Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1997002

ESAIM: Proceedings

Elasticite, Viscoelasticite et Contr^ole Optimal Huitiemes Entretiens du Centre Jacques Cartier URL:http://www.emath.fr/proc/Vol.2/

ESAIM: Proc., Vol. 2, 1997, 215{224

MODELING AND OPTIMIZATION OF A NON-SYMMETRIC PLATE.

L.J. ALVAREZ-VAZQUEZ AND J.M. VIA~NO

Abstract. In the rst part of this paper we present a limit model for non-symmetric elastic plates derived from three-dimensional elasticity by use of an asymptotic method and we prove existence and uniqueness of a solution of this model. In the second part we try the shape opti- mization of the plate, we obtain the existence of an optimal prole and we propose its numerical solution by means of penalty methods.

Key words:

Non-symmetric plates, asymptotic analysis, shape optimiza- tion, penalty methods.

Mathematics subject classication:

73K10, 73K40, 73C02, 35C20.

1. Introduction

The use of asymptotic methods for obtention and mathematical justica- tion of plate models in the framework of the theory of elasticity has been shown to be a successful technique during the last decades. First fundamen- tal contributions in this direction were obtained by Ciarlet - Destuynder 7]

with the justication of the classical bi-harmonic model of Kirchho - Love for the bending of symmetric plates. The application of this method to dif- ferent situations (linear and nonlinear elasticity, composite and anisotropic materials, static, dynamic and thermoelastic cases, homogenization and so on) has provided important contributions. A complete analysis of plate models with exhaustive bibliographic references may be found in Ciarlet 6].

Asymptotic methods have also reached an important development in the case of shells (Destuynder 11], Ciarlet - Paumier 9], Ciarlet - Miara 8]) and rods (Trabucho - Via~no 16], Cimetiere et al. 10]).

All of the above works have been exclusively devoted to the study of symmetric plates. The case of non-symmetric plates has so far remained, as far as we know, completely unreported. The rst aim of this paper is the obtention of a mechanical model for non-symmetric plates in the frame- work of linear elasticity. In section 2 we describe the physical problem, we obtain the mathematical model by use of an asymptotic analysis of the three- dimensional problem and we prove existence and uniqueness of a solution for such a model.

In section 3, we study the optimal design problem of minimizing the weight of the non-symmetric plate under some geometric and technological constraints, with some imposed bounds on the deection. In structural op- timization, these problems can be formulated as state constrained optimal control problems governed by an elliptic dierential equation, the control being a small parameter appearing in the coecients of the dierential op- erator (cf. Casas 3], Haug - Arora 13], Hlavacek - Bock - Lovisek 14], 15], Haslinger - Neittanmaki 12]). In this section we pose such a problem and obtain the existence of, at least, an optimal prole.

Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1997002

In order to carry out the numerical solution of the optimal shape prob- lem, we propose in the last section the use of penalty techniques, obtaining necessary optimality conditions and some convergence results.

2. Modeling of a non-symmetric plate

Let

"

be a positive real parameter and

!

be a domain in

R

² with coordi- nates axis

x

¹

x

²

:

Let

h

²

W

²¹(

!

) be a \thickness" function verifying

h

(

x

¹

x

²)

>

0

for all (

x

¹

x

²)²

!:

(1) We consider the non-symmetric elastic plate of thickness

"h

occupying the reference conguration ^" dened by

^"=^f(

x

¹

x

²

x

^"3) : (

x

¹

x

²)²

!

0

< x

^"3

< "h

(

x

¹

x

²)^g

:

(2) We denote by

@

_"

v

=

@

v

the derivative

@v=@x

and by

@

^3"

v

the derivative

@v=@x

^"3

:

Here and along the whole work we use, as it is customary in elas- ticity theory, the summation convention on repeated indices, supposing that the Latin indices range over ^f1

2

3^gand Greek ones over^f1

2^g

:

We study the physical problem corresponding to the mechanical behav- ior of a non-symmetric elastic plate, supposed to be clamped on the lateral surface and submitted to body and surface forces. We assume the constitu- tive material of the plate to be a homogeneous isotropic elastic material of Saint Venant-Kirchho's type with Young's modulus

E

and Poisson's ratio

:

Then, in the linearized elasticity framework, the displacement eld

u

^"and the Piola-Kirchho stress tensor

^" are the solution of the following mixed variational formulation (see Ciarlet 5]):

u

^"2

V

(^") ^"2(^") :

Z

"

(1 +

E

^"ij;

E

^"kk

ij)

_"ij

dx

^";Z

"

_"ij(

u

^")

_"ij

dx

^"= 0

⁸

^"2(^")

(3)

Z

"

_"ij

_"ij(

v

^")

dx

^"=

Z

"

f

_"i

v

_"i

dx

^"+

Z

;

"

+

;

"

;

g

_"i

v

_"i

da

^"

⁸

v

^"2

V

(^")

(4) where

;^"0 =^f(

x

¹

x

²

x

^"3) : (

x

¹

x

²)²

@!

0

< x

^"3

< "h

(

x

¹

x

²)^g

(5)

;^"+=^f(

x

¹

x

²

x

^"3) : (

x

¹

x

²)²

! x

^"3 =

"h

(

x

¹

x

²)^g

(6)

;^";=^f(

x

¹

x

²

x

^"3) : (

x

¹

x

²)²

! x

^"3 = 0^g

(7)

V

(^") =^f

v

^"(

v

_"i)²

H

¹(^")]³ :

v

_"i= 0 on ;^"0g

(8) (^") =

L

²(^")]⁹_S =^f

^"(

_"ij)²

L

²(^")]⁹ :

_"ij=

_"ji^g

(9) and

^"(

u

^") = (

_"ij(

u

^")) is the linearized strain tensor

_"ij(

u

^"_{) = 12(}

@

_"i

u

_"j+

@

_"j

u

_"i)

:

(10) We consider the reference symmetric plate of constant thickness 1 occupying the volume where

=^f(

x

¹

x

²

x

³) : (

x

¹

x

²)²

!

0

< x

³

<

1^g=

!

(0

1)

(11) and we dene

;⁰ =^f(

x

¹

x

²

x

³) : (

x

¹

x

²)²

@!

0

< x

³

<

1^g=

@!

(0

1)

(12)

;⁺=^f(

x

¹

x

²

x

³) : (

x

¹

x

²) ²

! x

³= 1^g=

!

^f1^g

(13)

;^;=^f(

x

¹

x

²

x

³) : (

x

¹

x

²) ²

! x

³= 0^g=

!

^f0^g

:

(14)

We can dene the change of variable from ^" to the xed domain :

^":

x

(

x

¹

x

²

x

³)^2;!

^"(

x

¹

x

²

x

³) =

(

x

¹

x

²

"x

³

h

(

x

¹

x

²))(

x

¹

x

²

x

^"3)

x

^"2"

:

(15) Now we scale the dierent elds appearing in the variational formulation (see Ciarlet - Destuynder 7]) and we dene

u

(

"

) and

(

"

) by

u

(

"

)(

x

) =

u

_"(

x

^")

u

³(

"

)(

x

) =

"u

^"3(

x

^")

(16)

(

"

)(

x

) =

_"(

x

^")

³(

"

)(

x

) =

"

^;1

_"³(

x

^")

³³(

"

)(

x

) =

"

^;2

^33" (

x

^")

:

(17) Also we assume that the applied forces are such that

f

_"(

x

^") =

f

(

x

)

f

^3"(

x

^") =

"f

³(

x

)

(18)

g

_"(

x

^") =

"g

(

x

)

g

^"3(

x

^") =

"

²

g

³(

x

)

(19) where

f

i ²

L

²()

g

i ²

L

²(;⁺ ;^;) are independent of

":

Then, we ob- tain that (

u

(

"

)

(

"

)) is the only solution of the following scaled variational problem posed in :

u

(

"

)²

V

()

²() :

; Z

h

_hij(

u

(

"

))

ij

dx

+

Z

h

^{f1 +}

E

⁽

"

)^;

E

⁽

"

)

^g

dx

+

"

^2Z

h

^f21 +

E

³⁽

"

)

^3;

E

⁽

³³(

"

)

+

(

"

)

³³)^g

dx

+

"

^4Z

h E

^{1 33(}

"

)

³³

dx

= 0

⁸

²()

(20)

Z

h

ij(

"

)

_hij(

v

)

dx

=

Z

hf

i

v

i

dx

+

Z

;

h

(

"

)

g

i

v

i

da

⁸

v

²

V

()

(21) where

V

() =^f

v

(

v

i)²

H

¹()]³ :

v

i = 0 on ;^0g

(22)

() =

L

²()]⁹_S

(23)

h

(

"

) = 1 +

"

²(

@

¹

h

)²+

"

²(

@

²

h

)²]¹⁼²

(24) and

^h(

v

)(

_hij(

v

)) is the generalized strain tensor dened by

_h(

v

_{) = 12}

@

v

+

@

v

^;

x

³

h

^;1

@

h@

³

v

^;

x

³

h

^;1

@

h@

³

v

]

(25)

_h³(

v

_{) = 12}

@

v

³+

h

^;1

@

³

v

^;

x

³

h

^;1

@

h@

³

v

³]

(26)

^33h(

v

) =

h

^;1

@

³

v

³

:

(27) In order to obtain the convergence of the scaled three-dimensional problem as

"

tends to zero we are going to show several technical results. In what follows, ^{j j0}D and ^{jj jj}mD denote the

L

²(

D

)-norm and

H

^m(

D

)-norm, respectively.

Let

E

^h() be the space

E

^h() =^f

v

(

v

i)²

L

²()]³ :

_hij(

v

)²

L

²()^g

endowed with the norm

jj

v

^jjh=^fj

v

^j20+^X3

ij⁼¹

j

_hij(

v

)^j20g1=2

for all

v

²

E

^h()

:

(28) We have the results whose complete proofs can be seen in 1].

Theorem 1.

H

()] =

E

^h() and the norms ^{jj jjh} and ^{jj jj1} are equivalent.

We introduce the space of displacements of Kirchho-Love

V

_hKL() =^f

v

(

v

i)²

V

() :

_hi³(

v

) = 0^g

:

(29) We have the following characterization for this space.

Theorem 2. The space

V

_hKL() is characterized by

V

_hKL() =^f

v

(

v

i) :

v

³(

x

¹

x

²

x

³) =

³(

x

¹

x

²)

³²

H

⁰²(

!

)

v

(

x

¹

x

²

x

³) =

(

x

¹

x

²)^;

x

³

h

(

x

¹

x

²)

@

³(

x

¹

x

²)

²

H

⁰¹(

!

)(30) Remark 3. We write (30) in the following shortened form

V

_hKL() = ^f

v

(

v

i) :

v

³ =

³

v

=

^;

x

³

h@

³

^{3 2}

H

⁰²(

!

) ²

H

⁰¹(

!

)^g

and we note that the following mapping is an isomorphism

j

:

(

¹

²

³)²

H

⁰¹(

!

)]²

H

⁰²(

!

)^;!

;!

j

(

) = (

^1;

x

³

h@

¹

³

^2;

x

³

h@

²

³

³)²

V

_hKL(

:

) (31) Theorem 4. The semi-norm^{j jh} dened by

v

²

V

()^;!j

v

^jh=^fX3

ij⁼¹

j

_hij(

v

)^j20g1=2 is a norm over

V

() equivalent to ^{jj jj1}

:

Corollary 5. The mapping

v

²

V

_hKL()^;!j

v

^jh=^{f X2}

⁼¹

j

_h(

v

)^j20g1=2 is a norm over

V

_hKL() equivalent to the norm ^{jj jj1}

:

Corollary 6. The mapping ^{jj jj}KL dened by (see (30))

v

²

V

_hKL()^;!jj

v

^jjKL=^fjj

^{1 jj21}_!+^jj

^{2 jj21}_!+^jj

^3jj22_!^g1=2 is a norm over

V

_hKL() equivalent to the norm ^{j jh} and, consequently, to the norm ^{jj jj1}

:

The rst step in order to obtain convergence of the sequence (

u

(

"

)

(

"

)) consists of the following a priori estimates.

Theorem 7. There exist

"

⁰

>

0 and a constant

C

=

C

(

h!

) such that for all 0

< " < "

⁰ we have

jj

u

(

"

)^jj1

C

^j

(

"

)^j0

C

j

"

³(

"

)^j0

C

^j

"

²

³³(

"

)^j0

C:

Corollary 8. The sequences^f

u

(

"

)^g">⁰ and^f

(

"

)^g">⁰ verify the following weak convergences

u

(

"

)

* u

in

V

()

(

"

)

*

in

L

²()

"

³(

"

)

*

0 in

L

²()

"

²

³³(

"

)

*

0 in

L

²()

where

u

is the unique element in

V

_hKL() solution of the problem

Z

1^;

E

²

h

^f(1^;

)

_h(

u

)

_h(

v

) +

_h(

u

)

_h(

v

)^g

dx

=

Z

hf

i

v

i

dx

+

Z

;

g

i

v

i

da

for all

v

²

V

_hKL()

(32) and

is given by

=

E

1^;

^2f(1;

)

_h(

u

) +

_h (

u

)

^g

:

(33) Corollary 9. The following strong convergences hold true:

u

(

"

)^!

u

in

V

()

(

"

)^!

in

L

²()

"

³(

"

)^!0 in

L

²()

"

²

³³(

"

)^!0 in

L

²()

:

Remark 10. We can make the limit problem (32) more explicit by taking

u

= (

^1;

x

³

h@

¹

³

^2;

x

³

h@

²

³

³)

(34)

v

= (

^1;

x

³

h@

¹

³

^2;

x

³

h@

²

³

³)

(35) for

²

H

⁰¹(

!

) and

³

³²

H

⁰²(

!

) and by letting

g

_i⁺(

x

¹

x

²) =

g

i(

x

¹

x

²

1)

g

_i^;(

x

¹

x

²) =

g

i(

x

¹

x

²

0)

:

Then the limit problem takes the following form:

1^;

E

^2f

Z

!

h

(1^;

)

(

)

(

) +

(

)

(

)]^;

Z

!

h

²

2 (1^;

)(

(

)

@

³+

(

)

@

³) +

(

(

)

@

³+

(

)

@

³)]

+

Z

!

h

³

3 (1^;

)

@

³

@

³+

@

³

@

³]^g=

Z

!

h

Z

1

0

f

i]

i^;

Z

!

h

²

Z

1

0

x

³

f

]

@

³+

Z

!

g

_i^;+

g

_i⁺]

i^;

Z

!

hg

⁺

@

³

for all

²

H

⁰¹(

!

) ^{3 2}

H

⁰²(

!

)

:

Remark 11. If we assume an asymptotic expansion of the form (

u

(

"

)

(

"

)) = (

u

⁰

⁰) +

"

²(

u

²

²) +

"

⁴(

u

⁴

⁴) +

:::

and if we substitute this expression into the scaled variational problem, we obtain that the rst term (

u

⁰

⁰) veries

u

⁰=

u

⁰ =

_i⁰³ =

x

³

_i⁰

@

h

^{;Z x3}

0

@

(

h

_i⁰ )^;

g

_i^{;;Z x3}

0

hf

i

:

Remark 12. We can return to the reference conguration ^"taking into ac- count that

u

is a limit approximation of

u

(

"

)

:

Performing the inverse change of variable and the inverse scaling, we obtain that the limit displacement eld

u

^0" in ^" is given by

u

^0"(

x

^") =

u

(

x

) =

(

x

¹

x

²)^;

x

³

h

(

x

¹

x

²)

@

³(

x

¹

x

²) =

=

(

x

¹

x

²)^;

"

^;1

x

^"3

@

³(

x

¹

x

²) =

_"(

x

¹

x

²)^;

x

^"3

@

^3"(

x

¹

x

²)

u

^03"(

x

^") =

"

^;1

u

³(

x

) =

"

^;1

³(

x

¹

x

²) =

^3"(

x

¹

x

²)

where

_"=

"

3=

"

^;1

³

:

Letting

g

_i^"+(

::

) =

g

_"i(

::"h

(

::

)) and

g

_i^";(

::

) =

g

_"i(

::

0) we have

g

^" =

"g

and

g

^3"=

"

²

g

³

:

Then, we obtain that

_"²

H

⁰¹(

!

) and

^3"2

H

⁰²(

!

) are the unique solution of the following problem which is the obtained model

1^;

E

^2f

Z

!

"h

(1^;

)

(

^")

(

) +

(

^")

(

)]^;

Z

!

"

²

h

²

2 (1^;

)(

(

^")

@

³+

(

)

@

^3")+

(

(

^")

@

³+

(

)

@

^3")]

+

Z

!

"

³

h

³

3 (1^;

)

@

^3"

@

³+

@

^3"

@

³]^g=

Z

!

Z "h

0

f

_"i+

g

_i^";+

g

_i^"+]

i^;

Z

!

Z "h

0

x

^"3

f

_"+

"hg

^"+]

@

³

for all

²

H

⁰¹(

!

) ^{3 2}

H

⁰²(

!

)

:

3. Optimization of a non-symmetric plate

We consider an homogeneous, isotropic, elastic non-symmetric plate oc- cupying the reference conguration

^h=^f(

x

¹

x

²

x

³)²

IR

³ : (

x

¹

x

²)²

!

0

< x

³

< h

(

x

¹

x

²)^g

clamped on its lateral surface and submitted to body forces of density

f

^h (

f

_hi) on ^hand surface forces of density

g

^h+(

g

_i^h+) on ;^h+and

g

^h;(

g

_i^h;) on ;^h; where

;^h+ =^f(

x

¹

x

²

x

³)²

IR

³ : (

x

¹

x

²)²

! x

³=

h

(

x

¹

x

²)^g

;^h;=^f(

x

¹

x

²

x

³)²

IR

³ : (

x

¹

x

²)²

! x

³ = 0^g

:

We denote by

u

^h (

u

_hi) : ^h;!

IR

³the displacement eld corresponding to the above conditions. In the sequel we identify the functions

u

^h

f

^h and

g

^h with the functions

u

(

h

) : ^;!

R

³

f

(

h

) : ^;!

R

³ and

g

(

h

) : ; ^;!

R

³ such that

u

(

h

)(

x

¹

x

²

x

³) =

u

^h(

x

¹

x

²

x

³

h

(

x

¹

x

²))

f

(

h

)(

x

¹

x

²

x

³) =

f

^h(

x

¹

x

²

x

³

h

(

x

¹

x

²))

g

⁺(

h

)(

x

¹

x

²

1) =

g

^h+(

x

¹

x

²

h

(

x

¹

x

²))

g

^;(

h

)(

x

¹

x

²

0) =

g

^h;(

x

¹

x

²

0)

:

For simplicity, we assume from now on that the applied forces are such that

f

(

h

) =

f

and

g

(

h

) =

g

where

f

and

g

are independent of

h:

As it was shown in the previous section the displacement eld

u

(

h

) can be approximated by the only solution of the following variational problem:

u

(

h

)²

V

_hKL()

a

h(

u

(

h

)

v

) =

l

h(

v

)

for all

v

²

V

_hKL()

where for all

uv

²

V

_hKL(),

a

h(

uv

) =

Z

1^;

E

²

h

^f(1^;

)

_h(

u

)

_h(

v

) +

_h(

u

)

_h(

v

)^g

dx

(36)

l

h(

v

) =

Z

hf

i

v

i

dx

+

Z

;

g

⁺_i

v

i

da

+

Z

;

g

_i^;

v

i

da:

(37)

The optimal design problem deals with minimizing the weight of the non- symmetric plate ^h

or equivalently its volume (since the material is homo- geneous), that we denote by

J

(

h

) =^R_!

h:

The plate must be designed in such a way that allowed deections be less than a given bound

e:

As

u

³(

h

)²

H

⁰²(

!

) we impose:

jj

u

³(

h

)^jj1!

e:

In addition, we impose to the thickness some constraints of technological nature:

i

) The plate is designed in order to be constructed later, and this con- struction would not be possible if the plate is too thick or too thin. This fact leads us to impose the constraint:

0

< a

h

(

x

)

b <

¹

for all

x

²

!:

ii

) We also need that the variation of the thickness be slow and progres- sive. Then, we require

jj

@

h

^jj1!

c

^jj

@

h

^jj1!

d:

These technological constraints lead us to work in the set of feasible thick- ness:

U

ad=^f

h

²

W

²¹(

!

) :

a

h

(

x

)

b

^jj

@

h

^jj1!

c

^jj

@

h

^jj1!

d

^g which is a convex, closed and bounded subset of

W

²¹(

!

)

:

If we dene the set

K

h = ^f

v

²

V

_hKL() : ^jj

v

^{3 jj1}!

e

^g which is a convex, closed subset of

V

_hKL()

we can formulate the following optimal design problem:

(

P

)

Find h

²

U

ad

such that

:

u

(

h

)²

K

_h

J

(

h

) =

inf

^f

J

(

h

) :

h

²

U

ad

u

(

h

)²

K

h^g

:

Our rst aim is to study the existence of a solution for problem (

P

)

:

In order to do so, we consider

J

as a function dened in the space

W

¹¹(

!

)

J

:

h

²

W

¹¹(

!

)^;!

J

(

h

) =

Z

!

h

²

IR

and we dene the set

B

=^f

h

²

U

ad :

u

(

h

) ²

K

h^g that we assume to be non empty. Then, problem (

P

) can be written in the following form:

(

P

)

Find h

²

B such that J

(

h

) =

inf

^f

J

(

h

) :

h

²

B

^g

:

Since

J

is continuous, to obtain the existence of a solution it should be enough to prove that

B

is compact in

W

¹¹(

!

)

:

In order to prove this fact, we write problem (

P

) in a more suitable form.

We recall that the denition of ^h implicitly assumes that

h

is in the subset

A

(

!

) of the space

W

¹¹(

!

) given by

A

(

!

) =^f

h

²

W

²¹(

!

) :

h

(

x

)

(

h

)

>

0

for all

x

²

!

^g

:

We denote by

L

²(

V

_hKL()) the space of continuous bilinear forms on

V

_hKL() and by

L

(

V

_hKL()

V

_hKL()⁰) the space of continuous linear forms from

V

_hKL() on its dual. We also introduce the functions

T

:

h

²

A

(

!

)^;!

T

(

h

) =

l

h ²

V

_hKL()⁰

F

:

h

²

A

(

!

)^;!

F

(

h

) =

a

h ²

L

²(

V

_hKL())

G

:

a

²

L

²(

V

_hKL())^;!

G

(

a

) =

A

²

L

(

V

_hKL()

V

_hKL()⁰)

where

A

(

y

)(

z

) =

a

(

yz

)

for all

yz

²

V

_hKL()

:

Of course,

G

is an isometric isomorphism.

With each bilinear form

a

h =

F

(

h

) we associate the operator

A

h =

G

(

a

h) =

G

(

F

(

h

))

:

Since

a

h is

V

_hKL()^;elliptic for all

h

²

A

(

!

), it fol- lows from Lax-Milgram lemma that for each

l

h ²

V

_hKL()⁰ there exists a unique element

u

(

h

) ²

V

_hKL() such that

A

h(

u

(

h

)) =

l

h

:

Then,

A

h is an isomorphism from

V

_hKL() onto

V

_hKL()⁰ and the following function is well dened and of class

C

¹:

u

:

h

²

A

(

!

)

W

¹¹(

!

)^;!

u

(

h

) =

A

^;1_h (

l

h)²

V

_hKL()

H

¹()]³

:

This is the key to prove that

U

ad and

B

are compact in

W

¹¹(

!

)

:

Conse- quently, if

l

h is of the form (37) then problem (

P

) has at least one solution.

Remark 13. For

hk

²

A

(

!

) we dene

m

hk =

DT

(

h

)(

k

)²

V

_hKL()⁰

a

hk =

DF

(

h

)(

k

)²

L

²(

V

_hKL())

:

Then we have from (37)

m

hk =

Z

kf

i

v

i

:

In the same way we obtain from (36) that for all

uv

²

V

_hKL() given by (34),(35)

a

hk(

uv

) =

E

1^;

^2f

Z

!

k

(1^;

)

(

)

(

) +

(

)

(

)]^;

Z

!

hk

(1^;

)(

(

)

@

³+

(

)

@

³)+

(

(

)

@

³+

(

)

@

³)]+

Z

!

h

²

k

(1^;

)

@

³

@

³+

@

³

@

³]^g

:

Let

A

hk =

G

(

a

hk)²

L

(

V

_hKL()

V

_hKL()⁰)

:

Then we have

p

=

Du

(

h

)(

k

) =

A

^;1_h

m

hk^;

A

hk(

u

(

h

))]

:

This equation will be helpful for the numerical computation of

Du

(

h

)(

k

) in the next section.

4. Penalty method

In order to obtain the numerical solution of problem (

P

) we will use penalty techniques, dening a family of problems (

P

) which approximate problem (

P

) and which have a suitable numerical solution. Besides, we will obtain necessary optimality conditions for the penalty problem (

P

)

:

For

>

0 we dene the function

J

:

h

²

A

(

!

)^;!

J

(

h

) =

J

(

h

_{) + 12}

^jj

u

(

h

)^;

P

K^h(

u

(

h

))^jj2_KL²

R

where

P

K^h :

V

_hKL() ^;!

K

h denotes the projection in the Hilbert space

V

_hKL() over the closed and convex subset

K

h

which is a continuous function (Barbu-Precupanu 3], Cea 11]). Thus,

J

is also a continuous function.

We consider the following problem:

(

P

)

Find h

²

U

ad

such that J

(

h

) =

inf

^f

J

(

h

) :

h

²

U

ad^g

:

Due to the compacity of

U

adand the continuity of

J

the problem (

P

) has, at least, one solution.

If for each

>

0 we note

h

a solution of problem (

P

) then we have the following result.

Theorem 14.

i

) If

¹

<

²

then

J

²(

h

²)

J

¹(

h

¹)_hinf

2B

J

(

h

)

(38)

jj

u

(

h

¹)^;

P

K^h 1

(

u

(

h

¹))^jjKL ^jj

u

(

h

²)^;

P

K^h 2

(

u

(

h

²))^jjKL

(39)

J

(

h

²)

J

(

h

¹)_hinf

2B

J

(

h

)

:

(40)

ii

) The family ^f

h

:

>

0^g is bounded in

W

²¹(

!

) and there exist sequences ^f

n^g

^f

h

^ng such that if

n

^!1

then

n^!0

h

^{n !}

h

in

W

¹¹(

!

)

:

Each of these limits

h

is a solution of problem (

P

)

:

iii

) It is veried that

lim^!0

J

(

h

) = lim

!0

J

(

h

) = inf_h

2B

J

(

h

)

:

As an immediate consequence, we obtain the following convergence result.

Corollary 15. If problem (

P

) has a unique solution

h

and, for each

>

0

h

denotes a solution of the problem (

P

)

then as

^!0 we have

h

^!

h

in

W

¹¹(

!

)

:

In order to obtain optimality conditions for the problem (

P

) we rst prove the following result.

Theorem 16. The function

J

:

A

(

!

)^;!

R

is of class

C

¹ and

DJ

(

h

)(

k

) =

J

(

k

) +

a

hk(

u

(

h

)

z

)^;

m

hk(

z

)

where

u

(

h

) and

z

verify

A

h(

u

(

h

)) =

l

h

(41)

p

=

I

^;

P

K^h

⁽

u

(

h

))

(42)

A

_h(

z

) +

p

= 0

(43) and where

A

_h:

V

_hKL()^;!(

V

_hKL())⁰ is the adjoint operator of

A

h

:

A point

h

²

U

ad is said to be a stationary point for problem (

P

) if and only if it veries

DJ

(

h

)(

h

^;

h

) 0

⁸

h

²

U

ad

:

Since

U

ad is a convex set, we have that if

h

²

U

adis a solution of (

P

) then

h

is a stationary point for (

P

) (cf. Cea 4]). This fact supplies us the following necessary optimality condition.

Theorem 17.

h

²

U

ad is a stationary point for (

P

) if and only if there exist

p

u

z

²

V

_hKL() verifying

A

h(

u

) =

l

h

p

=

I

^;

P

K^h

⁽

u

)

A

_h(

z

) +

p

= 0

J

(

h

)^;

J

(

h

) +

a

hh^;h(

u

z

)^;

m

hh^;h(

z

)0

⁸

h

²

U

ad

:

References

1] L.J. Alvarez-Vazquez and J.M. Via~no. Modeling and optimization of non- symmetric plates.To appear.

2] V.Barbuand T.Precupanu.Convexity and optimization in Banach spaces.Sijtho and Noordho, Bucharest, 1978.

3] E.Casas.Optimality conditions and numerical approximations for some optimal de- sign problems.Control Cibernet., 19, pp 73-91 (1990).