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1 TOPOLOGY OPTIMIZATION OF STRUCTURES USING GLOBAL STRESS MEASURE By VIJAY KRISHNA YALAMANCHILI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2012
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2 2012 Vijay Krishna Yalamanchili
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3 To my parents, Siva Rama Krishna and Geeta Kumary my brother Pavan Kumar my advisors, Dr. Ashok V. Kumar, Dr. Raphael T. Haftka, Dr. Nam Ho Kim, and other close family members, relatives and friends
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4 ACKNOWLEDGMENTS I would like express my gratitude and deepest appreciation to Dr Ashok V Kumar, Chairman and advisor for my t hesis committee for his guidance, patience and valuable insights throughout the period of my research work. Without his persistent help this thesis would not have been possible. I would also like to thank other members of the supervisory committee Dr. Raph ael T. Haftka, Dr. Nam Ho Kim for their time, willingness to help and suggestions during the review process. Finally, I would like to thank my family and friends for their support and University of Florida for providing me this opportunity.
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5 TABLE OF CONTE NTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF ABBREVIATIONS ................................ ................................ ........................... 10 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTI ON ................................ ................................ ................................ .... 13 Overview ................................ ................................ ................................ ................. 13 Objectives ................................ ................................ ................................ ............... 17 Outline ................................ ................................ ................................ .................... 17 2 TOPOLOGY OPTIMIZAT ION ................................ ................................ ................. 19 Topology Optimization Methods ................................ ................................ ............. 1 9 Homogenization Method ................................ ................................ ................... 19 Genetic A lgorithm Method ................................ ................................ ................ 20 Level Set Method ................................ ................................ ............................. 21 Solid Isotropic Method With Penalization ................................ ......................... 22 Objective Functions With Constraints ................................ ................................ ..... 23 Compliance Minimization ................................ ................................ .................. 23 Compliant Mechanism ................................ ................................ ...................... 24 Stress Constraints ................................ ................................ ............................ 24 Others ................................ ................................ ................................ ............... 31 3 NUMERICAL METHODS US ED TO SOLVE THE OPTIMIZATION PROBLEM ..... 32 Sequential Linear Programming ................................ ................................ ............. 32 Sequential Quadratic Programming ................................ ................................ ........ 34 Method Of Moving Asymptotes ................................ ................................ ............... 35 Moving Barrier Method ................................ ................................ ............................ 37 Comparison ................................ ................................ ................................ ............ 39 4 STRESS BASED TOPOLOGY OPTIMIZATION ................................ ..................... 40 Objective F unction ................................ ................................ ................................ .. 41 Singularity ................................ ................................ ................................ ......... 41 Global Stress Measure ................................ ................................ ..................... 42 Relation Between Mean Compliance And Von Mises Stress ........................... 45 Modified Optimization Problem ................................ ................................ ......... 46 Smoothing Scheme ................................ ................................ ................................ 46 Sensitivity Analysis ................................ ................................ ................................ 47
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6 5 RESULTS ................................ ................................ ................................ ............... 49 Bar ................................ ................................ ................................ .......................... 49 Michell Type Structure ................................ ................................ ............................ 51 Bracket ................................ ................................ ................................ .................... 54 Unconstrained Bracket ................................ ................................ ............................ 56 Bridge ................................ ................................ ................................ ..................... 59 L Shaped Structure ................................ ................................ ................................ 61 3 D Bracket ................................ ................................ ................................ ............. 63 3 D Example II ................................ ................................ ................................ ........ 64 6 CONCLUSIONS ................................ ................................ ................................ ..... 67 Summary ................................ ................................ ................................ ................ 67 Discussions ................................ ................................ ................................ ............. 67 Scope Of Future Work ................................ ................................ ............................ 68 APPENDIX A METHOD OF MOVING ASYMPTOTES ................................ ................................ 70 Selecting T he Move Limits ................................ ................................ ...................... 70 Solving T he Subproblem ................................ ................................ ......................... 70 B MOVING BARRIER METHOD ................................ ................................ ................ 73 Sele cting T he Move Limits ................................ ................................ ...................... 73 Solving T he Subproblem ................................ ................................ ......................... 73 LIST OF REFERENCES ................................ ................................ ............................... 75 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 79
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7 LIST OF TABLES Table page 2 1 Summary of the literature on topology optimization us ing stress cons traints .... 29 5 1 Configuration details used to obtain bar like structures ................................ ...... 49 5 2 Configuration details used to obtain Mitchell truss type structures ..................... 52 5 3 Configuration details used to obtain optimum bracket shapes ........................... 55 5 4 Configuration details used to obtain optimum shapes for unconstrained bracket ................................ ................................ ................................ ................ 57 5 5 Config uration details used to obtain b ridge type structures ................................ 59 5 6 Configuration details used to get optimal shapes for L shaped structure ........... 62
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8 LIST OF FIGURES Figure page 2 1 Unit cell of a microstructure with a rectangular void. ................................ .......... 20 2 2 ................................ .. 22 4 1 Plot of stress functions ................................ ................................ ....................... 44 5 1 Plane stress model of the design domain for axial loading problem ................... 49 5 2 Top ology optimization results for b ar problem ................................ .................... 50 5 3 Stress distribution of the optimal topologies for the b ar problem ........................ 50 5 4 Plane stress model of the design domain for Mitchell structure problem ............ 51 5 5 Topology optimization results for Mitchell truss type structure ........................... 52 5 6 Stress distribution of the optimal Michell Structures ................................ ........... 53 5 7 Convergence plots (i) Case b (ii) Case d ................................ ............................ 53 5 8 Plane stress model of a bracket ................................ ................................ ......... 55 5 9 Topology optimization results for bracket for different configurations ................. 55 5 10 Stress distribution of the optimal topologies for the beam problem .................... 56 5 11 Plane stress model of an unconstrained bracket ................................ ................ 57 5 12 Topology optimization results for unconstrained bracket for different configurations ................................ ................................ ................................ ..... 58 5 13 Stress distributions for the unconstrained bracket problem ................................ 58 5 14 Plane stress model of the design domain for the bridge problem ....................... 59 5 15 Top ology optimization results for b ridge problem ................................ ............... 60 5 16 Stress distribution of the optimal topologies for the b ridge problem ................... 60 5 17 Plane stress model of a loaded L shaped structure ................................ ............ 61 5 18 Topology optimization results for L shaped structure for different configurations ................................ ................................ ................................ ..... 62 5 19 Stress distribution of the optimal L shaped structures ................................ ........ 62
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9 5 20 Plane stress model of an 3 D bracket problem ................................ ................... 63 5 21 Topology optimization results for the 3 D bracket problem ................................ 64 5 22 Plane stress model of an 3 D example II ................................ ............................ 65 5 23 Topology optimization results for the 3 D example II ................................ .......... 65
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10 LIST OF ABBREVIATION S MBM Moving barrier method MBSLP Moving barrier sequential linear programming MMA Method of moving asymptotes SIMP Solid isotropic material with penalization SLP Sequential linear programming SQP Sequential quadratic programming
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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science TOPOLOGY OPTIMIZATION OF STRUCTURES USING GLOBAL STRESS MEASURE By Vijay Krishna Yalamanchili May 2012 Chair: Ashok V. Kumar Major: Mechanical Engineering Topology optimization is a mathematical approach to determine the optimal distribution of material within a given design space. Minimization of compliance for a given mass has been the predominant method of designing structures using topology optimization. Minimization of compliance is equivalent to maximizing the stiffness of the structure and mostly results in structures with approximately uniform stress es I n most cases a design with the lowest compliance is not always the same as the desig n with the lowest maximum stress and the amount of material that can be removed before stresses exceed the stress limits is often not known. Stress is the most important consideration optimiz ation which minimizes the mass would be a better choice However due to challenges such as singularity and localized stresses the problem has not been solved completely In this thesis, we propose an effective and low cost (computational cost) method of im posing stress constraints on continuum structures using a conservative global stress measure The shape is defined using a shape density function and the contours of this density corresponding to a threshold value (0.5) are defined as the
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12 boundaries of the design. The nodal values of shape density have been treated as the design variables and smooth density functions are obtained by interpolating the nodal values of density. Solid Isotropic Material with Penalization (SIMP) has been used to force the densit y towards 0 or 1. A relation is established between the mean compliance and Von Mises stress. Von Mises stress in the stress constraints is approximated by an equivalent compliance term. An expression which represents global stress measure is obtained by i ntegrating special functions of local stress violations. This expression is added to the mass of the structure to obtain the objective function. The optimization problem is then solved using Moving Barrier Method (MBM) to obtain the optimal design. Results indicate that this approach works well in the absence of highly localized stresses.
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13 CHAPTER 1 INTRODUCTION Overview Topology o ptimization is a mathematical approach that optimizes the distribution of material within a given design space while also meeting design and pe rformance requirements. Topology optimization has been used predominantly by structural designers and is the emphasis of this work. I t could also be used in other areas of physics such as micro electro mechanical systems, fluid and thermal systems. Topolog y optimization in its current form could be thought of as a tool that provides preliminary design which needs refinements to meet manufacturability and other performance parameters. It is most useful in problems where the design is not intuitive, it helps the designer by providing a conceptual design and thus reducing the cost in terms of design time and material used. For the reasons mentioned above t opology optimization has found applications in industries such as aerospace, civil, and automotive. A structural optimization problem definition includes a feasible domain, the boundary conditions and the objective function and the constraints. The objective of the optimization could be minimizing the weight of the structure or maximizing the stiffness o f the structure while constraints are specified on the maximum stress or displacement or weight. Sizing, shape and topology optimization are three different classes of structural optimization. In sizing optimization the goal could be to find the optimal thickness of a plate or cross sectional area of bar in a truss like structure. On the other hand, in a shape optimization problem the goal is to find the optimum shape of the domain. In topology optimization the objective is to find the optimal locations, shapes and sizes of voids in the design space. In some cases topology optimization is done to
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14 obtain a preliminary design and shape optimization is then performed starting from this initial design. This is because topology optimization has not yet reached the stage where it can handle all the design considerations to come up with a final design. Topology optimization of continuum structures is an established field with active research being carried out for more than two decades. In the early stages, finite element analyses were performed and regions with low stresses were removed and the process was repeated iteratively till the design was fully stress ed in most regions. In this approach the final design depends on the initial mesh discretization making it i nconsistent and unreliable. Since then several methods such as homogenization ( Bendse and Kikuchi 1988) Solid Isotropic Material with Penalization (SIMP) ( Bendse 1989, Bendse and Sigmund 2003) level set (Allaire et al 2004) and genetic algorithm ( Cha pman et al 1994, Ohsaki 1995, Kaveh and Kalatjari 2003, Wang and Tai 2005 ) have been proposed to identify the shapes, sizes and locations of voids in the final design All the mentioned methods are generally used in conjunction with finite element formula tion to obtain the optimal designs. The ideas behind each of the methods me ntioned above are explained in C hapter 2. The SIMP method converts the discrete feasible domain into a continuum setting which facilitates the use of gradient based optimization alg orithms which are very efficient and computationally less expensive. It is for this reason that the SIMP method has been gaining popularity and w e have used this approach in our work. Most methods which use the SIMP approach assume constant density within an element and treat density of element s as design variables. However, Kumar and Gossard (1992) have shown that treating density as nodal variables and using nodal densities as design
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15 variables yields smoother topologies with fewer elements when compared w ith constant element densities. Interpolation schemes are used to compute the density within an element and contours with density equal 0.5 are treated as boundaries of the design. This approach, which uses nodal values of density as the design variables, has been used for the purpose of this thesis. Topology o ptimization with minimization of compliance as the objective function has been understood very well and there are several methods which claim to do this efficiently ( Bendse and Sigmund 2003) In this technique compliance of the structure is minimized i.e. in other words the stiffness is maximized, for a given volume fraction. The optimization problem formulation for this technique is explained in detail in C hapter 2. F f view maximum stress in the structure is the most important consideration and it is not addressed by the minimization of compliance method. Pedersen (1998) demonstrated that compliance design is equivalent to stress design if the stress criterion is consi stent with the elastic energy measure. However, we consider the von Mises stress criterion which is consistent with the energy only when compliance method is that we need t o specify the amount of material to be removed. A better approach would be to minimize the volume of a structure without violating the limits on the maximum allowable stress. A l ot of research has been put into achieving topology optimization subject to stress constraints in the last decade and several methods have been proposed to solve this problem. The chief obstacles faced when tackling stress based topology optimization with the SIMP approach are
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16 ( Bendse and Sigmund 2003 Cheng and Jiang 1992, Cheng and Guo 1997, Rozvany 2001) and t he large number of nonlinear constraints resulting from the localized and nonlinear nature of stress The singularity phenomenon while dealing with stress constraints was first identifie d for truss problems It arises because the feasible design in n dimensional space contains degenerate subspaces of dimension less than n and the optimization algorithms canno t identify optimal designs which are an element of those degenerate subspaces In other words the optimization algorithm is unable to remove material completely to find true optimal designs as stress constraint would be violated in the se regions This can be overcome by relaxing the stress constraints to eliminate degenerate regions relaxation (and its variations) (Cheng and Guo 1997, Guilherme and Fonseca 2007) and qp approach (Bruggi 2008) are such methods which have successfully tackled the issue of singularity. However the i ssue of singularity allowable stress) of material with intermediate densities are modeled as real material with physic al consistency. In this work shape densities have been used, these densities do not represent real material properties and their main purpose is to compute sensitivity of shape changes of the topology in the optimization process. Another difficulty dealin g with stress constraints is the localized nature of stress which results in large number of constraints if local stress constraints are to be imposed strictly. This requires tremendous computational effort and algorithms which handle large constraints eff ectively. M ost of the methods proposed try to impose stress constraints at a local level (generally at the centroid of each element), thereby increasing the number of variables in the dual problem and adding to the complexity
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17 and computational expense. An a lternative method is to use global stress constraints which approximate the local constraints into a single constraint thereby reducing the computational effort required. A summary of different methods used to impose stress constraints are discussed and c ompared in C hapter 2. In this thesis we include a stress term (similar to the global stress constraint) in the objective function, which penalizes the design heavily when there is a stress violation Objectives The general objective of this research work i s to investigate the use of s tress b ased constraints for topology o ptimization using finite element formulation In particular, this research intends to accomplish the following: To implement topology optimization of 2D and 3D structures subject to stress constraints with minimization of mass as the objective. To study the effectiveness of a global stress measure to approximate local stress constraints. Outline The rest of the t hesis is organized as described below: Chapter 2, begins by discussing in brief various methods used to represent shapes in Topology Optimization, namely The Homogenization Method, Solid Isotropic Method with Penalization (SIMP). Following this, the previous work carried out in the field of Topology Optimization is discussed highligh ting the most popular objective functions, techniques to impose stress constraints. Chapter 3, discusses various numerical methods used to solve nonlinear optimization problems, structural optimization problems in particular and summarizes pros and cons of each method.
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18 Chapter 4, begins by highlighting the merits of using stress based topology optimization and then illustrates the relation between mean compliance, von mises stress and how stress constraints can be converted to equivalent compliance constrai nts. Finally the theory behind the objective function, smoothing scheme, constraints, and sensitivity is explained in detail. Chapter 5, demonstrates various 2D, 3D examples and compares them to the results obtained in previous work Chapter 6, Conclusions are drawn from the results obtained. Then the advantages, disadvantages of using stress based constraints are discussed and future scope of work is presented.
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19 CHAPTER 2 TOPOLOGY OPTIMIZATIO N The objective of topology optimization is to find the optimal distribution of material within a given design space. In an optimization problem, t he feasible design space, material properties and loading conditions are given as inputs and the goal is to design black and white structures (where black indicates presence of material and white indicates voids) which are optimal with respect to the objec tive function. Various methods used to achieve black and white designs are discussed in the following section. The most popular objective functions for topology optimization of structures are explained later in this chapter. Topology Optimization Methods Several methods have been proposed to represent topologies. The most popularly used methods are : Homogenization method, Genetic algorithm method, the level set method and Solid Isotropic Material with Penalization (SIMP). The SIMP method is used for the p urpose of this work. The others methods are discussed briefly below: Homogenization Method The idea of homogenization was established by Kohn and Strang (1986) when designing torsion bars. They proposed the use of three types of regions in the design: soli ds, voids and porous. Bendse and Kikuchi (1988) developed this idea and applied it to topology optimization. In this method, t he material i s modeled as being porous by assuming a microstructure with holes. The relation between material properties and poro sity is calculated for few hole sizes and the properties in between are interpolated. The porosity within each element is treated as constant in the finite element model and
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20 the porosity values are used as design variables in the optimization problem. A mi crostructure with rectangular void within a unit cell was used An example is shown in Figure 2 1. Figure 2 1 Unit cell of a microstructure with a rectangular void. In this method, the optimization algorithm tries to increase porosity where the material is underutilized and reduce the porosity where material is highly utilized. Genetic A lgorithm Method Genetic algorithm method has been gaining popularity with advances in high speed computing and parallel computing. In this method, each design is represented as a character string with each character representing a design variable. This is analogous to chromosomes in an individual. For the topology optimization problem, each character of the string could represent the density of an element in the fi nite element model which assumes discrete values 0 or 1. The algorithm starts with randomly generated individuals A fitness criterion is used to select parent individuals and next generation individuals are generated by genetic operations such as crossover, mutation, addition/ deletion or permutation The process is repeated iteratively until optimal designs are
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21 obtained. The main disadvantage of this approach is that for problems such as topology optimization where t he number of design variables is usually high, tremendous computational effort is required. Advantages of using this method are, it always yields a black and white design with no grey areas since the design variables only assume discrete values 0 or 1, glo bal optimum designs are obtained and computation of gradients is not required Level Set Method Level set method (Allaire et al 2004, Burger et al 2004) is similar to boundary variation method. The structure to be optimized is implicitly represented by a moving boundary embedded in a scalar function (the level set function) of a higher dimensionality In this method, each boundary can be split to form multiple boundaries and conversely several boundaries may merge to form a single boundary Wang et al (2 003 ) demonstrated that structural optimization can be described by letting the level set functions dynamically change in time. The dynamic model is expressed as shown below. ( 2 1 ) is the level set function corresponding to a contour with iso value k x(t) is the set of points on the contour boundary represented by A partial differential equation ( 2 2 ) is obtained by differentiating ( 2 1 ) The boundaries of the optimal structure are obtained by solving the PDE. ( 2 2 )
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22 Here is the movement of a point on the contour driven by the objective of the optimization. Solid Isotropic Method W ith Penalization In the SIMP method, the material properties are assumed to be related to the density of the material and an artificial relation between the two is assumed such that intermediate densities are penalized. The relationship between density and material properties is shown below: ( 2 3 ) Here is the density of the material, material and is the penalty. It has been observed that p= 3 or 4 yields good results. Figure 2 2. Variation of Young s modulus with density for p = 1, 3, 4. Figure 2 of p It can be noticed that intermediate densities have a penalty of the strength when
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23 the cost of the material is taken into accou nt. So the optimization algorithm tries to achieve densities closer to 0 1. Most topology optimization methods which use this method assume constant material density within each element of the finite element model and the element density is treated as a design variable (Rozvany et al 1994 Yang and Chuang 1994 ) An alternative to this is to treat density as nodal variable s and interpolate the density within an element In this thesis nodal densities are the design variables. The advantage of using this approach is that smoother topologies are obtained with fewer elements when comp ared to constant element density approach. Objective Functions W ith Constraints Topo logy optimization has been applied to a range of structural problems. The most widely used applications being compliance minimization and compliant mechanisms. The objectiv e functions used to solve these problems and other interesting problems in the field of structures are discussed below: Compliance Minimization Minimization of compliance for a given volume fraction has been the most popular topology optimization problem. In this problem the user specifies the amount of volume to be retained or removed and the algorithm obtains a structure with the least compliance utilizing the material specified. Compliance of the structure is the objective function and the material / vol ume constraint is either treated as an equality constraint (or inequality constraint in some cases). The optimization problem when finite element model is used can be states as: Minimize: ( 2 4 )
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24 Subject to: ( 2 5 ) ( 2 6 ) Here is the density function, F is the applied traction load and {U} is the displacement vector. [D] is the material properties matrix which depends on the density function M 0 is the amount of material to be retained (specified by the user) Compliant Mechanism Compliant mechanism is another popular area to which topology optimization has been applied. Compliant mechanisms are mechanical devices in which deformations produce mo tion as opposed to rigid body mechanisms which attain their motion from the use of hin ges, bearings and sliders. The goal is to make the output port move through a set of N desired points for N given input displacements. The objective function is formulate d as an error function: Minimize: ( 2 7 ) Subject to: ( 2 8 ) ( 2 9 ) Stress Constraints In this section, we will summarize previous work on topology optimization which use stress constraints. The class ic stress constrained problem optimizes the weight of the structure with out violating the stress constraint and satisfying elastic equilibrium.
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25 The optimization problem when using SIMP interpolation to predict intermediate material properties is shown belo w: Minimize: ( 2 10 ) ( 2 11 ) ( 2 12 ) ( 2 13 ) Here is the Von Mises stress, is the maximum allowable stress. has been introduced to account for attenuation of strength for m aterial with intermediate densities. Duysinx and Bendse (1998) state that q should be equal to p to maintain physical consistency and choosing a value less than p will lead to artificial removal of material. They used a variation of the stress constraint relaxation proposed by Cheng and Guo (1997) to solve the issue of singularity and used local stress constraints The modified stress constraints p roposed by Duysinx and Bendse (1998) can be written as: ( 2 14 ) Duysinx and Sigmund (1998) enhanced the stress constraint by multiplying a factor on to ensure the real constraint is imposed for [see equation ( 2 15 ) ] and used a global constraint to replace the local constraints thus saving computational effort. The global constraint is shown in equation ( 2 16 )
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26 ( 2 15 ) ( 2 16 ) Where Large values of q improve the global approximation of the local constraints but cause numerical issues. Computational experiments have shown that q = 4 is a good choice. Pereira et al (2004) used relaxation (stress constraint relaxation method) proposed by Duysinx and Sigmund [see equation ( 2 15 ) ] along with local constraints and concluded that this approach would require 10 times the computational effort when compared with the compliance minimization problem. Gu ilhe rme and Fonseca (2007) used a variant of the relaxation method [see equation ( 2 17 ) ] and a global stress constraint based on p norm. The global constraint and stress constraint relaxation are shown below: ( 2 17 ) ( 2 18 ) Svanberg and W erme (2007) minimized the volume of the structure weighted by the maximum stress and minimized the volume subject to local constraints. They solved the optimization problem using sequential integer programming and claimed that stress
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27 constrained problems can be solve d naturally without using any numerical tricks. A linear interpolation was assumed to compute intermediate material properties. Bruggi (2008) proposed a qp relaxation which is similar to relaxation in terms of results obtained. I n this approach, different penalties are used in the interpolation schemes for stiffness ( p ) and stress ( q ) such that q
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28 concentrations and produces results comparable to local constraints with only a small increase in the computational expense. The regions are defined by sorting the elements based on their stress level in the current design. The regional constraints for m regions are defined as: ( 2 21 ) Where is a normalizing parameter which uses previous iteration information to better approximate the maximum stress. Amstutz and Novotny (2010) minimized a linear combination of the area and the compliance of the structure while imposing the stress constraints via the penalty method. Their method uses topological derivatives as descent direction to find the optim um. The drawback of using this approach is that the user needs to specify weights associated with each term in the objective function. The objective function for this method is shown below: ( 2 22 ) Where are weights attached. The existing literature on stress constraints discussed above are summarized in Table 2 1
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29 Table 2 1 Summary of the literature on topology optimization using stress constraints ( Le et al 2010 ) Author Elasticity Tensor Relax ed stress constraint Probl em statement Optimization Algorithm / Remarks Duysinx and Bendse (1998) (SIMP) relaxation: Minimize volume subject to local stress constraints CONLIN Duysinx and Sigmund (1998) (SIMP) relaxation: Minimize volume subject to global stress constraint MMA (S vanberg 1987) Pereira et al. (2004) (SIMP) relaxation: Minimize volume subject to local stress constraint BOX QUACAN (Friedlander et al. 1994) Guilherme and Fonseca (2007) (SIMP) relaxation: Minimize volume subject to global stress constraint Sequential linear programming (SLP) Svanberg and W erme (2007) Minimize volume subjec t to local stress constraints, Minimize volume weighted by maximum stress Sequential Integer programming Bruggi and Venini (2008) Minimize Compliance, Subject to volume and local stress constraints MMA (Svanberg 1987) Paris et al. (2009) relaxation: Minimize volume subject to local stress constraints Sequential linear programming (SLP) Le et al. (2010) Minimize volume subject to regional normalized stress constraints MMA (Svanberg 1987)
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30 Table 2 1. Continued Amstutz and Novotny (2010) Does not use SIMP Minimize linear combination of mass and compliance with stress constraints imposed via penalty method Topology derivatives are used for descent direction Weights need to be chosen for compliance and stress penalty
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31 Others Other interesting problems that have been solved using topology optimization are free vibration, forced vibration problems in dynamics and buckling problems in structures ( Bendse and Sigmund 2003) For free vibration problems, maximization of the fundamental frequency is the most commonly used objective function. In forced vibration problems, the dynamic response could be maximized or minimized depending on the application. Maximization of response is used in applications such as sensors and minimization of response could be used when design aircrafts wh ere minimum vibrations are desired. Another important problem in structural optimization is the maximization of the fundamental buckling load.
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32 CHAPTER 3 NUMERICAL METHODS US ED TO SOLVE THE OPTI MIZATION PROBLEM It is important to understand the numerical methods used to solve the optimization problem. T opology optimization problem s have a large number of design variables and it is a good idea to use gradient based algorithms to solve the optimization problem Using topology optimization approaches such as the SIMP allow us to use gradient based methods. Since most of the objective fu nctions or constraints used for structural optimization problems are non linear functions in the design variables, sequential programming techniques are generally used to solve them. In this chapter we explain the basic sequential programming algorithms li ke the sequential linear programming (SLP), sequential quadratic programming (SQP). We also discuss algorithms such as Method of moving asymptotes (MMA) (Svanberg 1987) which has been used widely by the structural optimization community and the moving barr ier method (MBSLP) (Kumar 200) which has been used in this thesis. Sequential Linear Programming Sequential linear programming (SLP) ( Haftka and Gurdal 1990, Arora 2004 ) is a technique used to solve optimization problems with non linear objective functions and/or constraints. SLP algorithms use the first order Taylor series expansion to linearize the objective function and constraints about an initial guess point and then solve the linearized problem using standard linear programming techniques like the simplex method. The non linear problem is then linearized about the solution obtained and the process is repeated until convergence is reached. Let the optimization problem be defined as follows: Minimize: ( 3 1 )
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33 Subject to: ( 3 2 ) ( 3 3 ) H ere are non linear functions. Let be the estimate of the design variable at the k th iteration and be the change in design, then the above nonlinear optimization problem can be linearized about Minimize: ( 3 4 ) Subject to: ( 3 5 ) ( 3 6 ) It must be noted that the linear problem defined above may not have a bounded solution or the changes in the design variable ( x ) may be too large thus invalidating the linear approximation. To deal with this w e must impose limits on the design variable. These constraints are called the move limits, expressed as: ( 3 7 ) Where are the maximum allowed decrease and increase to the design variable s These move limits play an important role in the convergence of the SLP algorithm and need to be chosen carefully. The subproblem can be written as: Minimize: ( 3 8 ) Subject to: ( 3 9 )
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34 ( 3 10 ) ( 3 11 ) H ere ( 3 12 ) The convergence criteria for the problem can be stated as: ( 3 13 ) are small numbers specified by the user. The SLP algorithm can be summarized as follows: i. Estimate a starting design Set k=0 select two small numbers to define the convergence criteria and also select the move limits ii. Evaluate the objective function, constraints, the gradients of objective function and constraints at the current design and generate the linear subproblem. iii. Solve the subproblem for using standard LP sol ving algorithms like the simplex. iv. Check for convergence ( 3 13 ) If criteria is met then stop o therwise continue v. Update the design as k=k+1 Go back to step 2. Sequential Quadratic Programming Sequential quadratic programming ( Haftka and Gurdal 1990, Arora 2004 ) designed to overcome some of the limitations of SLP. There are many variations of the SQP, but most SQP methods have a subpro blem that has a quadratic objective function with linearized constraints. The basic functioning of the SQP is similar to SLP except that the objective function approximation is quadratic and the method used to solve the subproblem is different. The quadrat ic approximation of the objective function can be
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35 ( 3 14 ) H ere is the h essian of the objective function. The Hessian of the objective function is obtained using BFGS update. This method starts of by assuming Identity matrix for the hessian and then iteratively solves for the Hessian. The constraints of the SQP problem are linearized similar to SLP: ( 3 15 ) ( 3 16 ) The subproblem can be summar ized as: Minimize: ( 3 17 ) Subject to: ( 3 18 ) ( 3 19 ) Modifie d simplex method is used to solve the QP subproblem. Alternatively KKT conditions can be used when the number of design variables is small. Method O f Moving Asymptotes This method is based on a special type of convex approximation which works very well for structural optimization in particular and other nonlinear programming problems in general (Svanberg 1987) The general description of the method is presented below. Let the optimization problem be defined as: Minimize: ( 3 20 ) Subject to: ( 3 21 )
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36 ( 3 22 ) Where is the vector of design variables. is the nonlinear objective function. are m nonlinear inequality constraints. are the side constraints on the design variable This method solves the given problem iteratively as described below: i. Choose an initial point for the 0 th iteration (k=0) ii. At the given iteration point calculate iii. Generate a sub problem by using explicit approximating functions based on the above calculations. iv. Generate a dual problem for the sub problem Solve the dual problem and thereby the sub problem v. Update the iteration point Che ck for c onvergence else go back to ( ii ) The sub problem for the k th iteration is defined as: : minimize ( 3 23 ) Subject to: ( 3 24 ) ( 3 25 ) H ere
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37 ( 3 26 ) T he parameters are move limits chosen such that (to avoid division by 0) ( 3 27 ) Exampl e: The selection of the side constraints L, U and the method used to solve the subproblem are stated in Appendix A The above method works well only when the starting point is chosen such that the initial sub problem is feasible, but in most cases it may not be practical to choose such a point and so the concept of artificial variables is used. This issue can be solved by introducing extra variables in the objective function (Svanberg 1987) Moving B arrier Method Moving barrier method (Kumar 2000) is yet another sequential linear programming method in which the subproblem is generated by linearizing the objective function and imposing the move limits by using logarithmic barriers. The move limits are flexible and be modified at e very iteration. The reason behind using logarithmic barriers to impose move is limits is that the hessian is a diagonal matrix and it would enable us to use
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38 the exact sol ution of the subproblem and hence reduces the computational effort required. It has also been reported that it works well with ill conditioned problems. The moving barrier method has been used for optimization problems with linear equality constraints and side constraints. However this method can be extended to work with linear inequality constraints by including slack variables. Let the optimization problem be defined as follows: Minimize: ( 3 28 ) Subject to: ( 3 29 ) ( 3 30 ) is a nonlinear objective function. are the side constraints for our design variables. For the iteration, the linearized subproblem with logarithmic barrier can be written as: Minimize: ( 3 31 ) Subject to: ( 3 32 ) Where ( 3 33 ) And are flexible move limits at the k th iteration such that ( 3 34 ) The method suggested for select ing the move limits at the k th iteration and the method used to solve the sub problem are stated in Appendix B After the descent direction has been obtained by solving the subproblem, the variables are update as follows:
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39 ( 3 35 ) The step size is determined such that the new design variables obtained remain feasible. The step siz e is calculated as shown below: ( 3 36 ) After obtaining the next iteration point, the procedure is repeated till convergence like other sequential programs. To make sure that the starting point is feasible, the initial point is projected onto the plane t he projected point is obtained as follows: ( 3 37 ) Comparison SLP, SQP methods work we ll in general for non linear problems but it has been reported that MMA works well for structural optimization in particular and converges faster than SLP algorithms. For this reason, MMA has been extensively used in the structural optimization community e specially for topology optimization problems with non linear constraints. MBM has been used by kumar et al. ( 2011 ) for topology optimization problems and it has been reported that it out performs the MMA because this method does not require the sub problem to be solved completely. However, the advantage of MMA over MBM is that, it can handle non linear constraints. In this work, MBM method has been used and the constraints have been included in the objective function using the penalty method.
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40 CHAPTER 4 STRESS BASED TOPOLOG Y OPTIMIZATION Topology optimization via compliance minimization has been the dominant method used to design optimal structures. This method yields reasonable results; however there are several disadvantages of using compliance as the criteria to design structures. Stress is the most important criteria when designing a structure but the user does not have any idea about the stresses in the final design obtained using the compliance minimization method. So the user needs to perform a shape optimization with stress constraints after the topology has been obtained and the f inal design is generally not consistent, i.e. different designs might be obtained for different volume fractions used during topology optimization via compliance minimization. For the reasons mentioned above, it is beneficial to use minimization of mass as our objective function with constraints on the stress in the structure. This approach solves most of the issues stated above. However there are several challenges when using stress constraints namely the issue of singularity and the localized nature of st ress leading to large number of stress constraints. For more details on these issues refer C hapter 1 Solid Isotropic Material with Penalization (SIMP) approach (refer C hapter 2 ) with nodal densities as design variables has been used in this work. The depe ndence of ( 4 1 )
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41 Objective F unction Consider a domain in R 2 or R 3 in which an optimum design is to be found. The domain is the feasible region within which the structure must fit and the applied set of loads and boundar y conditions applied along the boundaries of this feasible region. The objective function should be defined to minimize the weight of the structure while not violating the maximum stress limit (often dictated by the material used). Ideally the problem is d efined as : Minimize ( 4 2 ) Subject to ( 4 3 ) ( 4 4 ) Here is the Von Mises stress, is the maximum allowable stress. Singularity been studied ext ensively fo r stress constrained problems ( Bendse and Sigmund 2003, Cheng and Jiang 1992, Cheng and Guo 1997, Rozvany 2001) The problem arises when optimal topologies belong to degenerate subspaces of the feasible domain i.e. some element s in the feasible region have zero density. In such cases gradient based algorithms fail to reach the true optimum. Table 2 1 shows different relaxation techniques used in the literature. modulus and yield stress (maximum allowable stress) of material with intermediate densities are
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42 modeled as real material with physical consistency. In this work we would like to design black and white structures using SIMP and we do not make an attempt to model materials with intermediate densities accurately. The main purpose of using s hape densities is to compute sensitivity of shape changes of the topology in the optimization process. We assume that the maximum allowable stress is a constant since the f inal topology does not contain material with intermediate density and so we do not encounter the issue of singularity. Global Stress M easure Localized nature of stress is another important issue that needs to be tackled to use stress constraints. In a cont inuum setting, the stress constraint should be considered at every material point I n a discrete setting, the number of points where the stress constraint is imposed is finite A popular approach has been to impose stress constraints at the centroid of eac h element in the FE model. However for topology optimization problems, the number of such constraints is still high and tremendous computational effort is required to solve the optimization problem. An alternative to replace the local stress constraints is to use a global stress constraint which can effectively capture the local stress behavior. The local stress constraints can be effectively stated as a single constraint as ( 4 5 ) Since the maximum function is not differentiable, we need to use a function to replicate similar behavior. P norm and Kreisselmeier Steinhauser (KS) function s (Kreisselmeier and Steinhauser 1979) have been used in previous works to construct global constraints. Please refer to Table 2 1 for further details
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43 In this work, we included a global stress measure which penalizes heavily local stress violations in the objective function and used the Moving Barrier Method (MBM) to solve the optimization. The modified optimization problem is stated as: Minimize ( 4 6 ) Subject to: ( 4 7 ) ( 4 8 ) The global stress measure needs to be chosen such that it assumes a value between 0 and 1 when the stress constraint is not violated and a very high value when the stress constraint is violated. Thus the optim ization algorithm would try to limit the stress at all integration points to the maximum allowable stress in the structure. On the other hand if too much material is present and the stress es are within the limiting value the optimization algorithm would tr y to reduce the mass of the structure without violating the stress limit. The b ehavior expected from function is summarized below: ( 4 9 ) We have tried us ing the following functions for : ( 4 10 ) ( 4 11 ) The plots of the above functions are shown in Figure 4 1. It can be noticed that these functions assume a value between 0 and 1 when x is less than 1 and the function
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44 value rises steeply thereafte r. It was observed that values between 2 and 5 yield good results. As the value of m increases the numerical instabilities increase and hence very high values of m are not used. Figure 4 1 Plot of stress functions The optimization algorithm requires us to compute the gradient of the objective function and hence the g radient of the Von Mises stress. Computing the gradient of Von Mises with respect to the nodal density is a complicated and difficult task. To ease our computations we re place the Von Mises stress with equivalent compliance term which is easier to work with. We make use of the below relation: ( 4 12 ) H ere is the compliance. In the above relation the compliance term is equal to the Von Mises stress when From the above relation it can be said that constraining the compliance term also constrains the Von Mises stress.
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45 Relation Between M ean C ompliance A nd Von Mises S tress ( 4 13 ) H ere The above relation can be derived as follows: ( 4 14 ) ( 4 15 ) Consider ( 4 16 ) ( 4 17 )
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46 ( 4 18 ) Modified O ptimization P roblem The modified optimization problem can be stated as shown below : Minimize ( 4 19 ) Subject to: ( 4 20 ) Here or m>>1 Smoothing S cheme Numerical issues such as m are encountered when solving the topology optimization problem. It is known that the discretized topology optimization problem does not have a unique solution. It has been observed that the efficiency of the structure generally increases with the introduction of more holes for the same volume fraction. This results in different solutions when the mesh discretization is changed. when using constant density elements with SIMP approach. This checkerboard problem arises due to finite element formulation which tends to overestimate the stiffness of checkerboards ( Bendse and Sigmund 2003) An overview of different techniques used to overcome these issues are presented in Sigmund and Peters so n (1998), Bendse and Sigmund (2003). In this thesis, we use the smoothing scheme proposed by Kumar and Parthasarathy ( 2011 ) A smoothing
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47 term is defined which is obtained by integrating the square of the gradient of density over the entire domain. This smoothing term multiplied by a weighting factor added to the objective function. Minimization of the smoothing term eliminates the numerical instabilities discussed above, when proper weights are se lected. The smoothing term is shown below : ( 4 21 ) It has been observed f rom several test cases that smoothing works well when the smoothing term is 10 20% of the objective function value. Hence the weight should be chosen accordingly. Sensitivity A nalysis Sensitivity analysis in optimization can be defined as evaluating the gr adient of the objective function and the constraints w ith r espect t o the design variables. This information is required by all gradient based optimization algorithm to move towards the optimal solution. The gradient of the objective function can be calcula ted as the sum of gradients of individual terms in the objective function as shown below: ( 4 22 ) Where The gradient of the mass term is computed as ( 4 23 )
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48 Where Nj are the shape functions and = Kronecker delta. The gradients of both the stress terms are shown below Stress function 1 ( ) : ( 4 24 ) Stress function 2 ( ) : ( 4 25 ) The gradient of the smoothing term is evaluated as shown below: ( 4 26 )
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49 CHAPTER 5 RESULTS Bar Our first example is a bar subjected to axial force. The feasible domain is a rectangle with dimensions 0.1m by 0.2m (see Figure 5 1 ). An axial force (P) of 6 000 kN is applied at the right end and the left end is fixed. The force P is distributed uniformly over a length of 0.03m to avoid stress concentration. The modulus of elasticity is 200 5 and unit thickness is assumed A topology with minimum mass is desired subject to a maximum stress of 200 MPa. This problem is similar to th e bar example demonstrated in Amstutz and Novotny (2009). Figure 5 1 Plane stress model of the design domain for axial loading problem The topologies obtained using different configurations are shown in Figure 5 2 The combination of variables used for each result is shown in Table 5 1. Table 5 1 Configuration details used to obtain bar like structures Case Stress function Mesh Type Mesh Density Stress penalty Smoothing weight fraction A 2 Quad4N 40 x 20 3 0.07 B 1 Quad4N 40 x 20 4 0.15 C 2 Quad4N 100 x 50 3 0.10 D 2 BSpline9N 80 x 40 3 0.10
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50 A B C D Figure 5 2 T op ology optimization results for b ar problem A B C D Figure 5 3 Stress distribution of th e optimal topologies for the b ar problem The theoretical optimum for this problem is a bar with uniform width of 0.03m. As expected the optimal topology resembles a bar with more or less uniform width
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51 throughout similar to the topology obtained by Amstutz and Novotny (2009). T he stress distributi on in the bar is also uniform and within the stress limit of 200MPa. The resulting topologies are slightly under stressed due to smoothing and the manner in which boundaries are extracted It was also observed that higher smoothing is required with stress function 1. Michell Type Structure Michell structure is a popular example used to demonstrate topology optimization. A rectangular domain with a circular hole (see Figure 5 4 ) is subjected to shear force of 10000 kN at the right end and the circular hole i s fixed. The modulus of elasticity is 200 desired subject to a maximum stress of 20 0 MPa. Figure 5 4 Plane stress model of the design domain for Mitchell structure prob lem The optimal topologies obtained using different configurations are shown in Figure 5 5 The combination of variables used for each result is shown in Table 5 2. Convergence plots for case B case D are shown in Figure 5 7 Objective function value is p lotted against the iteration number. It was noticed that the objective function
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52 value stabilizes after 30 40 iterations and further reduces by a small value before it converges. Table 5 2 Configuration details used to obtain Mitchell truss type structur es Case Stress function Mesh Type Mesh Density Stress penalty Smoothing weight fraction A 2 Quad4N 75 x 50 2 .5 0.05 B 2 Quad4N 75 x 50 3 0.05 C 2 BSpline9N 60 x 40 2.5 0.05 D 1 Quad4N 75 x 50 2.5 0.20 A B C D Figure 5 5 T opology optimization results for Mitchell truss type structure
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53 A B C D Figure 5 6 Stress distribution of the optimal Michell Structures (i) (ii) Figure 5 7 Convergence plots i) Case B. ii) Case D. An analytical optimal design for this problem was obtained by Michell (1904). The final topologies obtained resemble Michell structures and are similar t o topology designs
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54 by minim um compliance method ( Kumar 2000, Bends e and Sigmund 2003) The solutions presented here have fewer truss like members due to smoothing. The structures obtained are close to being uniformly stressed and are within the design stress limit of 200 MPa T he shapes obtained using both th e stress functions are similar and a s we increase the stress penalty the optimization algorithm tends to penalize grey regions which are under stressed and creates more holes. It can also be noticed that the desig ns are under stressed by 20 25 % this is because of the manner in which boundaries have been extracted and the conse rvative Von Mises approximation. Case C shows the structure obtained using B Spline elements. B Spline elements inherently induce smoothing effect into the designs, resulting in larger grey regions (intermediate densities) and hence designs which are more conservative. Bracket This example has a rectangular domain of dimension 0.2m x 0.3m. The left end has been fixed and a shear force (P) of magnitude 1000 kN is applied at the free end. ickness is 0.25 units. A topology with minimum mass is desired subject to a maximum stress of 100 MPa. The optimal topologies obtained are shown in Figure 5 9 and the stress distrib utions are shown in Figure 5 10 The combination of variables used for each result is shown in Table 5 3.
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55 Figure 5 8 Plane stress model of a bracket Table 5 3 Configuration details used to obtain optimum bracket shapes Case Stress function Mesh Type Mesh Density Stress penalty Smoothing weight fraction A 2 Quad4N 40 x 60 2.5 0.10 B 2 Quad4N 40 x 60 4 0.10 C 1 Quad4N 40 x 60 4.5 0.20 A B C Figure 5 9 Topology optimization results for bracket for different configurations
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56 A B C Figure 5 10 Stress distribution of the optimal topologies for the beam problem It can be observed that the topologies obtained using both the stress function s are similar but the stresses at the fixed ends exceed the maximum allowable stress limit of 100MPa. This is because the problem does not have a feasible solution. Howev er, a majority of the region is within the stress limit and has uniform distribution of stress. Unconstrained Bracket This example is the same as the previous example except that the feasible domain has been enlarged along the direction of loading. The dim ensions for this problem have been set to 0.2m by 0.6m. Table 5 4 shows the configuration settings; Figure 5 12 and Figure 5 13 show the optimal topologies and the stress distribution respectively.
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57 Figure 5 11 Plane stress model of an unconstrained bracket Table 5 4 Configuration details used to obtain optimum shapes for unconstrained bracket Case Stress function Mesh Type Mesh Density Stress penalty Smoothing weight fraction A 2 Quad4N 30 x 90 3 0.10 B 2 Quad4N 30 x 90 3.5 0.10 C 1 Quad4N 30 x 90 3 0.20 D 2 BSpline9N 20 x 60 3 0.10
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58 A B C D Figure 5 12 Topology optimization results for unconstrained bracket for different configurations A B C D Figure 5 13 Stress distributions for the unconstrained bracket problem In this problem, the feasible domain has been modified such that the optimal solution is not bounded. The optimal shape obtained after optimization is very close to the shapes obtained by Duysinx and Bendse (1998), Allaire et al. (2004), Bruggi and Duysin x (2012). Th e stresses in the structure for cases B C are mostly in the range 85 90 MPa with maximum stress close to the allowable stress limit of 100MPa. The results are conservative by 10 15% due to the manner in which the boundar ies are defined, the Von Mises approximation and smoothing It can be noticed that higher penalty leads to
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59 designs closer to t he optimal solution. In case D B S pline elements have been used and the final structure obtained is more conservative due to the smoothin g effect produced by B Spline elements. Bridge In this example, we optimize the topology of a bridge. The feasible domain is a rectangle of size 0.1m x 0.06m. The left bottom and right bottom edges are fixed. Force P of magnitude 6 00 kN is applied over a l ength of 0.35m as shown in F igure 5 14 The topology with minimum mass is desired subject to a maximum stress of 200 MPa. The thickness is 0.5 units. Figure 5 14 Plane stress model of the design domain for the bridge problem Table 5 5 Config uration details used to obtain b ridge type structures Case Stress function Mesh Type Mesh Density Stress penalty Smoothing weight fraction A 2 Quad4N 90 x 15 2.5 0.20 B 2 Quad4N 90 x 15 3 0.25 C 1 Quad4N 90 x 15 3 0.25 The optimal topologies obtained are shown in Figure 5 15 and the stress distributions are shown in Figure 5 16. The combination of variables used for each result is shown in Table 5 5.
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60 A B C Figure 5 15 T op ology optimization results for b ridge problem A B C Figure 5 16 Stress distribution of the optimal topologies for the b ridge problem
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61 The optimized topologies resemble bridge like structures. It can be noticed that a small region (in red) of the structure has stresses higher that the specified value. This is because the large penalty on the small region was negated by large regions (in blue) which are under stressed The stress violations coul d have been avoided by moving the truss like members closer to the center or by adding more truss like members at the center, but this leads to a significant increase in the mass of the structure. Using a very large penalty could force the algorithm to add material but lead to numerical instabilities. L Shaped Structure The feasible domain for this example is L shaped with dimensio ns as shown in F igure 5 17 The top edge highlighted in red is fixed and a shear load (P) of 10 00 kN is applied on the right edge. The modulus of elasticity of the material used is 20 0 GPa s ratio is 0.5 and unit thickness is assumed. A topology with minimum mass is desired and the maximum allowable stress in the material is 2 5 0 MPa. Figure 5 17 Plane stress model of a loaded L shaped structure Different configurations used are shown in Table 5 6, the optimal topologies and their stress distributions are shown in Figure 5 18 Figure 5 19 respectively.
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62 Table 5 6 Configuration details used to get optimal shapes for L shaped structure Case Stress function Mesh Type Mesh Density Stress penalty Smoothing weight fraction A 2 Quad4N 50 x 50 2.5 0.20 B 1 Quad4N 50 x 50 4 0.30 C 2 BSpline9N 35 x 35 2.5 0.10 A B C Figure 5 18 Topology optimization results for L shaped structure for different configurations A B C Figure 5 19 Stress distribution of the optimal L shaped structures The optimal topologies obtained for the problem are similar to those obtained by minimization of compliance. The optimization algorithm used could not get rid of the stress concentration at the reentrant corner This could be because the manner in which
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63 we tac kle the stress constraints is similar to the global approach and hence the higher stress at the reentrant corner was compensated by under stressed regions elsewhere in the structure. The optimal topologies obtained by Optistruct (Altair Engineering 2007), Paris et al. (2009) using global const raints also have stress concentration at the reentrant corner Pereira et al. (2004), Svanberg and Werme (2007) demonstrated that topologies without stress concentrations could be obtained by using local stress constra ints. Le et al. (2010) used regional stress constraints to achieve results similar to local stress constraints. Overall it has been observed that methods using global approaches do not remove stress concentrations if they are confined to very small regions 3 D Bracket This example has a rectangular domain of dimension 0.2m x 0.6m x 0.05m. The left end has been fixed and a shear force (P) of magnitude 100 kN is applied at the free end. The modulus of elasticity is 200 GPa minimum mass is desired subject to a maximum stress of 2 00 MPa. Figure 5 20 Plane stress model of an 3 D bracket problem
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64 Figure 5 21 Topology optimization results for the 3 D bracket problem A finite element mesh (Hexa8N) of size 15 x 45 x 4 was used and the power of density was set to 4. Stress function 1 ( ) with a penalty of 3 (m=3) was used to obtain the optimal topology. Smoothing fraction was set to 15%. The resulting opt imal topology is shown in Figure 5 21 The final topology resembles the results obtained for 2D bracket example. In the final design the stress term was observed to be less than the mass term which is an indication that most regions in the structure are un iformly stressed and within the maximum allowable stress. Smoother topologies could be obtained by increasing the mesh density. 3 D Example II This example has a rec tangular domain of dimension 1m x 1m x 1 m. The corners on the bottom face are fixed and a f orce (P) of magnitude 1 2 00 kN is is distributed into 4point loads and applied on the top face as shown in Figure 5 22 The modulus of elasticity is A topology with minimum mass is desired subject to a maximum stress of 1 00 MPa
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65 Figure 5 22 Plane stress model of an 3 D example II Figure 5 23 Topology optimization results for the 3 D example II A finite element mesh (Hexa8N) of size 16 x 16 x 16 was used and the power of density was set to 4. Stress function 2 ( ) with a penalty of 2.5 (m=2.5) was used to obtain the optimal topology. Smoothing fraction was set to 20%. The resulting optimal topology is shown in Figure 5 23
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66 The final topology obtained resembles a stool (seat) and the stress term was less than the mass term indicating that the stresses are within the desired stress limit.
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67 CHAPTER 6 CONCLUSIONS Summary An effective and simple method to tackle stress constraints for 2 D and 3 D problems has been presented and demonstrated to work reasonably well for problems without stress concentrations. The proposed method is able to estimate the maximum amount of material to be re moved such that stresses are within the specified limit and thus is more useful to the designer than the compliance minimization method. In the proposed method, Von Mises stress in the constraint was replaced with a compliance term ( which is easier to comp ute ) by expl oiting a relation between them and t hus also eliminating the evaluation of gradient of the Von Mises stress with respect to the nodal densities which is complex and cumbersome. The stress constraints were imposed indirectly by introducing a str ess term in the objective function. The stress term is a global measure of the stresses in the design and it penalizes the objective function by assuming large values when stress violation occurs. Finally, Moving barrier method, a low cost algorithm was us ed to solve the modified optimization problem. Discussions The method proposed in this thesis works well for problems without local stress concentrations and this has been demonstrated in the examp les such as b ar problem, Michell type structure, bridge pro blem and unconstrained bracket problem For problems without a feasible solution such as the bracket problem, the method provides a design which satisfies the stress constraints in most regions and which has the least weight. In problems such as the L shap ed structure with local stress concentrations the current method is unable to get rid of the stress concentration. Ideally we would want
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68 material to be removed around the edges with stress concentration. There could be several reasons why this does not hap pen with the current method. A possible explanation is that our approach of imposing penalty is similar to global stress constraint and the penalty imposed for stress violation is not large enough and hence the local stress violation in a very small region might not have sufficient impact at the global level. On the other hand, a higher penalty leads to numerical inaccuracies while computing the objective function and the gradients Numerical issues are discussed in more detail in scope for future work. Sco pe O f Future Work Topology optimization based on stress constraints has been implemented and works well in most cases However there are a few areas with scope for further improvement. In the present method, t he smoothing weight is an input given by the us er and is often chosen by trial and error method. In some cases a large smoothing weight might be desired in the final design but choosing such a value could prove counterproductive and hinder the progress of the optimization during the first few iteration s. It would be ideal if the algorithm is able to figure out the amount of smoothing required and start with a very small value and gradually increase as the algorithm converges to the optimal solution. Another area with scope for improvement is the comput ation of stresses and stress violations in the intermediate designs. We define designs by identifying boundaries which are iso contour lines or surfaces of density functions with a threshold value (0.5). But during the optimization process we do not extrac t intermediate designs to compute stress es and stress violations. Instead we calculate the stresses for the entire feasible domain with intermediate density values for ease of computation of
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69 gradients. Though it is not clear if the final designs would chan ge if exact stresses are computed for intermediate designs, it may be worth trying. Finally, topology optimization with stress constraints could be can be used for other applications such as the design of compliant mechanisms
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70 APPENDIX A METHOD OF MOVING A SYMPTOTES Selecting T he M ove L imits for the sub problem are defined as follow s For For If are opposite, i.e. If are of the same sign i.e. For some S olving T he S ubproblem t he sub problem is a convex separable problem, a dual method is used to arrive at the solution.
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71 The iteration index k is dropped in this section and we use instead o f instead of and instead of The sub problem can be written as follows: : minimize Subject to: The Lagrangian function for the sub problem or the dual objective function is: Where The above function is the dual objective function in the dual variable y (Lagrangian multipliers). An explicit relation between can be obtained as shown below:
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72 Now replace with the above expression and maximize the dual objective function to obtain and thereby The derivative of W(y) w.r.t dual variables y, can be written as:
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73 APPENDIX B MOVING BARRIER METHO D S electing T he M ove L imits If if Where if otherwise S olving T he S ubproblem The dual problem of the linear problem can be stated as: Where The optimality criteria for the subproblem can be derived using KKT conditions as follows:
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74 The above set of equations are nonlinear equations which can be solved using
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75 LIST OF REFERENCES Allaire G, Jouve F, Maillot H (2004) Topology optimization for minimum stress design with the homogeniza tion method. Struct Multidisc Optim 28(2 3):87 98 Allaire G, Jouve F, Toader A (2004) Structural Optimization using Sensitivity Analysis and a Level set Method. Journal of Computational Physics. 194(1): p. 363 393 Amstutz S, Novotny AA (2010) Topological optimization of structures subject to VonMises stress constraints, Struct Multidisc Optim 41:407 420 Arora JS (2004) Introduction to optimum design Second edition, Elsevier academic press. Bendse MP (1989) Optimal shape design as a material distribution problem. Struct Multidisc Optim 1(4):193 202 Bends Bendse and CA Mota Soares (eds.) p p. 159 205, Kluwer Academic Publishers, Netherlands Bendse MP, Kikuchi N ( 1988 ) Generating optimal topologies in structural design using a homogenization method. Computer methods in applied mechanics and engineering; 71: 197 224. Bendse MP, Sigmund O (20 03) Topology optimization: theory, methods and applicatio ns. Springer Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidisc Optim 36(2):125 141 Bruggi, M. and Duysinx, P. (2012 ), Topology optimization for minimum weight and stress constraints. Struct Multidisc Optim Online first Bruggi, M. and Venini, P. (2008), A mixed FEM approach to stress constrained topology optimization. International Journal for Numerical Methods in Engineering, 73 : 1693 1714. Burger M, Hackl B, Ring W ( 2004 ) Incorporating Topological Derivatives into Level Set Methods. Journal of Computational Physics 194(1): p. 344 362 Chapman CD, Saitou K, Jakiela MJ ( 1994 ) Genetic algorithm as an approach to configuration and t opology design. Journal of mechanical design; 116: 1005 1012. Cheng GD, Guo X (1997) Epsilon relaxed approach in structural topology optimization. Struct Multidisc Optim 13(4):258 266
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76 Cheng GD, Jiang Z (1992) Study on topology optimization with stress con straints. Eng Optim 20(2):129 148 Duysinx P, Bendse M (1998) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43(2):1453 1478 Duysinx P, Sigmund O (1998) New development in handling stress constraints in optimal material distribution. In: Proc. 7 th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization. A collection of technical papers (held in St. Louis, Missouri), vol 3, pp 1501 1509 Fleury C, Braibant V (1986) Structural optimization: a new dual method using mixed variables. Int J Numer Methods Eng 23(3):409 428 Friedlander A, Martnez JM, Santos SA (1994) A new trust region algorithm for bound constrained minimization. Appl Math Optim 30(3):235 266 Guilherme CEM, Fonseca JSO (2007) Topology optimization of continuum structures with epsilon relaxed stress constraints. In: Alves M, da Costa Mattos HS (eds) Solid mechanics in Brazil, vol 1. ABCM, Rio de Janeiro, pp 239 250 Haftka RT, Gurdal Z (199 0 ) Elements of structural opt imization (solid mechanics and its applications) Kluwer Academic Publishers Kaveh A, Kalatjari V. ( 2003 ) Topology optimization of trusses using genetic algorithm, force method and graph theory. International journal for numerical methods in engineering ; 58: 771 791 Kirsch U (1990) On singular topologies in optimum structural design. Struct Multidisc Optim 2(3):133 142 Kohn RV, Strang G ( 1986 ) Optimal design and relaxation of variational problems. Communications in pure applied mathematics; 39: 113 137 (Pa rt I), 139 182 (Part II), 333 350 (Part III). Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index, Proceedings of IFAC Symposium on Computer Aided Design of Control Systems, Zurich, Switzerland pp 113 1 17 Kumar AV (20 0 0) A Sequential optimization algorithm using logarithmic barriers: Application to structural optimization, Journal of mechanical design,122(3):271 277 Kumar AV, Gossard DC ( 1992 ) Geometric modeling for shape and topology optimization. In: P roc. of Geometric mo deling for product realization Kumar AV, Parthasarathy A (2011) Topology optimization using B spline finite elements, Struct Multidisc Optim (2011) 44:471 481
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77 Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress based topology optimization for continua. Struct Multidisc Optim 41:605 620 Michell AGM (1904), The limit of economy of material in frame structures, Philos. Mag., 8, No. 4. Ohsaki M ( 1995 ) Genetic algorithm for topology optimization of trusses. Computers and structures ; 57(2): 219 225 Paris J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints, Struct Multidisc Optim 39:419 437 Pedersen P (1998) Some general optimal design results using an isotropic, power law nonlinear elasticity, Structural Optimization 15: 73 80 Pereira JT, Fancello EA, Barcello CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidisc Optim 26(1 2):50 66 Rozvany GIN (2001 ) On design dependent constraints and singular topologies. Struct Multidisc Optim 21(2):164 172 Rozvany GIN, Zhou M, Sigmund O (1994) Topology optimization in structural design. In: Adeli H, editors. Advances in design optimization. Chapman and hall Sigmund O, Peters s on J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh dependencies and local minima. Struct Multidisc Optim 16:68 75 Sui Y K, Ye H L, Peng X R (2006) Topological optimization of continuum structure with global stress constraints based on ICM method, Computational Methods, First International conference on Computational Methods 1003 1014 Svanberg K (1987) The method of moving asymptotes A new method for structural optimization International methods for numerical methods in engineering, 24: 359 373 Svanberg K, Werme M (2007) Sequential integer programming methods for stress constrained topology optimization, Struct Multidisc Optim 34(4):277 299 nt generation of pareto optimal topologies for Engineering, vol. 87, pp. 1207 1228 Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Computer meth ods in applied mechanics and engineering; 192: 227 246
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78 Wang S Y, Tai K. (2005) Structural topology design optimization using genetic algorithms with a bit array representation. Computer methods in applied mechanics and engineering; 194: 3749 3770 Yang R J, Chuang C H (1994) Optimal topology design using linear progra mming. Computers and structures ; 52(2): 265 275.
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79 BIOGRAPHICAL SKETCH Vijay Krishna Yalamanchili is a graduate student at University of Florida, Gainesville. He g raduate d in Spring 2012, wi echanical e ngineering. He completed his b University College of Engineering, Osmania University in 20 08. His areas of interests are c omputational mechanics, s tructural o ptimization and solid mechanics.