Topology optimization of cylindrical shells for various ... · Topology optimization of cylindrical shells for various support conditions Mallika. A1, Ramana Rao. N ... Numerical

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 2, No 1, 2011

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

Received on June 2011 published on September 2011 11

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

[email protected]

doi:10.6088/ijcser.00202010089

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;

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 2, No 1, 2011

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

Received on June 2011 published on September 2011 12

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

Topology optimization of cylindrical shells for various support conditions

Mallika. A, Ramana Rao. N.V

International Journal of Civil and Structural Engineering

Volume 2 Issue 1 2011 12

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 (gj) defined. The design variables (ηi) 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 ηi 0 represents material

to be removed; and ηi 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 ηi (1)

Subjected to

0≤ ηi ≤ 1 where i=1,2,3……N (2)

gjl < gj < gju where j=1,2,3………M (3)

N=Number of finite elements

M=Number of constraints

gj=Computed j th

constraint value

gjl =lower bound for jth

constraint

gju = upper bound for jth

constraint





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

UC =a minimum w.r to ηi (4)

Subjected to

0≤ ηi ≤ 1 where i=1,2,3……N (5)

V≤ V0 –V* (6)

Where

V=Computed volume

V0=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 ith

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 ηi (7)

Subjected to





0≤ ηi ≤ 1 where i=1,2,3……N (8)

V≤ V0 –V* (9)

Where

i = ith

natural frequency computed

V=Computed volume

V0=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 (ωi ,……. m), the following weighted mean function (ΩW)

is defined:

ΩW = i

m

i

iW1

(10)

where

ωi = ith

natural frequency

Wi= weight for ith

natural frequency

The functional maximization equation (4) is replaced with

ΩW = a maximum w.r to ηi

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,





V= i

i

iV (11)

Vi = volume for element i

The pseudo densities effect the volume and the elasticity tensor for each element. That is,

iE = )( iE (12)

where the elasticity tensor is used to equate the stress and strain vector, designed in the

usual manner for linear elasticity:

σi= iE εi (!3)

where

σi = stress vector of element i

εi = 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 Marti16

, which was basically taken from Bletzinger and Ramm17

. 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/m2.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.

a)Right edges

supported

b)Curved edges supported c)All edges supported

Figure 1: Initial shapes of the concrete shell for various edge conditions

Case 1

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





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

Figure 2: Descritized model geometry of shell structure

1. Right edges supported

2. Curved edges supported

3. Right and Curved Edges supported

The shell structure is analysed and initial volume is found to be 5.3456m3 for all the

cases and the initially fundamental frequencies found from the modal analysis

are0.4269Hz ,0.9816 Hz,3.2972Hz for right edges simply supported, curved edges simply

supported and all the four edges simply supported respectively.

12 m

6 m

3m





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 3511.28 and after 31

iterations it was reduced to 1964.03 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 45.2509 and it

was reduced to 23.7568 with a percentage reduction of 47.5% in 31 iterations.

4.2Shell supported on curved edges

Case 1




Case 2



was reduced to 14.8309 with a percentage reduction of 49 % in 32 iterations.

4.3 Shell supported on four edges

Case 1




Case 2



was reduced to 89.58 with a percentage reduction of 42.62 % 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.





In all the cases the initial volume was 5.334 m3 and was reduced by 50% to 2.6728 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

1 3.2928 2.2249 0.97937 0.47828 0.42672 0.27524

2 4.1838 2.7023 1.1213 0.68301 1.6097 0.96984

3 6.1465 3.2190 2.4098 1.0452 2.8347 1.7259

4 6.2996 3.9834 2.4181 1.7318 3.6069 2.0829

5 6.4462 4.0440 3.4521 2.0111 4.0743 2.4801

minimum structural compliance minimum weighted frequency

Straight edges supported

Curved edges supported

both edges supported





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).





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.197–224.

2. K. Suzuki, N. Kikuchi, “A homogenization method for shape and topology

optimization”, Computer Methods and Applications in Mechanical

Engineering. 93, (1991), pp 291–318.

3. P. Duysinx, M.P. BendsHe, “Topology optimization of continuum structures

with local stress constraints”, International Journal of Numerical Methods in

Engineering, 43, (1998), pp 1453–1478.

4. G.K.Ananthasuresh, S. Kota, N. Kikuchi, “Strategies for systematic synthesis

of compliant”, MEMS, ASME Winter Annual Meeting, DSC- 55(2), (1994),

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5. O. Sigmund, “Design of multiphysics actuators using topology optimization-

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Mechanical Engineering, 190, (2001), pp 6577–6604.

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Mechanical Engineering, 190, (2001), pp 6605–6627.

7. 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 535–559.





8. J. Luo, H.C. Gea, “A systematic topology optimization approach for optimal

stiffener design”, Journal of Structural Optimization, 16 (4), (1998), pp 280–

288.

9. N.L. Pedersen, “Maximization of eigenvalues using topology optimization”,

Structural Multidisciplinary Optimization, 20, (2000), pp.2–11.

10. T.-Y. Chen, S.-C. Wu, “Multiobjective optimal topology design of structures”,

Computer Methods and Applications in Mechanical Engineering, 21, (1998),

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11. H.C. Gea, Y. Fu, “Optimal 3D stiffener design with frequency

considerations”, Advanced Engineering Softwares, 28, (1997), pp. 525–531.

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Springer, New York, (1995).

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15. Antonio Tomás_, Pascual Martí, (2010),”Shape and size optimization of

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Engineering Computations, 9, (1993), pp. 27-35

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Documents

Topology optimization of cylindrical shells for various ... · Topology optimization of cylindrical shells for various support conditions Mallika. A1, Ramana Rao. N ... Numerical