Some remarks on the optimal design of periodically reinforced structures

M2AN. M

- M

G ^IUSEPPE B ^UTTAZZO D ^OMINIQUE A ^ZE

Some remarks on the optimal design of periodically reinforced structures

M2AN. Mathematical modelling and numerical analysis - Modéli- sation mathématique et analyse numérique, tome 23, n^o1 (1989), p. 53-61

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T f a P Q MATHEMATICALMOOtLUNGANONUMERICALANALYSJS

: f / 1 A \ J,_J MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol. 23, n° 1, 1989, p. 53 à 61)

SOME REMARKS ON THE OPTIMAL DESIGN OF PERIODICALLY REINFORCED STRUCTURES (*)

by Dominique A Z E (:) and Giuseppe BUTTAZZO (2)

Communicated by E. SANCHEZ-PALENCIA

Abstract.—We present a naîural optimality criterion for periodically reinforced structures.

Some examples of optimal structures are given. More specifically we consider a non trivial optimal two dimensional honeycomb structure in dimension 2.

Résumé. — On donne un critère naturel d'optimalité pour les structures périodiques renforcées. Quelques exemples sont étudiés. En particulier on met en évidence une structure en nid d'abeille optimale non triviale en dimension 2.

1. INTRODUCTION

The aim of this paper is to give an optimality criterion for the design of a periodically reinforced structure and to make explicit calculations in some simple cases.

Our work is based on the paper by H. Attouch and G. Buttazzo [2]

concerning the homogenization of periodically reinforced structures, and we shortly indicate now the framework of this study. Consider a bounded open subset fi of Un (n = 2 or 3 in the applications) with a Lipschitz boundary,

(*) Received in December 1986.

O Université de Perpignan, Avenue de Villeneuve, 66025 Perpignan, France.

(2) Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56100 Pisa, ïtaly.

MAN Modélisation mathématique et Analyse numérique 0399-0516/89/015309/$ 2.90

54 D. AZE, G. BUTTAZZO

and assume that it is filled with many periodically distributed thin highly conducting layers. More precisely, let

Y = [0, 1 [n be the unit cube of R ".,

S ç Y be a piecewise smooth n — 1 dimensional surface ,

Ss = {e(x+y):xeS, y e Zn} ,

Se>r = : dist

if

a if x e IRn\5£jr

where e, r, \ , a are positive parameters.

The network Se (see ^g. 1) will be called a periodically reinforced structure, and we want to study its behaviour (in terms of the potential obtained with a given charge density g(x)) as e -• 0.

— - -=->- v

Figure 1.

To do this, define on the Sobolev space the functionals

and consider the solutions w^{e r x} of the variational problems

(1.1) min F

(w)+ J gudx:ueHl(n)\

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

ON THE OPTIMAL DESIGN OF STRUCTURES 55 where g e L²(fl) is a given function. In [2] (see also [C5-S]) H. Attouch and G. Buttazzo studied the asymptotic behaviour, as (e, r, X) - • (0, 0, + oo ), of the functions w^{e r x} ; by using the F-convergence theory, they proved that if Xr/e -^ k, then w^E)r>x tends in L²(fl) to the solution of the problem

(1.2) min i f f

(Du)dx+ | gu dx: u e Hù(Sl)\

where f^a(z) is the quadratic form in Rⁿ defined by

(1.3) /

( z ) = min fa j \Dv\

dy +

+ k f \D

v\

dH

-

:v- <z, .> e w\ .

Hère H"'1 dénotes the n - 1 dimensional Hausdorff measure, DT v is the tangential derivative

DT v = Dv - v (Dv7 v) (v is the unit normal vector to S) ,

and W is the space of ail functions in Hf;^0C(Rⁿ) which are Y-periodic. For the sake of simplicity, in the following we shall assume k = 1. The function f(z) defined by

(1.4) f(z) = lim fa(z) = inf fa(z)

CL->Q a>0

is a quadratic form on 0?n, and we have

(1.5) / ( z ) = i n f | J \D

v\

dH

~

:v- (z, .) e w\ .

Dénote by A the n x n symmetrie matrix

by T its trace, and by 15*1 the n — 1 dimensional measure of S. The main resuit of this paper is the following.

THEO REM 1.1: We have T^(n —1)|5|. Moreover, the eigenvalues

\t of A satisfy the inequalities

(1.7) o^x,.^ \s\ . m

56 O. AZE, G. BUTTAZZO

In this way, we say that a periodically reinforced structure is optimal if T= (n-l)\S\ .

In Section 2 we shall prove Theorem 1.1, and in Section 3 we shall give some examples of optimal periodically reinforced structures in dimension 2 and 3.

2. PROOF OF THEOREM 1.1

With the notations of Section 1, we define, for every p :> 0

^ ; » - (z, •> ^ w

It is proved in [6], [3], [1], that the n x n symmetrie matrix

a f i_ 1 32/ap

17 2 3z, 3z^y

is the G-limit as 8 -> 0 of the séquence of matrices

Dénote by \fp the eigenvalues of Aa^. In order to obtain informations on the

\f^p, we use a resuit of F. Murât and L. Tartar.

T H E O R E M 2.1 (see [7], [9]) : The eigenvalues \f verify the following estimâtes (for simplicity we omit the indices a, p) :

(2.1) ^_ ^ k^t ^ fx⁺

" 1 1 1

y ^ , ^ ~~ i . ^ X(. - a ^ n - - <* fx+ - a

(2.3)

- X,

6 \ - i )

0 = meas (Y n Sïti/$).

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

ON THE OPTIMAL DESIGN OF STRUCTURES 57

Proof o ftheorem 1.1 : We remark that

e = l^J- + co(p) with lim Po>(p) = 0 ,

P p-» + oo

so t h a t

(2.4) lim |x_ = a a n d lim |x+ = a + \S\ .

P - • + 00 p-^ + 00

By (2.3) it follows

O - |x

)(p - ^- ) £ n (

-

;) ^ [»P - M.+ - (» - 1) M-- ] f i O - h)

i = 1 j * i / = 1

which can be written in the form

(2.5) (p - n

)((3 - M._ ) £ [(3"-

- P"-

£ X, + P„_

where ^ „ - 3 0 ) , ^*n_2(P) a r e polynomials of degree n — 3, n — 2 respect- ively.

After some simple calculations, (2.5) becomes

(2.6) (3"[TrAH p-^_ - ( n - l ) M .+ H J \ - i ( P ) .

Multiplying both sides of (2.6) by 3~", passing to the limit as (3 -> + oo, and taking into account (2.1), (2.4) we get

(2.7) lim Tr (Aap) =s na + (H - 1 ) | S | .

K + 00

It is well k n o w n (see for instance [4], [8], [1]) t h a t

(2.8) fa(z)= lim /a p (z ) , for every a > 0 a n d zeUn ,

where /a is defined in (1.3).

Coming back to (2.7), by (2.8), we get Tr (Aa)^na+ (n

58 D. AZE, G. BUTTAZZO where Aa is the n x n symmetrie matrix

Passing to the limit in (2.7) as a - • 0, (1.4) yields T r A = s (n-l)\S\ . Finally, (1.7) follows from (2.1) and (2.4). •

Remark 2.2 : Theorem 1.1 provides only an estimate on the trace and on the eigenvalues of the matrix A. But in gênerai, if L :> 0 is fixed, the set M^L of all matrices given by formula (1.6), where S runs over all surfaces with

\S\ = L, is smaller than the set of all matrices with j (n-l)L JO as X,- *s L .

For instance, if n = 2 and L <: 1, it is easy to see that the set M^L reduces to the only null matrix, whereas the set M^x consists of the two matrices

(è l) - (o î) •

It would be interesting (and as far we know it is an open problem) to characterize explicitely the set ML, or at least the matrices of ML which are optimal.

3. SOME EXPLICIT CALCULATIONS

In this section we give some examples of optimal periodic reinforced structures in dimension 2 and 3. Let us assume first n = 2. If 5 is a curve y (s) parametrized by its curvilinear abscissa, then formula (1.5) becomes (see [2])

(3.1) (Az,z) =inf

where L = \S\ and the infimum is taken over all functions w(s) satisfying the periodicity conditions. By the first order necessary conditions, it is easy to see that the solution w(.) of (3.1) is such that w'(s) + (z, 7'C*)> is M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

ON THE OPTIMAL DESIGN OF STRUCTURES 59 constant on [0, L ] . A formula analogous to (3.1) holds if S is the union of finitely many curves 7, (s) (/ = 1, ..., N)

(3.2) (Az,z) - inf (z,y;(s))\2ds\ .

Consider now the structure whose elementary cell is represented in figure 2.

Figure 2.

It is easy to see that if ô = 0 or 5 = - the structure above is optimal ; in fact we have

if 8 = 0 (Azyz) = if 8 = 1 <^z;z> =

and | 5 | = 3 and | 5 | = 2 V2 •

PROPOSITION 3.1 : For 0 < 8 < - ?^e structure of figure 2 w optimal if and only if Ô = -— .

o

Proof: We want to use (3.2) to compute the quadratic form <^4z? z) . Let us define

Xo = 1 - 2 ô and X = ^ /ô' 12 + - ;

60 D. AZE, G. BUTTAZZO

in this way, if 7,(5) (i = 1, 2, 3, 4, 5 ) are parametrizations of the segments EF, AE, BE, CF, DF respectively, we have

yj = ( 5 / \ , 1/2X) i = (8/X, -1/2X) J = ( - S A , -1/2X)

Analogously, for the functions w,(s) (i = 1, 2, 3, 4, 5), denoting by 0, b the values on is, F respectively, we obtain

w[ = (b- 2=wï = a/X

= w's = & A , so that, by (3.2) we get

(3.3) {Az,z) =inf f l ( è - a

0,6 l ^

The optimality conditions for a, b give

a = — b = Xn 2 X - 2 8

hence, after some calculations, (3.3) becomes

Since \S\ = Xo + 4 X, the structure is optimal if (3.4) Xo + 4 X = T_ L _ + i .

X + Xo X Equation (3.4) can be written in the form

that is X = 2 ô, which gives ô = — . 6

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

ON THE OPTIMAL DESIGN OF STRUCTURES 61

Let us conclude by a simple example in dimension 3. Consider the case of the cross structure

S = | (xl9 x2, X3) G Y : x1 = - or x2 = - or #3 = - J . In this case we obtain

and this shows that the structure is optimal, because

REFERENCES

[1] H. ATTOUCH, Variational Convergence for Functions and Operators. Appl.

Math. Ser., Pitman, Boston (1984).

[2] H. ATTOUCH and G. BUTTAZZO, Homogenization of reinforced periodic one- codimensional structures, Ann. Scuola Normale Sup. Pisa, 14, 3 (1987).

[3] A. BENSOUSSAN and J. L. LIONS and G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).

[4] L. CARBONE and C. SBORDONE, Some properties of T-limits of intégral functionals. Ann. Mat. Pura Appl., 122 (1979), 1-60.

[5] D. CIORANESCU and J. SAINT-JEAN PAULIN, Reinforced and alveolar structures.

Publication 85042 du Laboratoire d'Analyse Numérique Paris VI, Paris (1985).

[6] P. MARCELLINI, Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl., 117 (1978), 139-152.

[7] F. MURÂT and L. TARTAR, Optimality conditions and homogenization. Proceed- ings of «Nonlinear Variational Problems», Isola d'Elba 1983, edited by A.

Marino and L. Modica and S. Spagnolo and M. Degiovanni, Res. Notes in Math. 127, Pitman, Boston (1983), 1-8.

[8] E. SANCHEZ-PALENCIA, Non Homogeneous Media and Vibration Theory, Lecture Notes in Phys. 127, Springer-Verlag, Berlin (1980).

[9] L. TARTAR, Estimations fines de coefficients homogénéisés. Proceedings of

« Ennio De Giorgi Colloquium », Paris 1983, edited by P. Kree, Res. Notes in Math. 125, Pitman, Boston (1985), 168-187.