh, for a given problem, will always give rise to a topological different and better solution, at least when optimizing the average stiffness, which is often the case. Instead of giving a better numerical approximation and a smoother boundary the structural members become smaller and smaller, a process that can continue indefinitely. Several techniques are available to avoid this problem, either by adding additional constraints or by applying some kind of filter which effectively sets a minimum length scale for the structural members. The additional constraints can be on the density gradients or the total perimeter. The issue with a lack of solution is thus resolved either by adding constraints to the original problem, or by allowing intermediate states. There are four types of methods described below.

1.3.1

Homogenization

The homogenization method uses a composite material model, consisting of infinitely small but infinitely many unit cells consisting of material and void or two different materials, although impossible to manufacture. This method is for ordinary engineering tasks superseded by the SIMP method described below which disregard the problem of using unreal material as long as the result consists of mostly black and white.[9] Homogenization based optimization treats the geometric parameters of a microstructure as design variables and homogenizes the properties in that microstructure. The microstructure as shown in Figure is called hole-in-cell or layered microstructure, which is anisotropic in general. The microstructure can be of any kind, but usually each cell is either layered or holds a quadratic or rectangular void, a so called ranked material uses two or more scales with one microstructure in each; each layer in the cell has another microstructure in itself.

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The design variables are defining the microstructure in each finite element and as the composite material is at least orthotropic, also the angular rotation of the microstructure becomes a design variable. From the design variables the stiffness tensor is calculated, which will be less than proportional to the effective density of the material except for all solid or all void elements. This is however not enough to drive the solution to all black and white, which is explained by the efficient use of material in the composite whilst ordinary homogenous material on the other hand, waste material in unstressed directions.[1] Depending on the number of load cases and whether the design domain is two or three dimensional there exists an optimal microstructure which will give good results considering the objective function but also a lot of grey material. By using a suboptimal microstructure or excluding the rotation from the optimization, grey material will be less efficient and the solution will contain less of it. If analytical expressions for the stiffness tensor exist, as is the case for layered structures, the stiffness tensors are calculated when needed. For other micro structures a database of stiffness tensors first has to be generated, e.g. by a finite element analysis of the unit cell, one analysis for each combination of admissible design variables.

Figure 1.2: Microstructures for two-dimensional continuum topology optimization problems: a) Perforated microstructure with rectangular holes in square unit cells, and b) Layered microstructure constructed from two different isotropic material. [1]

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1.3.2

SIMP

Solid Isotropic Microstructure with Penalization (SIMP) is the most widely used method for performing structural topology optimization. It is used in almost all the available commercial optimization technique. The design continuous variables in SIMP x  [0, 1] are one per element in the finite element discretisation and are commonly interpreted as a density ratio. This implies that before starting the optimization process, a design domain, in which the final structure must be enclosed is defined. At initialization, this domain contains elements with an arbitrary density distribution. Elements with a full density are commonly indicated as black. And element with zero density, is indicated by white. Intermediate densities are commonly indicated by different shades of grey. The element stiffness in the SIMP-model can be formulated as follows:

0 Eijkl (x) = xp Eijkl ,p > 1

(1.1)

0 Where Eijkl is a material property of the base material, the stiffness p is the penalization exponent and x is the design variable.

By varying the penalization it is defined how expensive grey material are A penalization of p = 1 means no penalization and increasing the penalization means intermediate densities get penalized i.e. the stiffness will no longer follow a linear relationship. This relationship between design variable and stiffness is showed in Figure. The reason for applying penalization is to get a more black and white solution, which from an engineering point of view is preferable. In order to get a true black and white solution, a penalization exponent greater than p = 3 is often required. It is however recommended, as pointed out to slowly increase the penalization, starting from p = 1. The reason for this is to not rush the optimization avoiding getting stuck at a local optimum. Despite the penalization, grey material can be so much more efficient that it is often seen in the optimized result. In equation 1.2, the variable p is an exponent but the p also appears as a factor when the expression is differentiated in the sensitivity analysis.[1] The weight is constrained or used as objective function and for single material structures the weight is equivalent to the density scaled volume. The weight W, is calculated by integrating the element densities(x) over the design domain Ω :

Z W =

ρ(X)dρ

(1.2)

Ω

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Figure 1.3: 3 Relative stiffness vs volume density for the SIMP material model for different values of the penalization power. [1] To update the density distribution a sensitivity analysis is performed during each iteration, they discovered that under certain conditions, SIMP would not reach the global, no matter how large the penalization, sometimes fails to present a black and white solution.[2]

1.3.3

ESO

One of the most commonly used method for structural optimization is Evolutionary Structural Optimization (ESO). ESO is based on the simple principle that a structure evolves towards an optimum if the least stressed elements in each iteration are removed. The starting analysis is over-dimensioned structure and its boundary conditions must be specified.[8] The stress criterion was replaced by an elemental strain criterion, also called element compliance, to find the elements with the least element strains, a sensitivity analysis is performed where a sensitivity number is calculated for each element, i, according to the following equation.

1 αi = uti ki ui (i = 1...n) 2 Department of Mechanical Engineering, R.K. University-Rajkot

(1.3) 7

Where ui the element displacement is vector and ki is the element stiffness matrix. After the sensitivity analysis, the elements are either fully removed or stays untouched based on the sensitivity numbers. No intermediate densities exist. This eliminates the need to penalize intermediate densities, which is necessary in other topology optimization methods such as SIMP. ESO, in its original form, is also a one-way method or a so called hard-killmethod.[5] Another form of ESO is the additive evolutionary structural optimization (AESO) method. Instead of removing elements with low sensitivity number or low stress as in ESO, elements are added next to the elements with high stress this is, similarly to ESO, a one-way method. After an element is added, it cannot be removed. At least a minimum ground structure is required from the beginning.By combining the original ESO and AESO, presented the bidirectional evolutionary structural optimization (BESO) method, where elements both can be added and removed.[6] BESO also offers the choice to start the iterations from a minimum ground structure to which material will be added, or to start with a maximum design domain from which material will be removed. Both starting points should end in the same result. Genetic ESO (GESO) is a method in which the concepts of survival-of-the-fittest and probability is introduced to the BESO-method. In GESO every element is given a binary string of arbitrary length, something that is not present in the BESO-method.[9] At start, all digits in the binary string are 1 for all elements which means that all elements are filled with material. After each iteration and sensitivity analysis, the genetic algorithms based on the concepts mentioned above, are used to update the binary string. Depending on whether the elements is likely or not likely to be needed in the structure further into the iteration process, one bit in the string will be changed to 1 or 0 respectively. An element with one 0 is preliminary removed and the element is permanently removed when the binary string contains only 0. The probability that the element is used in the iteration process is very low. Comparisons done by the method requires considerably more iterations to converge compared to other methods. The method in some cases also may not result in a globally optimal structure, but instead stops at a local optimum.The BESO method is more likely to present a result near the global optimum, it is pointed out that even for BESO, and the optimality of the solution cannot be guaranteed.[7] The evolutionary procedure for stiffness optimization is given as follows:

• Step 1: Discretize the structure using a fine mesh of finite elements. • Step 2: Carry out finite element analysis for the structure. • Step 3: Calculate the sensitivity number for each element. Department of Mechanical Engineering, R.K. University-Rajkot

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• Step 4: Remove a number of elements with the lowest sensitivity numbers according toa predefined element removal ratio ERR. • Step 5: Repeat Steps 2 to 4 until the mean compliance of the resulting structure reaches a prescribed limit.[8]

1.3.4

Level set

The methods for shape optimization can also be used in topology optimization if the boundary lines/surfaces are allowed to form new topologies. There is problem of re-parameterizations of the boundary when lines/surfaces merge, split, appears or disappears. Instead of defining a two dimensional geometry explicitly by its boundary lines a contour line of a surface in three dimensions, an isoline of a two dimensional scalar field can be used. As for a three dimensional structure its boundary surface can be defined by an iso surface of a three dimensional scalar field. This way of defining the structure implicitly through a higher dimension function is used in the level set method.[11]

1.4

Application of topology optimization

Many civil engineer structure like building, bridge have designed with extend topology optimization structure methods. A group company of Cube Construction Engineering Ltd, engaged in development of Central Bus Terminal of Vadodara, under PPP project with Gujarat State Road Transport Corporation. See Appendices-I. The airport terminal was designed with use of ESO methods. The layout and design with blend of airport terminal and commercial facilities is completely innovative and accommodates highest level of comfort and convenience for public. Seen figure to Appendices-II. The office building shown in Figure 9.2 was designed using an extended ESO method (Ohmori et al. 2005). In essence, the extended ESO method is a BESO method based on the stress level. The building had been planned as part of a large scale redevelopment of a shopping area near Takatsuki JR station in Japan. The land size for the building was approximately 10 m 6 m. The BESO procedure was applied to the south, west and north side walls simultaneously,while the east side wall and the floor slabs were kept unchanged. In the finite element model,both dead weight in the vertical direction and earthquake loading in the horizontal direction were included. The topology of the three walls evolved as material was gradually removed from regions with low stress and added to areas with high stress. Details of the optimization and design processes were presented by Ohmori et al. (2005). The inside and outside views of

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the building soon after the construction was completed in April 2004. See Appendices-III.

1.5

Objective of Research

The Objective of research is to familiarize with current development and methodologies of numerical methods of topology optimization of continuum structures.The detail objective describe below.

• To know structure optimization in detail. • Investigating all fours methods of topology optimization continuum structures. • To test the optimality of design results with the investigated methods. • Implementation of any one method for topology optimization continuum structure with genetic algorithm.

1.6

Thesis Outline

This thesis begins by covering these areas before moving on to showing theoriginal new work in the subsequent chapters. Hence the thesis is organised as follows. Chapter 2 contains a comprehensive literature review of the field of structural optimization. Topology optimization, optimization of composites and topological derivatives are detailed in the early sections, though are not investigated in this thesis. Chapter 3 contains the introduction about genetic algorithm and their applications. It includes the various Ingredients of GA like Chromosomes, Selection, crossover and Mutation. It also includes the advantages and disadvantages if genetic algorithm. Chapter 4 contains various methodologies used in the thesis. These methodologies includes explanation simple SIMP method and genetic algorithm with the help of GA matlab tool. Various methodologies describes the tool that is used to produce the optimum structure. Chapter 5 contains result and discussion produced with the help of GA tool.It includes various figure of optimal structures and graphs produced during the solution of GA with the help of matlab. Finally Chapter 6 concludes the overall thesis and Ideas for future work are set out as possible topics for investigation.

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Chapter 2 LITERATURE REVIEW In this section the research papers related with the present work has been discussed. Recently the topology optimization is the most ”popular” topic in the field of optimaldesign and number of papers indicate the importance of the topic. Among the papers published the related articles are highlighted in this subsequent section.

2.1

Literature review of homogenization

Hongwei Zhao, et al.[9] studied Homogenization Topology Optimization Method. The method of continuous approximation of material distribution for topology optimization is used, which is consistent with the mathematical models derived from the homogenization method. The spatial interpolation by low-order element shape function cannot ensure that the optimum topologies are free from checkerboard patterns and mesh dependence. To overcome this, MLS approximation and modified filter functions are adopted as interpolation function to construct continuous size or continuous density field. The method can be viewed as a nature extension of node-based homogenization method.

2.2

Literature review of SIMP

Niels Olhoff, et al.[1] Studied the topology or layout for given design objectives and constraints.The development of efficient and reliable procedures are used for solution of such problems. There are two different kinds of topology design processes; the Material or Microstructure Technique and the Geometrical or Macrostructure Technique. New areas of application of topology optimization is introduced. Basic concepts and knowledge concerning optimum topology design of continuum structures is presented.A direct way to obtain distinct solid-void topologies within the area of problem is to restrict the design space by including in the problem formulation an upper bound constraint on the perimeter or surface area of the structure. Approaches of this kind are advantageous because they admit application of simple material models like the SIMP model which by itself penalizes intermediate material densities. The model is very popular in commercial topology optimization codes, 11

and it is also very often used in directed toward extending the scope of topology design. Sigmund, et al.[2] Presented the freely available code for topology optimization written as a short MATLAB code. The code was based on the SIMP formulation and is used for Nested Analysis and Design (NAND) based approach to update the structure using an iterative method to converge to the given optimality criteria (OC) for the minimization of compliance subject to a volume constraint problem. Sigmund, et al.[3] Investigated the efficient 88 line MATLAB code for topology optimization. It has been developed using the 99 line code. The major difference with respect to the original 99 line code is the computational efficiency. An improvement in speed with a factor of 100 has been measured for an example problem with 7500 elements. This has mainly been accomplished by means of loop vectorization and memory pre allocation. Krishnan Suresh, et al.[4] Studied MATLAB Code for Pareto-Optimal Tracing in Topology Optimization. A 99-line topology optimization code written in MATLAB demonstrated that SIMP-based topology optimization can be easily implemented in less than hundred lines of MATLAB code. The method and code has been used even since by numerous researchers to advance the field of topology optimization. The notion of topological-sensitivity, one can generate pareto-optimal topologies in about twice the number of lines of MATLAB code. Simply,optimal topologies for various volume fractions can be generated in a highly efficient manner, by directly tracing the pareto-optimal curve. Future work will focus on: (a) non-compliance objectives since the topological sensitivity concept is well defined for a large class of problems, (b) further improving the efficiency of the proposed algorithm, and (c) considering arbitrary shaped features for the topological sensitivity field.

2.3

Literature review of ESO/BESO

X. Huang, et al.[5] Studied Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. This is an improved algorithm for the bidirectional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A meshindependency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. The accuracy of elemental sensitivity numbers is improved by its historical information. Examples shows the capability of the new BESO method starting from the full design to obtain the convergent, and mesh independent solutions. The results compare well with those of the SIMP method.The new BESO method may start from guess designs that are much smaller than the full design domain.

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X. Huang, et al.[6] Presented Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials.The generation of solid-void optimal topologies such as solid isotropic material with penalization method and evolutionary structural optimization (ESO). Utilizing the material interpolation scheme, a new BESO method with a penalization parameter is developed. However, the sensitivity number in the present BESO method is totally different from the strain energy criterion in the original BESO method when the material interpolation scheme utilized for topology optimization problems of structures with two or more Material. Figure 2.1 shows the final optimal topologies for both cases. the optimal topologies are totally different as different materials are used because the sensitivity numbers depend on the Youngs modulus ratio of two materials.

Figure 2.1: BESO optimal designs for a two material structure Numerical examples shows the for this type of topology optimization problems and the optimal designs are independent of the degree of penalization. The BESO method can only be applied to the self-adjoin optimization problems with simple volume constraints. Zheng Juan a, et al.[7] Studied the topology optimization design for continuum structures based on the element free Galerkin method. The element free Galerkin method, combined with evolutionary structural optimization method is applied to carry out the topology optimization of the continuum structures. Considering the deletion criterion based on the stresses, the mathematical formulation of the topology optimization is developed. The objective function of this model is the minimized weight.This formulates the continuum structure and how to solve steps in detail. The mathematical formulation of the topology optimization is developed based on the stress criterion. Several numerical examples shows the validity and feasibility of the present method. And the examples also show the simplicity and fast convergence of the proposed method.

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Xiaodong Huang, et al.[8] Analyzed ESO type methods for topology optimization its later version bi-directional ESO have gained widespread popularity among in structural optimization and practitioners in engineering and architecture. There are many critical comments on various aspects of ESO/BESO. To address those, they carried out extensive work to improve the original ESO/BESO algorithms in recent years. It is shown that the current BESO method stably converge to an optimal solution with high computational efficiency. The optimal solutions of soft-kill and hard-kill BESO methods are mainly discussed.they also demonstrates the current BESO method which can be easily extended to other constraint such as a displacement constraint.

2.4

Literature review of level-set

Vivien J. Challis, et al.[10] Presented a discrete level-set topology optimization codewritten in Matlab. Matlab implementation of the level-set method for topology optimization. The code can be used to minimize the compliance of a statically loaded structure. Simple code modifications to extend the code for different and multiple load cases are given. The code is inspired by a Matlab implementation of the solid isotropic material with penalization (SIMP) method for topology optimization including the finite element solver and comments, the code is 129 lines long. Despite having more lines than the SIMP implementation the level-set Matlab code is still very compact, andneglecting comments and the finite element code is only 63 lines long. Zhen Luo, et al.[11] Finds A level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. The key concept of this method is to represent the structural boundaryimplicitly as the zero level set of a higher-dimensional scalar function, and then the radial basis function with compact support is used to parameterize the level set function. The originally more difficult shape and topology optimization driven by the Hamilton- Jacobi PDE is fully parameterized into a relatively easier size optimization of the expansion coefficients, to which many well established optimization algorithms can be applied. The design of moving structural boundary is then equivalent to transporting the level set surface in an iterative numerical process by finding and updating the expansion coefficients and the level set surface. The present level set-based method can not only possesses the merits of the implicit free boundary method but also avoid some numerical difficulties in the standard level set method. Jian Hua Rong, et al.[12] Studied a continuum structure. It is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary. The design domains of practical structures are often limited and significantly affect thefinal optimal design of the structures. Structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set Department of Mechanical Engineering

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methods. Firstly, to overcome the limitations of current level set methods and the stopping issue of structural boundary movements, a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains.

2.5

Literature review of GA

Mark J. Jakiela, et al.[13] Analysedthe general aspects Continuum structural topology design with genetic algorithms. The genetic algorithm (GA), an optimization technique based on the theory of natural selection, is applied to structural topology design problems. The GA in structural topology optimization, describe a binary material/void design representation that is encoded in GA chromosome data structures. This representation is intended to approximate a material continuum as opposed to discrete truss structure. Specially, with regard to structural optimization, the included here demonstrate both the advantages and disadvantages of GAs. For relatively small problems, the GA-based solutions still require an arguably impractical amount of computation. Still, the versatility and ease of application of a GA is clear.The same representation and solution technique can be robustly applied to a variety of objective functions. Soon Yu Woon, el at.[14] Studied the success of GA-based topology optimization methods for continuum structures. Though a number of methods exist, the results are comparable to solutions from more selected methods in terms of quality and fidelity. This leads for many structural optimization researchers to dismiss the GA as a viable method for topology optimization of continuum structures. The work presented herein provides a new GA-based formulation based on the concept of design space separation through the simultaneous application of multiple genetic algorithms to a problem, and the use of structural response information to guide the GA through a high-level string to structure abstraction. Computational applications shows that the proposed method effectively generate results that are far exceed from existing GA-based methods, and importantly, compares favourably to solutions from more selected methods.No post-processing or additional algorithms are required to generate smooth well-defined solutions. Cuimin Li, el at.[15] Studied about genetic algorithm. GA is used to solve STOPs, because of its global stochastic search ability and its flexibility for various optimization problems. Crossover is the most important operator that controls global evolution direction. However, 1-point and 2-point crossover operator with GA to continuous STOPs often derives to disconnected phenomenon or checkerboard pattern. To obtain an applicable topology, an additional strategy must adopted to suppress the checkerboard pattern or eliminate the disconnected phenomenon. ESO is one formal approach to STOPs with which it is easy to obtain a connected geometric topology.

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Chapter 3 GENETIC ALGORITHMS 3.1

Overview of genetic algorithms

The science that deals with the mechanisms responsible for similarities and differences in a species is called Genetics. The word genetics is derived from the Greek word genesis meaning to grow or to become. The science of genetics helps us to differentiate between heredity and variations and seeks to account for the resemblances and differences due to the concepts of Genetic Algorithms and directly derived from natural heredity, their source and development.[13] A genetic algorithm (or GA) is a search technique used in computing to find true or approximate solutions to optimization and search problems. GAs are an optimization strategy in which points in the design space are analogous to organisms involved in a process of natural selection. The term ‘genetic’ is used because, along with the expected design representation, GAs employ a coded representation of design attributes that is analogous to a chromosome This code is commonly a character string, with each character position being analogous to a gene, andeach character assigned to a position being analogous to an allele. Organisms are generated and tested in generations, with offspring designs arising from parent designs. The creation of new designs for a new generation occurs with a process that is analogous to biological reproduction. Genetic crossover allows offspring designs to retain traits from parent designs, and infrequent mutations possibly yield radically improved designs, but almost always yield unsuitable configurations. The testing of new designs is done with a merit function, usually tailored to take the coded representation as input. In a given generation, designs with a higher merit are given a higher probability of creating offspring, and perhaps surviving themselves into the next generation.[14] Genetic Algorithms (GA) are direct, parallel, stochastic method for global search and optimization, which imitates the evolution of the living beings, described by Charles Darwin.

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GA are part of the group of Evolutionary Algorithms (EA). The evolutionary algorithms use the three main principles of the natural evolution: reproduction, natural selection and diversity of the species, maintained by the differences of each generation with the previous. Genetic Algorithms works with a set of individuals, representing possible solutions of the task. The selection principle is applied by using a criterion, giving an evaluation for the individual with respect to the desired solution. The best-suited individuals create the next generation.[16]

3.2

History of GA

The most popular technique in evolutionary computation research has been the genetic algorithm. In the traditional genetic algorithm, the representation used is a fixed-length bit string. Each position in the string is assumed to represent a particular feature of an individual, and the value stored in that position represents how that feature is expressed in the solution. Usually, the string is evaluated as a collection of structural features of a solution that have little or no inter actions. The analogy may be drawn directly to genes in biological organisms. Each gene represents an entity that is structurally independent of other genes.[17] The main reproduction operator used is bit-string crossover, in which two strings are used as parents and new individuals are formed by swapping a sub-sequence between the two strings. Another popular operator is bit-flipping mutation, in which a single bit in the string is flipped to form a new offspring string.

Figure 3.1: Bit string cross over of parents. [18] A variety of other operators have also been developed, but are used less frequently (e.g., inversion, in which a subsequence in the bit string is reversed).A primary distinction that may be made between the various operators is whether or not they introduce any new information into the population. Crossover, for example, does not while mutation does. All operators are also constrained to manipulate the string in a manner consistent with the structural interpretation of genes. For example, two genes at the same location on two strings may be swapped between parents, but not combined based on their values. Traditionally, Department of Mechanical Engineering

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individuals are selected to be parents probabilistically based upon their fitness values, and the offspring that are created replace the parents. For example, if N parents are selected, then N offspring are generated which replace the parents in the next generation.[16]

3.3

Main ingredients of GA

They are four types of GAs • Chromosomes • Selection • Crossover • Mutation

3.3.1

Chromosomes

During the division process of the human cells the chromatin (contained in the nucleus and built from DNA (deoxyribonucleic acid), proteins and RNA (ribonucleic acid)) become shorter and thicker and forms spiral strings chromosomes. In these chromosomes are the genes that carry the inherited cell information. Every gene codes particular protein and is independent factor of the genetic information, which determines the appearance of different peculiarities.For the genetic algorithms, the chromosomes represent set of genes, which code the independent variables. Every chromosome represents a solution of the given problem. Individual and vector of variables will be used as other words for chromosomes. From other hand, the genes could be Boolean, integers, floating point or string variables, as well as any combination of the above. A set of different chromosomes (individuals) forms a generation. By means of evolutionary operators, like selection, recombination and mutation an offspring population is created.[17]

3.3.2

Selection

In the nature, the selection of individuals is performed by survival of the fittest. The more one individual is adapted to the environment - the bigger are its chances to survive and create an offspring and thus transfer its genes to the next population. In EA the selection of the best individuals is based on an evaluation of fitness function or fitness functions. Examples for such fitness function are the sum of the square error between the wanted system response and the real one; the distance of the poles of the closed-loop system to the desired poles, etc. If the optimization problem is a minimization one, than individuals with small value of the fitness function will have bigger chances for recombination and respectively for generating offspring.[17]

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The population of the next generation is determined by n independent random experiments; the probability that individual xi is selected from the tuple (x1, x2. . . Xm) to be a member of the next generation at each experiment is given by[18] f (x ) >0 P (xi isselected) = Pm i j=1 f (xj )

3.3.3

(3.1)

Crossover

The first step in the reproduction process is the recombination (crossover). In it the genes of the parents are used to form an entirely new chromosome. The typical recombination for the GA is an operation requiring two parents, but schemes with more parents area also possible. Two of the most widely used algorithms are Conventional (Scattered) Crossover and Blending (Intermediate) Crossover.The point at which the chromosome is broken depends on the randomly selected crossover point.[17] This particular method is called single point crossover because only one crossover point exists. Sometimes only child 1 or child 2 is created, but oftentimes both offspring are created and put into the new population. Crossover does not always occur, however. Sometimes, based on a set probability, no crossover occurs and the parents are copied directly to the new population. The probability of crossover occurring is usually 60% to 70%.[17]

3.3.4

Mutation

The newly created by means of selection and crossover population can be further applied to mutation. Mutation means, that some elements of the DNA are changed. Those changes are caused mainly by mistakes during the copy process of the parents genes. In the terms of GA, mutation means random change of the value of a gene in the population.In order to ensure that the individuals are not all exactly the same, you allow for a small chance of mutation.[17] You loop through all the alleles of all the individuals, and if that allele is selected for mutation, you can either change it by a small amount or replace it with a new value. The probability of mutation is usually between 1 and 2 tenths of a percent. Mutation is fairly simple. You just change the selected alleles based on what you feel is necessary and move on. Mutation is, however, vital to ensuring genetic diversity within the population.[17]

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3.4

Flow chart

Figure 3.2: Flowchart of genetic algorithm. [18]

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3.4.1

The basic genetic algorithm is as follows:

• [Start] Genetic random population of n chromosomes (suitable solutions for the problem) • [F itness] Evaluate the fitness f(x) of each chromosome x in the population • [N ewpopulation] Create a new population by repeating following steps until the new population is complete • [Selection] select two parent chromosomes from a population according to their fitness (the better fitness, the bigger chance to get selected). • [Crossover] with a crossover probability, cross over the parents to form new offspring (children). If no crossover was performed, offspring is the exact copy of parents. • [M utation] with a mutation probability, mutate new offspring at each locus (position in chromosome) • [Accepting] Place new offspring in the new population. • [Replace] Use new generated population for a further sum of the algorithm. • [T est] if the end condition is satisfied, stop, and return the best solution in current population. • [Loop] Go for fitness evaluation.

3.5

Advantages and limitations of GA

The advantages of genetic algorithm includes: • Solution space is wider • Easy to discover global optimum • The problem has multi objective function • Easily modified for different problems. • Handles noisy functions well. • Handles large, poorly understood search spaces easily • Good for multi-modal problems. • Very robust to difficulties in the evaluation of the objective function. Department of Mechanical Engineering

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• They require no knowledge or gradient information about the response surface • They are resistant to becoming trapped in local optima • They perform very well for large-scale optimization problems • Can be employed for a wide variety of optimization problems[13] The limitation of genetic algorithm includes: • The problem of identifying fitness function • Premature convergence occurs • The problem of choosing the various parameters like the size of the population, • Mutation rate, cross over rate, the selection method and its strength. • Cannot use gradients. • Cannot easily incorporate problem specific information • Not good at identifying local optima • No effective terminator. • Not effective for smooth uni-modal functions • Have trouble finding the exact global optimum • Require large number of response (fitness) function evaluations • Configuration is not straightforward[13]

3.6

Application of GA

Genetic algorithms have been used for difficult problems (such as NP-hard problems), for machine learning and also for evolving simple programs. They have been also used for some art, for evolving pictures and music. A few applications of GA are as follows[14][18]:

• Nonlinear dynamical systemspredicting, data analysis • Robot trajectory planning • Evolving LISP programs (genetic programming) • Strategy planning

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• Finding shape of protein molecules • TSP and sequence scheduling • Functions for creating images • Controlgas pipeline, pole balancing, missile evasion, pursuit. • Designsemiconductor layout, aircraft design, keyboard configuration, communication networks • Schedulingmanufacturing, facility scheduling, resource allocation • Machine LearningDesigning neural networks, both architecture and weights, improving classification algorithms, classifier systems • Signal Processingfilter design • Combinatorial Optimizationset covering, traveling salesman (TSP), Sequence scheduling, routing, bin packing, graph coloring and partitioning

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Chapter 4 METHODOLOGY 4.1

The general scheme of topology optimization using SIMP method

At the first step of the process, the geometry, finite element mesh, and boundary conditions are set up followed by initialization of the density distribution ρ. SIMP usually starts with a uniform distribution of density which is equal to the specified volume fraction. Then the optimization loop begins with assembling and solving the equilibrium equations (Ku = F) using FEA. Next, in the sensitivity analysis, the Derivatives of the objective function with respect to design variables (ρj 0 s) are computed. Thereafter, an optional filtering technique is applied to remedy the checkerboard problem. The design variables are then updated in the next step using either the optimality criteria (OC). The updated design variables and the resulting topology will be analyzed again and the process of analysis and optimization is repeated until convergence is reached. In general, the optimization process may take many iterations to converge. In terms of computational efficiency, the most expensive part of the optimization is the finite element analysis (FEA) of each candidate design. Depending on the complexity of the structure (i.e., degrees of freedom), the cost of FEA could change. In addition, many topology optimization algorithms have some additional procedures, including the calculation of search direction in the gradient based methods (sensitivity analysis) and/or to filter the undesirable features of the final topology. These additional procedures tend to further increase the computational cost of topology optimization. The SIMP method has become a very popular approach as it is simple to implement, computationally efficient, and easy to integrate with general-purpose FEA codes. However, it suffers from several drawbacks including mesh-dependency of the final topology, undesirable checkerboard patterns requiring the use of filtering techniques, and entrapment in local minima due to its reliance on gradient-based optimization techniques. Recent improvements to the SIMP method include the use of mesh-independency filtering, higher-order 24

finite elements perimeter constraint on the density function and alternative density-stiffness interpolation schemes.

Figure 4.1: General scheme of topology optimization using SIMP SIMP usually starts with a uniform distribution of density which is equal to the specified volume fraction. Then the optimization loop begins with assembling and solving the equilibrium equations (Ku = F) using FEA. Next, in the sensitivity analysis, the Derivatives Department of Mechanical Engineering

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of the objective function with respect to design variables (ρj 0 s) are computed. Thereafter, an optional filtering technique is applied to remedy the checkerboard problem. The design variables are then updated in the next step using either the optimality criteria (OC). The updated design variables and the resulting topology will be analysed again and the process of analysis and optimization is repeated until convergence is reached. In general, the optimization process may take many iterations to converge. In terms of computational efficiency, the most expensive part of the optimization is the finite element analysis (FEA) of each candidate design. Depending on the complexity of the structure (i.e., degrees of freedom), the cost of FEA could change. In addition, many topology optimization algorithms have some additional procedures, including the calculation of search direction in the gradient based methods (sensitivity analysis) and/or to filter the undesirable features of the final topology. These additional procedures tend to further increase the computational cost of topology optimization. The SIMP method has become a very popular approach as it is simple to implement, computationally efficient, and easy to integrate with general-purpose FEA codes. However, it suffers from several drawbacks including mesh-dependency of the final topology, undesirable checkerboard patterns requiring the use of filtering techniques, and entrapment in local minima due to its reliance on gradient-based optimization techniques. Recent improvements to the SIMP method include the use of mesh-independency filtering, higher-order finite elements perimeter constraint on the density function and alternative density-stiffness interpolation schemes. Commonly used filtering techniques adjust either the sensitivity derivatives of the objective function with respect to the design variables or adjust the design variables themselves in order to eliminate the checkerboard effect. Introduced the sin (h) (pronounced ”cinch”) method remedy the drawbacks of both of these filtering approaches while capitalizing on the advantages of each approach. Unlike SIMP, sin (h) is not an acronym; instead, it merely references the use of the hyperbolic sine function. Using hyperbolic sine (sin h) functions, the intermediate density material is made less volumetrically effective than solid or void elements and consequently results in unambiguous and predominantly solid void designs. By adding a new constraint to the topology optimization problem, labeled the sum of the reciprocal variables (SRV), produced sharper 0-1 solutions than the SIMP with greater stiffness for the same amount of material. Efforts to produce better design topologies include relaxation or restriction of the design problem and discretization of the original topology optimization problem combined with heuristic rules to avoid unwanted effects such as checkerboards.

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4.2 4.2.1

MATLAB implementation with GA tool Introduction

Matlab provides an optimization toolbox that includes a GA-based solver. Tool box can be start the toolbox by typing optimtool in the Matlab’s command line and pressing enter. As soon as the optimization window appears, you select the solver GA - Genetic Algorithm.

4.2.2

Implementation of GA SIMP code in MATLAB

A number of simplifications are introduced to GA the Matlab code. First, the design domain is assumed to be rectangular and discretized by square finite elements. In this way, the numbering of elements and nodes is simple (column by column starting in the upper left corner) and the aspect ratio of the structure is given by the ratio of elements in the horizontal (nelx) and the vertical direction (nely). The Matlab code is built up as a standard topology optimization code. The main program is called from the Matlab prompt by the line top (nelx,nely,volfrac,penal,rmin) Where nelx and nely are the number of elements in the horizontal and vertical directions, respectively, volfrac is the volume fraction, penal is the penalization power and rmin is the filter size (divided by element size).

Figure 4.2: Optimization Tool Other variables as well as boundary conditions are defined in the Matlab code itself and can be edited if needed. For each iteration in the topology optimization loop, the code generates a picture of the current density distribution.

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4.2.3

Finite Element Analysis

FEM is incorporated in SIMP to model the optimization problem.in each problem, design domain will be given as 2D rectangle and divided with an orthogonal mesh into uniform square elements. These elements will be modelled as quadratic quadrilateral elements with four nodes and eight degrees of freedoms.

4.2.4

Element Stiffness Matrix

Each element has eight degrees of freedoms, as shown figure .in the element stiffness matrix, the eight DOFs are arranged in a counter- clockwise order and at each corner the horizontal DOF always goes first. Following this arrangement, the DOFs are denoted with letter a to h in an alphabetical order.

Figure 4.3: Typical Element in FEA Model

4.2.5

Global Stiffness Matrix

In global finite element system, degree of freedoms are systemized as figure. In this system, each node is labeled with two DOFs. Labeling starts from the left-most column, bottom to top. Then the labelling goes to the second column and start from the bottom as well. We define horizontal axis x and vertical axis y and then we’ll have nx columns and ny row. Therefore in total we have: Number of elements = (ny × nx) Number of DOFs = [2 × (ny + 1) × (nx + 1)]

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Size of global stiffness matrix = [2 × (ny + 1) × (nx + 1)] × [2 × (ny + 1) × (nx + 1)]

Figure 4.4: DOF in Global FEM Mesh

4.2.6

Optimality Criteria

The updated design variables are found by the optimizer. Knowing that the material volume (sum(sum(xnew))) is a monotonously decreasing function of the Lagrange multiplier (lag), the value of the Lagrangian multiplier that satisfies the volume constraint can be found by a bi-sectioning algorithm. The bi-sectioning algorithm is initialized by guessing a lower l1 and an upper l2 bound for the Lagrangian multiplier. The interval which bounds the Lagrangian multiplier is repeatedly halved until its size is less than the convergence criteria.

4.2.7

Mesh-Independency Filtering

The Matlab implementation that not all elements in the design domain are searched in order to find the elements that lie within the radius rmin but only those within a square with side lengths two times round(rmin) around the considered element. By selecting rmin less than one in the call of the routine, the filtered sensitivities will be equal to the original sensitivities making the filter inactive.

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4.3

Research field of Topology Optimization

The most common are the following: 1. Checkerboard patterns 2. Mesh dependency

4.3.1

Checkerboard patterns

Checkerboard pattern is one of the most common problems related to topology Optimization. Like solid and void Elements alternating themselves. This is a convergence problem caused by the incorrect evaluation of the strain energy by the finite element mesh. If one is solving the problem of minimum compliance with volume constraint, the topology on figure is really the minimum solution of the finite element problem, but not the continuum problem.

Figure 4.5: Checkerboard Pattern Therefore, this is not a desirable solution. To overcome this problem, we can use high order finite elements filtering techniques. The first solution leads to a more expensive computer problem and, sometimes, cannot even solve the problem if SIMP exponent higher than 3 Perimeter constraint is a good solution, because we are not only solving the checkerboard pattern but also the mesh dependency problem. Thus, constraining the perimeter, we can avoid the formation of several small holes. Two drawbacks can be noted in this formulation. The first and more direct is that we are adding a new constraint in the optimization problem, and manage with many constraints usually is not an easy task. Department of Mechanical Engineering

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The second we have no idea about which amount of the outer limits of an area we have to constraint. This can lead to different final topologies.

4.3.2

Mesh dependency

The mesh dependency problem comes from the fact that when the original discrete problem is relaxed each new mesh refinement leads to a new solution. There are several ways to overcome this problem. In fact, the same techniques used to avoid checkerboard can be used to control the mesh dependency.

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Chapter 5 RESULT AND DISCUSSION The proposed methodology is applied to minimize weight hence minimization of compliance structural topology design optimization problems to demonstrate the performance of the bit-array representation GA with the proposed design connectivity handling approach and population initialization method. Topology optimization using the present bit-array GA is also performed with the maximum number of generations being 100, a mutation rate of 0.001 and a random initialization, • The Best Topology by the Present GA is shown in Figure

Figure 5.1: Optimal Topology Structure with Objective Value: 413.3108

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Figure 5.2: Optimal Topology Structure with Objective Value: 229.5530

Figure 5.3: Optimal Topology Structure with Objective Value: 204.43 • Fitness function, variables, plots The first step is to provide the fitness function, that is, the function that calculates the quality of each member of the population (or in plain mathematics, the function you have to optimize).

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Figure 5.4: Plot between Fitness Function and Generation Set the Number of variables to 1800. The representation used is defined in the OptionsPopulation section. The default selection bit-string is fine. The algorithm starts, the plots are pop-up and soon have the results at the bottom left of the window. The best fitness function value (the smallest one since we minimize) and the termination condition met are printed, together with the solution (Final Point - it is very close to (0, 0)). The upper plot, which displays the best fitness at each generation, shows little progress in lowering the fitness value (black dots). The GA returns the best fitness function value of approximately and displays the plots.

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Chapter 6 CONCLUSION AND FUTURE SCOPE 6.1

Conclusion

Topology optimization of continuum structures is to find the optimal designs by determining the best locations and geometries of cavities in the design domains. Several numerical methods of topology optimization has practical applications including ’SIMP’ for intermediate densities. Literature review explores basic four methods for the optimization of continuum structure. The most useful methods used for solving a topology optimization problem is to use of SIMP, BESO and LEVEL SET method and a mathematical programming technique. An investigation into structural topology optimization using an enhanced GA is performed. Numerical results illustrate the efficiency, versatility and robustness of the present GA. It is shown that the present enhanced GA using problem specific knowledge can be more efficient and robust. The performance comparison using the present GA and the popular SIMP method is also carried out and it is obtained that the present GA generates acceptable with difficult checkerboard solutions with more computational cost due to the global search nature. To resolve the problem of representation degeneracy effectively, a recessive gene technique is developed for viable topologies with a structurally connected component. An efficient FEM-based function evaluation method is proposed to remove the void elements totally to reduce the computational time. It is suggested that the present enhanced GA using problem-specific Knowledge can be a powerful global search tool for structural topology optimization.

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6.2

Future scope

• Future work will be focused can removal of checkerboard pattern & mesh dependency of proposed method. • Proposed method of optimization can be used with different structures with the help of GA tool.

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Bibliography [1] Niels Olhoff, Hans A Eschenauer: Topology optimization of continuum structuresA; Review Appl Mech Rev vol 54, no 4, (2000) [2] Sigmund: A 99 line topology optimization code written in Matlab;vStruct Multidisc Optim 21, 120-127,(2001) [3] Erik Andreassen, Anders Clausen, Mattias Schevenels, Boyan S, Lazarov Ole Sigmund: Efficient topology optimization in MATLAB using 88 lines of code; (2010) [4] Krishnan Suresh: A 199-line Matlab Code for Pareto-Optimal Tracing in Topology Optimization; Structural and Multidisciplinary Optimization. , DOI: 10.1007/s00158010-0534-6 (2010) [5] X. Huang, Y.M.Xie: Convergent and mesh-independent solutions for the bidirectional evolutionary structural optimization method; Finite Elements in Analysis and Design 43,1039 - 1049(2007) [6] X. Huang, Y. M. Xie: Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials; Comput Mech 43:393-401,(2009) [7] Zhen Juan, LongShuyao, LiGuangyao: The topology optimization design for continuum structures base donth element free Galerk in method; Engineering AnalysiswithBoundaryElements34,666-672(2010) [8] Xiaodong Huang, Yi-Min Xie: A further review of ESO type methods for topology optimization; Struct Multidisc Optim , 41:671-683(2010) [9] Hongwei Zho, KaiLong: Homogenization Topology OptimizationMethod Based on Continuous Field; Article ID 528397, 7 pages(2010) [10] Vivien J. Challis: A discrete level-set topology optimization codewritten in Matlab; Struct Multidisc Optim , 41:453-464(2010) [11] Zhen Luo, Liyong Tong, Zhan Kang: A level set method for structural shape and topology optimization using radial basis functions; Computers and Structures 87 ,425434(2009)

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[12] Jian Hua Rong a, Qing Quan Liang: A level set method for topology optimization of continuum structures with bounded design domains; Comput. Methods Appl. Mech. Engrg. 197 , 1447-1465(2008) [13] Mark J. Jakiela, Colin Chapman, James Duda, Adenike Adewuya, Kazuhiro Saitou: Continuum structural topology design with genetic algorithms; Comput. Methods Appl. Mech. Engrg. 186 ,339-356(2000) [14] Soon Yu Woon, Liyong Tong, Osvaldo M. Querin, Grant P. Steven D: Effective optimization of continuum topologies through a multi-GA system; Comput. Methods Appl. Mech. Engrg. 194 ,3416-3437(2005) [15] Cuimin Li, Hiroyasu Tomoyuki, Mitsunori Miki: An Improved Stress-based GA for Multi-constrained Topology Optimization; APCOM’07 in conjunction with EPMESC XI,(2007) [16] M. P. Bendsoe, O. Sigmund: Topology Optimization Theory, Methods and Applications; ISBN 3-540-42992-i springer-verlag berlin heidelberg new York [17] Mitchell Melanie: An Introduction to Genetic Algorithms; First MIT Press paperback edition, (1998) [18] Randy L. Haupt, Sue Ellen Haupt: Practical Genetic Algorithms; Second Edition by John Wiley & Sons, Inc. (2004). [19] Cube Infrastructure (2014) Available: http://sphere-cube.com/index.html. [20] www. Missosology.info [21] X. Huang, Y.M. Xie: Evolutionary topology optimization continum structure; ISBN: 978-0-470-74653-0 A John Wiley and Sons, Ltd., Publication; (2010)

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APPENDICES-I

Figure 6.1: Baroda bus station.[19]

Figure 6.2: Baroda bus station.[19]

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Figure 6.3: Baroda bus station.[19]

Figure 6.4: Baroda bus station.[19]

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APPENDICES-II

Figure 6.5: Mumbai airport.[20]

Figure 6.6: Mumbai airport.[20]

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Figure 6.7: Mumbai airport.[20]

Figure 6.8: Mumbai airport.[20]

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APPENDICES-III

Figure 6.9: South-west view of an office building designed using BESO.[21]

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Figure 6.10: (a) second floor inside view; (b) another inside view of second floor; (c) first floor inside view; (d) ground floor outside view; (e) west side view; (f) south-west view BESO.[21]

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APPENDICES-IV

Publications Bhaumikkumar K.Amin, Ghansyam G.Tejani, Topology Optimization of Continuum Structure A Review, International Journal of Advanced Research and Technology., ISBN-23477469

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