Soko?owski J., Zolesio J.P. Introduction to shape optimization: shape sensitivity analysis

Jan

Sokolowski

Jean-Paul

Zolesio

lntroduction

to

Shape

Optimization

Shape

Sensitivity

Analysis

Springer-Verlag

Berlin

Heidelberg

Тokyo

Hong

Kong

NewYork

London

Paris

Barсelona

Budapest

П

ffit

бl

@

#

*.

чl

*rjф-.*u.d

Jan

Sokolowski

Systems

Research

Institute

PoIish

Aсademy

of

Sсlenсes

ul.

Newelska

6

01-447

Warszawa

Poland

Jean-Paul

Zolesio

Centre

National

de

|a

Recherche

Sсientifique

et Institut

non

Lineaire

de

Nice

B.

P.

7.l

Faоu|tё

des

Scienсes

06108

Niсe

Cedex

2

France

Contents

:.-

//)

6(i;

,.,

vi

M_aJhematiсs

Su

bjeсt

C|assifiсation

(

1

99 1

1

;

35Bз0,

49850,

73c6O,

73K4o,73т0ь

ISBN

3-540-S41Tl-2

Springer_Verlag

Berlin

Heidelberg

Newyork

|sBN

0-з87-54177-2

Springer-Vertag

шewYork

Ber|in

Heide|bero

Library

of

Congress

cаtaIoging-in-PubIiсation

Data

sokorowski,

Jan,

1949-

rntйou"tion

io.JllaЪЪ.*optimrzation:

shape

sensitivitУ

аna|ysis/J.

sokotowski,

J,

P zo|ёsio'

p

"'

lЬjing".

"",..""

in

сomputational

mathernatiсs;

16)

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bibIiographiсa|

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Contents

Chapter

1

Intгoduсtion

to

shape

optimization

Pгеfaсе

Chaptеr

2

Preliminaгies

and

thе

matеrial

сlerivativе

method

Domains

in

]RN

of

сlass

CA

Surfaсe

mеasuгes

on

Л

Funсtional

Spaсеs

Linеar

еlliptiс

Ьoundary

valuе

proЬlems

Shape

funсtionals

Shapе

funсtionals

for pгoblеms

gor,еrnесl

Ьy

linеar

еlliptiс

Ьoundary

valuе

proЬlеms

2.6.1.

Shape

funсtionals

for

transmission

pгoЬlems

2.6.2.

Approхimation

of

homogеnuous

Diriсhlеt

proЬlems

2.7.

Convеrgеnсе

of

domains

2.8.

Transformations

e

of

domains

2.9.

The

speed

mеtlrod

2.10.

AdmissiЬlе

spееd

vесtor

fiеlds

Уe(D.)

2.11.

Еulегian

derivativеs

of

shape

funсtionals

2.12.

}tron-diffеrеntiaЬlе

shapе

funсtionals

2.13.

Propегties

of

71 transformations

2.1

4.

Differ

entiaЬili

ty

of

transp

ort

еd

funсtions

2.15.

Derivativеs

for f

>

0

2.16.

Dеrivatives

of

domain

intеgгals

2.17.

Change

of vaгiaЬles

in

Ьourrdary

integrals

2.18.

Dегivativеs

of

Ьoundary

intеgrals

2.19.

Thе

tangеntial

divergеnсе

of

thе

fiеId

V

oп

Г

2.20.

Tarrgential

gradiеnts

and

Laplaсe.Beltrami

oреrators

on

Л

2.27.

Уarlational

proЬlепrs

on

Л

2.22.

The

transport

of

diffеrеntiai

opеrators

2.23.

Intеgтation

Ьy

parts

on

Г

2.24,

TЬe

tгanspoгt

of

Laplaсе-Beitrami

opегators

2.25.

Мatеrial

derivatives

1C

T4

1.7

1-

tl

20

29

31

32

Jb

4T

49

<2

o+

60

62

oo

to

77

IJ

81

0о

б.

89

90

o<

98

'

:.

:э.:.1

,:A

'..@и

2.30.

2.3I.

2.32.

2.33.

3.1.

3.2.

3.3.

cА

J.+. t

3.4.2

J.O-

J.O.

3.7.

3.8.

J.у.

C o

nten ts

2'26.

Мatetial

dегivativеs

on

Л

2.27.

The

material

dеrivative

of a

solution

to thе

Laplaсe

еquation

with

Diriсhlеt

Ьoundary

сonditions

2.28.

Strong

matеrial

dеrivativеs

for Diriсhlet

proЬlems

2.29.

Thе

matеrial

dеrivativе

of

a solution

to

the

Laplaсe еquation

with

Nеumann

boundary

сonditions

Shapе

derivativеs

Dеrivativеs

of domain

intеgrals (II)

Shape

dеrivativеs

on

Л

Dеrivativеs

of

Ьoundary

intеgrals

Chapter

3

Shapе

dеrivatives

for linеar pгoЬlems

Thе

shapе

dетivative

for

thе

Diгiсhlеt

Ьoundary

value proЬlem

The shapе

dеrivativе

for

the Neumann

Ьoundarv

valuе

proЬlеm

Nесеssary

optimality

сonditions

ParaЬoliс

еquations

Nеumann

Ьoundaгy

соnditions

Diriсhlet

Ьoundaгy

сonditions

Shape

sеnsitivity

in еlastiсity

Shapе

sеnsitivity

analysis

of the

smallеst

еigenvalue

Shape

sеnsitivity

analysis

of

the

Kiгсhhoff platе

Shapе

dеrivativеs

of Ьoundary

intеgгals:

the

rron-smooth

сasе

in

Shape

sеnsitivity

analysis

of

Ьoundary

value proЬlеms

with

singularities

3.10.

HypeгЬoliс

initia]

Ьoundaтy

valuе proЬlems

Chapter

4

Shapе

sensitivity

analysis

of

variational

inequalitiеs

4.l.

Diffеrential

staЬility

of

the

mеtгiс projесtion

in HilЬеrt

4.2. Sеnsitivity

analysis

of vaгiational

inеqualitiеs

in

IIilЬert

4'3.

The obstaсle

problеm

in н1(J7)

4.3.1.

Differеntiability

of thе

Nеwtonian

сapaсity

4.3.2.

Thе shapе

сontrolaЬility

of

the

frеe Ьoundaгy

4.4.

Thе

Signorini proЬlem

4,5' Variational

inequalitiеs

of

the seсond

kind

4.6.

Sensitivity

analysis

of the

Signorini proЬlem

in

4.6.1.

Differential

staЬility

of solutions

to vaгiational

in

HilЬеrt

spaсes

4.6.2.

Shapе

sеnsitivity

analуsis

4.7. Thе

Signorini proЬIеm

with given

friсtion

4.7.1.

Shapе

sеnsitivity

analysis

4.8.

Еlasto_Plastiс

torsion proЬlеms

4.9.

Elasto_Visсo_Plastiс

DгoЬlems

Rеfеrenсеs

spaсеs

100

101

r07

109

111

IT2

113

1t.f

111

rl I

118

119

12T

t27

128

133

135

r4l

146

lbtJ

153

10t

163

loJ

177

t79

184

l. бi)

193

196

205

20т

21.2

2r7

222

229

234

240

R,2

Notatior

Г

С].i

\r:

I

rl

elastiсity

inеqualities

100

Notation

]

iirт

1oп

..:.1t

loп

101

r07

.оll1епr

,гсlil1em

.i.

.:zrsе

in ]R2

D

a

Г:0{)

n

No

у@)

Xaotх

Q"

xn"

or

xc

lal

K

Char(D)

B

Bo

r(a)

po@)

.1\x),

B(x),

C (x)

-А(,)

Tt

v(,)(.)

DTt

It: 1(t)

M(T')

ott

:

a\t)

u(v)

dJ@;V)

G(n)

L;

s€)

I

1Г

ф

109

111

Lt2

113

115

1.1,7

118

119

'J.21

127

128

133

135

t4r

146

150

domain in ]RN

with

piесеwisе smоoth Ьoundary 0D

mеasuraЬlе sеt in

]RN or in D'

or domain of сlass Ce

Ьoundary

of f/

unit noгmal

vесtor

fiеld

on Л, outward to J.2

unitary

eхtеnsion of n to an

open

neighЬourhood

of Г in IRN

funсtion iл

W''p(О)

сharaсteristiс funсtion

of

o

:D\D(orп,N\D)

сharaсtеristiс funсtion of o"

N-dimensional

mеasure of o

mеan сurvature of Л

:

{X

с

L'(D)suсh

that

(t

-

x)x

:

0 a.е. on D}

unit

Ьall

in

]R,"

unit Ьall in IRN-I, Bo

С

B

domain funсtional

(or

сost funсtional)

pеrimeteг

of Q iп D

сontinuous matгiх funсtions

on

D

transpose of A(u

)

transformation of IRN or of D into IRN

:V(t,x)

spеed

veсtor

field

JaсoЬian of

Q

:

det(DTt)

М(T,)

:

1(t)-

DT,

1

:

||M(T1).n||р'

oп Л

:

+("

DV

+

DV)

Еulеrian dеrivative

shape

gгadient

dеnsity

of

thе

shapе

gradiеnt

the dепsity

gradiеnt

€

,1(lRN) a

(non unique) distгiЬuted reprеsеntation of

9(Л)

traсе operator on Л, e.8.

1г

€

L(H|

(o);/{+(г))

oЬstaсle funсtion

153

r57

:.:i)irсеs

:

S1)i1сеs

163

10t

1т7

179

184

lбD

193

196

205

207

21.2

217

222

229

234

240

Notation

Vд

a-

on

a

б"-

divд,

Ap

i(a;v)

уt(Q:V)

!

*Tt

уt(Г:V)

Cf

\И'

C.-(JZ;IRN;

tangеntial

gradient

on

Л

noгmal

dегivativе

on

Л

сonormal

dеrivativе

on

Г

assoсiatеd

to the

matгiх

,4

tangential

divеrgеnсе

on

Л

Laplaсe-Bеltrami

operator

on

Л

matеrial

deгivative

ot

у(a)

at

Q in

direсtion

of

thе

speed'

shapе

(domain)

derivative

of

у(a)

at

J-2

in

direсtion

of

the

spееd

fiеld

И

tгansportеd

distriЬution

Ьoundary

shapе

dеrivativе

of y(Л)

at

Л

in

direсtion

of

the

spеed

field

И

spaсe

of smooth

funсtions

(oг

of

vесtor

smooth

funсtions)

with

сompaсt

supports

in

o

fiеl

l.

Intг.

l.l

-

Рге

.1,-

--,r-

r='Ё

:L.

а

r

^<-

--,'--.--

.J]Е.

:-:.:..t:

'.

'

r:

.'-

"-!.

'

--

'.:f,

-,

:: -

:

--

.,.

;

j]

lIr сliгесtion

of

the

spеed

fiеli

l! atQin

l-

ztt

Л

in

i

:'t,.'161'

srnooth

funсtions)

1.

Introduсtion to

shape optimization

1.1. Pгefaсе

Тhis Ьook is motivatеd

largely Ьy a

dеsirе to solve shape

optimization

proЬ-

.еlns that arise

in appliсations,

partiсularly

in struсtural

mесhaniсs and in thе

,;ltimal

сontгol

of distriЬutеd paramеtеr

systems.

Many suсh

proЬlems

сan

Ье

:oгrnulated as thе

minimization of funсtionals

dеfined over a сIass of

admissiЬlе

iоrrrains.

Shapе optimization

is

quite

indispensaЬlе

in thе design and

сonstruсtion

of

llldustrial stгuсturеs.

For еxamplе,

airсгaft and spaсeсraft

havе to

satisfy, at

:hе samе time, veгy

stгiсt сriteria on

meсhaniсal

pеrformanсе

whilе

weighing

as

.itt1е as

possiЬlе.

The shapе optimization proЬlеm

for suсh a struсturе

сonsists

:l finding a

gеomеtry

of the stгuсturе whiсh

minimizеs a

given

funсtional

(е.g.

suсh as thе wеight

of

thе

struсture) and

yеt

simultanеously

satisfiеs

speсifiс

.:оnstraints

(likе

thiсknеss'

strain energy' oг

displaсеmеnt Ьounds).

Thе geometгy

of the struсtuге

сan Ьe

сonsidеrеd as a given

domain

in thе

:hгее

dimеnsional Еuсlidеan

spaсe.

The domain

is an opеn, Ьoundеd

sеt whosе

:opology

is

given'

e.g. it

may Ьe simply

or douЬly

сonnесted. The

Ьoundary

:s smooth or pieсewise

smooth,

so Ьoundary value proЬlеms

that

are defined

ln

the

domain and

assoсiatеd with

thе сlassiсal

partial

differеntial еquations

of

rlathematiсal

physiсs

arе well

posed.

Iп gеneгal

thе сost

funсtional

takеs thе

:огm of an

intеgral

ovеr the domain

oг its Ьoundary whеrе

thе intеgтand

depеnds

sпoothly on

the solution

of a Ьoundary value

pгoЬlem.

The

shape optimization

probl.еm

сonsists

in the minimization

of suсh

a funсtional with

respесt

to the

sеomеtriсal

domain whiсh

must Ьelong

to the admissiЬlе

family.

Мuсh

of the Ьook

is сonсеrned with

the shape

sensitivity anаJysis

for uni.

latеral

proЬlеms

desсribing

suсh

physiсal

phеnomеna

as

сontaсt

pгoЬlеms

in

еlastiсity, еlasto_plastiс

torsion proЬlems,

and thе oЬstaсle

proЬlеm.

An еlastiс

mеmЬranе in thе plarre

whiсh

is fiхed along

its Ьoundary

may

sеrvе as

a simplе ехamplе.

If a tгansvеrsal

foгсе

is applied, thеn

thе transveгsal

сlisplaсement

of thе

mеmЬranе in thе state

of statiс

equiliЬrium

is a sсalar funс-

tion dеfined

in thе

domain oссupiеd

Ьy thе mеmЬrane'

The speсifiс

funсtional

that has Ьееn

intensively

studied foг thе

purpose

of struсtural optimization

is

the сomplianсe'

i.е. the work

an external

forсe in that

proсеss.

That funсtional

*

1. Introduсtion to

shape optimization

pгovеs

to

be the integral of

thе

produсt

of

forсе aпd displaсemеnt

ovеr the

geomеtгiсal

domain.

If arЬitrarily

small vaгiations

of tlrе Ьoundary

of tlre geometriсal

domain

arе takеn

into aссount, Ьoth thе

funсtional

and thе foгсе's

displaсernеnt of

thе

mеmЬranе arе perturЬed.

To solvе

any shape optimization

pгoЬlеm

would

idеally mеan to

find the

minimum

_

whеnеvеr

it ехists

_

of a spесifiс

сost

frrnсtional over a

sеt of admis.

siЬlе domains. Howevеr,

it turns out that vеry

few adеquatе еxistenсe

rеsults

aге

availablе. In

genеral'

thе existеnсe

rеsults for suсh proЬlеrns

are oЬtainеd, pгo-

vidеd

that some unrеalistiс

сonstraints

arе

imposed on thе

family of admissiblе

domains. Thеse сonstraints

are

oftеn

not satisfaсtory

froпr the

physiсal point

of

viеw;

for examplе,

onе сannot peгform

tlre

modеIling of a

laгge oсеan wavе

by

a

grаph

without

exсluding the

interesting

phеnomenon

of a

Ьreaking wavе.

Thе сonstгaints undеr

disсussion also givе

no

гealistiс framework

foг the

dеvеl-

opmеnt of

optimization algorithms;

foг еxample,

onе is unaЬle

to

projeсt

onto

a tangent

сone to the

family of

domains having

a

pointwise

boundеd сurvature

of thе

Ьoundaгiеs.

In praсtiсe,

engineеrs

aге intеrеstеd

in

inсrеasing thе

stiffness of

a

plate'

impгoving

the dгag of

airсraft's wing'

dесrеasing

the weight

of a radiatoг,

еtс.

Мany studies

in thе field

of stгuсtural

optimization

have Ьееn

undеrtakеn

for

thе

last thirty

yеars.

Thе

results on

mесhaniсal formulation

of thе proЬlеms,

thеir funсtional

anа-lysis

and on сontrol

theory

havе rесently

Ьeеn сomЬined.

For a rеviеw

of suсh rеsults

thе rеadеr

is

rеfеrrеd to

(Haug

еt al. 1981).

Lеt us гесall

that a struсtuге

to

be optimizеd (сonsidered

in thе struсtural

optimization)

is

gеnегally

an assemЬlagе

of

different

parts

like Ьeams, piates,

shеlls, and

thгее-dimensional

Ьodies.

In this

Ьook we

rеstriсt our

attеntion

to

the

сontinuous

formulation of

suсh optimization

proЬlems

for

gеneгal

elastiс

struсturеs.

In suсh

proЬlеms

the sensitivity

analysis plays

a

сепtral ro1е

arrd was

in-

tеnsivеly

studied

in thе

1980s Ьy thе

authors.

Thе first

results сonсerning

the

diffеrentiaЬility

with

rеspесt

to

pеrturЬations

of a gеometriсal

domain wеrе

oЬtainеd

Ьy

Hadamard

for thе first

eigепvaluе

of a

mепrЬranе;

assuming that

tlrе Ьoundaгy

of a domain was

smooth,

hе

usеd

pегturЬations

of thе Ьound-

ary

along thе

normal dirесtion.

This

tесhnique

сannot

bе ехtended

to morе

gеnеral

situations'

in

paгtiсular,

to

domains with piесеwisе

smooth

boundariеs.

A

straightforward

approaсh

to

pеrturЬations

of

geometriсal

domain

сonsidеrs

hypographs

in the

Cartesian сoordinates.

In this сasе

linеar funсtion

spaсes

сarr Ье used

to

parametrize

thе

domains.

Мaking

usе of suсh

a

parametriza-

tion onе

сan oЬtain thе

eхistеnсе

rеsults

for thе shape

optimization problеms

in a standard

way.

A moге general

sеtting

of

pеrturЬations

of

geometriсal

do-

mains is to

сonsidеr a onе_parameter

family

of smooth

transformations

of

thе

N_dirnеnsional

Еuсlidеan spaсе.

That gеneral

setting

fails

to have

any linear

stгuсtuге

for transformations,

however

thе assoсiatеd

vесtor

field

(whose

flow

Н.

brl

I

f

a

t

b

lrr

rl{

&r

*.

tr

J

ta

Frh

ш{

at

;{r

t'l.nh.

tt.

*

iйrr

lr eЁ

l.1. Prefaсе

.iislllaсement

ovег

thе

].t, gеomеtriсal

domain

t,.s

сlisplaсemеnt

of

thе

..

: :.i"itllr'

lnеan

to

find

thе

:....,,iriL]

o\.еr

a

sеt

of

admis-

..

:

iii.(.ехlstenсе

rеsults

arе

:

: .r,ill.s

are

oЬtainеd'

pro-

.

:.

.]..,

farr.rily

of

admissiblе

:.' ::(]lll

the

physiсal

point

'

..-:.: Lrf

a

large

oсеan

wave

:...

:.

,]l

of

a

Ьreaking

wave.

..

::l.lllсrr'ork

for

thе

devel-

:.'

.t

iillal]le

to projeсt

onto

.:.:

l','i.е

Ьoundеd

сurvature

.:.:

.]r.:

stiffnеss

of

a

plate,

.'

.,.''.-lч}rt

of

a

radiator,

еtс.

. .^:.-''.,

llееn

undertakеn

for

;:.. ..l..iiitt

of

thе

problems,

...

:.-

.,

l]тh.

Ьeеn

сomЬined.

li ,

r. et

al.

1981).

.

:..-

..,:.t'сl

irr

the

struсtuгal

.i ]

':l..

likе

beams,

plates,

.i.-:',':гit:t

our

attеntion

to

:

...:ll.

for genегal

еlastiс

.l

'.:..l.ill

гolе

and

was

in-

:.:..

:.,sttlts

сonсегning

the

.' ::^-tt'iсai

domain

wеre

:...

:..:]l.i1llе:

assumin8

that

-:.:::ilTioпs

of

the

Ьound.

.|.

.'

Ilt'

ехtеndеd

to

morе

..'-ii

...,

stlrooth

Ьoundariеs.

::..-.

l:.'ir]

сlomain

сonsidегs

:.t'-

.:l..'аrг

frrnсtion

spaсes

: .l1..}1

a

Paramеtriza-

..

'-

,:,tiIlrization

proЬlems

.:..

:.:

()f

geometriсal

do-

'.. .:i1:.5foгmations

of

the

..

].:.. trl

have

any

linеar

.

..'

.:.)l.

fiеld

(whosе

flow

iЭ

i@)

:

ff@,'

:

у,@)

+

ffGo"'_

j#.h,).

:s the transformation)

doеs possess

the

lineaт

struсtuгe

of

an appropriatе

linеaг

spaсe. Usual

сonstraints

on

thе gеometriсal

dоmain

сan Ье

еasily

taken

into

.1ссount

Ьy

thе сhoiсe

of a

linеar

suЬspaсе

of

the veсtor

fiеlds.

In this

tесh-

:lique,

fluid meсhaniсs

dеvelopments

havе

Ьеen

of

great

importanсе

foг

dеriving

:hе shapе

analysis

used

in this

Ьook.

For

ехampiе,

the

prеsсriЬеd

volumе

of a

Eеometriсal

domain

is

prеserved

foг

thе

pеrturЬed

domains

providеd

that

the

...есtoг

field

is divеrgenсe-frее.

Thе

differеnсe

from

fluid

meсhaniсs

is that

thе

:llatеrial dеrivative

of

a solution

of

a

boundaгy

valuе proЬlem

is not

suffiсient

Ъг thе sеnsitivity

anаJysis

of

shapе

сost funсtionals.

In

thе Ьook wе

introduсе

the

shape

dеrivative

of solutions

to

linеaг

and

uni-

.ateral

boundary

valuе pгoЬIеms.

Thе

shapе

derivativе

dеpеnds

only

on

thе

nor-

:llal сomponеnt

of thе

vесtor

fiеld

on

thе Ьoundary

of

the gеometriсal

domain.

Тhеn thе

boundaгy

value proЬlem

whose

solutions

givе

the

shape

dеrivative

is

.:haraсterizеd.

In

paгtiсular,

foг

the

linеar

сasе

thе

ехistenсе

of thе

so-сalled

.}rapе gгadiеnt

is

proved

and

its

foгm

is dегived.

Lеt

us Ьegin with

a

гeviеw

of somе

сonсepts

usеd

in thе

Ьook,

suсh

as

thе

rlatеrial derivative,

thе

shapе

derivative

of

solцtions

to

PDE's,

thе

fiгst

and

:есond

ordеr

derivatives

of

domain

funсtionals.

For

thе sakе

of simpliсity

wе

геstriсt

our

сonsidеrations

to

the

domain

of

:ntе8ration

of ordinary

diffегentiai

equations.

Let

us сonsidеr

a simplе

eхample:

9

:

(0,с)

is a domain

in ]R

and

y

dеnotеs

thе

solution

to

thе

following

boundЪry

r.aluе

problеm

o

d-у,

,

-i;@

+

у@):

sinr

u

€

(0,

o), y(0)

:

У(o)

:

0.

It is assumed

that pеrturЬations

of

с are

in thе

foтm

o

|tu,

t

>

0, and

the

napping

71

fгom

J.2 onto

a1

:

(0,a

*

tu)

is dеfinеd

Ьу T1(a)

:

фс.

The

so]ution

у1

on

a1

is

givеn

Ьy

ut(z):-:":1"

*l']shz

+

lsinz

r

>

o.

Zsn(с+rDJ

2

_

Let

уt

Ье

the

solution

tгansportеd

to

the

fixed

domain

J.2:

tm

a

:ato!t

1sin(o*lu)

' ,a*tu.t,

1^:-uа*tu

:

-t

rL(.

+

tr)sh(-r)

*

,sin(J-r)'

Тhen thе

shapе

dегivative

takes

thе

foгm

tl t

0у,,

у\-l_

at\&l|t:o_

2

(sho)2

shu

u shс

сosс

_

sino

сho

rvhilе

thе

matеrial

dеrivativе

is given

Ьy

1.

Intгoduсtion

to

shape

optimization

It

сan

Ье

vеrifiеd

tЬal

у|satisfies

d,у,

t

-

d',

tу

:0,

y,(0)

:0'

у,(o)

:_ushс

сoso_sina

сho

,2shс

while y

is

the

solution

to

the

following

equation

d2i

u

.i7

+у

_

__1osx,

уФ):

y(с)

:0.

Both

!

and y,

depеnd

linearly

on

tl,

Ьut y,

dеpends

only

through

the

Ьoundary

valuе.

Let

us

сonsidег

thе

variational

inequality

assoсiated

with

minimization

of

thе

foliowing

quadratiс

funсtional

7lltv.1 /)6\,

,

I(ф):

Jo

L;|;)

-

fо]a'

ovеr

thе

сonvех

suЬsеt

Kt

:

{ф

с

11t|фФ)2

0}

of

thе

HilЬeтt

space,l71 :

{ф

с

f1l(0,

1

+

tu)|фQ*

fu)

:

61.

Thе

minimum

of

1(.)

ovеt

K1

is

aсhiеvеd

at

the

uniquе

elemеnt

z1

iп

K1

сharaсtегized

Ьy

the

following

vaгiational

inequality

zt

€

Kt

.

[,*,"

dzt

(

dф

_

d,,

\

"-

.

f7+ta

Jo

'(;_;)o"ZJo

f(ф_z1)dt

VфСh,

Thе

minimiz..t

21

СZII

Ье

evaluatеd

еxpliсitly

using

thе

solution

z1

to

thе

linеaг

proЬIеm

assoсiatеd

with

minimization

of

r(.)

ou"|н,'

d2u,

а,z(l):

f(.r)

t

€

Qt:

(0.1{iu1

suЬjeсt

to thе

Ьoundary

сonditions

ffO

-

o.

,rr(1

*

ru)

:

0

It

сan

be

vеrifiеd

that

d(r)

:

1_

ff,o,

an

elеmerrt

of,

Н1, is orthogonal

to

the

suЬspaсe

Hi@');

}/1

is

еquippеd

witlr

thе

sсalаr

produсt

['*,'

dф

,

J,

d'

d,d'

:

(o'ф)н,

Еaсh

еlement

ф

с,}7t takes

the

form

ф

:

Ф(0X1

_

тi.l

I

ф,

fotsome

ф

€

Hl@t).

Thе

mеtriс

projесtion

P1

irt

111onto

.I,(1

satisiе,"л,(,р) :

ф.

Thus

Pt(ф):o_min{o.

atoD(t-.+,

)

:

E- |oФl1-(.r-, )

\

1*ful

L'\-'r

\-

I*tu)

tlшr

!l

r*i,

t&Е

ilйЁ.Е

!,r'

t

l;

=,,

с т!-

Soko?owski J., Zolesio J.P. Introduction to shape optimization: shape sensitivity analysis

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