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Vol.49 No.4 199714) SEISAN KENKYU 239

l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

Application of Biological-Growth Strain Method on

TaketO UO 10TO*

Ef ct e Strcss Poillt

M a x M i l l l i l i D n n

Fig. 1 Illustration for curvature

based Baud's postulation

Design 'rotlle

- optilnai- : Intemrediate- Initial

variation on the design profile

demonstration3). ased on this validated Baud's postula-

ulated by Baud showed ts practical validation by analytical

demonstration3). ased on this validated Baud's postula-

tion, the stress optimal design can be achieved y varying

the outer-surface hape i.e. curvature of the design points

along the design profile) of the design structure. By

iteratively repeating he above process, he outer-surface

shape can be considered o be optimized once if the'

condition Eq. (1) is satisfied. The iterative optimization

process an be illustrated by Fig. 1. On the other hand, by

looking at the load carriers n animals r plants ike bones or

brunches, t is undoubtedly convinced hat the load carriers

develop heir optimum by well adapting hemselves o their

environment at a certain loading conditions4). With the

further inspect on those biological adaptation process'

several interesting aspects can be summarized n the

following points; (1) the state of constant stress can be

found on the surface of those well-adapted biological

structures nd (2) the adaptation process f these biological

tissues s carried out by growth or atrophy of their live

tissues ear the stress-concentrated urface. By comparing

these observed acts on the biological adaptation process

with stress optimal design process, t is not difficult to find

out the high simitarity between them on the optimality

Optimal Shape Design of Mechanical Components

Kai Lin HSU*and

1 . INTRODUCTION

By the notch stress heory developed by Baudl) and

Schack2), tc., the shape optimization problem of minimiz-

ing stress concentration n the continuous structures also

termed as stress optimal design) can be focused on the

adjacent area where stress oncentration ccurs n stead of

dealing with the whole design tructure. On the other hand,

from the comparative viewpoint on the mechanism of

minimizing stress ioncentration, it is found that the

adaptation rocess f biological issues y shape ariation o

their environment is very effective in reducing stress

concentration n the biotogical structures. The high simi-

larity between the biological adaptation process and the

stress ptimal design n the continuous tructures motivated

us to present one new optimal shape design method based

called biological-growth train method (BGS) by simulat-

ing biological adaptation process.

In the following, the characterjstics f BGS will be given

in section ; then, he numerical erification of the proposed

method and application n various ields of structural design

are llustrated n section 3. Finally, the concluding emarks

is included in section 4.

2. CHARACTERISTICS F BIOLOGICAL.GROWTH

STRAIN METHOD

(1) Similarity etween stress optimal design and biolo-

gical adaptation process

Through years of researches, uniform stress istribution

design (or equi-strength design) -the hypothesis post-* 5th Deputtment, Institute of Industrial Science, University of

Tokyo

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240 V01.49 No.4 1997.4) SEISAN KENKYU

condition and the optimization process. Here, it is worthmentioning hat the gradients or objective unction and heconstraint conditions s usually necessary or the conven-tional stress ptimal design while the biological optimizationprocess eeds no gradient nformation. As a result, it wasmotivated o propose one gradientless tress ptimal designmethod by simulating he biological adaptation process oupdate the design variables nstead of using the gradientoptimization techniques.

In fact, the derivation of BGS is similar to ,,initial stressmethod , i.e. the shape updating of the design structure san iterative process which consists of two stages: (i)generation of fictitious strain by some built-in rules (e.g.nonlinear constitutive elationship or the biological growthrules) and (ii) production of fictitious external oads by thegiven fictitious strain. In stage i), due to the definition ofthe fictious strain derived rom the simulation on biologicaladaptation, t is termed as the biological-growth train. Inthe followings, irst of all, the concept and the definition ofthe biological-growth train s given; hen, the algorithm orthe production of the fictious external loads termed as biological-growth strain analysis s derived.

(2) Brief Description on Biological-Growth train MethodIn order to optimally design he shape of the structure,

one optimal shape design method proposed y the authorss)was utilized to design he structures. Compared o otherconventional tructural optimization echnique, he featureof this proposed method was its ignorance on gradientcalculation, which could be regarded as he dominant actoron increasing he computational ost of structural optimiza-tion. The design objective of this rnethod s to minimize thestress oncentration within the design structure; which canbe expressed as

called biological growth strain was defined as follow

i0) , if element f *

_ 21

dghal ,hap0

3

{ Ph PL for dectile material:

#+

yh, if element e I*

Here, the design variables are the coordinates x1, 1)ofthe design oints selected long he design rofile -, where: 1..n. The reference tress s &; s nd the equivalent tressalong -is q. The algorithm of this method was proposed ysimulating the biological adaptation to their loadingenvironment like trees or bones; i.e. they change heirshapes y the growth or trophy of the living tissue near thehigh stressed rea. Based on this concept, one pararneter

Calcula

8Fig.2 COmputatiO,al low Of BGS

br b e matc 1 7h,r demcd jc121

Here j: jth design element within design domain f *; oi :

equivalent stress within jth design element; omear,t mean ofequivalent stress of all the design elements; o1: kth principal

stress, k:1..2; fr: uniaxial compressive or tensile strength;Vh: constant for search step. With the introduction of this

parameter, the shape of the design structure can be changed

by means of updating vectors of nodes coordinates (i.e. th e

shrinking or swelling of the design elements), which is

formulated as

[r]{ }: {4e} (3)

Here [K]: global stiffness matrix and {rr}, nodalcoordinate updating vectors. The equivalent nodal forcevector {dg} is

{ae} :/o

[B]' [D]{#}a O@)

That'is, he nodal coordinate updating vectors u} can beattained by solving he global governing equation superim-posed by Eq. (3) without figuring out the gradient of theobjective unction or constraint conditions. Therefore, he

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SEISAN KENKYUVol.49 No.4 1997.4)

new shape of the structure s obtained by adding he nodal

coordinate updating ectors o the old coordinates f nodes.

For brevity, the computational low of BGS is llustrated n

Fig.2. The termination of this terative computation an be

recognized when Eq. (1) is satisfied.

3. NUMERICAL ERIFICATION ON CORRECTNESS

OF BGS

After explaining haracteristics f BGS, the availability of

this method s necessary o be verified before applying his

method to general structures. To verify the availability of

this method, the design problem to find out the optimal

shape of the hole in a plate under biaxial stress was

considered. According to the elasticity theory6), the

tangential stress or the elliptical hole in an infinite plate

under biaxial stress s siven as

oe : o l

Here, k:

transversalintroduction

rewritten as

axis ratio (:bia; b: vertical axis and a:

axis) and 0: directional angle. With the

of the assumption 2/o1:k, Eq. (5) can be

follow

n, rigraF4

J

=2

J=

,

(b) rectangle hole

Fig.3 Sketch of FEM model for the hole in an infinite plate

under biaxial stress with different initial shapes

b)CaSe Bl

Fig.4 0ptimalshapes and EQS designed by BGS P/Q=1)

Table 2 Loading conditionand case abel

PiQ A B1 Al Bl

1:2 Z B2

= l 1+k)2 sin2 k 1+k)2 cos2

_k2_k

sin2 +k2cos2

: (1+k)o1

Thus, provided oz./or:b/a: k, the tangential tress

will be constant along the edge. According to Baud's

postulation, he shape with uniform strength distribution s

considered as optimum. Furthermore, to verify the

availability of BGS , different initial shapes of holes

including (u) square and (b) rectangle were used as

numerical examples as shown n Fig.3. Due to symmetry,

only one quarter of these structural models were analyzed.

The material property and oading condition with case abel

were given in Table.l. and 2 respectively. Due to ductile

material used n these ases, on Mises stress was selected s

the equivalent stress(abbr. s EQS). By applying BGS, the

optimized shape or different initial shapes under the equal

case 81

T: Dcsign Profilt

J

2

(a) square hole

Table 1 Material property

c* E*

1000 1000 7.00E+04x : urut kgul'cnr:)

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