INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN Topology optimization of cylindrical shells for various support conditions Mallika. A 1, Ramana Rao. N.V 2 1 Associate Professor, Dept. of Civil Engineering, VNR Vignana Jyothi Institute of Engineering and Technology,Hyderabad,Andhra Pradesh, India 2 Professor & Principal, Dept. of Civil Engineering, JNTU College of Engineering, Hyderabad,Andhra Pradesh, India mallikaala@yahoo.co.in doi: /ijcser ABSTRACT Topology optimization has been receiving unprecedented attention, due to the potential to automatically generate not only good, but also optimal designs. Since the introduction of topology optimization to the design of continuum structures, it has been successfully applied to many different types of structural design problems. Most FEM codes have implemented certain capabilities of topology optimization. In this paper topology optimization is studied for maximizing the static and dynamic stiffness of the shell structure. In static topology optimization, minimum compliance is considered as objective function to maximize the static stiffness with a constraint on volume and in dynamic topology optimization. Maximizing the Eigenfrequencies is considered to increase the dynamic stiffness of the structure. Numerical examples of shell structure with various boundary conditions are investigated and the results are presented. Keywords: Topology optimization, static stiffness, Dynamic stiffness, Minimum Compliance, Eigenfrequencies 1. Introduction In the present scenario topology optimization has been receiving unprecedented attention, due to the potential to automatically generate not only good, but also optimal designs. Since the introduction of topology optimization to the design of continuum structures (M.P.BendsHe, N. Kikuchi,1988), it has been successfully applied to many different types of structural design problems. Some authors (Suzuki and Kikuchi,1991) considered topology optimization of linear elastic plane structures for the stiffest design using the homogenization method. Topology optimization techniques are extended the to the optimization of continuum structures with local stress constraints (Duysinx and BendsHe,1998). Compliant mechanism design using topology optimization techniques has been studied extensively (G.K.Ananthasuresh, S. Kota, N. Kikuchi,1994;O. Sigmund,2001; S. Nishiwaki, M.I. Frecker, S. Min, N. Kikuchi,1991) by some researchers.the optimal stiffener design(t. Buhl, O. Sigmund,2001; J. Luo, H.C. Gea,1998;. Gea and Fu,1998) of shell structures with the small deformation was studied by some authors. Vibration problem using topology optimization (N.L. Pedersen, 2000; Received on June 2011 published on September

2 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN T.-Y. Chen, S.-C. Wu,1998; H.C. Gea, Y. Fu,1997)are studied to maximize Eigen frequencies with the assumption of linear elastic structural behavior. An extensive Received on June 2011 published on September

3 literature survey on topology optimization can be found in (M.P.Bendshe,1995).Mean while, many advances have been made in finite element technology(o. Sigmund,1997; T.Buhl, C.B.W. Pedersen, O. Sigmund,2000; Antonio Tomás_, Pascual Martí,2010; Bletzinger K.U, Ramm E,1993; H.C. Gea,J.Luo,2001; C.B.W. Pedersen et.,2001; M.P. BendsHe, J.M. Guedes, S. Plaxton, J.E. Taylor,1996; R.R. Mayer, N. Kikuchi, R.A. Scott,1996; K. Maute, S. Schwarz, E. Ramm,1998; Luzhong Yin and Wei Yang,2001), which have a direct bearing on structural topology optimization, since most of the applications in topology optimization employ the finite element method as an analysis tool. As mentioned previously however, little attention is usually paid to the actual finite element formulation in the application. Also a number of commercial topology optimization tools have been developed; either based on special Finite Element (FE) solvers or as add-ons to standard commercial FE packages. In the present paper topology optimization of shell structure using ANSYS, a commercial finite element software package. Shell-93 element is used for descretization of the shell structure. 2. General topology Optimization Problem Statement Topological optimization is a special form of shape optimization.it is sometimes referred to as layout optimization in the literature. The goal of topological optimization is to find the best use of material for a body such that an objective criterion (i.e., global stiffness, natural frequency etc.) takes out a maximum or minimum value subject to given constraints (i.e., volume reduction). Unlike traditional topological optimization does not require the explicit definition of optimization parameters (i.e., independent variables to be optimized). In topological optimization, the material distribution function over a body serves as optimization parameter. The theory of topological optimization seeks to minimize or maximize the objective function (f) subject to the constraints (g j ) defined. The design variables (η i ) are internal, pseudo densities that are assigned to each finite element (i) in the topological problem. The pseudo density for each element varies from 0 to 1; where η i 0 represents material to be removed; and η i 1 represents material that should be kept. Stated in simple mathematical terms, the optimization problem is as follows: f=minimize or maximize w.r.to η i (1) Subjected to 0 η i 1 where i=1,2,3 N (2) g jl < g j < g ju where j=1,2,3 M (3) N=Number of finite elements M=Number of constraints g j =Computed j th constraint value g jl =lower bound for j th constraint g ju = upper bound for j th constraint 12

4 In the present problem ANSYS software which is robust and with built-in topology optimization module is used to model, analyze and perform topology optimization. The topological optimization process consists of four parts: 1. defining optimization functions 2. defining objective and constraints 3. initializing optimization 4. executing topological optimization There are two options available in the ANSYS topology optimization module, optimality criteria (OC) approach which is the default choice and sequential convex programming (SCP) approach. 2.1 Maximum Static Stiffness Design (Subject to Volume Constraint) In the case of maximum static stiffness design subject to a volume constraint, which sometimes is referred to as the standard formulation of the layout problem, one seeks to minimize the energy of the structural static compliance (UC) for a given load case subject to a given volume reduction. Minimizing the compliance is equivalent to maximizing the global structural static stiffness. Minimum compliance topology optimization problems impose a constraint on the amount of material which can be utilized. In this case, the optimization problem is formulated as a special case of equation (1), (2) and (3) as U C =a minimum w.r to η i (4) Subjected to 0 η i 1 where i=1,2,3 N (5) V V 0 V* (6) Where V=Computed volume V 0 =Original volume V*=Amount of material to be removed 2.2 Maximum Dynamic Stiffness Design (Subject to Volume Constraint) In case of the "Maximum Dynamic Stiffness" design subject to a volume constraint one seeks to maximize the i th natural frequency ( i >0) determined from a mode-frequency analysis subject to a given volume reduction. In this case, the optimization problem is formulated as: i = a maximum w.r to η i (7) Subjected to 13

5 0 η i 1 where i=1,2,3 N (8) V V 0 V* (9) Where i = i th natural frequency computed V=Computed volume V 0 =Original volume V*=Amount of material to be removed Maximizing a specific eigenfrequency is a typical problem for an eigenfrequency topological optimization. However, during the course of the optimization it may happen that eigen modes switch the modal order. For example, at the beginning we may wish to maximize the first eigenfrequency. As the first eigenfrequency is increased during the optimization it may happen, that second eigen mode eventually has a lower eigen frequency and therefore effectively becomes the first eigen mode. The same may happen if any other eigenfrequency is maximized during the optimization. In such a case, the sensitivities of the objective function become discontinuous, which may cause oscillation and divergence in the iterative optimization process. In order to overcome this problem, several mean-eigen frequency functions (Ω) are introduced to smooth out the frequency objective. Hence in the present paper instead of maximizing the fundamental frequency minimization of weighted frequency is considered as the objective function in case 2 as mentioned in the following sections. 2.3 Weighted Formulation Given m natural frequencies (ω i,. m ), the following weighted mean function (Ω W ) is defined: m Ω W = Wi i i 1 where ω i = i th natural frequency W i = weight for i th natural frequency The functional maximization equation (4) is replaced with (10) Ω W = a maximum w.r to η i 2.4 Element Calculations While compliance, natural frequency, and total volume are global conditions, certain and critical calculations are performed at the level of individual finite elements. The shell element used for topology optimization in the present paper is shell 93 element. The total volume, for example, is calculated from the sum of the element volumes; that is, 14

6 V= ivi (11) i V i = volume for element i The pseudo densities effect the volume and the elasticity tensor for each element. That is, E = ( i ) i E (12) where the elasticity tensor is used to equate the stress and strain vector, designed in the usual manner for linear elasticity: {σ i }= E i {ε i } (!3) where {σ i } = stress vector of element i {ε i } = strain vector of element i 3. Numerical Example 3.1 Problem definition The geometry of the concrete shell is taken from the reference of Antonio Tomas, Pascual Marti 16, which was basically taken from Bletzinger and Ramm 17. The study has been extended for free vibration analysis and topology optimization for different boundary conditions. The concrete shell is subjected to its own weight and a vertical uniform load, for different design criteria. The shell thickness is 50 mm and the structure covers a surface of 6m x 12 m. Young's modulus of the material is 30 GPa and Poisson's modulus is 0.2. The structure is subjected to a vertical uniform load of 5 kn/m 2.The shell can be supported on the right edges, on the curved ones or on them all at the same time (Figure 1). Topology optimization of the shell has been carried out, under two different objective functions. Case 1 a)right edges supported b)curved edges supported c)all edges supported Figure 1: Initial shapes of the concrete shell for various edge conditions Maximization of static stiffness can be achieved by minimization of structural compliance, the constraint on the total material volume of the structure should be reduced 15

7 to 50% of the initial volume. The solution approach used for minimum compliance problem is optimality criteria approach, which is by default in ANSYS topology optimization module. Case 2 Maximization of Dynamic stiffness can be achieved by maximizing the weighted frequency (for first five frequencies) with a constraint that total material volume of the structure should be reduced to 50% of the initial volume. This obviously converts the problem into minimization of weighted frequency. The solution approach used for minimum weighted frequency problem is sequential convex programming approach (SCP). 3.2 Initial Geometry In the present analysis, the shell structure is modeled in ANSYS using nine key points, two straight lines for the right edges and the rest eight by segmented cubic splines (figure 2) Areas are generated and discretized using shell 93 elements. The height of the shell structure considered is 3m in the model. Various boundary conditions considered are 6 m 3m 12 m 1. Right edges supported Figure 2: Descritized model geometry of shell structure 2. Curved edges supported 3. Right and Curved Edges supported The shell structure is analysed and initial volume is found to be m 3 for all the cases and the initially fundamental frequencies found from the modal analysis are0.4269hz, Hz,3.2972Hz for right edges simply supported, curved edges simply supported and all the four edges simply supported respectively. 16

8 4.0 Results and Discussions 4.1Shell supported on right edges Case 1 With an objective function of minimizing the structural compliance with a constraint on volume reduction by 50%, initially the structural compliance was and after 31 iterations it was reduced to with a percentage reduction of 44.07%. Case 2 With an objective function of minimizing the weighted frequency with a constraint on volume reduction by 50%, initially the value of weighted frequency was and it was reduced to with a percentage reduction of 47.5% in 31 iterations. 4.2Shell supported on curved edges Case 1 With an objective function of minimizing the structural compliance with a constraint on volume reduction by 50%, initially the structural compliance was and after 18 iterations it was reduced to with a percentage reduction of 53.3%. Case 2 With an objective function of minimizing the weighted frequency with a constraint on volume reduction by 50%, initially the value of weighted frequency was and it was reduced to with a percentage reduction of 49 % in 32 iterations. 4.3 Shell supported on four edges Case 1 With an objective function of minimizing the structural compliance with a constraint on volume reduction by 50%, initially the structural compliance was and after 19 iterations it was reduced to with a percentage reduction of 30.87%. Case 2 With an objective function of minimizing the weighted frequency with a constraint on volume reduction by 50%, initially the value of weighted frequency was and it was reduced to with a percentage reduction of % in 32 iterations. The density plots of topology optimization for case 1 and case 2 for all the boundary conditions are presented in figure 3. The iteration histories of case 1 and case 2 for objective function are presented in figure 4. 17

9 In all the cases the initial volume was m 3 and was reduced by 50% to m 3. Table 1: Initial and final first five Eigen frequencies for a boundary condition of all edges supported all edges supported curved edges supported straight edges supported Mode Number Initial Frequency Final Frequency Initial Frequency Final Frequency Initial Frequency Final Frequency minimum structural compliance minimum weighted frequency Straight edges supported Curved edges supported both edges supported 18

10 Figure 3: Density plots for minimum structural compliance and weighted frequency(with 50% volume reduction) for various support conditions minimum structural compliance minimum weighted frequenc Straight edges supported Curved edges supported both edges supported Figure 4: Iteration history for minimum structural compliance and weighted frequency(with 50% volume reduction) for various support conditions 5. Conclusions 1. The goal of topology optimization is to find the best use of material for a body such that the objective criteria (stiffness, natural frequency) take out a maximum or minimum value subject to given constraints (volume or mass reduction). 19

11 2. Results are sensitive to the density of the finite element mesh. In general, a very fine mesh will produce clear topological results. A coarse mesh will lead to fuzzier results. 3. In the case of minimizing the weighted frequency, shell supported on curved edges the frequencies are low when compared to other boundary conditions and also the percentage reduction in weighted frequency is found to be almost 50%. 4. In the case of minimizing the structural compliance, shell supported on curved edges showed considerable reduction (53.3%) in compliance for 50% reduction in volume. 5. Un averaged density plots sometimes result into a truss like structure, which gives an idea of material distribution. 6. A large reduction in volume (up to 80%) as constraint can be studied for various cases. 6. References 1. M.P. BendsHe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods and Applications in Mechanical Engineering, 71, (1988), pp K. Suzuki, N. Kikuchi, A homogenization method for shape and topology optimization, Computer Methods and Applications in Mechanical Engineering. 93, (1991), pp P. Duysinx, M.P. BendsHe, Topology optimization of continuum structures with local stress constraints, International Journal of Numerical Methods in Engineering, 43, (1998), pp G.K.Ananthasuresh, S. Kota, N. Kikuchi, Strategies for systematic synthesis of compliant, MEMS, ASME Winter Annual Meeting, DSC- 55(2), (1994), pp O. Sigmund, Design of multiphysics actuators using topology optimization- Part I: one-material structures, Computer Methods and Applications in Mechanical Engineering, 190, (2001), pp O. Sigmund, Design of multiphysics actuators using topology optimization- Part II: two-material structures, Computer Methods and Applications in Mechanical Engineering, 190, (2001), pp S. Nishiwaki, M.I. Frecker, S. Min, N. Kikuchi, Topology optimization of compliant mechanisms using the homogenization method, International Journal of Numerical Methods in Engineering, 42, (1998), pp

12 8. J. Luo, H.C. Gea, A systematic topology optimization approach for optimal stiffener design, Journal of Structural Optimization, 16 (4), (1998), pp N.L. Pedersen, Maximization of eigenvalues using topology optimization, Structural Multidisciplinary Optimization, 20, (2000), pp T.-Y. Chen, S.-C. Wu, Multiobjective optimal topology design of structures, Computer Methods and Applications in Mechanical Engineering, 21, (1998), pp H.C. Gea, Y. Fu, Optimal 3D stiffener design with frequency considerations, Advanced Engineering Softwares, 28, (1997), pp M.P. BendsHe, Optimization of Structural Topology, Shape, and Material, Springer, New York, (1995). 13. O. Sigmund. On the design of compliant mechanisms using topology Optimization, Mechanics of Structures and Machines, 25, (1997), pp T. Buhl, C.B.W. Pedersen, O. Sigmund, Stiffness design of geometrically nonlinear structures using topology optimization, Structural Multidisciplinary Optimization 19, (2000), pp Antonio Tomás_, Pascual Martí, (2010), Shape and size optimization of concrete shells, Engineering Computations(accepted for publication in feb,2010) 16. Bletzinger KU, Ramm E. Form finding of shells by structural optimization, Engineering Computations, 9, (1993), pp H.C. Gea, J. Luo, Topology optimization of structures with geometrical nonlinearities, Computers and Structures, 79, (2001), pp C.B.W. Pedersen, T. Buhl, O. Sigmund, Topology synthesis of largedisplacement compliant mechanisms, International Journal of Numerical Methods in Engineering, 50, 2001, pp M.P. BendsHe, J.M. Guedes, S. Plaxton, J.E. Taylor, Optimization of structure and material properties for solids composed of softening material, International journal of Solids and Structures, 33 (12), 1996, pp R.R. Mayer, N. Kikuchi, R.A. Scott, Application of topological optimization techniques to structural crash worthiness, International Journal of Numerical Methods in Engineering, 39, (1996), pp

13 21. K. Maute, S. Schwarz, E. Ramm, Adaptive topology optimization of elastoplastic structures, Journal of Structural Optimization, 15, (1998), pp M. Bendsøe and O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer-Verlag, Heidelberg, (2002). 23. Luzhong Yin and Wei Yang, Optimality criteria method for topology optimization under multiple constraints Computers and Structures, 70 (20-21),2001, pp