Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 3, pp. 545-560, Warsaw 2003

OPTIMAL DESIGN OF MEMBRANE SHELLS.

HOMOGENIZATION-BASED RELAXATION OF THE

TWO-PHASE LAYOUT PROBLEM

Tomasz Lewiński

Institute of Structural Mechanics, Warsaw University of Technology

e-mail: T.Lewinski@il.pw.edu.pl

Józef Joachim Telega

Institute of Fundamental Technological Research, Polish Academy of Sciences

e-mail: jtelega@ippt.gov.pl

The aim of using shell structures instead of plates is to avoid bending,
hence the vital role of themembrane theory.Within this theory the classi-
cal optimumdesign problem is formulated: lay out two isotropicmaterials
such that the shell becomes the stiffest possible. The amount of both the
materials is fixed. The aim of the present paper is to reformulate this pro-
blem in a form assuring its well-posedness. Themembrane approximation
can be introduced from the very beginning or be imposed upon the rela-
xation. In the present paper it is shown that the latter modelling leads
to a better formulation. It does not lose its stability even if one material
degenerates to a void, thus leading to a well-posed shape design problem.

Key words: membrane shells, homogenization, minimum compliance
problem, relaxation by homogenization,Michell’s sphere

1. Introduction

Aprerequisite for solving the classical shape design problem is understan-
ding the followingmoregeneral problem: layout twonon-degenerated isotropic
materials in a feasible domain such that the two-phase bodyobtained becomes
the stiffest among all possible bodies transmitting a given surface loading to
a given support. The volume of one (or both)material must be fixed tomake
the problem solvable. It turns out that this problem requires a relaxation.



546 T.Lewiński, J.J.Telega

If applied to 2D or 3D elasticity problems, the relaxation by homogeniza-
tion is now well-understood and considered as a standard method for finding
stable optimal layouts, see Tartar (2000), Cherkaev (2000), Allaire (2002).
Two-phase compliance minimization problem of thin plates was solved by
Gibiansky and Cherkaev (1984) and further cleared up by Lipton (1994) and
Lewiński andTelega (2000, Sec. 26). New layouts have been recently reported
by Czarnecki and Lewiński (2001) and Kolanek and Lewiński (1999, 2003).
This latter work reports new results for the old problem of designing of circu-
lar and annular plates, for which the first relaxed results were found byCheng
and Olhoff (1981).

Optimization of shells is less developed, although the first relaxed numeri-
cal solutionswere already announcedbySuzuki andKikuchi (1991) andTenek
and Hagiwara (1994).

The relaxation requires the homogenization formulae for shells. Within
the bending theory of thin shells such formulae were derived in Lewiński and
Telega (1988) andTelega andLewiński (1998), see Lewiński andTelega (2000,
Sec. 17). These formulae are fairly complicated, since they couple the mem-
brane and flexural effects. The simplifying assumption of shallowness cancels
this coupling, hencemaking the final formulae similar to those for plates. Just
these formulae are usually used to relax the optimumdesign problemof shells.

Theaimof thepresentpaper is to consider a specific casewhen thebending
effects canbeneglected.This simplification canbe introducedat various stages
of the optimization process. In the present paper two possible methods of
modelling are discussed: neglecting the bending effects before relaxation and
then after relaxation. It is shown that only the latter method can lead to a
correct formulation of shape design of membrane shells.

2. Two-phase layout problem for a membrane shell formed on

a given middle surface

2.1. Equilibrium problem

Let Ω be a given plane domain whose image S ⊂ R3 is a middle surface
of the shell, the transformation of Ω into S being denoted by Φ, i.e.

Φ :Ω→S⊂ R3
(2.1)

ξ=(ξ1,ξ2)∈Ω→Φ(ξ)∈S



Optimal design of membrane shells... 547

The shell considered is formed around its middle surface S such that its
thickness h is kept constant. Assume that the shell is supported along ∂Su,
loaded by tractions T(s) along ∂ST and subjected to the surface loading
q(ξ), q = (q1,q2,q3). Here ∂S = ∂Su∪∂ST , ∂Su∩∂ST = ∅, ∂Su =Φ(Γu),
∂ST =Φ(ΓT), Γ =Γu∪ΓT and Γu∩ΓT = ∅.
The deformation of the shell is determined by the displacement field

(u1,u2,w) with u = (u1,u2) representing the tangent displacement and w
being the displacement normal to S.

To introduce the set of admissible displacements and then to formulate the
boundary-value problem, we define first

V =
{
(v,v) |vα ∈H1(Ω), v∈H2(Ω), vα =0 on Γu

}
(2.2)

and endow this space with the norm

‖(v,v)‖2V =
2∑

α=1

‖vα‖2H1(Ω)+‖v‖
2
H2(Ω) (2.3)

From now onward the small Greek indices: α, β, λ, µ will take values 1
or 2.

Let us recall the formulae for membrane strains

ǫαβ(u,w)=
1

2
(uα‖β +uβ‖α)− bαβw (2.4)

where (·)‖α represents the covariant derivative in the tangent plane and (bαβ)
is a curvature tensor, see Bernadou (1996) and Lewiński and Telega (2000,
Sec. 16). Assume that A(ξ) =

(
Aαβλµ(ξ)

)
represent a membrane stiffness

tensor which satisfies the usual symmetry and positive definiteness properties.
We define the bilinear form

a0(u,w;v,v) =

∫

S

Aαβλµ(ξ)ǫλµ(u,w)ǫαβ(v,v) dS (2.5)

and the norm

‖(u,w)‖0 = [a0(u,w;u,w)]
1

2 (2.6)

Now let V0 be the completion of V in the norm ‖·‖0.We observe that in
general

V 0 ⊂
{
(v,v) |vα ∈H1(Ω), v∈L2(Ω), vα =0 on Γu

}



548 T.Lewiński, J.J.Telega

and the inclusion is strong, see Sanchez-Hubert and Sanchez-Palencia (1997).
The space V0 plays the role of the space of kinematically admissible displa-
cements, within themembrane shell theory. Note that no boundary condition
can be imposed on w and v since in the norm of the space V 0 the derivative
of these fields does not intervene. Consequently the value (trace) of w and v
on ∂Γ cannot be determined.

The equilibrium problem of the membrane shell considered has the form

(P1)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

find (u,w)∈V 0 such that
a0(u,w;v,v) = f(v,v) ∀(v,v)∈V 0

with

f(v,v)=

∫

S

(qαvα+qv) dS+

∫

∂ST

Tαvα ds

(2.7)

and s parametrizes ∂S.

Themembrane stress resultants are given by

Nαβ =Aαβλµǫλµ(u,w) (2.8)

They satisfy the following local equations of equilibrium

N
βα
‖β
+qα =0 bαβN

αβ +q=0 (2.9)

in Ω.

The local equations (2.9) completed with boundary conditions on ΓT are
equivalent to the variational equation

∫

S

Nαβǫαβ(v,v) dS = f(v,v) ∀ (v,v)∈V 0 (2.10)

Remark 1. There exist problems for which the fields (Nαβ) can be found
by solving (2.10). Such problems are called statically determinate. For
these problems the set of statically admissible membrane forces

S(Ω)=
{
N∈L2(Ω,Es2) |N

αβ
‖β
∈L2(Ω) and (2.10) is satisfied

}
(2.11)

is a one-element set. In general, S(Ω) is an affine set. 2



Optimal design of membrane shells... 549

2.2. The layout problem

Assume that the membrane shell is composed of two isotropic materials,
thefillingbeing transverselyhomogeneous.Thus themembrane stiffness tensor
has the following representation

A(ξ)=χ1(ξ)A1+χ2(ξ)A2 (2.12)

where

Aα =2kαI1+2µαI2 (2.13)

Here k2 >k1, µ2 >µ1 are stiffnesses due to in-plane uniform stress and shear
stresses, respectively. Tensors I1, I2 are expressed in terms of themetric tensor
g=(gαβ):

I
αβλµ
1 =

1

2
gαβgλµ

(2.14)

I
αβλµ
2 =

1

2
(gαλgβµ+gαµgβλ−gαβgλµ)

they have properties of projection operators. The function χα(ξ) is a charac-
teristic function of the domain Ωα corresponding to the domain Sα around
which the α-th material is located. Thus Ω1∪Ω2 =Ω, Ω1∩Ω2 = ∅ and

χα(ξ)=

{
1 if ξ∈Ωα
0 otherwise

(2.15)

The tensor of compliances

a =A−1 (2.16)

is given by

a(ξ)=χ1(ξ)a1+χ2(ξ)a2 (2.17)

with

aα =2KαI1+2LαI2
(2.18)

Kα =(kα)
−1 Lα =(µα)

−1

The compliance C of the shell is defined in a standardmanner

C = f(u,w) (2.19)



550 T.Lewiński, J.J.Telega

The following equality

C = inf
N∈S(Ω)

∫

S

N : (aN) dS (2.20)

expresses the Castigliano theorem for membrane shells. It can be proved by
using the duality theory expounded in Ekeland and Temam (1976) or by a
hybrid approach developed by the second author, see Telega (2003, Sec. 3.2).
This problemwill be treated in a separate paper.

Note that if a shell problem is statically determinate, then inf in (2.20) is
redundant, since then S(Ω) is a one element set.

Now we are ready to formulate the layout problem

inf
{
C = f(u,w) |χ2 ∈L∞(Ω;{0,1}),

∫

S

χ2 dS=A
}

(2.21)

where (u,w) depend on χ2 and A is a given area occupied by thematerial 2.
To simplify further notation we put χ=χ2.

ByCastigliano’s theorem the layout problem (2.21) can be put in the form

(P)

∣∣∣∣∣∣∣∣∣∣

inf
χ∈L∞(Ω;{0,1})

inf
N∈S(Ω)

∫

S

N : (aN) dS

∫

S

χdS=A
(2.22)

Now we introduce the Lagrangian multiplier λ associated with the isope-
rimetric condition and change the position of supremum over λ before both
infima.Thiswas done in elasticity, seeKohn and Strang (1986), and is equally
justified in the case of membrane shells. For fixed λwe find

(Pλ)

∣∣∣∣∣∣
inf

χ∈L∞(Ω;{0,1})
inf

N∈S(Ω)

∫

S

[N : (aN)+λχ] dS (2.23)

The problems (P) and (Pλ) require relaxation. Indeed, imagine that the
directmethodof the calculusof variations is applied to solve theseproblems. In
this case such an approach will involve a sequence of characteristic functions,
say {χn}n∈N. Itsweak-∗ limit in L∞(Ω) is not a characteristic function but a
function 0¬m(ξ)¬ 1, ξ∈Ω.
Construction of the relaxed problem for (Pλ) will be the subject of the

next section.



Optimal design of membrane shells... 551

3. Relaxation of the layout problem (Pλ) for membrane shells

As we already know, relaxation means admitting weak limits of sequen-
ces {χn}n∈N of two-phase designs. These weak-∗ limits in L∞(Ω;{0,1}) are
understood by

lim
n→∞

∫

Ω

ϕχndξ→
∫

Ω

ϕmdξ ∀ϕ∈L1(Ω) (3.1)

The limit m belongs to L∞(Ω; [0,1]) and (L1(Ω))∗ =L∞(Ω) in the sense
of duality between L1 and L∞, cf. Ekeland and Temam (1976).
According to Remark 1, there exist statically determinate problems of

membrane shells in which S(Ω) is one-element set, say {N̂}= {(N̂αβ)}. This
element is independent of χn. The construction of the relaxation problem
for (Pλ) depends heavily on whether the equilibrium problem is statically
determinate or not.

3.1. Case of the equilibrium problem being statically determined

Let S(Ω)= {N̂}. Then (Pλ) assumes the form

inf
χ∈L∞(Ω;{0,1})

∫

S

[N̂ : (aN̂)+λχ] dS (3.2)

Note that N̂ does not depend on χ.We recall (2.17) and (2.18) and com-
pute

N̂ : (aN̂)= [1−χ(ξ)]N̂ : (a1N̂)+χ(ξ)N̂ : (a2N̂) (3.3)
Consider now a sequence {χn} such that χn

∗
⇀m∈L∞(Ω; [0,1]). Hence

a is replaced with an and

N̂ : (anN̂)→ 2W∗(N̂,m)= [1−m(ξ)]N̂ : (a1N̂)+
(3.4)

+m(ξ)N̂ : (a2N̂) weak-∗ inL∞

Thus the problem (Pλ) is replaced by

(P
1
λ)

∣∣∣∣∣∣
min

m∈L∞(Ω;[0,1])

∫

S

[2W∗(N̂,m)+λm] dS (3.5)

The result above holds irrespective of the isotropy assumption. If both the
phases are isotropic, the potential W∗ can be easily expressed in terms of



552 T.Lewiński, J.J.Telega

(trN̂)2 and trN̂
2
. Note yet that both the phases must be non-degenerated.

The relaxation does not pave the way for a formulation of the shape design
problem in which the shell is made of one material.

3.2. The statically indeterminate case

In the case considered the layout of both the materials influences the di-
stribution of the stress resultants (Nαβ). Therefore, the limit behaviour of the
sequence of functionals

inf
N∈S(Ω)

∫

S

[N : (a(χn)N)+λχn] dS (3.6)

where a(χn) is given by (2.17), should be consideredwithin the framework of
the Γ-convergence theory.
The above convergence problem has already been solved in a broader con-

text of the Koiter shell theory, see Telega and Lewiński (1998) and Lewiński
and Telega (2000, Sec. 17).We use here these results and conclude that (3.6)
must be replaced by

inf
N∈S(Ω)

∫

S

[2W∗(N,m)+λm] dS (3.7)

where the potential W∗ is defined as follows. First we assign a basic cell
Y = (0, l1)× (0, l2) to each point ξ of Ω. This cell is composed of both the
materials, their distribution being given by the characteristic functions

χY =χY2 χ
Y
1 =1−χY

χY =χY (y) y=(y1,y2)∈Y
Averaging over Y is denoted by 〈·〉 and defined by

〈f〉= 1|Y |

∫

Y

f dy |Y |= l1l2 (3.8)

Distributionof theflexibilities a =(aαβλµ)within Y is expressedas follows

a = [1−χY (y)]a1+χY (y)a2 (3.9)
Let us define the set

S
per(Y ) =

{
Ñ∈L2(Y,Es2)

∣∣∣
∂Ñαβ

∂yβ
=0 inY,

(3.10)

Ñαβνβ take opposite values at opposite sides of Y
}



Optimal design of membrane shells... 553

Here ν =(να) represents the unit vector normal to ∂Y .

Further, we introduce the set Gperm of tensors ah such that

N : (ahN)=min
{
〈Ñ : (aÑ)〉|Ñ∈ Sper(Y ), 〈χY 〉=m, 〈Ñ〉=N

}
(3.11)

where a is given by (3.9). Its closure is denoted by Gm, i.e.

Gm =G
per
m (3.12)

where the completion (·) is understood as admitting hierarchical microstruc-
tures within Y , like e.g. laminates of higher rank, see e.g. Cherkaev (2000).

Now we are ready to define the potential W∗

W∗(N,m)=min
{1
2
N : (aN) |a ∈Gm

}
(3.13)

A detailed analysis of the passage from (3.6) to (3.7), (3.13) will be given
elsewhere.

To put it briefly, the homogenization process retains the highest deriva-
tives of (2.9) at the microstructural level; hence the simplified form of the
equilibrium equations in the definition of the set S(Y ).

Let us note now that the definition (3.11) coincides with that of the plane
elasticity problem, see Lewiński and Telega (2000, Sec. 28.4). In the case of
the phases being isotropic, an explicit form of W∗ is known. It was found
by Gibiansky and Cherkaev (1987). Prior to recalling this expression let us
introduce some auxiliary notation. The invariants of N are chosen as

I(N)=
1√
2
trN II(N)=

1√
2
[(trN)2−4detN]

1

2 (3.14)

Next we set

ζN =
II(N)

|I(N)| (3.15)

If f takes two values f1 and f2, we define

〈f〉m =(1−m)f1+mf2 ∆f = |f2−f1|
(3.16)

[f]m =(1−m)f2+mf1



554 T.Lewiński, J.J.Telega

We assume here isotropy of both phases, as in Eqs (2.17) and (2.18). Let
us define the auxiliary quantities

Ǩ =
K1K2+L2〈K〉m
L2+[K]m

Ľ=
L1L2+K2〈L〉m
K2+[L]m

ζ1 =
K2+[L]m
m∆L

ζ2 =
m∆K

[K]m+L2
aL = Ǩ aR =K2 cL =L2 cR = Ľ

AL =
m∆L(L2+[K]m)

[K+L]m
AR =

m(1−m)(∆L)2[K]m
[L]m[K+L]m

(3.17)

Further, we introduce the function

H(ζ)=





HL(ζ) if ζ ∈ [0,ζ2]
Hi(ζ) if ζ ∈ [ζ2,ζ1]
HR(ζ) if ζ ­ ζ1

(3.18)

where
HL(ζ)= aL+ cLζ

2 HR(ζ)= aR+ cRζ
2 (3.19)

and
Hi(ζ)=HL(ζ)+AL(ζ− ζ2)2 (3.20)

or
Hi(ζ)=HR(ζ)+AR(ζ− ζ1)2 (3.21)

The potential W∗ assumes the following form, cf. Lewiński and Telega
(2000, Sec. 28.4)

2W∗(N,m)=






1

2
I2(N)H(ζN) if I(N) 6=0
1

2
ĽII2(N) if I(N)= 0

(3.22)

Let us note that W∗(·,m) is smooth along the interfaces ζN = ζ2 and ζN = ζ1
of three regimes occurring in (3.18). Thus W∗(·,m) is smooth for all N.
We conclude that the relaxation of (Pλ) gives

(P
2
λ)

∣∣∣∣∣∣∣∣∣∣∣∣∣

min
m∈L∞(Ω;[0,1])

min
N∈S(Ω)

∫

S

[2W∗(N,m)+λm] dS

and
∫

S

mdS=A
(3.23)



Optimal design of membrane shells... 555

This formulation looks like a similar problem for two-dimensional elasti-
city. The shell characteristics are concealed in S(Ω), where the differential
equilibrium equations involve the metric tensor and the curvature tensor of
the shell middle surface.
The sub-problem

min
N∈S(Ω)

∫

S

W∗(N,m) dS (3.24)

is a non-linear equilibriumproblem of a hypothetic physically nonlinearmem-
brane shell.

4. Shape design problem

Shapedesignmeans forminga shell fromonegivenmaterial, thusadmitting
some voids in S such that the isoperimetric condition (3.23)2 holds. In the
shape design problemwe usually assume that the surface loading q is absent,
to prevent from cutting out a loaded part of the shell.
Shape design formulation should emerge as a result of passing to zero:

k1 → 0, µ1 → 0 or K1 →+∞, L1 →+∞.
In the case of the shell problem being statically determined, the above

passage to the limit is not allowable. The definition (3.4) of W∗ cannot be
used if k1 or µ1 tend to zero.
In the statically indeterminate problems the passage to the limit k1 → 0,

µ1 → 0 is admissible, although the potential W∗(N,m) loses its smoothness.
Then it assumes the form

W∗(N,m)=W∗0(N)+
1−m
m
G(N) (4.1)

where

W∗0(N)=
1

4
KI2(N)+

1

4
LII2(N)

(4.2)

G(N)=
1

4
(K+L)(|NI|2+ |NII|2)

where K = K2, L = L2 and NI, NII represent the principal values of the
tensor N. Thepotential W∗0 refers to the one-phasematerial ofmoduli k= k2,
µ=µ2; K =1/k, L=1/µ. The expression (4.1) is the same as in the plane
elasticity case, see Allaire andKohn (1993).



556 T.Lewiński, J.J.Telega

Having found (4.1) one can consider the degenerated case when A is a
very small number. This corresponds to the case of the Lagrangian multiplier
λ being a large number. Following the arguments of Allaire andKohn (1993),
we conclude that the optimization problem reduces to

min
N∈S(Ω)

∫

S

(|NI|+ |NII|) dS (4.3)

see also Lewiński andTelega (2001). Theproblemabove is free of anymaterial
characteristics. It resembles the Michell formulation, see Strang and Kohn
(1983). Using the same duality arguments we can pass from (4.3) to a dual
formulation

max
ǫ(u,w)∈B

∫

∂ST

T ·u ds (4.4)

with

B=
{
ǫ∈ Es2 | |ǫI| ¬ 1, |ǫII| ¬ 1

}
(4.5)

Problem (4.4) can be viewed as a locking problem, while B can be treated
as a locking locus, see Telega and Jemioło (1998).
The literature onMichell’s structures, see e.g. Hemp (1973), concerns pla-

ne problems with one exception: the problem of forming the stiffest spatial
gridwork subjected to two opposite torques. Michell (1904) claims that the
stiffest network should be formed on a sphere, yet the proof of this property
has never been published. This famous Michell’s sphere problem can also be
put in the form: find the optimal layout of fibres forming the lightest spherical
gridwork capable of resisting two opposite concentrated torques applied at two
given points.Thesepoints are taken as poles of the optimal spherical gridwork,
see Michell (1904) and Hemp (1973). This Michell problem is formulated by
(4.4). Its solution can be found inHemp (1973). At first one shouldfind (u,w)
such that ǫI = 1 and ǫII =−1, uniformly on the sphere. The work done by
the tractions T (which replace the torques) determines the optimal weight.
This solution is conditioned by the assumption of the spherical shape of the
shell.

5. Final remarks

In the problem considered the shell middle surface S is taken as known.
More challenging problem is to admit certain variations of S and find its best



Optimal design of membrane shells... 557

shape. As indicated above, an open problem is whether just a sphere is the
stiffest among all shells of revolution subjected to two opposite torques, cf.
Michell (1904) and Hemp (1973). A general treatment of the problems with
varying middle surfaces seems to be a difficult task – let us remind mathe-
matical difficulties appearing in the classical problem of minimal surfaces, see
Nitsche (1975), Dierkes et al. (1992) and Pilz (1997).

Our considerations did not take into account prestressing of a membrane
to enforce a free of folding membrane behaviour. The role of prestressing is
described inBarnes (1988). Includingprestressing in the formulationpresented
here is a challenge for future work.

Acknowledgement

The work was supported by the State Committee for Scientific Research (KBN)

through the grant No. 7T07A04318.

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Optymalne projektowanie powłok w zakresie pracy bezmomentowej.

Relaksacja zadania optymalizacji rozkładu dwu materiałów

z wykorzystaniem metod homogenizacji

Streszczenie

Odpowiednie kształtowanie konstrukcji powłokowych zezwala na minimalizację
efektówzginania.Konstrukcje zaprojektowane idealnie powinnypracowaćbezmomen-
towo, co podkreśla szczególną rolę teorii powłok błonowych, czyli powłok nie podle-
gających zginaniu.W pracy rozpatrujemy klasyczne zadanie optymalizacji rozmiesz-
czenia dwu materiałów izotropowych w powłoce pracującej bezmomentowo w celu
maksymalizacji jej sztywności. Ilość obumateriałów jest z góry ustalona. Celem pra-
cy jest przeformułowanie tego zagadnienia do postaci dobrze postawionej. Założenie
bezmomentowej pracy powłoki może być narzucone od początku lub przyjęte już po
procesie relaksacji (w sensie rachunkuwariacyjnego).W tej pracy wykazujemy, że ta



560 T.Lewiński, J.J.Telega

ostatnia metoda modelowania jest bardziej korzystna. Otrzymuje się sformułowanie,
które zachowuje się stabilnie nawet wtedy, gdy jeden z materiałów degeneruje się
do pustek, co zezwala na otrzymanie dobrze sformułowanego zadania optymalizacji
kształtu.

Manuscript received December 3, 2002; accepted for print March 11, 2003