Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

3, 39, 2001

AUGMENTED LAGRANGIAN METHODS FOR A CLASS
OF CONVEX AND NONCONVEX CONTACT PROBLEMS

Józef Joachim Telega

Andrzej Gałka

Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw

e-mail: jtelega@ippt.gov.pl; agalka@ippt.gov.pl

Włodzimierz Bielski

Institute of Geophysics, Polish Academy of Sciences, Warsaw

e-mail: wbielski@igf.edu.pl

The aim of this contribution is threefold. First, we formulate unilateral
contact problems for three models of plates and the Koiter shell model.
Contact conditions have been formulated on the face being in contact
with an obstacle and not on the mid-plane of the plate or the middle
surface of the shell. Such a rigorous approach results in nonconvexmini-
mization problems even in the case of thin, geometrically linear plates.
Existence theorems are formulated for each model considered. Second,
the Ito andKunisch (1990, 1995) augmented Lagrangiansmethods have
been extended to nonconvex problems. Third, nonconvex duality theory
by Rockafellar and Wets (1998), valid for finite-degree-of-freedom sys-
tems has been extended to continuous systems. Specific examples have
also been provided.

Key words: unilateral contact problemswithout friction, plates, Koiter’s
shell model, augmented Lagrangianmethods, nonconvex duality

1. Introduction

Contact conditions for thin structures like plates and shells are usually
posed on the mid-plane of the plate or the middle surface of the shell, cf
Duvaut and Lions (1972, 1974), Panagiotopoulos (1985), Telega (1987). Such
an approach is unacceptable in the case of moderately thick structures and



742 J.J.Telega et al.

in the case of friction beetween the structure and obstacle. Also, rigorously
formulated contact problems shouldbe formulated on the face being in contact
with the obstacle.

Our considerations are confined to static frictionless contact problems. It
is then possible to formulate relevant boundary value problems in the form
of corresponding minimization problems. Since we are interested in the con-
tact conditions imposed on the face being in contact with an obstacle, the
resulting minimization problem is in general nonconvex even in the case of
a geometrically linear structure. We shall consider two geometrically linear
elastic plates, the vonKármán plate model and the linear Koiter shell model.
Amoderately thick nonlinear plate was studied by Bielski and Telega (1998),
cf also Bielski and Telega (1992, 1996). Other models of plates and shells,
including geometrically nonlinear models, can be studied similarly.

Ito andKunisch (1990, 1995) developedmathematically rigorous augmen-
ted Lagrangian methods valid for convex problems.We propose an extension
to the nonconvex contact problems, combining the approach of these two au-
thors with iterative procedures, cf also Bielski et al. (2000). An example has
also been provided. The papers by Telega and Gałka (1998, 2001) provide
many examples of usefulness of the method of the augmented Lagrangian.

The third topic studied in this paper concerns duality theory in the case
of nonconvex primal problems. In a series of papers we have shown that the
so-called Rockafellar’s theory of duality, as presented in the book by Ekeland
andTemam (1976), imposes restrictions on dual variables, cf Bielski andTele-
ga (1992, 1985a-d, 1986, 1996), Bielski et al. (1988, 1989), Telega et al. (1988),
Gałka et al. (1989), Telega (1989), Gałka and Telega (1990, 1992, 1995). For
instance, in the case of von Kármán’s plates, the matrix of membrane forces
has to be positive semi-definite, thus precluding compressed plates. Otherwise
the primal and dual problems will be characterised by a duality gap. Ano-
ther possibility is offered by so-called anomalous dual variational principles,
cf Gałka and Telega (1995), Telega (1995). However, their usefulness seems
to be of limited value, as prove the examples of compressed beems, studied
in these two papers. Recently, Rockafellar and Wets (1998) proposed a novel
approach to the formulation of dual problems in the nonconvex case where the
duality gap is possible. Their approach is confined to finite-degree-of-freedom
systems (discrete or discretized). In essence, this new approach exploits pro-
perly chosen augmented Lagrangians.We succeeded to extend theRockafellar
andWets (1998) duality theory to continuous systems.



Augmented lagrangian methods... 743

2. Geometrically linear plates

In Section 2 we shall formulate minimization problems in the case of the
obstacle problem for the linear Kirchhoff plate andReissner platemodel. The
obstacle is rigid and the contact occurs through the lower face of the plate.
An extension to the case where both the lower and upper facesmay come into
contact with rigid obstacles is straightforward.

2.1. Thin plates

Let Ω ⊂ R2 be a sufficiently smooth domain and Γ = ∂Ω its boundary.
Ω denotes themid-planeof anundeformedplate.Theplate occupies the region
Ω× (−h,h) ⊂ R3. The boundary Γ is decomposed into two parts: Γ0 and
Γ1 such that Γ = Γ0 ∪Γ1, Γ0 ∩Γ1 = ∅. Let v = [vi], vi = vi(xα,z) be
the displacement vector of a point (xα,z)∈Ω× (−h,h), α=1,2; i=1,2,3.
The axis z is directed downwards. We assume the classical Kirchhoff-Love
kinematical hypothesis

vα(xβ,z) =uα(xβ)−zw,α (xβ) v3(xβ,z)≡w(xβ) (2.1)

Here u=(uα) stands for the in-plane displacement vector whilst w denotes
the transverse displacement. By cijkl we denote the elasticity tensor of the
material of the plate. We assume that the plane z = 0 is the plane of the
material symmetry; hence cαβγ3 = c333α = 0. For a thin elastic plate the
constitutive relationship takes the form

σαβ =Cαβλµελµ(u) σα3 =2cα3λ3ελ3(u) σ33 =0 (2.2)

where σij are components of the stress tensor, and

Cαβλµ = cαβλµ− cαβ33c33λµc−13333
Here (cijkl) denotes the elasticity tensor of thematerial of the plate. As usual,
the strain-displacement relation is given by

εij(u)=u(i,j) =
1

2

(∂ui
∂xj
+
∂uj
∂xi

)
(2.3)

Let N=(Nαβ) and M=(Mαβ) be the membrane force tensor andmoment
tensor, respectively, defined by

Nαβ =

h∫

−h

σαβ dz Mαβ =

h∫

−h

zσαβ dz (2.4)



744 J.J.Telega et al.

The constitutive relations are given by

Nαβ =Aαβλµελµ(u) Mαβ =Bαβλµκλµ(w) (2.5)

Here εαβ(u) and καβ(w) are the strain measures defined by

εαβ(u)=
1

2
(uα,β +uβ,α) καβ(w) =−w,αβ

and

Aαβλµ =

h∫

−h

Cαβλµ dz Bαβλµ =

h∫

−h

z2Cαβλµ dz

The equilibrium equations of the plate (in the absence of the obstacle) are

Nαβ,β +pα =0 Mαβ,βα+p=0 in Ω (2.6)

Let the continuous function

f : Ω1 → R z= f(xα) Ω⊂Ω1

determines a rigid obstacle. The unilateral condition is specified by, cf Bielski
and Telega (1998), Dhia (1989)

w(xα)+h¬ f
(
xα+uα(xβ)−hw,α (xβ)

)
(2.7)

The lower face of the platemay come into contact with the rigid obstacle.We
introduce the set

K =
{
(u,w)∈H1(Ω)2×H2(Ω)

∣∣∣ (2.7) is satisfied for (xα)∈Ω
}

Remark 2.1. If K is non-empty, then, in general, it is a non-convex set.
K is a convex set provided that f is a concave function. 2

The boundary conditions are assumed in the form

w=0
∂w

∂n
=0 u=0 on Γ0 measΓ0 > 0

Here n denotes the outer unit vector normal to Γ . We set

V =
{
(u,w)∈H1(Ω)2×H2(Ω)

∣∣∣u=0, w=
∂w

∂n
=0onΓ0

}



Augmented lagrangian methods... 745

a(u,v)=

∫

Ω

Aαβλµ(x)εαβ(u)ελµ(v) dx

(2.8)

b(w,t) =

∫

Ω

Bαβλµ(x)καβ(w)κλµ(t) dx

where u,v∈H1(Ω)2 and t,w∈H2(Ω).
The functional of the external loading is assumed in the form

L(u,w)=

∫

Ω

(pαuα+pw) dx+

∫

Γ1

(
rαuα+ qw−M

∂w

∂n

)
dΓ (2.9)

where rα,q,M ∈ L2(Γ1), and pα,p ∈ L2(Ω). The functional of the total
potential energy is given by

J(u,w)=
1

2
a(u,u)+

1

2
b(w,w)−L(u,w) (2.10)

Now we are in a position to formulate the first, in general a nonconvex, mini-
mization problem.

Problem (P)

Find

inf
{
J(u,w)

∣∣∣ (u,w)∈K∩V
}

We observe that on account of unilateral condition (2.7) the in-plane and
transverse displacements are interrelated.Consequently, theproblem (P) can-
not be decomposed into membrane and plate problems. We recall that if the
contact condition is imposed on themid-plane of the plate then both problems
are independent and only the bending problem is of a unilateral type.

Theorem 2.2. The problem (P) possesses at least one solution (ũ, w̃) ∈
K∩V , provided that K 6= ∅. 2

For the proof the reader is referred to Bielski and Telega (1998).

Remark 2.3. The linearization of the r.h.s. of (2.7)was considered byBielski
and Telega (1998). In the same paper the linearization of the r.h.s. of
(2.7) has also been carried out. 2



746 J.J.Telega et al.

2.2. Reissner’s plate model

In a simple model of moderately thick plates accounting for transverse
shear deformations it is assumed that, cf Jemielita (1991), Lewiński (1987),
Reissner (1985)

vα(x,z)=uα(x)+zϕα(x) (x,z)∈Ω× (−h,h)

v3(xβ,z)≡w(xβ)

Here ϕα (α=1,2) denote the rotations of the plate transverse cross-sections.
The strain measures are given by

εαβ(u)=u(α,β) =
1

2

(∂uα
∂xβ
+
∂uβ
∂xα

)
ραβ(ϕ)=ϕ(α,β)

(2.11)

dα(w,ϕ)=w,α+ϕα

Let us denote by T =(Tα) the transverse shear force vector. The constitutive
relationships are given by

Nαβ =Aαβλµελµ(u) Mαβ =Bαβλµρλµ(ϕ)
(2.12)

Tα =Hαβdβ(w,ϕ)

where the elastic moduli Aαβλµ and Bαβλµ are specified in Section 2.1, and

Hαβ =

h∫

−h

cα3β3 dz

The equilibrium equations have now the form

Nαβ,β +pα =0 Mαβ,β −Tα+mα =0 Tα,α+p=0 (2.13)

provided that the obstacle is absent. The boundary conditions are

u=0 ϕ=0 w=0 on Γ0

where meas Γ0 > 0.
We set

V1 =
{
(u,w,ϕ)∈H1(Ω)2×H1(Ω)×H2(Ω)2

∣∣∣u=0, ϕ=0, w=0onΓ0
}

(2.14)



Augmented lagrangian methods... 747

Now, the impenetrability condition is given by, cf (2.7)

w(x)+h¬ f
(
xα+uα(x

)
+hϕα(x)

)
x∈Ω (2.15)

Consequently, the set of kinematically admissible displacements is defined by

K1 =
{
(u,w,ϕ)∈V1

∣∣∣w(x)+h¬ f(xα+uα(x)+hϕα(x)
)

x∈Ω
}

(2.16)
We assume that K1 6= ∅. The functional of the total potential energy is
expressed by

J1(u,w,ϕ)=
1

2

∫

Ω

[
Aαβλµεαβ(u)ελµ(u)+Bαβλµραβ(ϕ)ρλµ(ϕ)+

(2.17)

+Hαβdα(w,ϕ)dβ(w,ϕ)
]
dx−L1(u,w,ϕ)

where

L1(u,w,ϕ)=

∫

Ω

(pαuα+pw+mαϕα) dx+

∫

Γ1

(rαuα+qw+Mαϕα) dΓ (2.18)

We formulate the second minimization problem.

Problem (P1)

Find

inf
{
J1(u,w,ϕ)

∣∣∣ (u,w,ϕ)∈K1
}

In general, this problem is also nonconvex. The following existence results
are formulated as follows.

Theorem 2.4. The problem (P1) possesses at least one minimizer
(ũ, w̃,ϕ̃)∈K1. 2
For the proof the reader is referred to Bielski and Telega (1998).

3. Von Kármán’s plates

This model is still based on the Kirchhoff-Love kinematical hypotheses.
The strainmeasures are definedby, cf Fung (1965), Ciarlet andRabier (1980),
Lewiński and Telega (2000)

eαβ(u,w)= εαβ(u)+
1

2
w,αw,β καβ(w)=−w,αβ (3.1)



748 J.J.Telega et al.

where εαβ(u)=u(α,β).Wenote that only the first strainmeasure isnonlinear.
The constitutive equations have the form

Nαβ =Aαβλµeλµ(u,w) Mαβ =Bαβλµκλµ(w) (3.2)

As previously, N and M are themembrane forces tensor andmoments tensor,
respectively. In the absence of the obstacle the equilibriumequations are given
by

Nαβ,β +pα =0 Mαβ,βα+(Nαβw,β ),α+p=0 in Ω (3.3)

We impose the following boundary conditions

u=0 on Γ0 w=
∂w

∂n
=0 on Γ

An appropriate space for displacements is

V2 =
{
(u,w)∈H1(Ω)2×H20(Ω)

∣∣∣u=0 on Γ0
}

(3.4)

The functional of the total potential energy is now given by

J2(u,w) =
1

2

∫

Ω

[
Aαβλµ

(
εαβ(u)+

1

2
w,αw,β

)(
ελµ(u)+

1

2
w,λw,µ

)
+

(3.5)

+Bαβλµκαβ(w)κλµ(w)
]
dx−

∫

Ω

(pαuα+pw) dx−
∫

Γ1

rαuα dΓ

The nonlinear strain measure renders the functional J2 nonconvex on
H1(Ω)2 ×H20(Ω), and particularly on V2. This functional is weakly lower
semicontinuous and bounded frombelow, cf Bielski andTelega (1996), Ciarlet
andRabier (1980). For the obstacle problem the set of kinematically admissi-
ble fields is specified by

K2 =
{
(u,w)∈V2

∣∣∣w(x)+h¬ f
(
xα+uα(x)−hw,α(x)

)
, x∈Ω

}

Weassume that K2 6= ∅.We can now formulate the obstacle contact problem.

Problem (P2)

Find

inf
{
J2(u,w)

∣∣∣ (u,w)∈K2
}



Augmented lagrangian methods... 749

The existence to the solution to the Problem (P2) is ensured by the follo-
wing result.

Theorem 3.1. The functional J2 has at least oneminimizer on the set K2.
2

For the proof the reader is referred to Bielski and Telega (1998).

4. Obstacle problem for linear Koiter’s shell

Consider a shell of the thickness 2h. Let themiddle surface S of the shell
be specified by the equation

x=Φ(ξ) x=(xi)∈S i=1,2,3 S=Φ(Ω)
ξ=(ξα)∈Ω α=1,2

(4.1)

where Ω is a bounded sufficiently regular domain in R2 in the Cartesian
coordinate systemwith the base (e1,e2,e3). Let (uα,w) be the displacement
vector of a point belonging to S. Let rh(ξα) denote the position vector of a
point lying on the lower face of the deformed shell, cf Fig.1.

Fig. 1. Unilateral contact of a shell with a rigid obstacle



750 J.J.Telega et al.

We have

r
h(ξα)= r(ξα)+hN (4.2)

where
r(ξα)=Φ(ξα)+wN+uαaα

is the placement vector of a point lying on themiddle surface of the deformed
shell.Here (aα,N) formsa local base for themiddle surfaceof theundeformed
shell and N is given by, cf Koiter (1965)

N =

√
a

a

[
(ϕλl

λ
µ−ϕµlλλ)aµνaν +

1

2
(lλλl
µ
µ− lλµl

µ
λ)N
]

where
lκα = δ

κ
α+u

κ
|α−bκαw ϕα =w,α+ bκαuκ

Here a and a are the determinants of the first quadratic forms of themiddle
surfaces of the undeformed and deformed shells, respectively. After lineariza-
tion we get

a

a
=1+2εαα

Let Ω1 ⊂ R2 be such that Ω⊂Ω1. As previously, z= f(xα), (xα)∈Ω1,
defines a rigid obstacle. The impenetrability condition is now given by

r(ξα) ·e3+hN ·e3 ¬ f(r ·eα) (4.3)

After the linearization of N we get

n=−(w,µ+ bµσuσ)aµ+N

For the linear Koiter shell model the strain measures are

εαβ(u,w)=
1

2
(uα|β +uβ|α)− bαβw

καβ(u,w)=−w|αβ − bγα|βuγ − b
γ
αuγ|β − b

γ
βuγ|α+ bαβw

Here b=(bαβ) is the second quadratic form of the middle surface.
The total potential energy of the shell is expressed by

J(u,w)=L(u,w)+
(4.4)

+
1

2

∫

Ω

[
Aαβλµεαβ(u,w)ελµ(u,w)+Dαβλµκαβ(u,w)κλµ(u,w)

]√
a dx



Augmented lagrangian methods... 751

where Aαβλµ ∈ L∞(Ω), Dαβλµ ∈ L∞(Ω) and L(u,w) is the functional of
external loadings. The precise form of L is not needed, it suffices to assume
that it is weakly continuous in the topology of H1(Ω)2×H2(Ω).
For the linearKoiter shell being in unilateral contact with the obstacle the

set of constraints is given by

Ks =
{
(u,w)∈H(Ω)2×H2(Ω)

∣∣∣ r(ξα) ·e3+hn ·e3 ¬ f(r ·eα), (ξα)∈Ω
}

For a discussion of Sobolev’s spaces of functions defined on themiddle surface
of the shell the reader is referred toBernadou (1996) andLewiński andTelega
(2000). The set Ks is weakly closed. The proof is similar to the one given by
Bielski and Telega (1998) for plates, cf also Baiocchi et al. (1988).

Let the shell be clamped along ∂S0 ⊂ ∂S. Nowwe formulate the minimi-
zation problem.

Problem (Ps)

Find

inf
{
J(u,w)

∣∣∣ (u,w)∈Ks, u=0, w= ∂w∂n =0on ∂S0
}

Now we are in a position to formulate the existence theorem.

Theorem. The problem (Ps) has at least one solution. 2

Remark 4.1. The function f(r ·eα) can be linearized, compare the lineari-
zation in the case of plates byBielski andTelega (1998). The constraints
set Ks is then convex. We observe that only partial results concerning
unilateral contact problems for shells are available in the literature, cf
Floss and Ulbricht (1994), Telega (1987). 2

5. Augmented Lagrangian methods for nonconvex problems

In this section we propose augmented Lagrangian methods applicable to
nonconvex contact problems. To this end we extend the approach developed
by Ito and Kunisch (1990, 1995), cf also Bielski et al. (2000).



752 J.J.Telega et al.

5.1. Nonconvex set of constraints

Ito and Kunisch (1990) carefully studied the augmented Lagrangian me-
thod directly applicable to geometrically linear problems in the case of convex
sets of constraints, cf also Cea (1971).

Thisapproach isnowextended togeometrically nonlinear contact problems
in thepresenceof nonconvex constraints. First,weconsider thecasewhereonly
the set of constraints is nonconvex. The algorithm, we are going to present, is
applicable to geometrically linear structures where constraints are nonconvex.

The problem under investigation is

(P) min
{1
2
a(u,u)− l(u)

∣∣∣ g(u)¬ 0, u∈B
}

Here the following spaces and mappings are used: V is a Hilbert space;
B is a reflexive Banach space continuously embedded into V ; H is a Hilbert
latticewith the innerproduct 〈·, ·〉; a(·, ·) :V×V is a bilinear and continuous,
V -eliptic form, with a(u,u) ­ C0‖u‖2V , for some C0 > 0; l : V → R is a
continuous linear functional; g :B→H is in general anonconvex, continuous,
Gâteaux’s differentiable mapping.

From the practical point of view, the expression ”Hilbert lattice” merely
means that the constraint g(u)¬ 0 appearing in problem (P) is meaningful.
For a general definition of spaces being lattices the reader is referred toYosida
(1978). We assume that

g(u) =G(u)−G1(u) (5.1)

where themapping G is convexwhilst G1 is nonconvex. The Ito andKunisch
(1990) procedure can be extended by combining their augmented Lagrangian
technique with an iterative procedure:

— the mth step

G(u)¬G1(um−1) m=1,2, . . . (5.2)

Then the set

Km =
{
u∈B

∣∣∣G(u)¬G1(um−1)
}

(5.3)

is convex. At each step m we define a family of augmented Lagrangian pro-
blems by

(P)m,c,λ Lm,c(um,λm)=min
{
Lm,c(u,λ)

∣∣∣u∈B
}

where

Lm,c(u,λ) =
1

2
a(u,u)− l(u)+ 〈λ, ĝm(u,λ,c)〉+

c

2
‖ĝm(u,λ,c)‖2H



Augmented lagrangian methods... 753

and λ∈H, c> 0, c∈ R+. Moreover

ĝm(u,λ,c) = sup
(
gm(u),−

λ

c

)

Themapping gm is defined by

gm(u)=G(u)−G(um−1) (5.4)

The Algorithm

(1) Choose λm1 ∈H, λm1 ­ 0, and c> 0
(2) put n=1
(3) solve (P)m,c,λmn for u

m
n

(4) put λmn+1 =λ
m
n +cĝ(u

m
n ,λ

m
n ,c) = sup(0,λ

m
n +cg(u

m
n ))

(5) put n=n+1 and return to (3).

We observe that the parameter cmay also depend on m.

Applying Ito and Kunisch’s (1990) results we get

C0

∞∑

n=1

‖umn −um‖2V ¬
1

2c
‖λm1 −λm‖2H ¬ sup

m­1

1

2c
‖λm1 −λ∗m‖2H <∞ (5.5)

since c can be taken sufficiently large, such that for each m∈ N we have
1

2c
‖λm1 −λm‖2H <C1 C1 > 0 (5.6)

Let us pass to examples.

Example 5.1. As we already know, the sets of constraints given by K,K1,
and K2 are, in general, nonconvex. We can easily introduce sequences
of convex sets of constraints by

Km =
{
(u,w)∈H1(Ω)2×H2(Ω)

∣∣∣
∣∣∣ w(x)+h¬ f

(
x+um−1(x)−h∇wm−1(x)

)
, x∈Ω

}

and

Km1 =
{
(u,w,ϕ)∈V1

∣∣∣w(x)+h¬ f
(
x+um−1(x)+hϕm−1(x)

)
, x∈Ω

}

and similarly for the set Km2 ,m=1,2, . . .



754 J.J.Telega et al.

Example 5.2. To cope with geometrically nonlinear plates we aditionally
introduce a sequence of bilinear forms. For instance, in the case of von
Kármán’s plates we take

am(u,w;u,w) =

∫

Ω

[
Aαβλµ

(
εαβ(u)+

1

2
wm−1,α w

m−1
,β

)
·

·
(
ελµ(u)+

1

2
wm−1,λ w

m−1
,µ

)
+Bαβλµκαβ(w)κλµ(w)

]
dx m=1,2, . . .

Another possibility is to introduce the following sequence of the bilinear
forms

ãm(u,w;u,w)=

∫

Ω

[
Aαβλµ

(
εαβ(u)+

1

2
w,αw

m−1
,β

)
·

·
(
ελµ(u)+

1

2
w,λw

m−1
,µ

)
+Bαβλµκαβ(w)κλµ(w)

]
dx m=1,2, . . .

Then, instead of the problem (P), we have a sequence of the following
problems

(Pm) min
{1
2
am(u,w;u,w)−l(u,w)

∣∣∣ g(u,w)¬ 0, (u,w)∈K2
}

(5.7)
m=1,2, . . ., and similarly in the case of ãm.

Here l(u,w) is the loading functional. If K2 is a nonconvex set, in order to
use the previouly outlined augmented Lagrangianmethod, we have to replace
K2 by a sequence of convex sets of the constraints Km2 .

5.2. Nonconvexextensionof Ito andKunisch’s (1995)augmentedLagran-
gian method

Ito and Kunisch (1995) investigated an augmented Lagrangian method
for a significant class of nonsmooth convex optimization problems in infinite
dimensional Hilbert spaces. More precisely, let X, H be real Hilbert spaces
and K a closed convex subset of X. Consider the minimization problem
(Q) min

{
J(u)+ϕ(Λu)

∣∣∣u∈K
}

where J :X → R is a lower, semicontinuous differentiable, convex function,
Λ ∈ L(X,H) and ϕ : X → R is a proper, lower semicontinuous convex
function. The convex functional ϕ is not necessarily smooth; in applications
it can be an indicator function of a closed convex set. Several examples of the
linear and continuous operator Λ are provided by Ito and Kunisch (1995).



Augmented lagrangian methods... 755

For instance, in unilateral contact problems with constraints inposed on the
boundary, Λ is a trace operator (in the sense of value of a function on the
boundary).
A smooth approximation of ϕ yields the following problem:

(Q) min
{
Lc(u,λ)

∣∣∣u∈K
}

where

Lc(u,λ)= J(u)+ϕc(Λu,λ)
(5.8)

ϕc(v,λ)= inf
{
ϕ(v−u)+ 〈λ,u〉H +

c

2
‖u‖2H

}

Here (c,λ) ∈ R+ ×H. We observe that ϕ(·,λ) is (Lipschitz) continuously
Fréchet differentiable.
Ito and Kunisch (1995) developed the following augmented Lagrangian

method involving a sequential minimization:

Augmented Lagrangian Algorithm

Step 1: Choose a starting value λ1 ∈ H, a positive number c and
set k=1.

Step 2: Having given λk ∈H find uk ∈K by

Lc(uk,λk)=min
{
Lc(u,λk)

∣∣∣u∈K
}

Step 3: Update λk by λk+1 = ϕ
′
c(Λuk,λk), where ϕ

′ denotes the
Fréchet derivative of the functional ϕ(·,λ).

Step 4: If the convergence criterion is not satisfied then set k= k+1
and go to Step 2.

Under suitable, physically plausible assumptions, the just sketched au-
gmented Lagrangian algorithm converges.
Obviously, this algorithm is not directly applicable to nonconvex contact

problems of, say, finitely deformed elastic bodies and geometrically nonlinear
structures. There are three basic sources of nonconvexity:
(i) a nonconvex functional J,
(ii) a nonconvex functional ϕ,
(iii) nonlinear operator appearing in the functional ϕ.

Such an operator is denoted by N. Obviously, in practice, various combina-
tions of the cases (i)-(iii) are important.



756 J.J.Telega et al.

For geometrically nonlinear problems the functional ϕ is usually an indi-
cator function of a (weakly) closed andnonconvex set, cf Examples (5.1), (5.2)
and He et al. (1996). We already know how to generate a sequence of convex
sets of constraints. A large class of geometrically nonlinear problems leads to
the functional J of the form, cf Bielski andTelega (1985b), Gałka andTelega
(1992)

J(u) =G(Λ̃u)+F(u) (5.9)

where G represents the functional of the total internal energywhilst F is the
loading functional, usually a linear one. The functional G is nonconvex. For
nonlinear structures it can often be written as follows

G(Λ̃u)=G(Λ1u,Λ2u) (5.10)

where the functional G(·,Λ2u) is convex whilst G(Λ1u, ·) is nonconvex. To
use the augmented Lagrangian method we combine the approach by Ito and
Kunisch (1995) with the iterative procedure. To this end we set

Gm(Λ̃u)=G(Λ1u,Λ2u
m−1) m=1,2... (5.11)

and consider a sequence of regularized minimization problems

(Q)m,c,λ min
{
Gm(Λ̃u)+F(u)+ϕc(Λu,λ)

∣∣∣u∈K
}

Now we have a sequence of the convex problems (Q)m,c,λ, m = 1,2, . . ., to
which we can apply the augmented Lagrangian method developed by Ito and
Kunisch (1995).
Consider now a more specific case of a body made of the Saint-Venant

Kirchhoff material, cf Benaouda and Telega (1997). Let F stand for the de-
formationgradient, F =∇χ, χ=(χi), i=1,2,3.The stored energy function
of isotropic Saint-VenantKirchhoffmaterial is expressedby, seeCiarlet (1988),
Benaouda and Telega (1997)

W(F)=
µ

4
‖F⊤F −I‖2+ λ

8
(‖F‖2−3)2 (5.12)

where λ and µ are the Lamé moduli. The function W is not of even rank-
one convex, consequently it is neither quasiconvex nor polyconvex. However,
defining the sequence of the convex function

Wm(F)=
µ

4
‖F⊤Fm−1−I‖2+

λ

8
(‖Fm−1‖2−3)2 (5.13)

where Fm−1 =∇χm−1, m=1,2, . . ., one can apply the outlined augmented
Lagrangian method to various frictionless contact problems for bodies made
of the Saint-Venant Kirchhoffmaterial.



Augmented lagrangian methods... 757

Remark 5.1.

(i) It seemspossible toapply theapproach sketched for theSaint-Venant
Kirchhoff stored energy functions to other, well-known hyperelastic
materials. Such stored energy functions were discussed by Ciar-
let (1988) and Ogden (1984). Obviously, the choice of the sequen-
ce Wm, m=1,2, . . ., is not unique and depends on the particular
case.

(ii) The paper byTelega andGałka (2001) reviews various applications
of augmented Lagrangian methods, including contact problems, cf
alsoTelega andGałka (1998), Telega and Jemioło (2001). However,
the presented approach seems to be novel.

(iii) Our study is confined to frictionless contact problems. Unilateral
contact problems with friction are still more complicated. It seems
possible to extend the Ito and Kunisch (1990, 1995) augmented
Lagrangianmethods to contact problemswith frictionbycombining
these methods with time discretization. 2

6. Specific one-dimensional nonconvex contact problem

In this section we are going to study a simple one-dimensional nonconvex
contact problem. Consider the following minimization problem. Find u∈K
such that

J(u)= inf
v∈K
J(v)

where

J(u)=

1

2∫

−1
2

a
(
u,x+

1

2
u,xu,x

)2
dx+

1

2∫

−1
2

bu(x) dx a> 0

K =
{
u∈W1,4(0,1)

∣∣∣u
(
−1
2

)
=u
(1
2

)
=0, g(u)¬ 0

}

Particular forms of the function g are given below. Anyway, we assume that
the set K is convex. To solve this problem we introduce the sequence of
functionals, cf the previous section

Jm(u)=

1

2∫

−1
2

a
(
u,x+

1

2
u,xu

m−1
,x

)2
dx+

1

2∫

−1
2

bu(x) dx m=1,2, . . .



758 J.J.Telega et al.

and the family of augmented Lagrangians

Lm,c,λ(u)= Jm(u)+
1

2c

1

2∫

−1
2

[
(sup{0, λ+ cg(u)})2 −λ2

]
dx

To apply the augmented Lagrangianmethodwe consider two cases of constra-
ints

g(u)=





−u(x)− 1
64

case (a)

−u(x)− 65
64
−
√
1−x2 case (b)

In the first case we put: b=−1, u0(x)= 0, λ0 =1, c=200.
In case (b) we take b=−1, u0(x)= 0, λ0 =1, c=50.
The results of calculations are presented in Fig.2-Fig.4. They have been

obtained by using FEM.

Fig. 2. The function u(x) in case (a), steps 1,2 and 14; c=200, c – the parameter
in the augmented Lagrangian

Fig. 3. The function u(x) in case (b), steps 1,2 and 3; c=50

Weobserve that theLagrangianmultiplier λ represents the contact forces.
The augmented Lagrangian solutions tend to the problemwith the obstacle.



Augmented lagrangian methods... 759

Fig. 4. Lagrangianmultiplier λ for cases (a) and (b)

7. Augmented Lagrangians and nonconvex duality

In a series of papers we studied dual problems for nonlinear elastic solids
and structures, cf Bielski andTelega (1985a,c,d, 1986, 1988, 1992, 1996), Biel-
ski et al. (1988, 1989), Telega et al. (1988), Gałka et al. (1989), Telega (1989,
1995), Gałka andTelega (1990, 1992, 1995).We derived the dual problems by
using the duality theory expounded by Ekeland and Temam (1976). Unfortu-
nately, this theory is more appropriate for convex problems, since in the case
of nonconvex problems it imposes restrictions on dual variables. For instance,
in the case of von Kármán’s plates thematrix formed of themembrane forces
N=(Nαβ), α,β =1,2, has to be positive semi-definite. Without this type of
restriction the duality gap

infP > supP∗ (7.1)
arises. Here (P) denotes the primal problem and (P∗) is its dual.
Rockafellar andWets (1998) developed the duality theorywhich avoids the

duality gap like that given by inequality (7.1). This duality theory, however,
is confined to finite dimensional spaces. It means that its applicability is re-
stricted to discrete or discretized problems, including contact problems of this
type.
The aim of this section is to extend the nonconvex duality theory byRoc-

kafellar and Wets (1998) to infinite dimensional spaces. Consequently, it will



760 J.J.Telega et al.

be possible to apply it to nonlinear solids and structures, thus extending the
range of applicability of our previous results concerning the duality. Our ap-
proach combines some results presented by Ekeland and Temam (1976) with
the developments of Rockafellar andWets (1976).
Let V and Y be locally convex topological spaces, and V ∗, Y ∗ their

duals, cf Ekeland and Temam (1976). Onemay think of Sobolev’s spaces and
Lp-spaces. The space V is usually the space of kinematically admissible di-
splacements. The primal problemmeans evaluating

(P) infF(u) = inf
{
Φ(u,0)

∣∣∣u∈V
}

where
Φ(u,p)= J(u,Λu−p) (7.2)

and Λ :V →Y is a linear and continuous operator.

Definition 7.1. For a primal problem of minimizing F(u) over u∈ V and
any dualizing parametrization F =Φ(·,0) for a choice of Φ :V ×Y →
R = [−∞,+∞], consider any augmenting functional f; by which a
proper, lower semicontinuos, convex functional is meant

f : Y → R with minf =0 arg minf = {0}

ThecorrespondingaugmentedLagrangianwith thepenaltyparameter c>
0 is then the functional

L(u,p∗,c) := inf
p∈Y

{
Φ(u,p)+ cf(p)−〈p∗,p〉

}
(7.3)

The corresponding dual problem consists of maximizing over all (p∗,c) ∈
Y ∗× (0,∞) the functional

G(p∗,c) := inf
{
Φ(u,p)+ cf(p)−〈p∗,p〉

∣∣∣ (u,p)∈V ×Y
}

(7.4)

Here 〈·, ·〉 : Y ∗×Y → R denotes the duality pairing, cf Ekeland and Temam
(1976).
To formulate the duality theoremwe set

h(p) := inf
{
Φ(u,p)

∣∣∣u∈V
}

(7.5)

hc,f(p) := inf
{
Φc,f(u,p)

∣∣∣u∈V
}

where
Φc,f(u,p) :=Φ(u,p)+ cf(p) (7.6)



Augmented lagrangian methods... 761

The notion of the augmented Lagrangian in the nonconvex duality arises from
the idea of replacing the known inequality in the convex duality

infP =supP∗
p∗ ∈ arg max(P∗)

}
⇔
{
h(p)­h(0)+ 〈p∗,p〉 ∀p,
with p(0) 6=−∞ (7.7)

with one of the form

h(p)­h(0)+ 〈p∗,p〉− cf(p) ∀p

Whatmakes the approach successful inmodifying the dual problem to get rid
of the duality gap is that the last inequality is identical to

hc,f(p)­hc,f(0)+ 〈p∗,p〉 ∀p

Indeed, hc,f(p)=h(p)+cf(p) and hc,f(0)=h(0), because f(0)= 0.

The Lagrangian associated with Φc,f is Lc,f(u,p
∗)=L(u,p∗,c), where L

is defined by (7.3). The resulting dual problem consists ofmaximizing Gc,f =
−Φ∗c,f(0, ·) over p∗ ∈Y ∗. We have

Gc,f(p
∗)=G(p∗,c)

We can apply the theory developed by Ekeland andTemam (1976) to this
modified formulation, where Φc,f replaces Φ, and in that way capture new
powerful features.

Theorem 7.1 (duality without convexity).

For the problem of minimizing F on V consider the augmented
Lagrangian L(u,p∗,c) associated with the dualizing parametrization
F = Φ(·,0), Φ : V × Y → R, and a certain augmented functional
f : Y → R. Suppose that Φ(u,p) is level-bounded in u locally uni-
formly in p, and let h(p) := inf{Φ(u,p) |u∈V}. Suppose further that
infuL(u,p,c)>−∞ for at least one (u,c)∈V × (0,∞). Then

F(u)= sup
p∗,c
L(u,p∗,c) G(p∗,c) = inf

u
L(u,p∗,c)

where actually F(u) = sup
p∗
L(u,p∗,c) for every c> 0, and in fact

inf
u∈V
F(u)= inf

u
[sup
p∗,c
L(u,p∗,c)] = sup

p∗,c
[inf
u
L(u,p∗,c)] = sup

p∗,c
G(p∗,c)



762 J.J.Telega et al.

Moreover, the optimal solutions to the primal and augmented dual pro-
blems are characterized as saddle points of the augmented Lagrangian

u∈ arg minF(u)
(p∗,c)∈ arg max

p∗,c
G(p∗,c)

}
⇔





inf
u
L(u,p∗,c)=L(u,p∗,c)=

= sup
p∗,c
L(u,p∗,c)

the elements of arg maxp∗,cG(p
∗,c) being precisely the pairs (p∗,c)

with the property that

h(p)­h(0)+ 〈p∗,p〉− cf(p) ∀p 2

The proof will be given elsewhere.
Let us recall the definition of the level boundedness, cf Rockafellar and

Wets (1998).
A functional g : V → R is (lower) level bounded if for every α ∈ R

the set level¬αg :=
{
α ∈ V

∣∣ g(u) ¬ α
}
is bounded (possibly empty). This

requirement can be replaced by coercivity.

Specific case

Consider now the case where

f(p)=
1

2
‖p‖2 = 1

2
‖p‖2L2 (7.8)

Then, since f is finite we have

L(u,p∗,c)= sup
q∗

{
L(u,q∗)− 1

2c
‖q∗−p∗‖2

}
=sup
q∗

{
L(u,p∗−q∗)− 1

2c
‖q∗‖2

}

where L is the standard Lagrangian

L(u,q∗)= inf
{
Φ(u,p)−〈p∗,p〉

∣∣∣ p∈Y
}
=−sup

{
〈p∗,p〉−Φ(u,p)

∣∣∣ p∈Y
}

For Φ being given by (7.2) we get, cf Ekeland and Temam (1976)

L(u,q∗)=−〈q∗,Λu〉−J∗u(−q∗)

where Ju denotes the functional p→J(u,p) and J∗u is its dual defined by

J∗u(q
∗)= sup

{
〈q∗,q〉−J(·,q)

∣∣∣ q∈Y
}

After some calculations we obtain, cf (7.4)

G(p∗,c) = inf
u
L(u,p∗,c) = sup

q∗

{
−J∗(Λ∗q∗,−q∗)− 1

2c
‖q∗−p∗‖2

}
(7.9)



Augmented lagrangian methods... 763

Here Λ∗ is the adjoint (dual) operator of Λ.
For the practically important case where

J(u,Λu) =G(Λu)+F(u)

we calculate

G(p∗,c)= sup
q∗

{
−G∗(−q∗)−F∗(Λ∗q∗)− 1

2c
‖q∗−p∗‖2

}
=

(7.10)

=− inf
{
G∗(−q∗)+F∗(Λ∗q∗)+ 1

2c
‖q∗−p∗‖2

}

Remark 7.1. From relations (7.9) we conclude that, at least for the augmen-
ted functional given by Eq. (7.8), the dual functional G consists of the
standard term J∗(Λ∗q∗,−q∗) and the regularizing term ‖q∗−p∗‖2/(2c).
According to the terminology given by Rockafellar andWets (1998) the
dual function G(p∗,c) is then the minus of the Moreau envelope of
J∗(Λ∗q∗,−q∗). 2

Remark 7.2. Indicator functions of a set determining constraints can be in-
cluded into the functional F. 2

Example 7.1. Consider a simple case of the nonconvex functional G in the
one-dimensional case of an elastic nonlinear rod. Then

G(Λu) =
1

2

l∫

0

a
(
u,x+

1

2
u2,x

)2
dx a> 0

and

‖q∗−p∗‖2 =
l∫

0

(q∗−p∗)2 dx

The primal problemmeans evaluating

(cP) inf
{
G(Λu)−

l∫
0
r(x)u(x) dx

∣∣∣u∈W1,4(0, l), u(0)=u(l)= 0
}

Now Λu=(u,x,u,x), and

G(q1,q2)=

l∫

0

W(q1,q2) dx



764 J.J.Telega et al.

We recall that the operator Λ has to be linear. Standard calculation yields,
cf Bielski and Telega (1985b), Gałka and Telega (1995)

W∗(q∗1,q
∗
2)=

1

2a
(q∗1)

2+






0 if q∗2 =0 ∧ q∗1 ­ 0
1

2q∗1
(q∗2)

2 if q∗1 > 0

+∞ otherwise

(7.11)

Thephysicalmeaning of the dual variable q∗1, q
∗
2 is: q

∗
1 =N :=σx, q

∗
2 =Nu,x,

where σx is the normal stress. FromEq. (7.11) we conclude that N has to be
non-negative, i.e., the classical duality theory admits only tension. To include
compression we use the developed nonconvex duality theory. Now we have

G(N,Q,c) = inf
(Ñ,Q̃)∈[L4/3(0,l)]2

{ l∫

0

[
W∗
(
Ñ(x), Q̃(x)

)
+

(7.12)

+
1

2c
[(Ñ −N)2+(Q̃−Q)2]

]
dx+ IS(Ñ,Q̃)

}

where

S =
{
(N,Q)∈ [L4/3(0, l)]2

∣∣∣ (N +Q),x ∈L4/3(0, l),

(N +Q),x+r=0, x∈ (0, l)
}

provided that the rod is clamped at x=0 and x= l. Here r(x) (x∈ (0, l))
denotes the loading distributed along the rod. It can be shown that

G(N,Q,c) =−
l∫

0

W∗c (N,Q) dx− IS(N,Q) (7.13)

where

W∗c (N,Q)=






W∗(N,Q) if Q=0 ∧ N ­ 0 orN > 0
1

2c
(N2+Q2) otherwise

(7.14)

Nowwe conclude that the normal force N in the problem (P∗) is not neces-
sarily non-negative, due to the regularization given by Eq. (7.14).
The augmented dual problem takes eventually the form

sup
{
G(N,Q,c)

∣∣∣ (N,Q)∈S, c> 0
}



Augmented lagrangian methods... 765

Remark 7.3. Theaugmented dual problem is bynomeansunique.There are
a lot of problems related to the augmenting functionals f(p), satisfying
the conditions specified in Theorem 7.1. 2

Acknowledgement

Theworkcarriedoutby thefirstauthorwaspartially supportedby theStateCom-

mittee for Scientific Research (KBN, Poland) through the grant No. 7T07A04316.

References

1. Baiocchi C., Buttazzo G., Gastaldi F., Tomarelli F., 1988, General
Existence Theorems for Unilateral Problems in Continuum Mechanics, Arch.
Rat. Mech. Anal., 100, 149-189

2. Benaouda, M.K.-E., Telega J.J., 1997, On Existence of Minimizers for
Saint-Venant Kirchhoff Bodies: Placement Boundary Condition, Bull. Pol.
Acad. Sci., Tech. Sci., Tech. Sci., 45, 211-223

3. Bernadou M., 1996, Finite Element Methods for Thin Shell Problems, John
Wiley & Sons, Chichester; Masson, Paris

4. Bielski W., Telega J.J., 1985a, A Contribution to Contact Problems for a
Class of Solids and Structures,Arch. Mech., 37, 303-320

5. BielskiW.R.,Telega J.J., 1985b,ANote onDuality for vonKármánPlates
in the Case of the Obstacle Problem,Arch. Mech., 37, 135-141

6. Bielski W.R., Telega J.J., 1985c, On the Complementary Energy Princi-
ple in Finite Elasticity, in: Proc. of the Int. Conf. on Nonlinear Mechanics,
Shanghai, China Science Press, Beijing, 211-218

7. Bielski W.R., Telega J.J., 1985d,On theObstacle Problem for Linear and
Nononlinear Elastic Plates in: Variational Methods in Engineering, edit. by
C.A. Brebbia, Springer-Verlag, Berlin, (3-55)-(3-64)

8. Bielski W.R., Telega J.J., 1986, The Complementary Energy Principle in
Finite Elastostatics as a Dual Problem, in: Finite Rotations in Structural Me-
chanics, Lecture Notes in Engineering, 19, Springer-Verlag, Berlin, 62-81

9. Bielski W., Telega J.J., 1992, On Existence of Solutions andDuality for a
Model ofNon-LinearElasticPlateswithTransverse ShearDeformations, IFTR
Reports, 35/1992

10. Bielski W.R., Telega J.J., 1996, Non-Linear Elastic Plates of Moderate
Thickness: Existence, Uniqueness andDuality, J. Elasticity, 42, 243-273

11. Bielski W.R., Telega J.J., 1998, Existence of Solutions to Obstacle Pro-
blems for Linear and Nonlinear Elastic Plates, Math. Comp. Modelling, 28,
55-66



766 J.J.Telega et al.

12. Bielski W.R., Gałka A., Telega J.J., 1988, The Complementary Energy
Principle and Duality for Geometrically Nonlinear Elastic Shells – Part I. A
Simple Model of Moderate Rotations Around Tangent to the Middle Surface,
Bull. Polish Acad. Sci., Tech. Sci., 36, 415-426

13. Bielski W.R., Gałka A., Telega J.J., 1989, The Complementary Ener-
gy Principle and Duality for Geometrically Nonlinear Elastic Shells – Part IV.
SimplifiedFormofNonlinearTensor ofChangesofCurvature.TheComplemen-
taryEnergyPrincipleExpressed inTermsof InternalForcesandDisplacements,
Bull. Pol. Acad. Sci., Tech. Sci., Tech. Sci., 37, 391-400

14. Bielski W.R., Gałka A., Telega J.J., 2000, On Contact Problems for
Linear and Nonlinear Elastic Plates: Existence of Solutions and Application
of Augmented Lagrangian Methods, in: Mutifield Problems-State of the Art.,
A.-M. Sänding, W. Shiehlen, W.L. Wenland (edit.), 237-245, Springer-Verlag,
Berlin

15. Cea J., 1971,Optimization: Théorie et Algorithme, Herrmann, Paris

16. Ciarlet P.G., 1988,Mathematical Elasticity, North Holland, Amsterdam

17. Ciarlet P.G., Rabier P., 1980, Les Equations de von Kármán, Springer-
Verlag, Berlin

18. Dhia H.B., 1989, Equilibre d’une Plaque Mince Elastique avec Contact Uni-
latéral et Frottement de Type Coulomb,C. R. Acad. Sci. Paris, Série I, 308,
293-296

19. DuvautG., Lions J.-L., 1972,Les Inéquations enMécanique et en Physique,
Dunod, Paris

20. Duvaut G., Lions J.-L., 1974, Problémes Unilatéraux dans la Théorie de
la Flexion Forte des Plaques, J. Méc., 13, 51-74; II. Le cas d’evolution, ibid.,
245-266

21. Ekeland I., Temam R., 1976, Convex Analysis and Variational Problems,
North -Holland, Amsterdam

22. FlossA., UlbrichtV., 1994,Elastic-Plastic ShellswithFiniteDeformations
and Contact Treatment,Arch. Mech., 46, 449-461

23. Fung Y.C., 1965,Foundations of Solids Mechanics, Prentice-Hall, Englewood
Cliffs, New Jersey

24. Gałka A., Telega J.J., 1990, The Complementary Energy Principle for a
Model of Shells with an Independent Rotation Vector, Zeitschr. Ang. Mech.,
70, T253-T256

25. Gałka A., Telega J.J., 1992, The Complementary Energy Principle and
Duality for a SpecificModel of Geometrically Nonlinear Elastic Shells with an
IndependentRotationVector: GeneralResults,Eur. J.Mechanics, 11, 245-270

26. GałkaA., Telega J.J., 1995,Duality and theComplementaryEnergyPrin-
ciple for a Class of Non-Linear Structures. Part I. Five-parameter ShellModel,
Arch.Mech.,47, 1995, 677-698;Part II.AnomalousDualVariationalPrinciples
for Compressed Elastic Beams, ibid., 699-724



Augmented lagrangian methods... 767

27. Gałka A., Telega J.J., Bielski W.R., 1989, The Complementary Energy
Principle and Duality for Geometrically Nonlinear Elastic Shells – Part III.
Nonlinear Tensor of Changes of Curvature, Bull. Pol. Acad. Sci., Tech. Sci.,
37, 375-389

28. HeQ.-C., Telega J.J., CurnierA., 1996,Unilateral Contact ofTwoSolids
Subject to Large Deformation and Existence Results, Proc. R. Soc., London,
A452, 2691-2717

29. Ito K., Kunisch K., 1990, An Augmented Lagrangian Technique for Varia-
tional Inequalities,Appl. Math. Optim., 21, 223-241

30. Ito K., KunischK., 1995,Augmented LagrangianMethods for Nonsmooths,
ConvexOptimization inHilbert Spaces, in:Control of Partial Differential Equ-
ations and Applications, edit. by E. Casas, 107-117,Marcel Dekker

31. Jemielita G., 1991, On the Windings Paths of the Theory of Plates, Pol.
Warszawska,Prace Naukowe, Budownictwo, 117,Warszawa, in Polish.

32. Koiter W. T., 1965, On the Nonlinear Theory of Thin Elastic Shell, Proc.
Kon. Nederl. Acad. Wetensch.,B69, 1-54

33. Lewiński T., 1987, On Refined PlateModels Based on Kinematical Assump-
tions, Ing.-Archiv., 57, 133-146

34. Lewiński T., Telega J.J., 2000, Plates, Laminates and Shells: Asymptotic
Analysis and Homogenization, Series on Advances inMathematics for Applied
Sciences, vol. 52,World Scientific, Singapore

35. Naniewicz Z., Panagiotopoulos P. D., 1985,Mathematical Theory of He-
mivariational Inequalities and Applications, Marcel Dekker, NewYork

36. Nečas J., Hlavaček I., 1981, Mathematical Theory of Elastic and Elasto-
Plastic Bodies: An Introduction, Elsevier, Amsterdam

37. Niordson F.I., 1985, Shell Theory, North-Holland, Amsterdam

38. OgdenR.W., 1984,Non-Linear Elastic Deformations, EllisHorwood,Chiche-
ster

39. Panagiotopoulos P.D., 1985, Inequality Problems in Mechanics and Appli-
cations, Brikhäuser Verlag, Basel

40. Reissner E., 1985, Reflections on the Theory of Elastic Plates, Appl. Mech.
Rev., 38, 1453-1464

41. Rockafellar R.T., Wets R.J.-B., 1998, Variational Analysis, Springer-
Verlag, Berlin

42. Telega, J.J., 1989, On the Complementary Energy Principle in Non-Linear
Elasticity, Part I: Von Kármán Plates and Three-Dimensional Solids, C.R.
Acad. Sci. Paris, Serie II, 308, 1193-1198; Part II: Linear Elastic Solid and
Non-Convex Boundary Condition, Minimax Approach, ibid., 309, 951-956

43. TelegaJ.J., 1987,VariationalMethods inContactProblemsof theMechanics,
Uspekhi Mekhaniki (Adv. in Mech.), 10, 3-95, in Russian



768 J.J.Telega et al.

44. Telega J.J., Jemioło S., 2001,Macroscopic Behaviour of LockingMaterials
withMicrostructure. Part III. Stochastic Homogenization and Augmented La-
grangianMethods for SolvingLocalProblems,Bull. Pol. Acad. Sci., Tech. Sci.,
in press

45. Telega J.J., Gałka A., 1998, Augmented Lagrangian Methods and Appli-
cations to Contact Problems, in: Theoretical Foundation of Civil Engineering,
edit. by W. Szcześniak, 335-348, Oficyna Wydawnicza Politechniki Warszaw-
skiej,Warsaw

46. Telega J.J., Gałka A., 2001, Augmented LagrangianMethods for Contact
Problems,OptimalControl and ImageRestoration, in:FromConvexity toNon-
convexity, Kluwer

47. Telega J.J., Bielski W.R., Gałka A., 1988, The Complementary Ener-
gy Principle and Duality for Geometrically Nonlinear Elastic Shells-Part II.
Moderate Rotation Theory,Bull. Pol. Acad. Sci., Tech. Sci., 36, 427-439

48. Yosida K., 1978,Functional Analysis, Springer-Verlag, Berlin

Metody rozszerzonego lagranżianu dla pewnej klasy wypukłych
i niewypukłych zagadnień kontaktowych

Streszczenie

Cel pracy jest trojaki. Po pierwsze, sformułowane zostały jednostronne zagadnie-
nia kontaktowedla trzechmodeli płyt oraz liniowegomodelu powłokKoitera.Warun-
ki kontaktu zostały sformułowane na powierzchni będącej w kontakcie z podłożem,
a nie na powierzchni środkowej płyty lub powłoki. Takie ścisłe podejście prowadzi do
niewypukłych zadań minimalizacji, nawet w przypadku płyt cienkich. Dla każdego
zagadnienia sformułowano twierdzenie o istnieniu rozwiązań. Po drugie, metody roz-
szerzonego lagranżianu Ito i Kunischa (1990, 1995) uogólnione zostały na przypadek
zagadnień niewypukłych. Po trzecie, teoria dualności Rockafellara i Wetsa (1998),
opracowanadla skończeniewymiarowych zagadnień niewypukłych, została rozszerzo-
na na przypadek układów ciągłych. Podano również kilka przykładów.

Manuscript received January 26, 2001; accepted for print March 26, 2001