The parabolic KS function for the structural topology optimization

Tie Jun College of Mechanical Engineering and Applied

Electronics Technology Beijing University of Technology

Beijing, China E-mail: tielaoshi@sina.com

Sui Yun-kang College of Mechanical Engineering and Applied

Electronics Technology Beijing University of Technology

Beijing, China E-mail: ysui@bjut.edu.cn

Abstract—This paper is devoted to the parabolic KS function for the structural topology optimization and presents a different strategy to deal with the minimum stress design problem. We transform the local stresses to global integrated stress of structure through the parabolic KS function. A new optimization model is introduced which provide quality solutions and the minimum stress design to obtain the final design. Finally, the parabolic KS function is important because we firstly introduce a different KS function to study the structural topology optimization problem.

Keywords- structural topology optimization; parabolic KS function;stress

I. INTRODUCTION

During the past few decades the subject “structural topology optimization” has attracted increasing attention from researchers all over the world (see [1,2,3,4,5]). The structural topology optimization is classified as skeleton-type structure topology optimization and continuum structure topology optimization. By contrast with the topology optimization of skeleton-type structures, the topology optimization of continuum structures is more difficult in formulating the mathematical model. Therefore, it is currently regarded as the most challenging task in the field of the structural optimization.

In fact, most of these papers referred to above have been focused on global objectives, such as compliance, displacement, frequency and so on . It is fewer about the min-max problem of the stress of the structural elements in the structural topology optimization. Since the essence of topology optimization lies in searching for the optimal path of transferring loads, the stress of the structural elements in the structural topology optimization has to be considered to obtain a reliable design. Clearly the stress of the structural elements is a local quantity, there are two difficulties for solving the local problem as follows: (a) the dimension number of the design variable is too extensive and it is too difficult to be solved. However, every the structural element in the structural topology optimization is just with the corresponding one design variable. Even for one stress at a point in each element

region is only restricted as a stress constraint of the region, the whole structure with N elements will be associated with N L× stress when there are L load cases. (b) Since the stress of the structural elements is highly nonlinear with respect to the design variables, a great quantity of stress sensitivity analysis has to be computed so that the computation cost of stress sensitivity analysis is too expensive to be accepted.

This paper is devoted to the parabolic KS function for the structural topology optimization and presents a different strategy to deal with the minimum stress design problem. The parabolic KS function may effectively transform the local stresses to global integrated stress of structure. A new optimization model is introduced which provide quality solutions and the minimum stress design to obtain the final design.

II. THE PARABOLIC KS FUNCTION To transform the local stress function to the global stress

function, Yang and Chen introduced firstly the KS function as follow

1 max

( )1ln exp[ ]

( )

ni

ksi

fG p

p f

σσ=

= (1)

J.París and F.Navarrina, etc. (2008) introduced a new KS

function as follows:

max

max

ˆ( )

ˆ

1

ˆ1

( ) ln( )jn

ksj

G e

σμ

σ

σ

ρμ

−

=

= (2)

where ( ( , ))ˆˆ h joj rj σ ρσσ = is the local stresses and maxσ̂ is

the maximum allowable stress.

In fact, (1) and (2) are linear KS functions in nature. Since

the KS function [6,7,8,9,10] plays an important role in this research, now we firstly give the parabolic KS function in Theorem 2.1.

2009 Second International Workshop on Computer Science and Engineering

978-0-7695-3881-5/09 $26.00 © 2009 IEEE

DOI 10.1109/WCSE.2009.85

385

2009 Second International Workshop on Computer Science and Engineering

978-0-7695-3881-5/09 $26.00 © 2009 IEEE

DOI 10.1109/WCSE.2009.85

385

2009 Second International Workshop on Computer Science and Engineering

978-0-7695-3881-5/09 $26.00 © 2009 IEEE

DOI 10.1109/WCSE.2009.693

385

Theorem 2.1. Let 2 2

,u pv e= if the following conditions are satisfied:

1 0, 0u p> >

(2) ( ) 0, 1,2, , , mjf x j n x R> = ∈ .

and let 2 2

1

( ) 2

1

1( ( ), ) [ln( )]i

np f x

i

i

G f x p ep =

= then we have that

a1

lim ( ( ), ) max ( )i ip i n

G f x p f x→+∞ ≤ ≤

= (3)

b1

min max ( ) lim min ( ( ), )m mi ipi nx R x R

f x G f x p→+∞≤ ≤∈ ∈

= (4)

Proof. (a) If 2 2

,u pv e= then we have that 121 (ln )u v

p= .

For ( ) 0, 0if x p> > , 2 2 ( )[ ( ), ] ,ip f x

iv f x p e= (5)

let 1max ( ) ( )i

i nf x f x

≤ ≤= , since 0, 0u p> > , then we have that

1

[ ] [ ( ), ] [ ]( ), ( ),n

ii

v v f x pf p nvx f x p=

≤ ≤ (6)

i.e. 2 22 2 2 2( )( ( )) ( ( ))

1

i

np f xp f x p f x

i

e e ne=

≤ ≤ (7)

2 21

( ) 22

1

1 ln( ) ln[ ] [ ( )]i

np f x

i

nf x e f xp p=

≤ ≤ + (8)

Taking the limit in (8) as p → +∞ , we get that

1lim ( ( ), ) ( ) max ( ).i ip i n

G f x p f x f x→+∞ ≤ ≤

= = (9)

Now we prove that ( ( ), ) 0iG f x p

p∂ <

∂.

2 2

2 2 2 2

2 2

( ) 2

1 1

( ) ( ) 12 2

2( )1 1

1

( ) ( )( ( ), ) 1

(ln[ ]) (ln[ ])

i

i i

i

np f x

in np f x p f xi i

np f xi i

i

e f xG f x p

e ep p e

−=

= =

=

∂= − + ⋅

∂

2 2 2 2 2 2

2 2 2 2

( ) ( ) ( )2 2

1 1 1

1( ) ( )2 2

1 1

( ) ( ) ( ) ( ln [ ] )

( ln [ ] )

i i i

i i

n n np f x p f x p f x

ii i i

n np f x p f x

i i

p e f x e e

p e e

= = =

= =

−=

2 2 22 2

2 2 2 2

( )( )

11

1( ) ( )2 2

1 1

( ) ( l n [ 1 ] )

0

( l n [ ] )

ji

i i

n np f x pp f x

ji

i j

n np f x p f x

i i

e e

p e e

−

==

≠

= =

− +

= <

(10)

Since ( , ) 0G x pp

∂ <∂

, then the function

2 21

( ) 2

1

1( ( ), ) [ln( )]i

np f x

ii

G f x p ep =

=

is descent for variable p .

Let 2 2

1( ) 21( ( ), ) ln[ ]p f xG f x p ne

p= , then

lim ( ( ), ) ( )p

G f x p f x→+∞

= .

For an increasing ordered series of numbers

1 20

kp p p< < < < < < +∞

we have that

1 1 2

1

( ( ), ) ( ( ), ) ( ( ), ) ( ( ), ) lim ( ( ), ) max ( )i i i k i i

p i n

G f x p G f x p G f x p G f x p G f x p f x→+∞ ≤ ≤

≥ ≥ ≥ ≥ ≥ ≥ = (11)

1 1min ( ( ), ) min ( ( ), ) min ( , ) minlim ( ( ), )

m m m mi k i

px R x R x R x R

G f x p G f x p G x p G f x p→+∞∈ ∈ ∈ ∈

≥ ≥ ≥ ≥ ≥ (12)

Taking the limit in (12) as 1p → +∞ , then the result (b) follows.

Corollary2.1 Let σ be the stress and ( )if σ is the von Mises stress for each finite element where N is the number of finite elements in the design domain, then we have the parabolic KS function as following

2 2

1( ) 2

1

1( ( ), ) [ln( )]i

np f

ii

G f p ep

σσ=

= (13)

Moreover, a1

lim ( ( ), ) max ( )i ip i nG f p fσ σ

→+∞ ≤ ≤=

(14)

b1

min max ( ) lim min ( ( ), )m mi ipi nx R x R

f G f pσ σ→+∞≤ ≤∈ ∈

=

(15)

Proof. It is clear that the results follow from Theorem 2.1.

III. FORMULATION OF THE NEW STRUCTURAL TOPOLOGY OPTIMIZATION MODELS

In this section, we choose the maximum local stress as the objective function and solve the stress minimization problem.

The original optimization problem is as follows:

386386386

PI: Find topology variables nE∈t

minimize ,

max iji j

σ

subject to: 0 1it≤ ≤

{1, , }ni I n∈ = , {1, , }mj J m∈ = .

(16)

where ijσ is the von Mises stress for the i th− element under the j th− load case.

In fact, by Corollary2.1 the local stress objective function

,max ij

i jσ can be expanded to the global stress function

( , )ijG pσ and the transformed topology optimization model PII can be obtained as following:

PII: Find topology variables nE∈t

minimize 2 2

12

1

1( , ) [ln( )]ijn

pij

i

G p ep

σσ=

=

subject to: 0 1it≤ ≤

{1, , }ni I n∈ = , {1, , }mj J m∈ = . (17)

where ijσ is the von Mises stress for the i th− element under

the j th− load case and 0, 0u p> > .

IV. CONCLUSIONS The transformed topology optimization model PII greatly

reduces the scale of stresses of the structural elements in the global sense and it can help obtain optimal topology paths of transferring loads more easily. Moreover, we can choose suitable parameters p to the transformed topology optimization problem. Finally, the parabolic KS function is important because we firstly introduce a different KS function to study the structural topology optimization problem.

ACKNOWLEDGMENT

This paper is supported by National Natural Science Foundation of China (10872012), Beijing Natural Science Foundation 3093019 Foundation of National Key Laboratory for Structural Analysis of Industral Equipment Dalian University of Technology GZ0819 Open Research Fund Program of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body ( Hunan University ) 30715002 and Scientific Research Foundation of Doctoral Subjects in Chinese Universities (20060005010) Foundation Postdoctoral Science Foundation of Beijing University of Technology (X0001015200803)Doctoral Science Foundation of Dalian Nationalities University (20086103), State Ethnic Affairs Commission Science Projects (2009).

REFERENCES

[1] Eschenauer H A, Olhoff N. Topology optimization of continuum structures: A review. Appl Mech Rev, 2001, 54(4): 331~390.

[2] Sui Y K. Modeling, Transformation and Optimization: New Developments of Structural Synthesis Method . Dalian University of Technology Press, 1996.(in Chinese) ).

[3] Behrooz Hassani, Ernest Hinton. Homogenization and Structural Topology Optimization. Berlin:Springer, 1998.

[4] Scott A. Burns, etc. Recent Advances in Optimal Structural Design.Reston, Virginia: ASCE and American Society of Civil Engineers, 2002.

[5] Bendsoe M.P.,Sigmund O. Topology Optimization Theory,Methods and Applications. Berlin:Springer,2003.

[6] Yang R.J.and Chen C.J.. Stress-based topology optimization. Structural Optimization 1996,12,98~105.

[7] J.París and F.Navarrina, etc. Topology optimization of continuum structures with local and global stress constraints. Struct Multidisc Optim, 2008,4:511-529.

[8] Kreisselmeier G, Steinhauser R. Systemic control design by optimizaing a vector performance index. Proceedings of IFAC Symposium on CAD of Control Systems . Zurich,

[9] Switzerland on CAD of Control Systems. Zurich, Switzerland,1979:113-117.

[10] Li Xing-si. An effective solution to nonlinear minimax problems. Chinese Science Bulletin,1991,24(19):1448-1450.

[11] Sui Yun-kang, Yu Xin. Uniform of K-S function and norm function. Journal of Dalian University of Technology, 1998(9),3(5):502-505.

[12] Yunkang Sui, Jiazheng Du and Yingqiao Guo. Independent continuous mapping for topological optimization of frame structures. Acta Mech Sin, 2006,22:611-619.

[13] Sui Yun-kang,etc. Topological optimization of continuum structure with globalization of stress constraints. Journal of Beijing university of technology,2006,32 Supp:1-8.

387387387