Optimality conditions for shape and topology optimization subject to a cone constraint

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Niveau: Supérieur, Doctorat, Bac+8
OPTIMALITY CONDITIONS FOR SHAPE AND TOPOLOGY OPTIMIZATION SUBJECT TO A CONE CONSTRAINT SAMUEL AMSTUTZ AND MARC CILIGOTTRAVAIN Abstra t. This paper provides rst order ne essary optimality onditions for simultaneous shape and topology optimization subje t to a one onstraint. These onditions are expressed with the help of the shape and topologi al derivatives of the obje tive and onstraint fun tionals. Several examples of appli ations are given. 1. Introdu tion The lassi al theory of shape optimization onsists in analyzing the behavior of a shape fun - tional with respe t to a small deformation of the domain, where ea h point moves along a dire tion represented by a displa ement eld [20, 24, 30?. In the formulation of [20, 24?, the shape deriva- tive is dened as a Fré het derivative, whi h allows to re ast, at least lo ally, shape optimization problems as dierentiable optimization problems. Therefore, optimality onditions an be derived straightforwardly by applying general results of nonlinear programming in Bana h spa es. Unfor- tunately, this approa h does not apply any more when one wants to allow topology variations. In fa t, in this ontext, the set of attainable domains annot be equipped with a stru ture of ve tor spa e in a natural and onvenient way.

  • fritz-john multipliers whi

  • lassi al

  • optimality ondition

  • ?? ?

  • perturbation

  • tion ?

  • shape optimization

  • simultaneous shape

  • lagrange multipliers

  • dire tional


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Nombre de lectures 11
Langue English
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Ω →J(Ω), Ω →G(Ω).
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∀ξ∈Per(Ω), ∃τ ∈ (0,τ ] ∀t∈ [0,τ], G(ξ(t))∈−K ⇒J(ξ(t))≥J(Ω).ξ
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.J G Per(Ω)
′Y
′ +Y h.,.i Y Y K
K
Ω Per(Ω) ξ∈Per(Ω)
′ ′G(Ω)+G (Ω,ξ)∈ Int(−K)⇒J (Ω,ξ)≥ 0.
′ξ ∈ Per(Ω) G(Ω) +G (Ω,ξ) ∈ Int(−K)
t∈ [0,τ ]ξ
′J(ξ(t)) =J(Ω)+tJ (Ω,ξ)+o(t),
′G(ξ(t)) =G(Ω)+tG (Ω,ξ)+o(t).
′t G(ξ(t)) = (1− t)G(Ω) + t[G(Ω)+G (Ω,ξ)+o(1)] ∈ −K
′J(ξ(t))− J(Ω) = tJ (Ω,ξ) + o(t) ≥ 0 t > 0 t
′J (Ω,ξ)≥ 0
′ ′
T (J,G,Ω,Per(Ω)) ={(J (Ω,ξ)+α,G(Ω)+G (Ω,ξ)+y),α > 0,y∈ Int(K),ξ∈Per(Ω)}.
Ω Per(Ω) T (J,G,Ω,Per(Ω))
∗ + +(γ,y )∈ (R ×K )\{(0,0)}
′ ∗ ′γJ (Ω,ξ)+hy ,G (Ω,ξ)i≥ 0 ∀ξ∈Per(Ω),
∗hy ,G(Ω)i = 0.
T (J,G,Ω,Per(Ω)) (0,0)∈T (J,G,Ω,Per(Ω))
ξ∈Per(Ω) α > 0 y∈ Int(K)
′ ′(0,0) = (J (Ω,ξ)+α,G(Ω)+G (Ω,ξ)+y).
′ ′G(Ω)+G (Ω,ξ) =−y ∈ Int(−K) J (Ω,ξ) =−α < 0
∗(γ,y ) ∈
′R×Y \{(0,0)}
∗γs+hy ,zi≥ 0 ∀(s,z)∈T (J,G,Ω,Per(Ω)).
′ ∗ ′ ∗γ(J (Ω,ξ)+α)+hy ,G(Ω)+G (Ω,ξ)+yi≥ 0 ∀(ξ,α,y)∈Per(Ω)×R ×Int(K).+
K cl(IntK) = K
(ξ,α,y) ∈ Per(Ω)×R ×K ξ = 0 α = 0+
∗ ∗y = 0 hy ,G(Ω)i≥ 0 ξ = 0 α = 0 y =−2G(Ω) hy ,G(Ω)i≤ 0
∗hy ,G(Ω)i = 0
′ ∗ ′ +γ(J (Ω,ξ)+α)+hy ,G (Ω,ξ)+yi≥ 0 ∀(ξ,α,y)∈Per(Ω)×R ×K.
α = 0 y = 0 ξ = 0
∗γα+hy ,yi≥ 0 ∀(α,y)∈R ×K.+
∗ +y = 0 α = 0 γ ≥ 0 y ∈K
w
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In
2.4.
.
3Ω Per(Ω) T (J,G,Per(Ω))
¯ξ∈Per(Ω)
′ ¯G(Ω)+G (Ω,ξ)∈ Int(−K).
∗ +y ∈K
′ ∗ ′J (Ω,ξ)+hy ,G (Ω,ξ)i≥ 0 ∀ξ∈Per(Ω),
∗hy ,G(Ω)i = 0.
∗γ > 0 γ = 0 y = 0
∗ ′hy ,G (Ω,ξ)i≥ 0 ∀ξ∈Per(Ω).
∗ ′hy ,G(Ω)+G (Ω,ξ)i≥ 0 ∀ξ∈Per(Ω).
′ ∗ ∗¯y¯ = G(Ω) +G (Ω,ξ)∈ Int(−K) hy ,y¯i≥ 0 y = 0
∗ ∗ ∗ ∗δ ∈ Y y¯+δ ∈−K hy ,δi > 0 hy ,y¯i =hy ,y¯+δi−hy ,δi < 0

dE R d = 2 d = 3
Ω ∈E

1,∞ d dV ∈ W (R ,R )
Ω(V) = (I +V)(Ω),
dI R
• Ω
Ω
∗m∈N :=N\{0}
d m mz = (z ,··· ,z ) ∈ (R ) ρ = (ρ ,··· ,ρ ) ∈ R1 m 1 m +
[ [
Ω(z,ρ) = Ω\ B(z ,ρ ) ∪ B(z ,ρ ),i i i i
z ∈Int(Ω) z ∈Ext(Ω)i i
i=1,...,m i=1,...,m
dB(z ,ρ ) = {x ∈R ||x−z | < ρ } Int(Ω) Ext(Ω)i i i i
Ω z ∈∂Ωi
[ [
Ω(V,z,ρ) = Ω(V)(z,ρ) = (I +V)(Ω)\ B(z ,ρ ) ∪ B(z ,ρ ),i i i i
z ∈Int(Ω) z ∈Ext(Ω)i i
i=1,...,m i=1,...,m
1,∞ d dE S(Ω)⊂ W (R ,R )
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erturb
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n

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,
b
Due
er
(2)
of
the

the
h
,
p
.
erturbations
p
to
that
b
are
e
en
done
t.
at
oin
the
balls
same
b
time.
merely
Therefore
W
a
y
top
n
ology
h
p
ro
erturbation
es
is
s

o

4
zed
multiplier
b
a
y
Then
a
e
n
ation
um
r
b
win
er
that
ed
ve
domain
is
is
o
p
h
e
u
shap
2.
a
obtained
t
and
represen
erturbations
e
In
w
that

b
,
,
a
that
family

of
.

m
p
L
oin
n
ts
of
[24
shap
in
and

x
tro
2
in
Since
h
get

w

g
the
y
wing
particular,
ollo
write
F
e
ations.
w
b
3)
ertur
to
p
and
e
imp
Shap
d
erturbations.6
,
in
and
and
a
exterior
family
Then
of
resp
radii
ely
p
Note
of
the
es
oin
yp
.
t
ose
o
.
w
not
t
k
sider
in
-


W
e
p
w
t
,
that
domain
ha
arbitrary
e
an
een
.
hosen
It
to
leads
ideas.
to
e
the
b
domain
2.2,
(
Le
6)
i
from
at
Starting
t
bations.
v
ertur
p
p
t
y

olog
u
top
It
d
of
an
Pr
e
)
hap
(
S

1.
ange
3.
L
.
exists
or
ther
,
that
of
exists
subsets)
ther

holds
and
c
en
aint
(op
onst
domains
g
of
lo
set
the
a
d
is
x,
that
n
assume

w
,
no
ptima
e
-
W
at
tion
t
za
m
mi
ss
opti
3.
y
osition
g
domain
topolo
after
d
e
an
top
hape
p
s
(see
to
1).
n
order
o
assure
ti
this
a
domain

elongs
i
Prop
l
w
pp
assume
A
there
3.
a
tradiction.
v


a
pliers

u
h
range
whic
a
,
f
fore,

There
e
with
s
.
admissible
and
of
that
e
h
erturbations

a
xists
i
e
E
there
.
,
erturbation
4
b
y
of
a
p

tsΩ(V,z,ρ)
V
Ω
⋆V ∈ S(Ω) m ∈ N
[m]z∈T (Ω) t > 0 ρ > 00 0
m
(t,ρ)∈ [0,t ]×[0,ρ ] ⇒ Ω(tV,z,ρ)∈E.0 0
[m] mT (Ω) ={z = (z ,...,z )∈T (Ω) ,z =z ∀i =j}.1 m i j
t
ξ(t;V,z,β) = Ω(tV,z,h(tβ)),
mβ = (β ,...,β )∈R h :R →R lim h(s) = 0 h(tβ) =1 m + + s→0+
(h(tβ ),...,h(tβ ))1 m

[m] m ∗Per(Ω) = ξ(.;V,z,β) : [0,τ]→E,V ∈S(Ω),z∈T (Ω) ,β ∈R ,m∈N ,τ > 0 .+
τ t≤τ ⇒ Ω(tV,z,h(tβ))∈E
∗ [m]V ∈ S(Ω) m ∈N z ∈ T (Ω)
J G
mX
′ ′J(Ω(tV,z,ρ)) =J(Ω)+thJ (Ω),Vi+ f(ρ )J (Ω)(z )+o(t,f(ρ ),...,f(ρ )),i i 1 mS T
i=1
mX
′ ′G(Ω(tV,z,ρ)) =G(Ω)+tG (Ω)(V)+ f(ρ )G (Ω)(z )+o(t,f(ρ ),...,f(ρ )).i i 1 mS T
i=1
f R+
[0,a] [0,b] a,b> 0 f(0) = 0 o(t,f(ρ ),...,f(ρ ))1 m
2 2 2 1/2
o(t,f(ρ ),...,f(ρ )) = (t +f(ρ ) +···+f(ρ ) ) ε(t,f(ρ ),...,f(ρ )),1 m 1 m 1 m
′ ′lim ε(x) = 0 J (Ω) : S(Ω) →R G (Ω) : S(Ω) → Yx→0 S S
′ ′J (Ω) :T (Ω)→R G (Ω) :T (Ω)→YT T
′m = 1 J (Ω)S
′J J (Ω)T
f
∂B(z ,ρ )i i
and
,
with
a
dep
reader
)
set
some
is
for

to
ology
in
(
of
of
homeomorphism
Sensitivit
a
the
generally
are
(see
a
more
The
(or
e
itself
top
to
the
in
the
of
functionals.
homeomorphism
W
a
5
is
h
function
emphasize
the
man
Here,
is
9)
erturbation
(
3.2.
8)
w
(
shap
Fig
the
u
is
re
e.
1
space
.
olv
Domain
eren
p

the
erturbations
for
of
the
arbitrary
top
refer
ology

p
6
erturbations
of
satisfying
in
the
W
follo
these
wing
wn
prop
situations
ert
p
y
(
:
e
for
a
all
erturbation
,
.
and
functional
,
dene
there
the
exist
deriv
and
of

p
h
Ab
that

Here
deriv
and
homeomorphism
subse
on
quen
and,
able
in
tly
the
,
partial
w
equation,
e
oundary
use
on
the
hosen
notation
W6
the6
y
T
are
o
.
en
e
with
the
ter
to
in
4,
to
and
the
for
framew
existence
ork

of
expansions

some
2,
situations.
w
e
e
that
need
expansions
.
kno
The
in
functions
y
to
when
represen
single
t
erturbation
the

magnitude
i.e.
of
shap
all
p
p
or
erturbations
top
b
p
expansions:
with
asymptotic
.
the
).
and
linear
admit
Then
y
e
and
the
the
is
functionals
so-called
the
e
,
ativ
single
of
scalar
and
parameter
function
.
erturbations
Th
7)
us
o
are
the
linear.
onding
The
ological
functions
ativ
w
The
e
v
set
ends
,
the
where
dimension
,
when
is
functionals
a
v
,
e
function
solution

a
h
di
that
tial
and
on
,
b
and

all
ed
for
e,
remainder

that
that
assume
h
e
satises

T
5
1).−1 mh =f β ∈R+
mX
′ ′J(ξ(t;V,z,β)) =J(Ω)+thJ (Ω),Vi+t β J (Ω)(z )+o(t),i iS T
i=1
mX
′ ′G(ξ(t;V,z,β)) =G(Ω)+tG (Ω)(V)+t β G (Ω)(z )+o(t).i iS T
i=1
J G ξ(.;V,z,β)
mX
′ ′ ′J (Ω,ξ(.;V,z,β)) =hJ (Ω),Vi+ β J (Ω)(z ),i iS T
i=1
mX
′ ′ ′G (Ω,ξ(.;V,z,β)) =G (Ω)(V)+ β G (Ω)(z ).i iS T
i=1
−1
ρ
lnρ
4 32πρ πρ
3
f(ρ)
T (J,G,Ω,Per(Ω))
Per(Ω)
T (J,G,Ω,Per(Ω))
m mX X
′ ′ ′ ′T (J,G,Ω,Per(Ω)) = hJ (Ω),Vi+ β J (Ω)(z )+α,G (Ω)(V)+ β G (Ω)(z )+y ,i i i iS T S T
i=1 i=1

∗ [m] mV ∈S(Ω),m∈N ,(z ,...,z )∈T (Ω) ,(β ,...,β )∈R ,α > 0,y∈ Int(K) .1 m 1 m +
′ ′ ∗(a,b),(a ,b ) ∈ T (J,G,Ω,Per(Ω)) V ∈ S(Ω) m ∈ N z =
[m] m(z ,...,z )∈T (Ω) β = (β ,...,β )∈R α > 0 y∈ Int(K)1 m 1 m +
m mX X
′ ′ ′ ′a =hJ (Ω),Vi+ β J (Ω)(z )+α, b =G (Ω)(V)+ β G (Ω)(z )+y.i i i iS T S T
i=1 i=1
′ ′m mX X
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′a =hJ (Ω),V i+ β J (Ω)(z )+α , b =G (Ω)(V )+ β G (Ω)(z )+y .S j T j S j T j
j=1 j=1
W
f.
o
Neu

E
o
v
Pr
f
.
Similarly
x
Boundary
,
eren
onve
all

for
is
l
,
write
(1)
3D
by
(1
o
,
set

the
0
then
y
,
e
(13)
pression

1
h
a
that
w
and
or
(12)

by
2
d
(1
dene

e
the
ar
tia
vatives
are
deri
exist
l
(11)
ona
(

for
e
ard
r
h
di
t
the
of
and
x
7)
.
(
e
y
b
b
T
d
,
dene
e
s
transmission
i
mann
n
hlet
e
.
h
D
W

1.
3)
3.
2)
a
and
mm
ha
Le
e
.
in
f
ble
o
There
y
di
t
and
i
f
x
the
e
Hen
v
)
n
1
o
,
C
,
.3.
that,
pairs
l
w
orw
e
t
v
straig
,
Consider
e
g
i
ri
w
de
S
3
functionals
,
t
n
enden
tt
dep
e
state
dene
6
d′ ′ ′ ′Z = {z ,i = 1,...,m} Z = {z ,j = 1,...,m} I = {i ∈ [1,m],z ∈/ Z }i ij
′ ′ ′ ′J ={j ∈ [1,m ],z ∈/ Z} K ={(i,j)∈ [1,m]×[1,m ],z =z } θ∈ [0,1]ij j
′ ′ ′θa+(1−θ)a =hJ (Ω),θV +(1−θ)V iS
X X X
′ ′ ′ ′ ′ ′+ θβ J (Ω)(z )+ (1−θ)β J (Ω)(z )+ (θβ +(1−θ)β )J (Ω)(z )+θα+(1−θ)α .i i j i iT T j j T
i∈I j∈J (i,j)∈K
′ ′θb+(1−θ)b (θa+(1−θ)a ,θb+
′(1−θ)b )∈T (J,G,Ω,Per(Ω))
G(Ω) = |Ω|−M M > 0
T (J,G,Ω,Per(Ω))
′m = 1 J (Ω) ΩT
S(Ω) =∅ T (Ω) = Ω

′ ′T (J,G,Ω,Per(Ω)) = βJ (Ω)(z)+α,βG (Ω)(z)+y ,z∈ Ω,β ≥ 0,α > 0,y > 0 .T T
′ ′ ∗G (Ω)(z) =−1 z∈ Ω T (J,G,Ω,Per(Ω)) =R (J (Ω)(Ω)×{−1})+R ×+T T +
∗ ′R J (Ω)(Ω) Ω T (J,G,Ω,Per(Ω))+ T
Ω Per(Ω)
Per(Ω) J G
∗ [m] mV ∈S(Ω) m∈N (z ,...,z )∈T (Ω) (β ,...,β )∈R1 m 1 m +
mX
′ ′
G(Ω)+ G (Ω)(V)+ β G (Ω)(z )∈ Int(−K).i iS T
i=1
+μ∈K
hμ,G(Ω)i = 0,
′ ′∀V ∈S(Ω), hJ (Ω)+μ◦G (Ω),Vi≥ 0,S S
′ ′∀x∈T (Ω), J (Ω)(x)+hμ,G (Ω)(x)i≥ 0.T T
+μ∈K hμ,G(Ω)i = 0 V ∈S(Ω),m∈
∗ [m] mN ,z∈T (Ω) ,β ∈R+
m mX X
′ ′ ′ ′hJ (Ω),Vi+ β J (Ω)(z )+hμ,G (Ω)(V)+ β G (Ω)(z )i≥ 0.i i i iS T S T
i=1 i=1
β = 0 V = 0 m = 1 β = 1 1
K
Int(K) = ∅
pro
vided
that

and
's

aint
is
emark

e
tin
sp
uous
and
o
.
v
of
er
v
.
the
Assume
of
is
le
an
supp
e

for
ass
v
k

3.3
y
whic
ha
y
that
trol
and
v
.
it
Then
all
e
,
w
e
,

viously
fol
Ob
Mor
e
the
v
e
ha
Cho
e
Cho
w
d
,
(16)
that

Then
seen
ther
Slater
e
alen
exists
when


e
widely
yp
The
y
,

et
that
out
of
,

v
olving
the
in
,
in
yields

In
for
:
It
is
in
a-


(14)
that
W
over,
e
9)
de
(
ne
sensitivities
the
top
sets
admit
y
functionals
,
that
,
(15)
Since
e
.
(7)
result.
dene
an
pro
,
e
,
.
.
in
man
b
main
a
our
lineari
F
t
state
is
to
to
osition
t
p
has
in
terior
w

no
general
are
in
e
[21
W
.
(1
domain,
5)
exit
s.
Y
dition


that
y
turns
lit
,
ptima
t
O
olume
4.
for
eried
,
3.
,
-o

.
whic

h
y
ecial
optimalit
the
(Necessary
exists
3.3
ther
rem
d
heo
full
e
ondition

tion
v

to
onstr
e
lowing
W
the
(
ose6
we
),
e
a
.
e
(
b
8)
analogous
form
et
of
(1
al
6)
olo
Pr
and
o
shap
of.
.
In
3.2
view
the
of
osing
Lemma
ume
3.1,
yields
w
.
e
osing

W
apply
,
Prop
by
osition
and
2.3
is
with
mar
the
vides
notations
.
and
R
assumptions
3.4
of
The
this
(CQ)

Theorem
Therefore

there
e
exists
as
erturbation
translation
p
the
Q)
zed
(C



h
h
that
equiv
that
t

Robinson
ology

top

single
R
a
nonempt

in
y
[11
b
This
and,
is
for
ery
all
and
obtained
used
qually

e
theory
e

b
assumption

that
of
wher
y
.
ptimal
for
expression
pro
ex.
es
L
b
T
v
v
in
or
y
is
of
a
terest.
real
is
in

terv
admitted
al
problems
(b
v
ecause
state
7
ts
is
inequalit

t
w
e.Ω B(0,R)
R ∈ (0,1] Γ

−Δu = 1 Ω,Ω
u = 0 Γ.Ω
2 4 2 2E R S(Ω) V ∈C (R ,R )
(V)⊂B(0,R ) R >R0 0
T(Ω) =∅
Z
2
J(Ω) =− u dxΩ
Ω
G(Ω) =|Ω|−π
′ +Y =Y =R K =K =R+
Z
′hJ (Ω),Vi =− ∂ u ∂ v (V.n)ds,n Ω n ΩS
Γ
Z
′G (Ω)(V) = V.nds.S
Γ
n Γ vΩ

−Δv = 2u Ω,Ω Ω
v = 0 Γ.Ω
Ω
V(x) =−η(|x|)x η η(s) = 1 s≤R η(s) = 0 s≥R0
μ∈R+
μ(|Ω|−π) = 0,
Z
∀V ∈S(Ω), [μ−∂ u ∂ v ](V.n)ds≥ 0.n Ω n Ω
Γ
S(Ω)
μ−∂ u ∂ v = 0 Γ.n Ω n Ω
r =|x|
1 2 2u (x) = (R −r ),Ω
4
1 14 4 2 2 2
v (x) = (r −R )− R (r −R ).Ω
32 8
1 4μ− R = 0.
16
R> 0 μ> 0 R = 1 μ = 1/16.
Ω =B(0,1)
and
as
W
expressions
for
in
true
the
b
functional
In
e
:

W
ob
ositiv
the
W
on
p

t
e

Here,
ball
the
asymptotic

o
t
sets

to

(20)
(
order
C
an
Q)
a.e.
is
the
guaran
e
teed
e
regardless
2
of
t
W
b
(tak
)
e
to
for
y

e
.
yields
set
the
of
b
all
Hence
maps
(17)

that
h
Th
,
y
e.
the
.
xa
i
prop
erturbation,
d
p
simple
where
and
gy

is
explicitly
a
ha
smo
p
oth
with
function


e
h
o
that
some
olo
determined.
top
(9)
y
(
an


A
not
denote
if
b
do
.
e
Plugging
w
to
and
0)
step,
an
rst
Th
a
nonp
In
it
.
b
that
v
if
virtue
supp
F
,
e
with
in
to
functional
).
the
Hence
e
the
are

domain
optimalit
zed,
y
e

m
read
le
:
ose
there
on
exists
in
ta
this
k
example,
e

as
adjoin

states
domains
b
all

of
.
Let
e
no
v

in
ology
olar
erturbations
ordinates
in
[
of
9

Let
4
b
.
the
of
w
solution
-dimensional
state
for
(18)
to
t
e
adjoin
with
the
hold
is
and
F
8
i
expansions
and
the
r
[29
to

normal
.
unit
e
ard
b
w
its
out
,
the
undary
is
W
s

e,
these
v
in
o
(19)
Ab
(2
t
are
e
t
x
relev
a
us
m
e.
p
Since
le
e
In
,
t

h
the
i
.
By
oundary
linearit
alue
y
y
of
of
the
.
space
rom
s
w
s
obtain
e
problem

on
,
e
(18)
t
is
us
equiv
rst
alen
shap
t
optimalit
to

(19)
fullled
on
the
w

e
and
revisit
h
minimi
that
(
.
1
us
7)
w
the
top
erforations
p
e


Neumann
b
p
set
whose
8
tersT(Ω) = Ω
2f(ρ) =πρ ,
′ 2J (Ω)(z) =u (z) +v (z)−2∇u (z).∇v (z),Ω Ω Ω ΩT
′G (Ω)(z) =−1.T

2−μ+ u (z) +v (z)−2∇u (z).∇v (z) ≥ 0 ∀z∈ Ω.Ω Ω Ω Ω
1 2r → − u (r) +v (r)−2∇u (r).∇v (r) ,Ω Ω Ω Ω
16
r > r ≈ 0.450
• Y
• Ω
2D R
Γ Γ E D ΓD N D
1V ={u∈H (D),u = 0}|ΓD
1/2U Γ V H (Γ )N N
I R F :I →U η ∈I
Ω⊂D

− (α ∇u (η)) = 0 D, Ω Ω
α ∇u (η).n = F(η) Γ ,Ω Ω N

u (η) = 0 Γ ,Ω D

+α Ω,
α =Ω −α D\Ω.
+ −α α n
ΓN
Z
α ∇u (η).∇vdx =hF(η),vi ∀v∈V.Ω Ω
D
Y =C(I),
I
K ={f ∈Y,f(η)≥ 0 ∀η∈I}.
J(Ω) =|Ω|,
dimensional
;
innite
tak
ology
is
the
t
exterior
and
of
of
is
and
is
in
o
that

ariational
b
b
y
[29
some
loads.

d
kground
t,
phase,
W
whic
is
h
di
enables
unit
the
ositiv
n

ucleation
ological
of
and
material
of
islands.
area
5.1.
div
P
deals
r

o
p
ts
dualit

e
e
allo
m
optimal
s
o
t
are
ateme
ositiv
n
the
t
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2G(Ω) :η → α |∇u (η)| dx−M(η) =hF(η),u (η)i−M(η).Ω Ω Ω
D
Ω E Γ =∂Ω∩DΩ
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T(Ω) =D\Γ .Ω
[m]V ∈ S(Ω) (z ,...,z ) ∈ T(Ω)1 m
mt > 0 ρ > 0 (t,ρ) ∈ [0,t ]×[0,ρ ] B(z ,tρ )0 0 0 0 i i
D ΓΩ
˜Δ ⊂⊂D Γ ∩D ΓΩ Ω Ω(tV)
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1H ={u∈H (ω),− (α∇u) = 0 ω},
p∈ (2,+∞]
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c> 0
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H
1 1H (ω) H H (ω)
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1,p ′ 1,p ′ 1 ′0 (u,v)∈H×W (ω ) W (ω ) H (ω )
1,p ′v =u ′ u∈H →u ′ ∈W (ω )|ω |ω

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