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

(rs(]

-

sino

сho

.

:].гough

thе

Ьoundary

r

'.','ith

minimization

of

l,r'гt

spaсe

,]1,:

{ф

с

.;liе

е]ement

z1

in

K1

...

r1r

Yф

с

Kt

.

..:.ion

u1

to

thе

linеaг

orthogonal

to

the

-

r.'.

for

some

ф

6

i..

1:

ф.

Thus

_.-t

lItu/

sho

1.1.

Prеfaсе

.1:1d lve

Ьave

z1

:

Pt(ut).Hеnсe

zt(x)

-

ut(x)

-

[r,(o)]-

(r

- -{-)

\

!+tu/

Тirеrеfore'

thеrе

еxists

the

shapе

dегivativе

t, , ,.

1.

z

t"t:

u)

:

,iftr

7(21(r)

_

zs(-r))

i.s rr.еI1

as thе

matеrial

derivative

2(t;u)

=

ЧP]t(..

oT)(x)

-

zo(x))

rl0

r

^.

is assumed

tЬat

z{x):

0

for

r

2

1,

+

tu.

To

makе

ouг

сonsideгations

moгe

spесifiс

we

supposе

that

z9(0)

:

0.

If

ut

i.,lrotеs

thе

shape

dеrivativе

foг

the

linеaг

proЬle-,

tЬ",,

u,

is

linear

with

respeсt

:,]

l.whеre

U:1,

or

u:

_1.

It

foliows

that

,, \ ..

1

-.

(;г;u)

:

ti,17(,,{,)

_,o(.))

:

Цlll}((л,"

t)(r)

-

(Pgug)(r))

,to

I"

:

u*](,,(,)

-

[,,(о)]_

(t

- -'

\

,rot'*.

\

1+t')_ug(a)1-["o(о)]-(1_,))

:

u'(x')

-

[u'(0)]-(1

_

r)

.

:':or.idеd

that

we

assume

u,(t)

_

0,

for

с

2

t

+

tu.

. ..

Si'.:

thе

tегm

[",(0)].

:.min.{,O'_",(0)}

is

not

linеar

with

respесt

to

r,l,

it

:

,jlorvs

that

thе

shapе

dегivаtive

z|(x;r,)

faii'

to

Ье

linear

with

respесt

to

u.

The

.аme

argumеnt

appliеs

to

the

matеrial

derivativе

2(xiu),

on

the

other

hand

wе

':)Ialn

z'(x;u):

u'(r)

for

u6(0)

<

0

z'

(r;

u)

:

u'

(x)

-

u'

(o)(t

-

x)

for

z6 (0)

>

0

.

.еt

us

сonsidеr

a

сost

funсtional

to

Ье

minimized

rvith

respесt

to

the

domain

I].

Тhe

simplеst

example

is

as

follows:

J(t\

-

L

['*'",",

(t):;J,

@1

-уa)2dr

l>0

Тhеn

the

semi_dеrivativе

is givеn

Ьy

r,(o)

=rix,]r.rrtl

-

r(o))

,+:'

ш

10

1.

Intгoduсtion

to shape

optimization

гl

:

(u(1)

-уd(I))zu+

l

@_aa'1u'dx

Jo

It is

oЬvious

that

"r,(0)

is

linеar

with respесt

to u. Let us introduсе

the adjoint

state

d'p,

in

_

-(f,)

:

u(.т)-Уd(.r).

#(0)

:p(1) :0

.

drz'

dt'

lЕ*:

i

5Еt

hЕ

tsr

*;rsf

тh.

_

Е{*Еt

|rfв.tт

щ'::

t-:

Hеnсе

wе have

.1

.r,(0)

:t("(r)

_уdFD2*

| #pl!p1а'1".

Jo

сII

aI

Thе

sесond

dеrivative

is given by

d2J

.,. ..

r.

-,

;(o)

:

lim;(J'(r)

-

/'(0))

dt"

t10

т

:

u|_.2уaп)u,1t1

+

"*,,lfftл

*

|,,

(

*Ц

#

-

##)

o",

Hеrе

p, dеnotеs the

shapе dеrivative

of the solution

to thе adjoint

еquation. In

the сasе

of the unilatеral

proЬlem

with thе funсtional

гl+fU

JG)

:

Jo

(z1

_

у1)2

dr.

.I,(0)

:

u(z(I)

-

уа(\))'

+

[,

pp1

-

у1(x))z,(r;u)dr

Jо

If

thе mapping

,r,'

--+

",(.,u)

fails to

Ьe

linеaг,

thеn thе funсtional

.I(.) is

not

diffегentiaЬlе

at

f

:

0' and thе adjoint statе

cannot Ье introduсеd.

Lеt us

сonsidеr the

gеnеral sеtting of the shapе optimization

proЬlеm. Suсh

a

pгoЬlеm сan

Ье formulatеd as

thе

following

miniпrization

proЬlеm:

9*

с'Uoa

: J(Л-):

nlд!""r(a)

(1.1)

whеrе thе

set

|/o4

of admissiЬle

domains in IRN' l{

:

1,2,3,...,

inсludеs

the

сlasses

of all admissiЬle

geomеtriеs

for the

proЬlem

undеr сonsidегation.

Usually

the

сost funсtional

J(.) takеs thе form

J(a): h(a,у(a))'

where thе еlеmеnt

y(JZ)

is

givеn

as the solution

of a Ьoundaгy

value

proЬlеm

wеll

posed

in thе

domain

J./. In

general, thе elеmеft

у(a)

Ьеlongs to a funсtional spaсe

W(n)'

usually a Sobolev

spaсе of funсtions definеd

in thе domain o. Thе

existеnсе of

a solution

to the

problem (1.1)

is еnsuгed,

provided

thе

sеt

Иoa

is endowеd

with

a

topology

(сonvergеnсe

of domains) suсh

that

(i)

the mapping

9

--+

J(О) is lower

semiсontinuous'

and

(ii)

thе setUoа is сompaсt.

Efrs.

:

Пк

::-

E::

I

pв:в.

:

tЬ

с-o:..

f

:":г

:.:

Lr

:".

',*r

3..:с

:

l}

}т..iо

lЁ

q;

':.

it#'т.::..

.}..:Ё

:..:'

Э

С:.а

l*.*з:-.

1.

Lr*:.:::-l

'

llrlЕ.iъ....

I

'r-'

]rrq.:а_:.Ё

i:е

aсljoint

. 1S not

',::r.

Suсh

(1.1)

l.

it.s

thе

.

:.'-

.,]епlеnt

.

..,'i

irr

thе

'',,'.:

I1-(J7),

.-:':::tепсе

of

:.

1

,.'r.t.сl

rvith

l.l' Prеfaсe

l1

Roughly

speaking,

thе сompaсtnеss

of

thе sеt

of admissiЬlе

domains

|,!oа f'ol-

lows' if for eхample pointwisе

сonstraints

on

thе aЬsolutе

value

of thе сuтvaturе

of thе Ьoundaг}

Л:

0a are

presсгiЬеdfor

any

admissiblе

domain

Q

с|,loa.

In

ordеr to ensurе

thе

ехistеnсе of

an optimal

solution

to thе

proЬlem (1.1),

thе

standard approaсh

of сontrol

thеory

is

proposеd,

i.е. a regularizing

term

сan

Ье

introduсеd.

Therеfore

the сost funсtional

takеs

the form

J"(a):J(n)+аE(a)

o)0,

rvhere

thе rеgularizing

tеrm

the

sеt

Е(o) еnjoys

the

following pгoperty:

thе сlosure

of

{alE(a)

<

M}

is сompaсt;

M

>

0 is a

сonstant.

Thе

main

part

of this

Ьook is

сonсеrnеd

with

thе shapе

sensitivity

analysis

of

thе mapping

9

--+

J(9).It

is supposеd

that

thе Ьoundary

Л of thе

domain

J2 is

piесеwisе

smooth.

Thе infinitеsimal

peгturЬation

J-ll of thе

domain

J7 is dеfinеd

as follows:

a'

:

T49), whеге

Q

is a smootlr

onе_to_onе

пrapping

dеfinеd

in

a

nеighЬourhood

of D.

Thе

mapping

T1 сan Ье

сonsid.eгеd

as thе

flow of thе veсtor

field и(t)

:

*

oT;',t>

0; this point

of viеw

has Ьeen

introduсеd

in

(Zolesio

1976). Thе

Еulerian

dеrivativе

of

"r(.)

in

dirесtion

У(.,

.) at t

:

0 definеd

Ьy

u-]1-l1о,1

-

J@))

tl0

t

depеnds

on the

vесtor

fiеld и(0).

All the tгansformations

[

сonsideгеd

in

Chapt.

2, can

Ье

сonstruсtеd

using vесtor

fields

in

C0([0,e); и&(D))

(seе

Sесt. 2.10).

Foг a

diffеrеntiaЬlе

doпrain

funсtional

the struсtrrrе

thеorеm (Theo-

rеm

2.5)' whiсh

definеs

thе general

form

of

thе shape

gradiеnt,

is

givеn

(following

(Zolеsio

1979))

Ьy

G(9):

tik")

.

Thе

gradiеnt

is an

еlemеnt

of

thе spaсе

of

distгiЬutions

2,(IRN;

IRN),

supported

on

thе Ьoundary

Г

=

ОQ'

In order

to oЬtain

the eхpliсit

form

of G(o)

one

has to

dеfi.ne:

thе

matеrial

derivativе

Й(a;V),

thе shapе

derivativе

у,(Q;V),

and thе

Ьoundary

shapе

dеrivative

уl(Г;У).

The foгms

of

thosе

dеrivativеs

are dеrived

for elliptiс,

paгaЬoliс'

and

hypeгЬoliс proЬlепrs.

The

material

and

the shapе

dегivativеs

arе givеn

as the

uniquе

solutions

to

thе assoсiated

partial

diffeгential

equations.

In

partiсulaг,

thеsе

еquations

involve

tangеntial

opеrators

on thе

boundary'

е.g.

thе Laplaсе_Bеltгarni

operator.

In

Chapt. 3

thе neсessary

optimality

сonditions

for

proЬlem

(1.1)

arе stated.

Мoгеovеr, related

геsults

on thе

shapе

sеnsitivity

analysis

foг

iinеar

proЬlems

inсlrrding

systеms

of equations

of

linеar

elastiсity,

the

Kiгсhhoff plate,

multiplе

eigenvalue proЬleшrs'

heat

transfеr

еquations'

and wave

equations

are

prеsentеd.

Chapt.

4

is сonсегned

with

thе

shape

sеnsitivity

anаJysis

of variational

inequalitiеs.

We present

rеlated

rеsults

on

thе diffеrеntial

staЬilitv

of

the

Г

|2 1. Introduсtion to shapе optimization

mеtriс

projeсtion

in HilЬert sPaсes due to

Harauх

(1'977),

Мignot

(1976)'

Sokolowski

(1981с;

1985a,Ь,с; 1986Ь; 1987Ь;

1988Ь,с,d)'

Sokolowski

and Zole-

sio

(1985a'Ь;

1987a), and Zolеsio

(1985Ь).

For other

rеsrrlts

on

the shape optimization thе reader

is rеfеrred to thе

monographs by Baniсhuk (1983)' Haslingег et

al.

(1988)'

Haug et al. (1981),

Haug еt al.

(1986)'

Pironnеau (1984)'

Troiсki et al.

(1982)

and

Zolеsio

(1988).

Wе

also

provide

a list of rеferеnсеs at

the end of this book.

Thе геsеaгсh of the

fiгst

author was

sponsored Ьy thе Systеm

Rеsearсh

Institutе of thе Рolish Aсademv of

Sсiеnсes undeг thе

Researсh Prosramme

сPBP 02.15.

P]€fli

tr

f

il

fh{

Lсt

(l

s

Dlx

}к

!(

drlr

I

L

S:r

Пt{

Пii

h*т

'

ht.rr

!Ппi

.r;r

Г

L

Sra"t

*ar

'llш

|'ш

ilЦ:l

aki

tit

.lrB

l.l:*пot

(1976),

'',

til

aпd Zolе-

,:r,ггеd

to thе

,.:

al.

(1981),

.,.sio

(i988).

',:ll

Rеsеaгсh

- Progгamme

2. Prеliminaries

and

the

material

derivative

method

This сhapter

is сonсегnеd

with

mathеmatiсal

mеthods

usеd

in the shapе

sensi-

tivity analysis.

In

partiсular

in Seсt.

2.9 thе

so_сalled

matеrial

dеrivativе

and

thе

speеd

mеthod

arе

introduсеd.

The

iattеr

is appliеd

in

Chaps. 3

and

4

for

the

shape

sеnsitivity

anаlysis

of

the Ьoundary

valuе proЬlеms

of

elliptiс

type

as

wеll

as for

thе initial

-

boundaгy valuе proЬlems

of

paraЬoliс

and

hypеrЬoliс

typеs.

In

Chap.

4 the spееd

mеthod

is

usеd for

thе

shape

sеnsitivity

anаlysis

of nonlinеar proЬlеms

of elliptiс

type.

In Sесt.

3.3

of Chap.

3, the

nесеssaгy

optimality

сonditions

for a

modеl

shape optimization

proЬlem

aте

derivеd

using

thе

speеd

mеthod.

In this

сhaptеr

wе

desсгiЬe

mathеmatiсal

toois

that

сan

bе

usеd

to

pгovе

the

ехistenсе

of solutions

to

rеlatеd

shape optimizatioл

problеms,

е.g. thе

notion of

the

pегimeter

of

a Ьoundеd

domain

in

]RN (see (DеGiorgi

еt

aI,

|972))

is disсussed.

In Sесt.

2.7 we

intгoduсе'

following (Мiсhеlеtti

1972),

thе

notion

of

the сonvеrgеnсе

of

domains

that

еnsurеs

е.g.

thе

сonver8enсe

of

nor_

mal

vесtoг

fiеlds

on the

Ьoundaries, аs

wеll

as the сurvatuгes

of

thе Ьoundariеs,

еtс.

In

Sесt.

2.1 thе

domain

o

с

IRN with

thе

Ьoundary

Г

:

О9

is defined.

The notion

of an

intеgral

on the

marrifold

Г

:

ОQ is

disсussed

in Sесt.

2.2.

Funсtional

spaсes

usеd

in thе

Ьook

are

ехaminеd

in Seсt.

2.3,

in

paгtiсulaг

thе SoЬolеv

spaсеs

(sее

е.g. (Adams

1975;

Lions еt

al.

1968))

are

introduсеd.

In Sесt.

2.4 tЬe

notion

of weak

solutions

to еlliptiс

Ьoundary valuе

proЬlеms

is

investigated

using

Stampaссhia's

theorеm,

and

thе

Laх_Milgram

lemma

in

thе

symmetriс

сasе.

Several

eхamples

of

the

sесond

orсiеr

and

thе

fourth

or-

dеr еlliptiс

proЬlems

rеlatеd

to appliсations'

е.g.

in struсtural

rnесhaniсs,

are

given.

In Seсt.

2.5

the

notion

of a

funсtional

dеpеnding

on

thе domain

o

с

IRN

is introduсed.

In Sесt.

2.6 a

shapе

funсtional

for

an elliptiс

Ьoundary

value

proЬlem

is dеfinеd.

The notion

of

thе perimeter

is usеd

to

dеfine

a regul.arizing

term

oссurring

in the

eхprеssion

of

thе shapе

funсtional (seе (2,46))

for the

transmission proЬlem.

Thе сonvеrgеnсе

of

minimizing

sequеnсеs

for

a

modеl

shapе

optimizatioп proЬlem

is

studied.

Seсt.2.7

is

сonсеrnеd

with

the

anаlysis

of thе

сonvеrgеnсе

of domains

in IRN;

for

this

Purposе

the

mеthod

intгoduсеd

Ьy (Мiсhеletti

1972)

is applied.

Furthеrmoгe'

undеr

some

spесifiс

assumptions,

the expliсit

form

of onе-to_one

transformations

of

domains

in

]RN

is derived.

In Seсt.

2.8 familiеs

of pеrtuтЬations

of

a

givеn

domain

in ]RN

are

definеd.

Suсh

a family

сan

be dеfined

in a

numЬeг

of ways'

in

partiсular

that proposed

by

b

|4 2. Prеliminariеs and thе matеrial dеrivative method

Hadamard

(1908)

is

pгеsentеd.

In Sесt. 2.9 a

geneгal

mеthod

of

dеfining suсh a

family is desсriЬеd. Using thе speеd mеthod, the shape gradiеnt of

a

given shapе

funсtional is dеfinеd in Sесt. 2'11. In Sect.2'|2 thе сasе of

multiplе eigеnval-

uеs

is

studiеd

with

thе use of non-smooth optimization

tесhniquе. Diffеrеntial

propеrtiеs

of thе mapping T1 assoсiatеd with thе speеd

mеthod aге oЬtainеd in

Sесt. 2.13. Sect, 2.I4 dеals with thе

diffеrentiaЬility'

with

rеspeсt to the

spaсe

variaЬlеs,

of the сomposеd funсtions

/

o

T1 for

given

mappings

"f

. ]RN

--+

IR or

given

distriЬutions. In Sесt. 2.15 additional

properties

of thе mapping

T1 are

oЬtainеd. In Seсt. 2.16 thе speеd method

is used to dеfinе thе dеrivatives

of

domain

integгals

in

thе

dirесtions of

vесtoг

fiеlds. In Seсt. 2.18 thе dеrivatives

of

Ьoundaгy intеgrals arе dеrivеd. Tangential

diffеrential opегators on Л

are

dеfined in Sесts. 2.19 and 2.20.

Sесt. 2.29 is сonсernеd with еlliptiс

pгoЬlems

on

thе marrifold Г

:

0Q. Seci.

2.22 deals

with

the transformation of

differеntial

opеratoгs, aссomplished Ьy

means of thе mapping

T1 assoсiatеd

with

the spеed

mеthod. Formulае usеful

in

thе

shapе sеnsitivity analysis of partial

differеntial

equations are

givеn.

Thе notion of tlre

matеrial dегivativеs of funсtions

dеfined:

(i) on thе domain

o,

(ii)

on thе Ьoundarу

Г

of

Q

is introduсеd in Seсts.

2.26 and 2.26' rеspeсtivеly.

Thе notion of thе

shape deгivative

is

prеsented

in

Sесt. 2.30. Finally'

in

Sесt. 2.31 thе

notion of thе shape

derivatives of

funсtions dеfinеd on thе

marrifold

Г

:

0Q is introduсеd.

In Chaps. 3 and 4 the

shapе dеrivatives of solutions

to

speсifiс

Ьoundary

value proЬlems

will

Ьe

сonsideгеd. It

will

Ье

shown that thеsе

shapе derivativеs aсtually

depend on

thе normal сomponent

of thе spеed vесtor

field on Г

:

0a.

This

property

of the shape derivativеs

is сruсial for the

shapе

optimization.

2.1. Domains

in IRN

of сlass C&

We

dеnotе Ьy J2 an open

set in IRN whiсh

is

gеnеrally

assumеd

to Ье Ьoundеd;

hеnсe

D

is сompaсt.

Г denotes

the Ьoundarу

of Q : Г

:

Q\

o. Morеover

it is

assumed that

o is a smooth

domain of сlass СA:

Г is a Ck mаrrifold

ald a is loсatеd

on one side

of Л; ioсal сoordinatеs

are

dеfined as follows:

thеrе ехists

a family

Оt,...,О-, of opеn sets

in ]RN and

mappings с; from

(Эi

oл|o

B

:

{€

:

(€r,...'(N_r,€n.,)

с

]RN suсh that

||(||n"

S

r},

с; is a onе_to_onе

mapping,

c;

с

Ck(О;;п"N; with

с11

с

сь(B;п.N)

Тr

a-"т

3

}*lш.'*

lrs

S',:

}Ь

=

*,'.

(-:*: i

fli:fr

*Еr?.

i

=

}ш

э..*''-

lв

т!:

t

*

ъ'*:

Ll.

.Т:

]Ъ.

:,:r:

Г*.t:*:

rlГ

]l:r-

*-i

.r.'":

ir*.

*,-16*

i.:

lс..

"t'

,

*:.:

-

'":,

!'s.'.!,,,|j'fr.i:.1+у;'..J''!,?;'/|./}t',t/.r./|,'i,1,:.'€^ё'1'?:y.:'.:."-:..'

J

3l).

Гinally'

in

i

'':.

tile

пranifold

I :olutions

to

.:..

.'.'lr

t]rat thesе

:

.'.'-

tlrеес1

r'есtor

-:,. :

,г

the shape

.

'

i,.'}lсlundеd;

'

].ILlгeоr'er

it is

..

,l;гсiinat€S

oГе

-,'-

irr

IRN and

:

f

.

2.1.

Domains

in IRл

of сlass

Ce

15

and

с;(О;О

Q)

:

B+ =

{(

с

Bl1ш

}

0},

c;(О;o

Л):

Bo =

{€

с

Bl€ш

:0}

.

The

family

О;,

i

:

I,2,.

, .

'

m)

сovеrs

D,

it

mеans that

aс

U

О;

i=1.....m

It is also

supposed

that thеre

is

givеn

a partition

of

unity rt

с

Cf;(O;)

suсh

that

0

S,;-S

1 and

!},

ri:I

on Л.

Thе

loсal сoordinates

on

Л

arе

definеd

Ьу h;:

сn

,.

Hеnсe

foг any

e Ьelonging

to

ЛП

Оi

we

can write

u

:

h;(€)

whеrе

С

:

cl(x).

The tangent

linеar

spaсе

T'Г

to Г

at a

is spanned

Ьy

the

N

_

1

vесtoтs

Dh(().e;,

i:1,2,...,N

-

1,

I

whетe

e;, i

:

7,2,.

.

., N, is

thе

сanoniсal

Ьasis

of

]RN,

"n

:

(ilij,

0' .

. ., 0).

It

сan Ьe

еasily verifiеd

that

thе

vесtor

-Dh(1;-t.ед',

is orthogonal

to

thе tangеnt

space

T'Г:

\Dh(с)

.

ei,- Dh(()-l

.

e,.r)в'

:

(e1,

eд)р,v

:

0

it might

Ье

well

to point

out

that

thе l/

_

1

veсtors

тl

:

Dh(€)

,

e;, i

:

1,2,

. . .,

N

-

1,

whiсh

form

a

Ьasis

of thе

tangеnt

spaсе

T'Г

,

are

not oгthogonal:

(т.,тi)р.

:

(-Dь(()

.

Dh(€).

e;' e1)р'

The

normal

veсtor

fiеld

m on

Л сan

Ьe

dеfinеd'

as

follows

m(x)

:

i,i

Dь(,o('))-'

.

",

i=

1

and thе

unitary

noгmal

fiеld

on

Л is givеn

Ьy

n(r)

:

||m(с)||*''m(r)

.

Furthеrmorе

wе

assumе

that

the vесtor

field

n(t,),

z

€

Л, is

outward pointing

on Л.

It is

impoгtant

to oЬservе

that the

vесtor

fiеlds

m and

n

are

not only

dеfined

on thе

Ьoundary

Л,

Ьut

m is a].so

dеfined

in

(/:

\Jtt,...'*О;,

[/

is an

open

neighЬourhood

of

Л. Wе

dеnote

Ьy

,Ц/g an

unitary

extension

of thе

noгmal

fiеld n.

,A/6 is dеfinеd

in a

neighЬourhood.of

D in

IRN.

In

faсt,

n(с)

is only

definеd

in

the

nеighЬourhood

tJ|

of' Г

in IRN'

U,

:

{x

€

[/

suсh

tЬat

m(l)

l

o)

bb.

16

2. Prеliminariеs

and

thе matеrial dеrivativе

mеthod

Lеt us

сonsidеr

ro

€

C-(]RN)' 0

{

16

S

1' 16

=

0

on'IRN\U, andГo

:

1

on(J|t

1

[/,, is

an opеn sеt' U|

)U|| )

Л; lеt a

vесtor е

с

IR," Ьe

suсh that

||e||pш

:1.

Thеn

the unitаry

vесtor fiеld,^/o is dеfined

as follows

М(,)

:

(1

_

16(r))е

*

rg(c)n(l) .

A

vесtor fiеld

У dеfinеd on Л is

said to Ье in Cl(Л)

,0

<

|

<

fr, if and only

it

(r;V) o

h; is ir\

Ct(Bo) for all i.

]f

f/ is a smooth

opеn sеt

with

C} Ьoundary'

}

>

1,

(in

suсh a сasе с1

€

cn(on) and

h;

с.

Ck(B)), then

the fiеlds m and n dеfined

on Л are only еlemеnts

of Ch-l(Гiп,N;.

тьis loss of regularity

of thе noгmal fiеld

on

Л

is duе to thе

сontriЬution

of thе

tеrm Dh in thе еxprеssions foг

m and n. The samе сonсlusion

is

valid for the unitаry

eхtension

Л'6

of thе noтmal fiеld

n:

,A/o

is in Cь-'(п,N) .

Sinсe сoh is

the idеntity, using thе сhain гulе it

сan Ье shown that

(Db)-1

:

(Dс)

o

h. Thегеforе

we сan

aiso

writе that

ё/l*n

m(r)

:

|r;(t)-

Dс;(r)

.е"

i- I

l\{ean

сurvatuгe к of the Ьoundaгy Г

__

aо.

Let, Q

С

IR3 Ье a domain of сlass C2 in lR3; Г/ is loсated on one side of

Г

:0t7

and Л is a

mалifold of сlass C2. With eaсh

point

16 of Л thе sесond

fundamental

form of Г with thе еigеnvaluеs

(&1

,&2)

is assoсiatеd. Thе mean

сurvaturе

of

Г

at zg is dеfinеd as follows (DaCarmo

i976,

p.146)

к(rg):

*(*'

*

*,)

.

Thе еigеnvaluеs

fr1 and k2 are assoсiatеd

with

the eigenveсtors т1(u0) and т2(zq)'

thе so_сallеd

prinсipal

сurvaturе dirесtions, whiсh are oЬtained

in

thе following

way.

Lei

rg

€

Л Ье a givеn

point

and dеnotе Ьy n6 thе unit normal field at сg, it

is assumed that

ng

is

outward

pointing

on

Л. For any fixed unit tangent

veсtor

T, T

€T'oГ,

one has to сonsidеr

thе two_dimеnsional

linеar marrifold

spanned

by

the vесtoгs np, т

(this

marrifold is

of thе

form ,o(т)

:

ro

*

(IR.n

Ф

]R.

,))'

and thе

plarrе

сurvе

ц(т):

ГОE(т), i.е. a сurvе in.Е(r) suсh that xo

С

ц(т).In

thе loсal сoordinatеs

(,,no),4(т)

is thе

graph

of afunсtion

/(.).(_.'e)

-*

IR'

e

>

0, suсh that

/(0)

:0

(i.е.

ro

ец(т)).

Morеovеr thе сurvaturе

_

f,,(o)(l

+

f,(o),)-З

a

€

(_с,

e)

!,a*f,'t

::-

:.

itir1-: r

D"].

*'ir'

-: ::

llr

.'-:tr.:

-trt:': -

-Ё3*

.?

.

ft

*.-*-r.:-:

l:

.:.:

:-

if"

-;

ri :1 I

Рtr?Е-:.:--

t=;

i

-:

aз.

Surf;

TЬ

"'.:.]...

.

]lEliJ:с.

i.

.с

*,:б

.:j:

tЙ.rт.J..

1n,

Г.r

a:.'

-

Л^С::.j

.aЁ.

:

lliЁ:a.:

:

-.

:Эl

.l

TLн..:*

-'

la':.

t*3.

Гunсt

!.,

r'ln?.."' r

Э'

"!+|r'

l ooUtt,

р-т

:

1.

;

:: фпd only

^. i1

('аsе

сi

t

:.^''. еlеrnents

:t,irrе

to

thе

:..''()nС1usion

Dh1-t

-

:. :.e siсle

of

^-

.i..:

sесond

. Тjlе mean

..:..l

12(э9),

':.'

l':rllorving

:....,1 at ro

r

it

::.j.Ilt \:eсtoг

.: ^.1 .spannеd

.

IR.т)),

./\т

'

*

|J\т),

In

-..:)

+[{,

2.3.

Funсtional

spaсеs

17

is

dеfined

in

a

neighЬourhood

of

z6

on

ry(т).

The

sign

of

this

ехprеssion

is

сhosеn

in

suсh

a way

that

thе

сuгvature

is positivе'

pгoviсled

tirat

the

osсulating

сirсlе

to

ц(т)

in

thе plа.тrе

t(т)

is

loсated

in

Г/,

othегwise

ihe

сurvaturе

is

negativе.

Thе

сurvature

of the.

pla,Irе

сurve

ц(т)

at

thе point

e6

is

dеnote

d Ьу

i(т1

:

_!,,(0)1t,+

f,Фr)_Ё.;

wе.lravе

also

(Da

Carmo

1976)

that

k(т):

R_,('),

whеrе

Л(т)

is

the

radius

of

сurvature

of

ц(т)

at

zg'

i.е.

lл(?)l

i;

thе

rad'ius

of

the

osсulating

сirсle

at

z6

to

ц(т)

tn

Е(').

It

сan

Ье

shown

(Da

Carmo

1976)

that

there

eхist

two

tangential

direсtions

т1,

т2

at

,,6

suсh

that

т1

minimizеs

(т2

mа-хimizes)

l}(r)|.

ThЪy

are

сalled

thе

p'i"",'у'"}

счrva!1rre'

{iгесtions

of

Л

at :16

,

aDd

the

mеan

".'..,,ut,,.Ь

takes

thе

form

к

:

i|

tr( г1

)

+

k\т2)).

2.2.

Suгfaсe

mеasuгes

on

г

Thе surfaсe

measuгe

on

Л

сan

Ье

definеd

with

the

usе

of

thе

сofaсtor

matrix

notation.

For

a

mapping

A: IRN

*

IRN,

we

dеnotе

ьу

М(h)

thе

matriх

of

сofaсtoгs

of

thе

matriх

Dh.

i.e.

M(h):

dеt(DA)-(Dh)-1,

whеге

*Dt

dеnotеs

the

transposе

of

Dh.

It

is

known

that

(-DЬ)-l

:

-((Dh)-,

),

hеnсе

thе

form

-(Dь1-.'

is

wеll

dеfined.

Foг

any

сontinuous

funсtion

/

dеfined

on

Л with

сompaсt

suppoгt

on

Л'

_

Г

o

О; thе

following

rеlation

holds

[

.ro,

:

I

r

o

h;||M(h;).e,.||ршd(,,

Jг

Jвo.

wherе

(,

:

(4r'

€z,...,(ш-').

The

term

|!tvt(hn).e"llп,'

is

сontinuous

with

геsptсt

to

x''

€

Л (sinсe

(оь1_,

:

(Dс)

o

h,

h u''d

"*.,"

Cl)

and

Ьounded

* Bo,n,

1

<

'

s

m,

1vhег3

tl"

::.:

Bo/

:-с(spt(r;

n

Л))

arе

inсludеd

in

jl..

Therеforе

f

еrlq1if

andonlyif

f

ohсr,1".(вo)

forallintеgеrsz,

1{

iЗm.

Henсе

2.3.

Funсtional

spaсes

Let

О

Ье

an

opеn

set

in

]RN

diffеrеntiaЬle

funсtions

with

(,l"f)

o

h.l|M(hi)

.

е.||p*

d(,

D(Q

is

thе

lirrеar

spaсе

of

infinitеly

many

times

сompaсt

st.tpports

in

f/;

a

sequеnсе

фд

сonvеrges

[-ror:t[-

Jl

i=l

JБo

L>

18 2. Prеliminariеs and thе matеrial dеrivativе

mеthod

to

ф

in D@) if and only if therе ехists

a сomPaсt seto1o

с

Q, suсh that for

all }

:

\,2,.'., sPtdь

С

О,

aпd

a]I deгivativ."

(*)"фk

сonverge to

(*)'ф

uпifoгmly on

(J

as

}

---+

oо.

The following

notation is introduсеd

/а\",

ao'...0o,

\а,/

Ф

=

6';'

...6";,"

Ф'

whеre o

=

(@r,...,oN)

and o;, 1

<'

<

N, are integеrs.

Thе dual space

D|(a) is thе spaсe

of distriЬutions

on a.

We

denotе

Ьy

(,.Io,@),o1о; thе

duality Ьilinear

foгm Ьеtwеen spaсеs

D,@) and D(o) (for

thе

sakе

of simpliсity we shall writе (.,

.)

whеnеvеr possiЬle).

Foт any еiement

F

\лD,(a) onе сan

dеfinе, foilowing (Sсhwartz

1966), thе

derivativе (*,)"

F

as an еlеmеn| of

D,(О) suсh

that

Yф

с

D(a):

(

r+)

"

,,^):

(_1)lo|

<',

(

!\" oi,

\

\dnl

/

\dn )

wherе

|o|

:

ot

*

...*

oш.

It should Ьe еmphasizе

that

thе spaсе

ifu.(o) сan Ье

idеntified as a suЬspaсe

of D'(A) in the following way:

Foг any

еlеmеnt

f

с

LЬ"@) wе

dеfine thе distriЬut\on

T1of

thе

form

(Tt,ф):

I

fro'

vфсD@) .

JQ

Wе shall

idеntify the distributior,

F1with

thе еlemеnt

/.

Foг

any

mеasuraЬlе

sеt J-l in R,N'

the

сharaсteristiс

funсtion

Xо

(for

the

sakе of simpliсity

wе shall write

1

whеnеvег possiЬle)

is defined

Ьy

,,,,,,

=

{i

i::;

:

3"

=*,1n

o" is the сomplemеnt

of J7 in R"N.

If the

LeЬеsguе

mеasurе of'

a

С

]RN

is

finitе, then

19

is in.t,(IR,").

In

general

117

is

in,LЬ.(RN), th-еrefoге

117

сan Ьe

identified with

an еlemеnt

of thе spaсе

of distгiЬutions

D,(IRN

).

Let us

сonsidег

thе

gradient

(a

a

\

\,хл

:

\u.,

*"'

....

a^\'

)

whiсh

is an еlеmеnt

of D,(IRN,IRN)

definеd

as follows

(Yхо,ф)o,@),o(о):

_

[

divфdx

Ja

Yф:

Фt,.., ,фм)

с

D(IRN;IRN)

nal

i3

qj

fT

с{&

Тlл

* ж.,

L*o

lr:;

щдr

5щ

rgт

{Itu

:el

Е

Jlrг*;

rrr'.

tЁ

I

fur-r

Гlrt

=

I

?"д'

!!

iПirЕ

.-}

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

Подождите немного. Документ загружается.