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

t'.

...

Ll

]

o. suсh

that

for

'

]

:

i'

(]oli\'егg

e to ($).

ф

2.3.

Еunсtional

spaсes

l9

Let

Q Ьe

a

domain

of

сlass

Ck,

lс >

1,

using

thе

rvеll_known

Stokеs,formula

I

o'"оo,:

IE.nаr

Jn

Jг

rvе

gеt

(Yхо,ф):_

[6.no,,

,

Jг

rvheте

n

is

thе

outward

unit

normal

veсtor

o^Ч

Л

:

0o.

Making

usе

of

thе

tгaсе

opеratoг-

1г

с L(Dk-'(D;tR,}

Dk_|(Г;IRN;)

introd...";

;;йwarz

(1966),

lr'here

De(Л)

dеnotes

the

sеt

of

funсtions

,6

аeАn"a."

г'""ь1i

Iti;,1,1,;61.t,;

is

in

CЁ(B,),

we

have

(Ухrl,ф)

:

_(lг

,n,ф)

that

is

thе

gгadient

of

x9

takеs

the

foгm

Yхл:_*^|Г.n

V1g

is

an

еlеmеnt

o1

2t-t1п1N;]RN):

(Dь-r1рш.IR,N)),,

whеrе

*7д

denotes

thе

tгansposed

operator,

and

wе

havе

*l

r

с

L1Dk-'

(IR,,;

IRN1;

at_*1]R,;

ц1"))

.

Lеt

us

сonsider

a

domain

J2

of

сlass

Ck,

lс >

L.

-For

any

P,

I

S

P

{

*оo,

and

foг

s

>

iAdams

1975)

is

dеfined

as

thе

сlosurе

of

th1

noгms:

if

s

is

an

intеgеr

:.: OIl

12. Wе

dеnotе

Ьy

',.s

D'rO)

and

D(A)

$or

.t:illе

l.

-:.s

.

Sсhrr.aгtz

1966),

thе

'-:(

J

\' ,

. I

a),

Lltl

/

,

jrrltifiесl

as a

suЬspaсe

.

'r^

tJ

of

the

form

0

thе

SoЬolеv

spaсе

W"'p(a\

spaсе

C-(D)

in

the

followlns

:.l:r.'tioll

\я

(for

thе

:t-ll^есl

Ьr.

;lstlr.е

of

o

с

]R.N

i'

. thегеforе

X12

сan Ьe

гt.\'1.

Lеt

rrs

сonsideт

=

I

RN;lRN)

|Iф||,w'',rq:

D

lolSs

l"lG)"

ф(z)

and

for

aгЬitrary

s,

s

2

0,

||ф||ow"

,tnl:

||ф|I,wьl',tn,*

/

/',"ffiffi,"o,,

lvhеre

[s]

is

thе

integеr

suсh

that

[s]

Ss<[s]+r.

For p

-

2,

W"'2(a)

is

a

HilЬеrt

spaсе;

it

is

dеnoted

Ьу

H"(a).

Let

a

C

]RN

Ьe

a

mеasuгaЬle

set.

Lеt

us

assumе

that

0

S

s

<

},

and

сonsidег

the

lИ,'Р(IRN)

,,o..

of

thе

сharaсteгistiс

fu,'.tio;;;р".

ьy

||хn!1*",,.**):2

[^ [^

p_

иll[!I+,ll

dxd'у

.

JA

JA.

20

2.

Prеliminariеs

and

thе

matегial

dеrivatrvе

mеthod

Ё

,*t''ffi,

#,цl-

rc-

2.4.

Linear

elliptiс

boundаry

value problems

Let

us

сonsidеr

an

еlеmеnt

у())

of

w"'P(Q)

whiсh

depеnds

on

thе

domain

o

с

]RN.

In gеnегal

this

еlemЬnt

i. gi..,",'

in the

foгm

of

a so]ution

to

a

boundaгy

value proЬlеm

definеd

in

o.

Thе

sensitivity

anаJysis

of

thе

mapp

iпg

a

---

у(n1

wiil

Ье

сarried

on

in

aтr

aЬstraсt

way.

However'

we

shall

stari

with

a

simplе

eхamplе

of the

linеar

еlliptiс

Ьound.ary

lalue

pгoЬlеm

that

сan

serve

as

a

mod.еl.

Two

Ьasiс

linear

proЬlеms'

to

Ье

formulated

latеr

on'

arе

usеd

as

mathеmatiсal

models

for

the

small

displaсements

of

a

mеmЬгanе

and

foг

the

Кiгсhhoff

plate

in

thе

statе

of

statiс

equiliЬгium.

First

we

сonsidеr

the

Lapiaсe

еquation

in .Г2

with

two

diffеrеnt

Ьoundaгy

сonditions:

the

Diriсhlet

Ьo.,.j.,y

сonditions

on

a

paгt

Лg

of

thе

boundary

Л

_

aQ,

rvhеrе

thе

displaсemеnt

ofihе

memЬгanе

is

pгеsсribed,

and

thе

Nеumann

Ьoundary

сonditions

orr

Л1

-

г\Л6.

Thе

Кirсhhoff

modеi

of the plate

lеads

to

the

fourth

ordеr

Ьiharmoni.

ЪquatiЬn

with

Ьoundary

сonditions

of

diffеrеnt-

typеs.

Suсh

examples

of

mathеmatiсal

Ьoundary

value

proЬlems

are

used

only

as

models.

The

use

of

mеmЬrane

as

a

modеl

of

thе

Laplaсе

еquation

is

not

of

сruсial

importanсе

in

ouг

сonsiderations;

many

otheг

eхamplеs'

arising

in

the

shape

optimization,

сan

Ьe

rеlated

to

the

samе

boundary

value

problеm.

For

eхamplе'

wе

сan

сonsidеr

thе

stеady

state

heat

tгansfer

еquation,

or

thе

steady

irrotational

flow

of

an

inсomprеssiЬlе

perfeсt

fluid

giving

risе

to

thе

similar

mathematiсal

modеl.

.

Let

А(t)

:

сIij(I),

t

с

Q,

I

<

i,j

S

i{,

Ье

a iy'

x

iy'

matгix

funсtion

suсh

that

д(

)с

с(D;п-N,)

and

The

following

result

holds

truе

(Baioссhi

et

al.

Proposition

2.I

If

a is

о

domain

of

сIass

C|

function

уg

Ьelongs

to

W",p(RN

)

foт

s

<

llp,

for

some

сv

>

0

and

a|I

r

с

9,

Let

f

€

L,(a),

g

с

H+(Л)

Ье

givеn'

to

the

following

proЬlеm

(1(")

1,()п..

>"ll1llfo,

V4: (4r,.'.,(ш)

с

IRN

1e84).

iz

IRN,

then

the

сhа,ractеristiс

1{p<+-.

and

denotе

Ьу

у

с

H1

(И

a

solution

(2.1)

(2.2)

*r

iЕ

tffi

i&

6J

sг".r

ltfltr".ъ

tlrt

j,'

rt

!€

l

'l":r

lrbG'r

lr

-'.:*

tri'шЕЕ-

-.

]

F'.r

'.:-

-

i8;::

rr

fur*i

3

-:

J

I

h{o-i

St:r--

r.:-

t=#

.

x

{I;

-.'--

9:

!-

ls+i-*д

.-

Jr<

,r-.

а

*.

ll

s:A-:

:

.

-div(А

.Yу)

+у:

f

inD,(a),

у=g

ir-H+(Г).

We

introduсe

the

Ьilinear

form

oп

H|(a)

x

H|(a)

..|-." .i

t сharaсteristic

'

lе ttls

:. ..

,:,

tIrе domain

'

...

:.

:()

a Ьoundary

:..1::::ig О-у({l)

...:..

.,''.itlr

a simplе

.:.

t,

]..''t' :rs a

modеl.

..

:

'1t

]ll':1t}rematiсal

.:..

Ii:г.:1rhoff

platе

.

,',.

,'rlrration

in J2

.

.:-.'

:.,ttсlitions

on a

:

.]..,

1llеПЬгanе

is

.- ._

'

Тht.Iiirсhhoff

-

.,.:.1-

l-^,,,-l^--.

.l rr

|,UullцotJ

.

..

]

.'lltiсlaгv

valuе

-.

'.

:l'odеl of

the

:

..-

:.t: rriа:rv other

....-

t11:llе

Ьoundary

....,-

]lеаrt transfer

,

.

...

,'..

i] ' ' irl oirli n о

'

..

: - -': :l lltсt ion suсh

a solution

(2.\)

tt t\

\-,- J

2.4. Linеaт

еlliptiс

Ьoundary

valuе pтoЬlems

21

а(ф,z)

:

[

Ф{').VФ(,),Vz(r))рш

а'

+

[

6,а'

Ja

Jо

and

thе

сlosеd сonvеx

suЬset

I{

of

H|(a)

assoсiatеd

with

thе

Diriсhlеt

data g

K:{oсH1(a)|1Гф:g\

To

provе

thе ехistenсе

and

uniquenеss

of

a solution

to (2.1),

(2.2)

thе

usе

is

madе

of Stampaссhia's

theorеm.

Theorem

2.2

Let

H be a

Hilbert

spaсe,

K о,

сIosеil

сonleI

set

in

H, а(,,,)

a сontinuous

bilinear

forrn

on

H x

H such

thа,t

fс

>

0 .

a(ф,ф

2

"l|ф||,н

Vф

с

ll

Then

for

anу

continuous

linеar

forn

L(.)

on

H there

erists

the шnique

eleтnent

у

С.

K

suсh thаt

а(у,ф

*

у)

>:

L(ф

_

у)

Yф

с

к

Remark'

In

gеnеral

it is not

assumed

that

thе Ьilinear

form

o(.,

.)

is symmetriс.

If it is thе

сasе,

then y

is also

the

unique

solution

in К of

thе

minimization

proЬlem

J(у)

<

J(ф)vф

€

1{, wheгe

J(.)

is thе quadratiс

еnеrgy

funсtional

J(ф)

:

L,<o,ol

-

L(ф)

.

For the

linеar

form

on

H|({))

definеd

ьу

L(ф):

Inf

фd',

whеrе

/

is givеn

h

L2(a)' wе

саn provе

using

Thеorem

2.2 tЬe

ехistйсе

and

uпiquеness

of an

еlеment

у

с

Hl(o)

with

1pу

_

g

and

/",^

Yу,Y(ф_

y))п-'

o'*

|na(ф

_

фdx

for

all

ф

in Hl(a)

with

ф

_

9

on

Л.

Sinсе

an

elеment

z ofthе

spaсe

f1j(J2)

сan

Ье сonsidered

as thе

diffеrеnсе

,:ф-

y,wherе

ф:у-|zСK,

wеhavе

ffr

Jn6

Yy.Vz)ршd'*

Jnуzdt

2

Jnf

,n,

vz

с

Hi(9)

.

Hi(a)

is

a linеar

spaсе'

therеforе

z сan

Ьe

replaсed

ЬУ

-"

and

wе

oЬtain

frr

уСI(:

|^(s.vu,Vz)рлfu+

|уzdr_ l

f,а'

vzсHi(a).

Q.3)

Ja

Jо-

\4aking

usе

of thе

сlassiсal

..vaгiatiorral

intеrpretatiori''

of

(2.3) y

сan

bе iden-

tifiеd

as a wеak

solution

in

H|(a)

to

the proЬlеm

(2.1)

and

(2.2),

To

this

еnd

rvе

seleсt

z

сD@)

and then (2.i)

follows

from

(2.3).

.

I

rro_фd,

JA

2. Prеliminariеs

and thе material

dеrivativе

mеthod

Anothеr appгoaсh to

the soiution of

thе Ьoundary value proЬlеm (2.1)

and

(2.2)

can Ье formulatеd

as follows,

first thе

homogenous

Diгiсhlet Ьoundary

valuе

proЬlem

is сonsidered

(i.е.

with

9

:

0) usin8 the wеll_known

Laх_Мilgram

lеmma.

Thеorеm 2.3

Let H, а(.,.) and,

L(.) ьe

such as in

Theorem

2'2. There

erists

thе unique

element

у

е

H uhich satisfi,es

the bounilarу

aаIue problem

a(у'ф):

L(ф)

Vф

с

rl

Proof,

ItissuffiсienttoapplyThеorem2.2with

K: H andfoг

9_

уiz

(with

z

с

i/) in thе vaгiational

inеquality

of Thеorem

2.3.

П

Let us suppose

now that g

is

givеn

in the spaсе

f11(o),

and сonsidеr

the

ш

с

Hё(о)

i-П

lrЁ'

Ц

'"

&ir

ш*

.rh-

]Е*

FLf,B

r

Fт*'..

li(ffiтЕ

lll

-:r"n

ъцtй...

D

3l-

r

|Д'\.

l€

i

зr.;

tlш.

з,T

0

following

linеar form defined

on the

spaсe

f/o1

(J2)

L(z)

:

I

v,

_

(А,Уg,Vz)рш

_

gz)dx

JA

From Theorem

2,3 it

follows that

therе eхists

the

uniouе elеmеnt

suсh

that

_div(А.Vu)*

u:

f

*

div(А.vg)-

g

|nD,(Q)

u:0

onЛ

\2.4)

(2.б)

Assumе that

y

:

u*

9,

henсe

y

is an

elemеnt of

K and satisfiеs (2.1)

and

(2.2).

Non_smooth

right_hand

sides

of elliptiс

equations

It is сlear

that it

nееd not Ье

assumеd

in

(2.1)

and (2.a)

that thе еlеmеnt

/

is

in L,@).Foт

an еlemеnt

f

е

H-L

@),

the

еxistеnсе

of a solution

to

(2.1)

and (2.2)

сan Ье proved

using thе

same

argument

as aЬove

for

/

€

Lz(a).

тor

example,

if

/

is given

in L2(Q),

then we

сan provе

thе

eхistеnсе

of thе

uniquе

solution

у

с

H1(о)

to thе proЬlem

-div(A.Vy)+

у

:

#

iлD,(Q)

у:g

in,FIi(Л)

.

To еliminatе

the

lowеr ordеr

term appeaгing

on thе

lеft-hand

sidеs

of еqua.

tions

(2.1),

(2.a)

and (2.6),

one

has to

makе

usе of thе

Poinсarё

inequality.

It

might bе wеll

to

point

out

that the

fiгst (positive)

еigеnvaluе

of the

sесond

ordеr

еlliptiс

opегatoг

in f/j(o)

is definеd

as follows

I

)r

:min{

l

Ф.vф,Vф)р'dс| Ф

с

н!1tl1,

llФllн'191

:

t1

Jn

(2.6)

(2.7)

(2.8)

'

r'€^'F,!"1.

.i. . :i-''

-..;'lrrе

1lгoЬlеm

(2.1)

and

i.

]..

;'-:^.,'Ls

Diгiсlr]еt

Ьoundary

:.:

.]..

..'''.,..

ktiorr.n

Laх_Мilgram

:,

:.,

Т;,,,тttгt

2.2.

There

erists

.-..:

:.

.

сittе

problem

....

^]..

-

11

iLrlсl

for

ф

:

у

* z

.'.

.,

:..

],3'

П

.:

.

..i:

Г],.

alrd

сonsideг

the

сiепlent

u

€

Ht(a)

(,

4\

(2.5)

:...jрr

r2.1)

and

(2.2).

(2.6)

(2.7)

:.

.-.'

.. :.

iiand

sidеs

of

equa-

:

.....].,lllс:rтё

inеquality.

It

-.:.

:..':...:.,

оf

thе

sесond

ordеr

rr

:1].

(2.S)

2.4.

Linеar

elliptiс

Ьoundary

value

proЬlеms

Henсе

Ьy

the

definition

of

)1

wе

havе

thе

following

inequality

foг

any

ф

HJ@),

ф

+

0,

х,

S (J"t,l.

VФ.

Vd)п"

а,)

/|IФ||,",tnl

whеnсе

it

follows

that

I|фI|L,rnl

=

*

l"*.Yф,Yф)o,t"dx

23

in

As

a

сonsequеnсе

of

(2.9)'

without

loss

of

thе

genеrality,

the

loweг

order

term

u

on

the

left_hand

sidе

of

the.еquation

(2.4)

canЬ"

n"gle"i"d.

н..,""

the

following

proЬlem

is

to

Ье

сonsidеrеd:

Find

u

с

Hl@)

suсh

that

-div(,

.V2)

:

#.div(,4.

Vg)_

s

inD,(A),

u:0

inЕ*(.).

From

thе

inеquality

(2.9)

it

follows

that

thе

Ьilinеaг

form

аo(ф,z) :

[

@.vф,Vz)ц,'dr

JA

is

сoеrсivе

on

H}@)

equipped

with

thе

following

norm

||,I|,н:tnl

=

|

1v

,1,

a"

.

Ja

foг

wе

havе

аo(ф,

ф)

z

!t|a|f,,.n,

*

;

/'Уф

.Уфd,с

\ /^, o\

<

m'n

\т,т

)

|Iv||'n,,n,

vф

с

Hl(a)

.

Therеfore

Theoгem

2.3

ensures

thе

eхistenсе

and

uniqueness

of

thе

solution

u

to (2.10)

and

(2.i1)

in

the

spaсе

Hl@).

\:'o -

u*

9i

Ьy

dirесt

".t"..,tui;o,,

with

thе

use

of (2.10)

we

oЬtain

that

у

с

K

is

thе

solution

to

thе

following

pгoЬlеm:

_div(,4

.vу'l:

9f

inD,(t)),

'

0"t

у:g

inHь(Г).

Finallу,

let

us

сonsidег

the

Neumann

Ьoundary

сonditions

for

thе

tt 1t\.

(2.e)

(2.10)

(t

11\

(2.r2)

(2.13)

pгoЬlem

I

ttl

L--.

24

2. Prеliminaтiеs

and thе

matеrial

dеrivativе

mеthod

0u

й:

o i.е.

the

homogеnous

Nеumann

сondition

on

Л.

As faг as

appliсations

in struсtrrral

mесhaniсs

are сonсerned,

the

сondition

y

_

0

on Л

means

that the

displaсement

is presсriЬеd

on

thе

Ьoundary

Г:0Q,

e.в.

thе

mеmЬranе

is сlamped.

Wе

introduсe

thе vесtor

field

nд

:

A.n oл

Г; nд

is

a C&-1

tгansvеrsal

field

on

Л

(here

wе

assumе

that

thе

mapping

r

-+

Aц(а)

is C}(Л)),

thе

transvеrsality

of

nд is

deгivеd

fгom

thе positivе

definitenеss

of,4

(nд,n)цш

_

{A,

n,n)р'

2

o||n||fo.

:

o

)

0 .

In

paгtiсular'

if n

is outwaгd

pointing

on

Л, thеn

nд

is also

outward pointing

on Л.

Thе сonditi

"":&

:

0 on

Л means

that

thе

displaсеment

y

is not

presсriЬed

on thе

boundary

Л. In order

to ensurе

thе

eхistеnсе

of a

solution

to the

boundary

valuе proЬlеm

undеr

сonsidеration,

it is oЬvious

that

somе

additional

сonditions

on thе

right_hand

sidе

tегm

of

the еquation

(the

sourсe

term)

should

Ье imposеd.

Let

f

€

L"(9)

Ье

an

е]emеnt

suсh

that'

tn

f

(z)dx:

0, th".,

the

mapping

!.

-

I.n

f фdx

i2

a linеar

сontinuous

form

dеfinЁd

on

ti.е quotient

spaсe

.Е/1

rji tй

(in othег

woгds,

one

has

to

idеntify

in the

spaсе

1/1(J7)/IR

еlemеnts

ф

aлd,

z

suсh that

thеrе

ехists

a сonstant

с with

ф

:

,

f

с)

equippеd

with

the

norm

llФll

=

@o(ф,ф))Ь.

Hеnсe

it

follows

from

Theorеm

2.3

tlrat

thете

еxists

the

unique

solution

у

€

Hl

(a)llР.'

to

the

proЬlem

-div(.,{.Vy):

f

hQ,

Оu

;::0

on

Л

onА

In thе

сasе

of

non_homogenous

Neumann

Ьoundary

сonditions,

i..*:

g,it

is supposеd

that g

с

H|(a)

is

an

elеmеnt

with

Мoreovеr

wе

intгoduсе

thе

linеar

foгm

whiсh

is

сontinuous

on

thе

spaсе

f/l

@)lF..Using

Theoreo-

2.3 we

сan prove

thе existenсе

and

uniquenеss

of

a solution

to

thе еlliptiс

Ьoundaгy

val'uе pгoЬlеm

in

the spaсе

H\@)lP''

-div(А.Vy)

:

f

inQ,

Оu

^.:0

onЛ

ont

t, 14\

/o r < \

'h

{..

;!-r::

- --

',.

.,: _

-;-'.

:

l;: : -;

a-:

':-.

: :

t=

i-:-;

:

::-:

Тrаllslrrissi

Io,]г:o

JГ

Гf

L(фI:

I

f'а"+

|

ga(]Г

J{l

JГ

(2.16)

(2.L7)

..

:.

llll

Л.

-':.,.r'onditiony-0

.:. ]i1ri. Г

:

ОQ,

e,8,

-

,

.]lе

transversality

..

.

l:ttwaГd

pointing

:..

.

ls lrot

РrеsсriЬеd

..

:. iO the Ьoundary

.

:. :: t

lсllra] сonditions

'

.]..;'..r]сl

Ьe

imposed.

'irt.n

the

mapping

-.:.: sl)aсe

H1@)llР,'

..

,.l,lrlcnts

ф

arrd

z

'

i

ц.it}r

the

norm

..

.'

i]tеге ехists

the

(2.t4)

(2.15)

,,,

fl;:

g,it

]'3

r''.е

сan

pгove

,:.'-

''.а]rtе

problеm

(2.16)

(2.r7)

2.4. Linеaт

elliptiс

Ьoundarу valuе

proЬlеms

25

The sourсе

teгm

of

thе equation

(2.16)

сan Ье trеated

in the

samе way

as it

lvas

donе

for thе

Diriсhlet proЬlem'

e.8.

one сan

make use of

a distribution

f!

сo,1a1,

whеге

f

с

L,(t7).

Let us

йt.oduс"

thе

linеar

foгm

L(ф):-

[

r*o"+

[(s+fnl)фdг

Jn

oxl

Jг

and assumе

that

thе сondition

,(1)

:

0 is

satisfied,

henсе

(2.18)

Thе еquation (2.18)

is wеll

definеd

providеd

that

/

с

H"(a)

for some

s

>

].

This rеquiremеnt

results

from

thе сondition

that

tlre traсe

of

f

on

Г has

to

Бe

in L2(Г)

(sее

(Adams

1975)).

Thе

сompatiЬility

сonditions

/,,

f

dx

:

О,

!,

gdГ:

0

for thе proЬlem

(2.16)

and (2.17)

сan

Ье rеplaсed

with

thе

еquivalеnt

сondition

(2.1e)

In thе

nехt sесtions

wе shall

сonsideг

the

Ьoundary

valrrе proЬlems

definеd

in

thе

pеrturЬеd

domain

J21.

It wili

Ьe

assumеd

that

for thе

Neumann

proЬlеm

definеd on

the

family

of domains'

thе

funсtions

/

с

L2(IRN)

and

9

с

й11п.N;

are

given.

In

gеneral

it is not

supposed

that

the

сondition

(z.rg)

is satisfiеd

for

any domain

J2.

Howеver

for any

givеn

domairr

J7 with

thе

Ьoundarу

Г

:

0Q

wе

defi.nе

the

еlemеnts

f,rn*fn)d,Г=o.

rl

0:Z(r)=

l

fdrI

I

gdг

J{')

JГ

fГ

l

f(оlа'=

|

g(9)dГ:0

Jr2

JГ

f@):il"

-*"Ь

/n,o'

g(Г):9|.

- --l='

/,oo,

Undег

thеse

assumptions,

thе

funсtions

f

(Q'i

and

9(Л)

satisfy

the

сonditions

and

thе proЬiem

(2.i6)

and (2.17)

with

thе data

/(J7)

and

9(Л)

is wеll posеd

in any

domain

J7.

TYansmission

proЬlems

Lеt us

сonsider

thе

transmission

сonditions

геlated

to

the

proЬlеm (2.16).

In this

сasе' thе

matrix

А(r)

is not

supposеd

to Ьe

сontinuous

with

rеspeсt

to

r, Ьut

it сan

possеss

thе

disсontinuity

linеs

oг intеrfaсеs.

Let

D

с

]R2 Ье

a fiхed

domain,

B(r)

and

C(r)

two

сontinuous

matriх

funсtions

dеfined

on

D. Wе

dеfine

on

D thе

follorvins

matriх

funсtion

(2.20)

(2.21)

2.

Prеliminaгies

and

thе

matетial

dеrivativе

mеthod

A(a):хn.B+XлC

foг

a

given

mеasurable

sеt

12'

a

с

D.It

should

Ьe

rесallеd

that

a":

D

\

o is

the

сomplemеnt

of

{l in

D,19"

and

1p

denotе

the сharaсteristiс

funсtions

of

sets J.2"

and d),

rеspeсtivеly.

Lеt us сonsider

thе following

proЬlem:

йт:{

-i1

l*й:.:rе-

^т*:a

}iЬ.

'.l..;-д

;.

}

l

3*в:.=r".,:

:.

.dн.

.':-с

-_':|

Jtjь

';G.]Ё

(-.:

!l:

grа..

i :::.

*i;:

'-,

:i-*

trтr#:rr

::.с

*l

tш3

Ц-:

:

т

llryrь

ordеr

..а,

.

j

}Нв.:l.-.

'ъ.

.,-

Jl

eeJ,

1

.-:.

:.

Ц

!1

]:..

-.....

ItLr

:r:*:

.

-'

i3

.l

э::.. ]:-

i.д"l

:. ::.::

.

t*ir:-д

.*

:

.':-

*"-rr::

r

j

:

Е;с..*

l'.ё

.1

{i

'r:

li

:'.

.-

d';-

i.::-:-:. i

f}rr,з::

.j.i]

and

with

thе

pгеsсriЬеd

transmission

сonditions

on

tlre

intеr{aсe

Г

=

Оa

Gt

is

assumеd

that

Л is smooth):

\(C-B).n,Yу)gz:g

onЛ,

where

g

€

Hl (D)

is a given

elemеnt.

In addition'

sinсе y

r

Hl@),

prеsсriЬed

thе

homogеnuous

Diriсhlеt

Ьoundaгy

сondition

on

ОD

!:О

oп

0D '

l'or thе proЬlеm

formulatеd

we

introduсe

the Ьilinеar

form

'(Л;ф,,)

:

[

И@).Yф,Уz)у1,сlr

JD

dеfinеd

on

f/ol

(D;

IR2)

and

thе

linеar

form

Find y

in

Е0t(D)

suсh

that

_

div(,a(J7)

.vЙ

=

#

in н_1(,)

Find y

с

D,(n)

suсlr

that

_div(,.t(o)

.Уу):

f

inD,({))'

a

a"oу

_

Aгa

:9

on

Л,

{t rt\

(2.23)

wе

havе

(2.24)

(2.25)

(2.26)

L(ф:-

[

'#"d'+

[gфdГ

JD

oxt

Jг

It

is assumed

that

thе

matriсеs

B(u)

and C(l)

are

uniformly

positivе

dеf-

initе,

B(u),

C(')

>

aI

for some

0

>

0,

then

А(x)

:

А(о)(I)

Z-oI,

x

сD.

Мaking

usе

of Thеorеm

2.3 and

Green's

foгmula,

onе

сan provе

the

еxistеnсe

and

uniqueness

of

thе solution

у

с

Hl(D)

to

thе prоЬlеrn

(2.22)

and

(2.23).

Finally

lеt

us сonsidег

the

Ьoundary

сonditions

on

Л inr,olving

the

tangеntial

derivativеs

of an

unknown

solution.

For

any

еlеmеnt

Ф

с

с,1a1,0

с

dN,

we

dеnotе

Ьу

У

гф

the

tangеntial

gгadient

of

ф

on

Л, i.e.

vr.Ф

i;

thе

tangеntial

сomponent

of

thе gradient

Vф on

the

Ьoundaтy

Л

Y7Ф:ve_P,

olrЛ

on

Lеt

us сonsidеr

tЬе

Ьoundary

valuе proЬlеm:

E.

I2':D\Ois

ll.l

funсtions

of

\2.22)

Г:0o

(it

is

(2.23)

J?).

we

have

{2.24)

....'.

;lositivе

dеf-

::

сJ.

r

СD.

.

,-

:]ll,

ехistenсe

r]

aпсI (2.2З).

.

':-c

tangential

'

!!

-

lRN,

we

-

':.(.

tangential

(2.25)

\2.26)

2.4.

Lineaг

elliptiс

Ьoundary

valuе

proЬlem

s

27

,r.hеrе,4(.)

is

a

сontinuous

matriх

funсtion

on

Г,

47

is

thе

so-сalled

Laplaсe_

Bеltгami

opеrator

on

Л (defined

in

Seсt.

2.20).Inpu'ti.,,l..

fo.

.'y

ф

in

Cl(Г)

thе

following

identity

holds

_

/"o"o,o,

:

/,Y

7у

,Y

рфdГ

It

is

assumеd

that y

is

suсh

tЬat

Aру

€

L2(Г)

and

Vдy

с

L2(Г;]RN).

Lеt

us

dеfinе

thе

energy

spaсeW@)

for

thе proЬlеm

(2.25)

and

(z.zо')

as

thе

сlosuге

of

the

spaсe

Cl(a)B'

in

thе

norm

||oу;_

[^to

oa'Vd)п'd,

+

[(v,o.vГф).,,Nd'Г

Ja

Jг,

'/п.*-

тh3

]in91г

foгm

J(ф)

;

I.n.,f

Фd".*

!,gФdГ,

with

thе

сompatiЬility

сonditions

satisfiеd

Ьy

thе

data

ln

ji.r

+'I,gd'T-:0,

is

сontinuo,,,

o.,

thе

spaсe

W@).

Тherеforе

thе

ехistеnсе

and

uniquenеss

of

thе

solution

у

с

W@i1"the

problem

2.25)

and

(2.26)

results

fгom

Theorеm

2.3.

Гourth

oгdeг

elliptiс

pгoЬlems

.

L:.

f

с

L,@)

Ье

a givеn

elemеnt,

lеt

us

сonsideт

thе

following

еlliptiс

Ьoundary

value

pгoЬlеm:

(2.27)

(2.28)

(2.2e)

(2.30)

(2.31)

rr.hеге

A(hAу):

f

inD,(Q),

a:0

onЛ,

Оu

;:

:0

on

Л.

on

hсC(a)'

A(r)2oq

)0

V"rс

Q

.\ weak

solution

to

thе'

system

(2.2s)_(2.30)

is

an

elеment

of

thе

Sobolеv

spaсе

H3@).It

is assumеd

that

th"

dЬmain

iz

i.

or

сlass

C}'

,+

!

i.

ть"

space

H!(9)

is the

сlosed

suЬspaсе

ot

H2 (a)

dеfined

by

two

сonditions

(2'29)

and

(2.30),

see

е.g.

(Adams

1975)

for

thе

dеtails.

Lеt

us

introduсе

the

Ьilinеar

form

o(ф,,):

[-ьa6ма'

Vф,z

€

H3@)

.

JГ

\Iaking

use.of

the

еlliptiс

гegulaгity

for

the

Laplaсe

еquation,

onе

сan

show

that

t}rе

тr_oгm

"(ф,ф)ь

is

еquivalеnt

oп

the

"pu""

н31q;;1;;;.Ь;

,'o,*.

Hеnсе

using

thе

linеar

foгm

L(ф):

Iofiа'',

Wе

сan

u,o.l,,Th"o,еm

2.3

and

prove

thе

ехistеnсе

of

thе

uniquе

,ol..iй,.

h

H&@)

t.

tiJp,.ьrem

(2.28)_

?

3.0)

лoг

h(z)

:g6n51,

thе

proЬlem

(2.2s)*(2.3q

; Ь"

.o,'.;й"a

as

a

modеl

of

thе

Кirсhhoff

plate

of

thе

сonstu.,i

thi"k,'",s'

сlampеd

on

Л

:

Ef2'

sее

e.g.

Washizu

19S2).

28

2. Pтеliminaries

and thе material

dеrivativе mеthod

Now, lеt us сonsider

the Ьoundaгy сonditions

for thе Kirсhhoff

plate whiсh

is free

on thе Ьoundary Г

:

ОЛ. The

proЬlеm

is to find a

weak solution

in

H,@)

to thе еlliptiс

еquation:

:Ё.

r-

. .

i::

-:-

A(hAу):

f

inD,(a)

with

thе following boundaгy сonditions:

Aу

:0

on Л,

!ду:

o on Л

on

In order

to assurе the uniquеness of solutions to

(2.32)_(2.3a)

thе HilЬert spaсе

H: H2(Q)lfu1

is takеn into сonsidеration,

whеге,М

is thе

сlosеd suЬspaсe of

H"@) dеfinеd Ьy

fu1

:

{ф

€

H2

(n)|Aф

:

o in oi

i.е. thе suЬspaсе

of thе harmoniс funсtions in i12iJ7). It is oЬvious that for thе

spaсe .Е1

with

the

quotiеnt

noгm,

thе Ьilinеar form o(ф,z)

is сoеrсive

on fI.

Let

us сharaсterizе

linеar forms on ,F1. Thе linеar form L(ф)

:

Inf

фd"

is dеfinеd

oл

Lz(a); this form is definеd oп I/ if and only if thе еlеmеnt

/

satisfies the

orthogonality сondition

|

1Eal:o

vфе,^/1 ,

JA

In oгder to havе

the orthogonality

сondition in an еxpliсit foгm' lеt us сonsidеr

an еlеmеnt

F

€

H2@)

О Hё@),

depending on

/

€

L,(9), suсh that thе

following

equation

holds

AF:f inГ2, .Р:0 onЛ

The

orthogonality сondition сan Ье writtеn

as follows

I

lrФа':o,

J!)

Applying

Green's foгmula to this сondition wе

сan show that fot

ф

е

fu1

f ra^

а

-\

J,\*'"

_

ыo'

)dГ

:

О

Thus wе have thе auхiliary сondition for F,

flF:

0 on Л.

Therеforе

/

satisfies

the oгthogonality сondition if and only if

f

=AF

with

FсH?@),

(2.35)

Finally

wе сan сonсlude

that for any

/

с

L,(a),

/

satisfying

(2.35),

thе

proЬlem

(2.32), (2.33)

and

(2.3a)

is

wеll

posеd

on the

quotiеnt

spaсе 1l

:

H2(a)l/v|.

(2.32)

(2.33)

(2.34)

,.5.

Shа

t

'!t

-.

,яi-'.a'.:

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

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