TOPOLOGY OPTIMIZATION OF STRUCTURES USING GLOBAL STRESS MEASURE

Transcription

1 OPOLOGY OPIMIZAION OF SRUCURES USING GLOBAL SRESS MEASURE By VIJAY KRISHNA YALAMANCHILI A HESIS PRESENED O HE GRADUAE SCHOOL OF HE UNIVERSIY OF FLORIDA IN PARIAL FULFILLMEN OF HE REQUIREMENS FOR HE DEGREE OF MASER OF SCIENCE UNIVERSIY OF FLORIDA

2 2012 Viay Krishna Yalamanchili 2

3 o my parents, Siva Rama Krishna and Geeta Kumary, my brother Pavan Kumar, my advisors, Dr. Ashok V. Kumar, Dr. Raphael. Haftka, Dr. Nam-Ho Kim, and other close family members, relatives and friends 3

4 ACKNOWLEDGMENS I would like express my gratitude and deepest appreciation to Dr. Ashok V. Kumar, Chairman and advisor for my thesis 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. Raphael. 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. 4

5 ABLE OF CONENS page ACKNOWLEDGMENS... 4 LIS OF ABBREVIAIONS ABSRAC CHAPER 1 INRODUCION Overview Obectives Outline OPOLOGY OPIMIZAION opology Optimization Methods Homogenization Method Genetic Algorithm Method Level Set Method Solid Isotropic Method With Penalization Obective Functions With Constraints Compliance Minimization Compliant Mechanism Stress Constraints Others NUMERICAL MEHODS USED O SOLVE HE OPIMIZAION PROBLEM Sequential Linear Programming Sequential Quadratic Programming Method Of Moving Asymptotes Moving Barrier Method Comparison SRESS BASED OPOLOGY OPIMIZAION Obective Function Singularity Global Stress Measure Relation Between Mean Compliance And Von-Mises Stress Modified Optimization Problem Smoothing Scheme Sensitivity Analysis

6 5 RESULS Bar Michell ype Structure Bracket Unconstrained Bracket Bridge L-Shaped Structure D Bracket D Example II CONCLUSIONS Summary Discussions Scope Of Future Work APPENDIX A MEHOD OF MOVING ASYMPOES Selecting he Move Limits Solving he Subproblem B MOVING BARRIER MEHOD Selecting he Move Limits Solving he Subproblem LIS OF REFERENCES BIOGRAPHICAL SKECH

7 able LIS OF ABLES page 2-1 Summary of the literature on topology optimization using stress constraints Configuration details used to obtain bar like structures Configuration details used to obtain Mitchell truss type structures Configuration details used to obtain optimum bracket shapes Configuration details used to obtain optimum shapes for unconstrained bracket Configuration details used to obtain bridge type structures Configuration details used to get optimal shapes for L-shaped structure

8 Figure LIS OF FIGURES page 2-1 Unit cell of a microstructure with a rectangular void Variation of Young s modulus with density for p = 1, 3, Plot of stress functions Plane stress model of the design domain for axial loading problem opology optimization results for bar problem Stress distribution of the optimal topologies for the bar problem Plane stress model of the design domain for Mitchell structure problem opology optimization results for Mitchell truss type structure Stress distribution of the optimal Michell Structures Convergence plots (i) Case b (ii) Case d Plane stress model of a bracket opology optimization results for bracket for different configurations Stress distribution of the optimal topologies for the beam problem Plane stress model of an unconstrained bracket opology optimization results for unconstrained bracket for different configurations Stress distributions for the unconstrained bracket problem Plane stress model of the design domain for the bridge problem opology optimization results for bridge problem Stress distribution of the optimal topologies for the bridge problem Plane stress model of a loaded L-shaped structure opology optimization results for L-shaped structure for different configurations Stress distribution of the optimal L-shaped structures

9 5-20 Plane stress model of an 3-D bracket problem opology optimization results for the 3-D bracket problem Plane stress model of an 3-D example II opology optimization results for the 3-D example II

10 LIS OF ABBREVIAIONS MBM MBSLP MMA SIMP SLP SQP Moving barrier method Moving barrier sequential linear programming Method of moving asymptotes Solid isotropic material with penalization Sequential linear programming Sequential quadratic programming 10

11 Abstract of hesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science OPOLOGY OPIMIZAION OF SRUCURES USING GLOBAL SRESS MEASURE Chair: Ashok V. Kumar Maor: Mechanical Engineering By Viay Krishna Yalamanchili May 2012 opology 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 stresses. In most cases, a design with the lowest compliance is not always the same as the design 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 from a structural designer s viewpoint and stress based topology optimization 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 imposing stress constraints on continuum structures using a conservative global stress measure. he 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 11

12 boundaries of the design. he 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 density 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 integrating special functions of local stress violations. his expression is added to the mass of the structure to obtain the obective function. he 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. 12

13 CHAPER 1 INRODUCION Overview opology optimization is a mathematical approach that optimizes the distribution of material within a given design space while also meeting design and performance requirements. opology optimization has been used predominantly by structural designers and is the emphasis of this work. It could also be used in other areas of physics such as micro electro mechanical systems, fluid and thermal systems. opology 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 topology 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 obective function and the constraints. he obective of the optimization could be minimizing the weight of the structure or maximizing the stiffness of 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 obective is to find the optimal locations, shapes and sizes of voids in the design space. In some cases topology optimization is done to 13

14 obtain a preliminary design and shape optimization is then performed starting from this initial design. his 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. opology 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 stressed in most regions. In this approach the final design depends on the initial mesh discretization making it inconsistent and unreliable. Since then several methods such as homogenization (Bendsøe and Kikuchi 1988), Solid Isotropic Material with Penalization (SIMP) (Bendsøe 1989, Bendsøe and Sigmund 2003), level-set (Allaire et al. 2004) and genetic algorithm (Chapman et al. 1994, Ohsaki 1995, Kaveh and Kalatari 2003, Wang and ai 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 conunction with finite element formulation to obtain the optimal designs. he ideas behind each of the methods mentioned above are explained in Chapter 2. he SIMP method converts the discrete feasible domain into a continuum setting which facilitates the use of gradient based optimization algorithms which are very efficient and computationally less expensive. It is for this reason that the SIMP method has been gaining popularity and we have used this approach in our work. Most methods which use the SIMP approach assume constant density within an element and treat density of elements as design variables. However, Kumar and Gossard (1992) have shown that treating density as nodal variables and using nodal densities as design 14

15 variables yields smoother topologies with fewer elements when compared with 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. his approach, which uses nodal values of density as the design variables, has been used for the purpose of this thesis. opology optimization with minimization of compliance as the obective function has been understood very well and there are several methods which claim to do this efficiently (Bendsøe 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. he optimization problem formulation for this technique is explained in detail in Chapter 2. From a designer s point of 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 consistent with the elastic energy measure. However, we consider the von Mises stress criterion which is consistent with the energy only when the material is incompressible, i.e. Poisson s ratio is 0.5. Another issue with the compliance method is that we need to 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 lot of research has been put into achieving topology optimization subect to stress constraints in the last decade and several methods have been proposed to solve this problem. he chief obstacles faced when tackling stress based topology optimization with the SIMP approach are the so called singularity phenomenon 15

16 (Bendsøe and Sigmund 2003, Cheng and Jiang 1992, Cheng and Guo 1997, Rozvany 2001) and the large number of nonlinear constraints resulting from the localized and nonlinear nature of stress. he singularity phenomenon while dealing with stress constraints was first identified 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 cannot 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 these regions. his 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 issue of singularity arises only when properties such as Young s modulus and yield stress (maximum allowable stress) of material with intermediate densities are modeled as real material with physical 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 dealing 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. his requires tremendous computational effort and algorithms which handle large constraints effectively. Most 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 16

17 and computational expense. An alternative 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 compared in Chapter 2. In this thesis we include a stress term (similar to the global stress constraint) in the obective function, which penalizes the design heavily when there is a stress violation. Obectives he general obective of this research work is to investigate the use of stress based constraints for topology optimization using finite element formulation. In particular, this research intends to accomplish the following: o implement topology optimization of 2D and 3D structures subect to stress constraints with minimization of mass as the obective. o study the effectiveness of a global stress measure to approximate local stress constraints. Outline he rest of the thesis is organized as described below: Chapter 2, begins by discussing in brief various methods used to represent shapes in opology Optimization, namely he Homogenization Method, Solid Isotropic Method with Penalization (SIMP). Following this, the previous work carried out in the field of opology Optimization is discussed highlighting the most popular obective 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. 17

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 constraints. Finally the theory behind the obective 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. hen the advantages, disadvantages of using stress based constraints are discussed and future scope of work is presented. 18

19 CHAPER 2 OPOLOGY OPIMIZAION he obective of topology optimization is to find the optimal distribution of material within a given design space. In an optimization problem, the 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 obective function. Various methods used to achieve black and white designs are discussed in the following section. he most popular obective functions for topology optimization of structures are explained later in this chapter. opology Optimization Methods Several methods have been proposed to represent topologies. he most popularly used methods are : Homogenization method, Genetic algorithm method, the level set method and Solid Isotropic Material with Penalization (SIMP). he SIMP method is used for the purpose of this work. he others methods are discussed briefly below: Homogenization Method he idea of homogenization was established by Kohn and Strang (1986) when designing torsion bars. hey proposed the use of three types of regions in the design: solids, voids and porous. Bendsøe and Kikuchi (1988) developed this idea and applied it to topology optimization. In this method, the material is modeled as being porous by assuming a microstructure with holes. he relation between material properties and porosity is calculated for few hole sizes and the properties in between are interpolated. he porosity within each element is treated as constant in the finite element model and 19

20 the porosity values are used as design variables in the optimization problem. A microstructure 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 Algorithm 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. his 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 finite element model which assumes discrete values 0 or 1. he 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. he process is repeated iteratively until optimal designs are 20

21 obtained. he main disadvantage of this approach is that for problems such as topology optimization where the 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, global 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. he 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. (2003) demonstrated that structural optimization can be described by letting the level set functions dynamically change in time. he dynamic model is expressed as shown below. S( t) { x( t) : ( x( t), t) k} (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). he boundaries of the optimal structure are obtained by solving the PDE. ( x, t) dx ( xt, ) 0 t dt (2-2) 21

22 Here dx dt is the movement of a point on the contour driven by the obective of the optimization. Solid Isotropic Method With 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. he relationship between density and material properties is shown below: p E( x) ( x) E, p ( x ) 1 (2-3) Here ( x) is the density of the material, E is the Young s modulus of the original 0 material and p 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 2-2 shows the variation of Young s modulus with density for different values of p. It can be noticed that intermediate densities have a penalty of the strength when 22

23 the cost of the material is taken into account. 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 variables and interpolate the density within an element. In this thesis nodal densities are the design variables. he advantage of using this approach is that smoother topologies are obtained with fewer elements when compared to constant element density approach. Obective Functions With Constraints opology optimization has been applied to a range of structural problems. he most widely used applications being compliance minimization and compliant mechanisms. he obective 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 obective function and the material / volume constraint is either treated as an equality constraint (or inequality constraint in some cases). he optimization problem when finite element model is used can be states as: Minimize: L( ) F { U} (2-4) 23

24 Subect to: M( ) d M0 (2-5) 0 { } [ D( )]{ } d { u} { f } d 0 1 (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 ( x). 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 motion as opposed to rigid body mechanisms which attain their motion from the use of hinges, bearings and sliders. he goal is to make the output port move through a set of N desired points for N given input displacements. he obective function is formulated as an error function: N Minimize: * 2 ( ) [ uout, n uout, n ] (2-7) n1 Subect to: M( ) d M0 (2-8) 0 { } [ D( )]{ } d { u} { f } d 0 1 (2-9) Stress Constraints In this section, we will summarize previous work on topology optimization which use stress constraints. he classic stress constrained problem optimizes the weight of the structure without violating the stress constraint and satisfying elastic equilibrium. 24

25 he optimization problem when using SIMP interpolation to predict intermediate material properties is shown below: Minimize: ( ) d e (2-10) 0 { } [ D( )]{ } d { u} { f } d 0 1 (2-11) VM (2-12) q max p C( ) C 0 (2-13) Here VM is the Von-Mises stress, max is the maximum allowable stress. q has been introduced to account for attenuation of strength for material with intermediate densities. Duysinx and Bendsøe (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. hey 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. he modified stress constraints proposed by Duysinx and Bendsøe (1998) can be written as: VM ( 1) q max 2 min (2-14) Duysinx and Sigmund (1998) enhanced the stress constraint by multiplying a factor (1 ) on to ensure the real constraint is imposed for 1[see equation(2-15)] and used a global constraint to replace the local constraints thus saving computational effort. he global constraint is shown in equation(2-16). 25

26 VM 1 (1 ), p max 2 min (2-15) 1/ q q N VM max 0, 1, p e1 max (2-16) Whereq 1. 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. Guilherme and Fonseca (2007) used a variant of the -relaxation method [see equation(2-17)] and a global stress constraint based on p-norm. he global constraint and stress constraint relaxation are shown below: VM max p 1 (2-17) 1/ q q N v e VM max 0, 1, p e1 V t max (2-18) Svanberg and Werme (2007) minimized the volume of the structure weighted by the maximum stress and minimized the volume subect to local constraints. hey solved the optimization problem using sequential integer programming and claimed that stress 26

27 constrained problems can be solved 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. In this approach, different penalties are used in the interpolation schemes for stiffness (p) and stress (q) such that q<p. he author shows that this approach yields similar results to -relaxation as q approaches p and approaches 0. Bruggi and Venini (2008) minimized compliance subect to volume and local stress constraints. he relaxed stress constraints are shown below: 2 VM 2q 2 max 1 (2-19) Paris et al. (2009) used stress constraint relaxation similar to Duysinx and Bendsøe (1998) and presented both local and global constraint methods. Kreisselmeier- Steinhauser (KS) function (Kreisselmeier and Steinhauser 1979) was used to create a global constraint. he global constraint is shown below: N i 1 1 e 1 max i ln e ln( Ne ) 0 (2-20) i 1 Where i 1 Le et al. (2010) proposed using different interpolation schemes for the stiffness, stress and volume. hey ustify choosing different interpolation schemes by arguing that the ultimate obective is to design black and white structures and they are not interested in representing the material behavior at intermediate density accurately. he authors divide the design domain into regions (regions need not be connected) and implement regional constraints. hey claim this method effectively gets rid of the stress 27

28 concentrations and produces results comparable to local constraints with only a small increase in the computational expense. he regions are defined by sorting the elements based on their stress level in the current design. he regional constraints for m regions are defined as: 1/ p p e c k ve e k max 1, (2-21) k 1,2,..., m Where ck 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. heir method uses topological derivatives as descent direction to find the optimum. he drawback of using this approach is that the user needs to specify weights associated with each term in the obective function. he obective function for this method is shown below: 2 M d f. ud p d 2 max (2-22) N 1/ Where ( ) p p (1 t ) 1,, are weights attached. p t he existing literature on stress constraints discussed above are summarized in able

29 able 2-1. Summary of the literature on topology optimization using stress constraints (Le et al. 2010) Author Elasticity ensor Relaxed stress Problem statement Optimization Algorithm / constraint Remarks Duysinx and p C( ) C -relaxation: Minimize volume subect CONLIN 0 Bendsøe (1998) (SIMP) to local stress constraints VM ( 1) p max Duysinx and p C( ) C 0 -relaxation: Minimize volume subect MMA (Svanberg 1987) Sigmund (1998) (SIMP) to global stress constraint VM 1 (1 ) p max Pereira et al. p C( ) C 0 -relaxation: Minimize volume subect BOX-QUACAN (2004) (SIMP) to local stress constraint (Friedlander et al. 1994) VM 1 (1 ) p max Guilherme and p C( ) C 0 -relaxation: Minimize volume subect Sequential linear Fonseca (2007) (SIMP) to global stress constraint programming (SLP) VM p 1 Svanberg and Werme (2007) Bruggi and Venini (2008) Paris et al. (2009) Le et al. (2010) C( ) C 0 p C( ) C C( ) C 0 0 C( ) ( ) C c 0 max VM max 2 VM 1 2q 2 max -relaxation: VM 1 (1 ) p max ( ) VM max ( ) c Minimize volume subect to local stress constraints, Minimize volume weighted by maximum stress Minimize Compliance, Subect to volume and local stress constraints Minimize volume subect to local stress constraints Minimize volume subect to regional normalized stress constraints Sequential Integer programming MMA (Svanberg 1987) Sequential linear programming (SLP) MMA (Svanberg 1987) 29

30 able 2-1. Continued Amstutz and Does not use Novotny (2010) SIMP VM max Minimize linear combination of mass and compliance with stress constraints imposed via penalty method -opology derivatives are used for descent direction -Weights need to be chosen for compliance and stress penalty 30

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 (Bendsøe and Sigmund 2003). For free vibration problems, maximization of the fundamental frequency is the most commonly used obective 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 where minimum vibrations are desired. Another important problem in structural optimization is the maximization of the fundamental buckling load. 31

32 CHAPER 3 NUMERICAL MEHODS USED O SOLVE HE OPIMIZAION PROBLEM It is important to understand the numerical methods used to solve the optimization problem. opology optimization problems 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 obective functions 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 like 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 barrier 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 obective functions and/or constraints. SLP algorithms use the first order aylor series expansion to linearize the obective function and constraints about an initial guess point and then solve the linearized problem using standard linear programming techniques like the simplex method. he 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: f ( x ) (3-1) 32

33 Subect to: 0, 1,2,..., h x i m (3-2) 0, 1,2,..., i g x n (3-3) f x, h x, g( x ) are non-linear functions. Here Let k x be the estimate of the design variable at the k th ( ) iteration and Δx k be the change in design, then the above nonlinear optimization problem can be linearized about k x using first order aylor s series to obtain the linearized subproblem: Minimize: k Δx Δ k k k k f x f x f x x (3-4) Subect to: k x k k k k i i i h x h x h x Δx 0, i 1,2,..., m (3-5) k x k k k k g x g x g x Δx 0, 1,2,..., n (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. o deal with this we must impose limits on the design variable. hese constraints are called the move limits, expressed as: L U Δx Δx Δx (3-7) Where Δ x L, Δ U x are the maximum allowed decrease and increase to the design variables. hese move limits play an important role in the convergence of the SLP algorithm and need to be chosen carefully. he subproblem can be written as: Minimize: c Δx (3-8) Subect to: n Δx h 0 (3-9) i i 33

34 a Δx g 0 (3-10) L U Δx Δx Δx (3-11) Here c f, n h, a g (3-12) i i he convergence criteria for the problem can be stated as: g ε, h, Δ k 1 i 1 2 x (3-13) 1, 2 are small numbers specified by the user. he SLP algorithm can be summarized as follows: i. Estimate a starting design x (0). Set k=0, select two small numbers 1 2 the convergence criteria and also select the move limits Δ L U x, Δx., to define ii. Evaluate the obective function, constraints, the gradients of obective function k and constraints at the current design x and generate the linear subproblem. ( ) iii. Solve the subproblem for Δx k using standard LP solving algorithms like the simplex. iv. Check for convergence(3-13). If criteria is met then stop, otherwise continue v. Update the design as k x 1 x k Δx k, 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. here are many variations of the SQP, but most SQP methods have a subproblem that has a quadratic obective function with linearized constraints. he basic functioning of the SQP is similar to SLP except that the obective function approximation is quadratic and the method used to solve the subproblem is different. he quadratic approximation of the obective function can be written using aylor s series expansion as: 34

35 x k k k k 1 k k f x Δ f x c Δx Δx HΔx (3-14) 2 2 Here H f Δx k is the hessian of the obective function. he Hessian of the obective function is obtained using BFGS update. his method starts of by assuming Identity matrix for the hessian and then iteratively solves for the Hessian. he constraints of the SQP problem are linearized similar to SLP: k x k k k k i i i h x h x h x Δx 0, i 1,2,..., m (3-15) k x k k k k g x g x g x Δx 0, 1,2,..., n (3-16) he subproblem can be summarized as: 1 k k Minimize: c Δx Δx HΔx (3-17) 2 Subect to: n Δx h 0 (3-18) i i a Δx g 0 (3-19) Modified simplex method is used to solve the QP subproblem. Alternatively KK conditions can be used when the number of design variables is small. Method Of Moving Asymptotes his 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). he general description of the method is presented below. Let the optimization problem be defined as: n Minimize: f0, R x x (3-20) f x fˆ for i 1,, m (3-21) Subect to:, 2, i i 35

36 x x x, for 1,2,, n (3-22) Where x1, x2,, x n x is the vector of design variables. f0 f i x is the nonlinear obective function. x f, are m nonlinear inequality constraints. i x, x are the side constraints on the design variable x. his method solves the given problem iteratively as described below: 0 i. Choose an initial point x for the 0 th iteration (k=0) ii. At the given iteration point k k k x calculate i x i x f and f for i 0,1,2,, m k iii. Generate a sub problem P by using explicit approximating functions based on the above calculations. iv. Generate a dual problem for the sub problem P k. Solve the dual problem and thereby the sub problem v. Update the iteration point x k, k k1. Check for convergence else go back to (ii) he sub problem for the k th iteration is defined as: k P : minimize p q k k k n Σ 1 k k U x x L r (3-23) k k pi q k n i Subect to: ri Σ 1 fi, for i 1,2,..., m k k U x x L (3-24) k k x x x max, max, (3-25) Here 36

37 p q k i k i 2 k k fi fi U x, if 0 x x fi 0, if 0 x fi 0, if 0 x 2 k k fi fi x L, if 0 x x r f x Σ k k n i i 1 k U p k i x q k i k k k x L (3-26) k k he parameters, are move limits chosen such that (to avoid division by 0) L x U (3-27) k k k k k Example: 0.9 L 0.1 x and 0.9 U 0.1 x k k k k k k he selection of the side constraints L, U and the method used to solve the subproblem are stated in Appendix A. he 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. his issue can be solved by introducing extra variables in the obective function (Svanberg 1987). Moving Barrier Method Moving barrier method (Kumar 2000) is yet another sequential linear programming method in which the subproblem is generated by linearizing the obective function and imposing the move limits by using logarithmic barriers. he move limits are flexible and be modified at every iteration. he 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 37

38 newton s method to solve the subproblem. his method does not require us to calculate the exact solution of the subproblem and hence reduces the computational effort required. It has also been reported that it works well with ill conditioned problems. he 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: f x (3-28) Subect to: AX b (3-29) l x u, i 1,2,, n (3-30) i i i f is a nonlinear obective function., l u are the side constraints for our design variables. i i th For the k iteration, the linearized subproblem with logarithmic barrier can be written as: Minimize: Subect to: Where f n k n k k k ln i i k ln i i i1 i1 F x c x x l u x (3-31) Ax b (3-32) k x c (3-33) And k k l, u i are flexible move limits at the k th iteration such that i k k l l x u u (3-34) i i i i i he method suggested for selecting 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: 38

39 k1 k x x Δx, k1 k z z Δ, z k1 k λ λ Δλ k k k (3-35) he step size is determined such that the new design variables obtained remain feasible. he step size is calculated as shown below: ( ˆ k k k k li xi ) / xi, if xi 0 k min ( ˆ k k k li xi ) xi, otherwise (3-36) After obtaining the next iteration point, the procedure is repeated till convergence like other sequential programs. o make sure that the starting point is feasible, the initial point x 0 is proected onto the plane Ax b, the proected point is obtained as follows: 1 x x A AA ( Ax b ) (3-37) Comparison SLP, SQP methods work well 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 especially 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 obective function using the penalty method. 39

40 CHAPER 4 SRESS BASED OPOLOGY OPIMIZAION opology optimization via compliance minimization has been the dominant method used to design optimal structures. his 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 final 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 obective function with constraints on the stress in the structure. his 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 stress leading to large number of stress constraints. For more details on these issues refer Chapter 1. Solid Isotropic Material with Penalization (SIMP) approach (refer Chapter 2) with nodal densities as design variables has been used in this work. he dependence of Young s modulus and density is as shown below: p E( x) ( x) E, p 1 0 ( x) 1 0 (4-1) 40

41 Obective Function Consider a domain Ω in R 2 or R 3 in which an optimum design is to be found. he domain Ω is the feasible region within which the structure must fit and the applied set of loads and boundary conditions applied along the boundaries of this feasible region. he obective 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 defined as: Minimize Subect to dv (4-2) v, ( x ) VM max, x 0 th 1 (4-3) D dω u f dγ (4-4) Ω0 Γ Here VM is the Von-Mises stress, max is the maximum allowable stress. Singularity he Singularity problem has been studied extensively for stress constrained problems (Bendsøe and Sigmund 2003, Cheng and Jiang 1992, Cheng and Guo 1997, Rozvany 2001). he problem arises when optimal topologies belong to degenerate subspaces of the feasible domain i.e. some elements in the feasible region have zero density. In such cases gradient based algorithms fail to reach the true optimum. able 2-1 shows different relaxation techniques used in the literature. he issue of singularity arises only when properties such as Young s modulus and yield stress (maximum allowable stress) of material with intermediate densities are 41

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. he main purpose of using shape 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 final topology does not contain material with intermediate density and so we do not encounter the issue of singularity. Global Stress Measure Localized nature of stress is another important issue that needs to be tackled to use stress constraints. In a continuum setting, the stress constraint should be considered at every material point. In 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 each 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. he local stress constraints can be effectively stated as a single constraint as VM max 1, q max (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) functions (Kreisselmeier and Steinhauser 1979) have been used in previous works to construct global constraints. Please refer to able 2-1 for further details. 42

43 In this work, we included a global stress measure which penalizes heavily local stress violations in the obective function and used the Moving Barrier Method (MBM) to solve the optimization. he modified optimization problem is stated as: Minimize VM dv dv v v max (4-6) Subect to: D dω u f dγ (4-7) Ω0 Γ 0 1 (4-8) th he 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. hus the optimization 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 stresses are within the limiting value the optimization algorithm would try to reduce the mass of the structure without violating the stress limit. he behavior expected from function is summarized below: VM VM 0 1, [0,1] max max VM VM 1, 1 max max (4-9) We have tried using the following functions for : ( m 1 (4-10) m 1 x) x, ( ) mx ( 1), 1 2 x e m (4-11) he 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 43

44 value rises steeply thereafter. 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 he optimization algorithm requires us to compute the gradient of the obective function and hence the gradient of the Von-Mises stress. Computing the gradient of Von-Mises with respect to the nodal density is a complicated and difficult task. o ease our computations we replace the Von-Mises stress with equivalent compliance term, which is easier to work with. We make use of the below relation: 2 3E VM 21 C (4-12) Here σ C σ is the compliance. In the above relation the compliance term is equal to the Von-Mises stress when 0.5. From the above relation it can be said that constraining the compliance term also constrains the Von-Mises stress. 44

45 Relation Between Mean Compliance And Von-Mises Stress 2 3E VM 21 C (4-13) 1 1 Here[ C] 1 E, 1 1 { } 2 3 he above relation can be derived as follows: VM (4-14) 1 [ ] (4-15) E C 2 21 Consider VM C 3E E E E [ E (4-16) E , E 0, 0, LHS RHS 21 i. e. { } M { } C (4-17) 3E 45

46 2 3E VM 21 C (4-18) Modified Optimization Problem he modified optimization problem can be stated as shown below: Minimize dv v v 3 E0 C( ) dv (4-19) 2 21 max Subect to: { } [ D( )]{ } d { u} { f } d (4-20) th m Here ( x) x or( x ) ( 1) e mx, m>>1. Smoothing Scheme Numerical issues such as mesh dependency and Islanding 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. his results in different solutions when the mesh discretization is changed. Islanding is analogous to the checkerboard problem when using constant density elements with SIMP approach. his checkerboard problem arises due to finite element formulation which tends to overestimate the stiffness of checkerboards (Bendsøe and Sigmund 2003). An overview of different techniques used to overcome these issues are presented in Sigmund and Petersson (1998), Bendsøe and Sigmund (2003). In this thesis, we use the smoothing scheme proposed by Kumar and Parthasarathy (2011). A smoothing 46

47 term is defined which is obtained by integrating the square of the gradient of density over the entire domain. his smoothing term multiplied by a weighting factor added to the obective function. Minimization of the smoothing term eliminates the numerical instabilities discussed above, when proper weights are selected. he smoothing term is shown below: n npe 2 e 2 dv dv (4-21) v N x e1v 1 e i It has been observed from several test cases that smoothing works well when the smoothing term is 10-20% of the obective function value. Hence the weight should be chosen accordingly. Sensitivity Analysis Sensitivity analysis in optimization can be defined as evaluating the gradient of the obective function and the constraints with respect to the design variables. his information is required by all gradient based optimization algorithm to move towards the optimal solution. he gradient of the obective function can be calculated as the sum of gradients of individual terms in the obective function as shown below: () t 2 dv dv w dv (4-22) i v i v i v i Where t 0 2 max 3 E C( ) 21 he gradient of the mass term is computed as ne npe dv N, i dv (4-23) v i e1v 1 e 47

48 Where N are the shape functions and, i i = Kronecker delta. he gradients of both the stress terms are shown below Stress function 1 ( 1): m dv 2 2 [ C]{ } dv (4-24) v m () t m 3E [ C]{ } i 2 21 ν max v i Stress function 2 ( 2 ): 2() t 3E mt ( 1) m [ C]{ } dv e dv 2 21 ν 2 * t (4-25) v i max v i he gradient of the smoothing term is evaluated as shown below: 2 ne N i dv 2 dv 2. dv (4-26) x x v i v i e 1ve 48

49 CHAPER 5 RESULS Bar Our first example is a bar subected to axial force. he feasible domain is a rectangle with dimensions 0.1m by 0.2m (see Figure 5-1). An axial force (P) of 6000 kn is applied at the right end and the left end is fixed. he force P is distributed uniformly over a length of 0.03m to avoid stress concentration. he modulus of elasticity is 200 GPa, Poisson s ratio is 0.5 and unit thickness is assumed. A topology with minimum mass is desired subect to a maximum stress of 200 MPa. his problem is similar to the bar example demonstrated in Amstutz and Novotny (2009). Figure 5-1. Plane stress model of the design domain for axial loading problem he topologies obtained using different configurations are shown in Figure 5-2. he combination of variables used for each result is shown in able 5-1. able 5-1. Configuration details used to obtain bar like structures Case Stress Mesh ype Mesh Stress Smoothing function Density penalty weight fraction A 2 Quad4N 40 x B 1 Quad4N 40 x C 2 Quad4N 100 x D 2 BSpline9N 80 x

50 A B C Figure 5-2. opology optimization results for bar problem D A B D C Figure 5-3. Stress distribution of the optimal topologies for the bar problem he 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 50

51 throughout similar to the topology obtained by Amstutz and Novotny (2009). he stress distribution in the bar is also uniform and within the stress limit of 200MPa. he 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 ype Structure Michell structure is a popular example used to demonstrate topology optimization. A rectangular domain with a circular hole (see Figure 5-4) is subected to shear force of kn at the right end and the circular hole is fixed. he modulus of elasticity is 200 GPa, Poisson s ratio is 0.3 and thickness is 0.25m. A topology with minimum mass is desired subect to a maximum stress of 200 MPa. Figure 5-4. Plane stress model of the design domain for Mitchell structure problem he optimal topologies obtained using different configurations are shown in Figure 5-5. he combination of variables used for each result is shown in able 5-2. Convergence plots for case B, case D are shown in Figure 5-7. Obective function value is plotted against the iteration number. It was noticed that the obective function 51

52 value stabilizes after iterations and further reduces by a small value before it converges. able 5-2. Configuration details used to obtain Mitchell truss type structures Case Stress Mesh ype Mesh Stress Smoothing function Density penalty weight fraction A 2 Quad4N 75 x B 2 Quad4N 75 x C 2 BSpline9N 60 x D 1 Quad4N 75 x A B C D Figure 5-5. opology optimization results for Mitchell truss type structure 52

53 A B C Figure 5-6. Stress distribution of the optimal Michell Structures D (i) Figure 5-7. Convergence plots. i) Case B. ii) Case D. (ii) An analytical optimal design for this problem was obtained by Michell (1904). he final topologies obtained resemble Michell structures and are similar to topology designs 53

54 by minimum compliance method (Kumar 2000, Bendsøe and Sigmund 2003). he solutions presented here have fewer truss like members due to smoothing. he structures obtained are close to being uniformly stressed and are within the design stress limit of 200 MPa. he shapes obtained using both the stress functions are similar and as 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 designs are under stressed by 20-25%, this is because of the manner in which boundaries have been extracted and the conservative 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 his example has a rectangular domain of dimension 0.2m x 0.3m. he left end has been fixed and a shear force (P) of magnitude 1000 kn is applied at the free end. he modulus of elasticity is 200 GPa, Poisson s ratio is 0.3 and the thickness is 0.25 units. A topology with minimum mass is desired subect to a maximum stress of 100 MPa. he optimal topologies obtained are shown in Figure 5-9 and the stress distributions are shown in Figure he combination of variables used for each result is shown in able

55 Figure 5-8. Plane stress model of a bracket able 5-3. Configuration details used to obtain optimum bracket shapes Case Stress Mesh ype Mesh Stress Smoothing function Density penalty weight fraction A 2 Quad4N 40 x B 2 Quad4N 40 x C 1 Quad4N 40 x A B C Figure 5-9. opology optimization results for bracket for different configurations 55

56 A B C Figure Stress distribution of the optimal topologies for the beam problem It can be observed that the topologies obtained using both the stress functions are similar but the stresses at the fixed ends exceed the maximum allowable stress limit of 100MPa. his is because the problem does not have a feasible solution. However, a maority of the region is within the stress limit and has uniform distribution of stress. Unconstrained Bracket his example is the same as the previous example except that the feasible domain has been enlarged along the direction of loading. he dimensions for this problem have been set to 0.2m by 0.6m. able 5-4 shows the configuration settings; Figure 5-12 and Figure 5-13 show the optimal topologies and the stress distribution respectively. 56

57 Figure Plane stress model of an unconstrained bracket able 5-4. Configuration details used to obtain optimum shapes for unconstrained bracket Case Stress Mesh ype Mesh Stress Smoothing function Density penalty weight fraction A 2 Quad4N 30 x B 2 Quad4N 30 x C 1 Quad4N 30 x D 2 BSpline9N 20 x

58 A B C D Figure opology optimization results for unconstrained bracket for different configurations A B C D Figure Stress distributions for the unconstrained bracket problem In this problem, the feasible domain has been modified such that the optimal solution is not bounded. he optimal shape obtained after optimization is very close to the shapes obtained by Duysinx and Bendsøe (1998), Allaire et al. (2004), Bruggi and Duysinx (2012). he stresses in the structure for cases B, C are mostly in the range MPa with maximum stress close to the allowable stress limit of 100MPa. he results are conservative by 10-15% due to the manner in which the boundaries are defined, the Von-Mises approximation and smoothing. It can be noticed that higher penalty leads to 58

59 designs closer to the optimal solution. In case D B-Spline elements have been used and the final structure obtained is more conservative due to the smoothing effect produced by B-Spline elements. Bridge In this example, we optimize the topology of a bridge. he feasible domain is a rectangle of size 0.1m x 0.06m. he left bottom and right bottom edges are fixed. Force P of magnitude 600 kn is applied over a length of 0.35m as shown in Figure he modulus of elasticity is 200 GPa, Poisson s ratio is 0.3 and unit thickness is assumed. A topology with minimum mass is desired subect to a maximum stress of 200 MPa. he thickness is 0.5 units. Figure Plane stress model of the design domain for the bridge problem able 5-5. Configuration details used to obtain bridge type structures Case Stress Mesh ype Mesh Stress Smoothing function Density penalty weight fraction A 2 Quad4N 90 x B 2 Quad4N 90 x C 1 Quad4N 90 x he optimal topologies obtained are shown in Figure 5-15 and the stress distributions are shown in Figure he combination of variables used for each result is shown in able

60 A B C Figure opology optimization results for bridge problem A B C Figure Stress distribution of the optimal topologies for the bridge problem 60

61 he 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. his is because the large penalty on the small region was negated by large regions (in blue) which are under stressed. he stress violations could 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 he feasible domain for this example is L-shaped with dimensions as shown in Figure he top edge highlighted in red is fixed and a shear load (P) of 1000 kn is applied on the right edge. he modulus of elasticity of the material used is 200 GPa, Poisson 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 250 MPa. Figure Plane stress model of a loaded L-shaped structure Different configurations used are shown in able 5-6, the optimal topologies and their stress distributions are shown in Figure 5-18, Figure 5-19 respectively. 61

62 able 5-6. Configuration details used to get optimal shapes for L-shaped structure Case Stress Mesh ype Mesh Stress Smoothing function Density penalty weight fraction A 2 Quad4N 50 x B 1 Quad4N 50 x C 2 BSpline9N 35 x A B C Figure opology optimization results for L-shaped structure for different configurations A B C Figure Stress distribution of the optimal L-shaped structures he optimal topologies obtained for the problem are similar to those obtained by minimization of compliance. he optimization algorithm used could not get rid of the stress concentration at the reentrant corner. his could be because the manner in which 62

63 we tackle 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. he optimal topologies obtained by Optistruct (Altair Engineering 2007), Paris et al. (2009) using global constraints 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 constraints. 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 his example has a rectangular domain of dimension 0.2m x 0.6m x 0.05m. he left end has been fixed and a shear force (P) of magnitude 100 kn is applied at the free end. he modulus of elasticity is 200 GPa, Poisson s ratio is 0.3. A topology with minimum mass is desired subect to a maximum stress of 200 MPa. Figure Plane stress model of an 3-D bracket problem 63

64 Figure opology 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 ( 1) with a penalty of 3 (m=3) was used to obtain the optimal topology. Smoothing fraction was set to 15%. he resulting optimal topology is shown in Figure he 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 uniformly stressed and within the maximum allowable stress. Smoother topologies could be obtained by increasing the mesh density. 3-D Example II his example has a rectangular domain of dimension 1m x 1m x 1m. he corners on the bottom face are fixed and a force (P) of magnitude 1200 kn is is distributed into 4point loads and applied on the top face as shown in Figure he modulus of elasticity is 200 GPa, Poisson s ratio is 0.2. A topology with minimum mass is desired subect to a maximum stress of 100 MPa 64

65 Figure Plane stress model of an 3-D example II Figure opology 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 ( 2 ) with a penalty of 2.5 (m=2.5) was used to obtain the optimal topology. Smoothing fraction was set to 20%. he resulting optimal topology is shown in Figure