Bi-Complementarity and Duality: A Framework in Nonlinear … · 2017-02-15 · the well-known books...

Ž .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 221, 672]697 1998ARTICLE NO. AY985935

Bi-Complementarity and Duality: A Framework inNonlinear Equilibria with Applications to the Contact

Problem of Elastoplastic Beam Theory*

David Yang Gao

Department of Mathematics, Virginia Polytechnic Institute and State Uni ersity,Blacksburg, Virginia 24061

Submitted by A¨ner Friedman

Received July 21, 1995

Nonlinear complementarity problems and variational inequalities in nonlinearequilibrium problems are studied within a unified framework. Based on thegeneralized Rockafellar]Tonti diagram, a bi-complementarity problem with bothinternal and external nonlinear complementarity conditions is proposed. A generalduality theory in variational inequality is established and the Mosco dual varia-tional inequality has been generalized to the nonsmooth systems. In order to studythe frictional contact problem of beam theory, a two-dimensional elastoplasticbeam model is proposed. The external complementarity condition provides the freeboundary of contact region, while the internal complementarity condition givesthe interface of the elastic]plastic regions. Our results shown that in nonsmoothequilibrium problems, the dual approaches are much easier than the primalproblems. Q 1998 Academic Press

Key Words: complementarity problems; duality; variational inequality; non-smooth analysis; contact problem; beam theory.

1. INTRODUCTION

Complementarity theory has become a rich source of inspiration in bothmathematical and engineering sciences. Since the nonlinear complemen-

Ž . w xtarity problem NCP was introduced in Cottle’s 4 Ph.D. dissertation,many different kinds of complementarity problems have been proposed,and the theory has been extended and generalized in various directions to

* This research was supported in part by National Science Foundation under grantDMS-94-00565. The main result of this paper was announced at the International Symposiumon Complementarity Problems, Nov. 1]4, 1995, Baltimore, MD.

672

0022-247Xr98 $25.00Copyright Q 1998 by Academic PressAll rights of reproduction in any form reserved.

BI-COMPLEMENTARITY AND DUALITY 673

study a wide class of problems arising in optimization and control, mechan-ics, operation research, fluid dynamics, economics, and transportationequilibrium. Several monographs have documented the historical details

Ž w x.about the origin of these problems and their evolution cf., e.g., 5, 22, 28 .The comprehensive, up-to-date treatment of the NCP in finite dimensionsand the extensive documentation of applications in engineering and equi-

w xlibrium modeling were given recently in survey articles 9, 33 .It is known that the complementarity problems are special cases of the

w xvariational inequality 9 . Variational methods and direct approaches forunilateral variational problems and free boundary value problems havebeen studied extensively during the last 30 years. Here we only mention

w xthe well-known books by Duvaut and Lions 7 , Kinderlehrer and Stampac-w x w x w xchia 25 , Glowinski, Lions, and Tremolieres 19 , Panagiotopoulos 31, 32 ,

w x w x w xRodrigues 36 , Kikuchi and Oden 24 , Hlavacek et al. 21 , and Friedmanw x10 . From the point of view of optimization, the direct methods for solvingprimal variational inequality problems provide only upper bound ap-proaches of the problems. Meanwhile, the solutions of dual variationalinequalities will give lower bound approaches of the problems. For nonlin-ear, nonsmooth systems, dual approaches have more favorable properties.Recent research results shown that some fully nonlinear equilibriumproblems in finite deformation theory can be solved by using a nonlinear

Ž w x.dual transformation method see 15 . The dual variational inequality wasw xfirst studied by Mosco 29 in 1972. Since then many papers have been

Ž w x.published in this field see, for example, 1, 20, 26, 27, 34 . Recently, theŽ .duality in geometrically nonlinear i.e., weakly nonlinear variational in-

w xequality was studied for von Karman plates 39 and for nonlinear elasticw xbeam theory 13, 14 . Actually, by introducing a so-called superpotential,

the variational inequalities in most conservative systems are equivalent toŽ w x.certain nonsmooth variational problems cf., e.g., 31, 32 . The fields of

nonsmooth optimization and of inequality mechanics have experiencedsignificant development in recent decades. Complicated phenomena inengineering, mechanics, economics, and many other fields require theconsideration of nondifferentiability for their accurate modeling. A novelapproach to this important area was presented recently by Dem’yanov

w xet al. 6 .The purpose of the present paper is to establish a unified framework

and duality theory for nonlinear, nonsmooth complementarity problems.Based on the generalized Rockaffelar]Tonti diagram, a nonlinear bi-com-plementarity problem with both internal and external complementarityconditions is proposed, and a classification for various complementarityconditions is given in Section 2. In Section 3, a duality theory is establishedfor variational inequality problems. The Mosco dual variational inequality

DAVID YANG GAO674

is generalized to the nonsmooth systems with internal complementaritycondition.

The contact problems in elasticity have been studied extensively fromŽ w x.both theoretical and numerical aspects cf., e.g., 24, 32 . It is known that

Timoshenko beam theory is an extension of classical beam theory andallows for the effect of transverse shear deformation by relaxing thenormality assumption. But in this theory, the shear deformation does notvary in the lateral beam direction, i.e., the plane section remains planeŽ .but not necessarily normal to the longitudinal axis after deformation.This beam model is not well adapted for studying frictional contactproblems. Recently, an extended Timoshenko beam model has been estab-

w xlished to take account of shear variation in the lateral direction 16 . In thepresent paper, this extended beam model is generalized to elastoplasticbeam theory with different displacement variables. A second order partialdifferential inequality system is established in Section 4. The externalcomplementarity condition defines the free boundary of the contactregion, while the internal complementarity condition gives rise to theinterface between the elastic and plastic domains. Our results shownthat in nonsmooth mechanics, dual problems are much easier than primalproblems.

2. CLASSIFICATION OF COMPLEMENTARITYPROBLEMS

Ž . Ž .Let VV , VV * and EE, EE* be two pairs of real vector spaces, finite orinfinite dimensional, in duality with respect to certain bilinear functionsŽ . ² :), ) and ), ) , respectively. In mathematical physics, if we call VV the‘‘configuration’’ space, its dual space VV * should be the ‘‘source’’ space. EE

Ž wand its dual space EE* are called intermediate variable spaces see 30, 35,x.37, 38 . Let the geometrical operator L be a continuous linear transfor-

mation from VV to EE. Its adjoint L* : EE* ª VV * is defined by

² :Lu , e* s u , L*e* ;u g VV , e* g EE*. 1Ž . Ž .w x w xAccording to Rockafellar 35 and Tonti 38 , three equations can be

defined in mathematical physics:

Geometrical Equation: e s Lu ,

Constituti e Equation: e* s C e , 2Ž . Ž .Equilibrium Equation: u* s L*e*,

where the constitutive mapping C : EE ª EE* could be a nonsmooth, mono-Žtone operator. The problem is called physically nonlinear or strongly


.nonlinear if the constitutive operator C is nonlinear. In terms of u only,this physical nonlinear equilibrium problem can be written in the funda-mental form

L*C Lu s u*. 3Ž . Ž .

If C is a linear symmetric operator, this fundamental form can be writtenas

Au s L*CLu s u*,

where A [ L*CL is a symmetric operator. In the celebrated textbook byw xStrang 37 , this nice symmetrical structure can be seen from continuous

theories to discrete systems. The problem is called geometrically nonlinearŽ .or weakly nonlinear if the geometrical operator L is nonlinear. In thiscase, the geometrical equation and constitutive equation are the same as

Ž .in 2 . But the equilibrium equation should be

u* s LU u e*, 4Ž . Ž .t

where LU : EE* ª VV * is the adjoint operator of L , the Gateaux derivativeˆt tŽ w x.of e s Lu see 17 . The duality theory for geometrically nonlinear

variational problems as well as applications in finite deformation mechan-Ž w x.ics have been discussed in a series of papers see 16]18 .

In complementarity problems, two kinds of general complementarityŽ . Ž .conditions can be classified on the paired spaces VV , VV * and EE, EE* ,

respectively:

Ž .i External complementarity condition,

B u G 0, B* u* F 0, B u , B* u* s 0Ž . Ž . Ž . Ž .Ž .5Ž .

; u , u* g VV = VV *,Ž .

where B: VV ª VV and B*: VV * ª VV * could be linear or nonlinear opera-tors. In the present paper, we consider only the affine operators

B u s u x y c x , B* u* s yu* x q u* x . 6Ž . Ž . Ž . Ž . Ž . Ž . Ž .

c g VV and u* g VV * are given functions. In physics and engineeringŽ .applications, condition 5 usually controls the boundary complementarity

condition. For example, in contact problems, where c is the obstaclefunction, u* is the given external load. This complementarity condition

Ž .gives the free boundary of the contact region see Section 4 .

DAVID YANG GAO676

Ž .ii Internal complementarity condition,

Tg e G 0, g* e* F 0, g e g* e* s 0 ; e, e* g EE = EE*,Ž . Ž . Ž . Ž . Ž .7Ž .

where g : EE ª Rn and g*: EE* ª Rn are vector-valued functions. Inelastoplasticity, this complementarity condition models the interface ofelastic]plastic regions. Figure 1 shows the inner relations of the frame-work of the bi-complementarity problems.

Ž .Let VV be a feasible or kinetically admissible configuration space, inawhich the essential boundary conditions are prescribed. Then the bi-

FIG. 1. Framework for bi-complementarity problems in geometrically linear systems.


ŽŽ . .complementarity problem BCP for short for geometrical linear systemscan be proposed as follows:

Ž . Ž . Ž .Problem 1 BCP . For the given functions c x and u* x , find u g UUasuch that

Ž . Ž .1 L*C Lu s u*,Ž . Ž . Ž . Ž Ž . Ž ..2 B u G 0, B* u* F 0, B u , B* u* s 0,Ž . Ž . Ž Ž .. Ž .T Ž Ž ..3 g Lu G 0, g* C Lu F 0, g Lu g* C Lu s 0.

In continuum mechanics, the configuration variable u is a displacementvector; e s Lu is a strain tensor. Its dual variable e* s s should be a

Ž .stress tensor. For elastoplastic material, the vector-valued function g e isŽ .a so-called plastic flow factor; its dual function g* s is the plastic yield

function, which is a convex function. In the elastic region V , we haveeŽ . Ž . Ž . Ž .g e F 0, g* s - 0, while in the plastic region V , g e ) 0, g* s s 0.p

Ž .The internal complementarity condition 7 gives the interface G s V lep eV of the elastic]plastic regions. For the so-called nonassociated plasticpmaterials, such as soil, rocks, etc., the plastic flow factor and the yieldfunction are independent on the constitutive equation. However, for thoseso-called associated plastic media, like engineering metals and compositematerials, the complementarity conditions depend on the constitutiveequation. In this case, we can introduce a plastic superpotential W: EE ª

[ � 4R [ R j q` , such that the constitutive equation and the internalcomplementarity conditions are equivalent to the subdifferential form

e* g W e , 8Ž . Ž .

where W: EE ª EE* is a set-valued mapping:

² : W e [ e* g EE* N e*, e y e F W e y W e ;e g EE .� 4Ž . Ž . Ž .

Ž . Ž w x.The variable e* g W e is called the subgradient see 8 . If W: EE ª RŽ . � Ž .4is Gateaux differentiable, then W e s DW e . Here DW denotes theˆ

Gateaux derivative of W with respect to e.ˆThe conjugate function of W can be given by the Legendre]Fenchel

transformation:

² :W * e* s sup e*, e y W e . 9� 4Ž . Ž . Ž .egEE

Obviously, W *: EE* ª R[ is convex, l.s.c. If W: EE ª R[ is convex, then

² :e* g W e m e g W * e* m W e q W * e* s e*, e .Ž . Ž . Ž . Ž .

DAVID YANG GAO678

For example, in the extension of an elastoplastic bar, the internalcomplementarity condition and the constitutive equation between the

Ž w x.strain e s Lu s durdx x g 0, L and the stress s s e* can be writtenas

s s Ce if g e s e y e - 0,Ž . b

g e G 0, g* s F 0, g e g* s s 0,Ž . Ž . Ž . Ž .Ž .where e is a positive constant; the real-valued function g* s s s y sb b

is the plastic yield function, s s Ce . By introducing the step function d ,b b

1, if g G 0,d g sŽ . ½ 0, if g - 0,

the superpotential energy function can be defined as

1 1L 2 c 2W e s Ce d g q Ce d g q s g e d g dx ,Ž . Ž . Ž . Ž . Ž .H b b2 20

cŽ . Ž .where d g [ 1 y d g . The conjugate superpotential energy is

LW * s s sup s e dx y W eŽ . Ž .H½ 5

0e

1 1L y1 2 c y1 2s C s d g* q C s d g* dxŽ . Ž .H b2 20

0, if g* s F 0,Ž .q ½ q`, if g* s ) 0.Ž .

So the constitutive equation and the internal complementarity conditionfor this associated elastoplastic bar can be simply written in the equivalentsubdifferential inclusions,

s g W e m e g W * s .Ž . Ž .

The idea of associated plasticity can be generalized to complementarityproblems. In conservative systems, if we can define a superpotential Wsuch that its subdifferential is equivalent to the constitutive equation andinternal complementarity conditions, then the so-called associated bi-

Ž .complementarity problem ACP can be proposed as follows:

Ž .Problem 2 ACP . For the given c and u*, find u g VV such thata

u* g L* W Lu , 10Ž . Ž .B u G 0, B* u* F 0, B u , B* u* s 0. 11Ž . Ž . Ž . Ž . Ž .Ž .


If the system has the internal complementarity condition only, thenŽ .Problem 2 ACP is reduced to the associated internal complementarity

Ž .problem AICP ;

Ž .Problem 3 AICP . For a given source u* g VV *, find u g VV suchathat

u* g L* W Lu . 12Ž . Ž .

In convex analysis, this nonlinear equilibrium problem can be consid-Žered as a Euler]Lagrange equation of a primal variational problem see

w x.8 .

3. VARIATIONAL INEQUALITY AND DUALITY THEORY

Let CC be a closed convex subset of VV :

CC [ ¨ g VV N B ¨ G 0 . 13� 4Ž . Ž .

Ž .If B ¨ s ¨ , CC is a convex cone with vertex at the origin. The dual spaceCC* can be defined as

CC* [ ¨* g VV * N ¨*, B ¨ F 0 ;¨ g CC . 14� 4Ž . Ž .Ž .

The indicator function C of the convex set CC is a convex, l.s.c. functionCC

from VV to R[:

0, if ¨ g CC ,C ¨ s 15Ž . Ž .CC ½q`, otherwise.

It is easy to prove the equivalent relations

u* g C u m u* F 0, B u G 0, u*, B u s 0. 16Ž . Ž . Ž . Ž .Ž .CC

In many boundary value problems with given external source u* g VV *, we] � 4can define a concave, u.s.c. function F: VV ª R [ R j y` such that

u* g F u , 17Ž . Ž .

Ž w x.where F: VV ª VV * is the so-called superdifferential see 2 , defined asŽ . F s y yF . For example, if VV is a feasible subspace of VV , in whicha

the essential boundary conditions are prescribed, then F can be defined as

u*, u , if u g VV ,Ž . aF u s u*, u y C s 18Ž . Ž . Ž .VV ½a y`, otherwise.

DAVID YANG GAO680

So the total potential energy P: CC ª R[ in nonlinear equilibrium prob-lems can be defined as

P u s W Lu y F u . 19Ž . Ž . Ž . Ž .

Ž .The primal äriational problem PVP can be proposed as follows: Findu g CC such that

PVP P u s inf P ¨ ;¨ g CC . 20Ž . Ž . Ž . Ž .

We can prove that this problem is equivalent to the following primalŽ .äriational inequality problem PVI :

PVI L* W Lu , ¨ y u G F ¨ y F u ;¨ g CC . 21Ž . Ž . Ž . Ž . Ž .Ž .

If P is Gateaux differentiable on VV , the directional derivative of P at uîn the direction ¨ g VV is

P u q t y P uŽ . Ž .dP u; ¨ s lim s DP u , ¨ ,Ž . Ž .Ž .

q ttª0

Ž .where DP u : VV ª VV * is the Gateaux derivative of P at u defined byˆ

DP u s L*DW Lu y DF u .Ž . Ž . Ž .

Then the PVI can be written as

PVI9 DP u , ¨ y u G 0 ;¨ g CC . 22Ž . Ž . Ž .Ž .

Since CC is a convex cone, CC* is its polar cone; it is obvious that theŽ .associated bicomplementarity problem ACP can be written as the primal

Ž .complementarity problem PCP

PCP u g CC , P u g CC*, P u , B u s 0. 23Ž . Ž . Ž . Ž . Ž .Ž .

LEMMA 1. If L: VV ª EE is a linear, continuous operator, P: CC ª R[ isconëx, l.s.c., then

PVP m PVI m PCP . 24Ž . Ž . Ž . Ž .

The proof of this lemma is similar to the proof of Theorem 1 givenbelow.

Direct methods for solving nonsmooth variational problems with bi-complementarity conditions are very difficult. In this section, we areinterested in establishing the dual problems. To find the conjugate func-tion of P: CC ª R[, we should extend its domain CC to VV by defining

J u , e s W e y F u q C u s W e y F u ,Ž . Ž . Ž . Ž . Ž . Ž .CC c


Ž . Ž . Ž . Ž . Ž .where F u s F u y C u . It is obvious that on CC, P u s J u, Lu .c CC

For geometrical linear operator L, the conjugate function J*: VV * =] Ž w x.EE* ª R of J can be easily given as cf., e.g., 8

² :J* u*, e* s inf inf u*, u q y e*, e q J u , e� 4Ž . Ž . Ž .ugVV egEE

s FU L*e* y W * e* ,Ž . Ž .c

[ Ž . U ]where W *: EE* ª R is given by 9 . F : VV * ª R is given byc

FU u* s inf u*, u y F u q C u .� 4Ž . Ž . Ž . Ž .c CCugVV

Let u s f q c . Then we have

UF u* s inf u*, u y u*, u� 4Ž . Ž . Ž .cugCClVVa

s inf u* y u*, c q u* y u*, f� 4Ž . Ž .fG0

F* u* , if yu* q u* F 0,Ž .s 25Ž .½ y`, if yu* q u* ) 0,

UŽ . Ž .where F* u* s u* y u*, c . Let EE ; EE* be a dual feasible space, inawhich the natural boundary conditions are prescribed, and DD* is a convexsubset of EE

U :a

DD* [ e* g EEU N B* L*e* F 0 .� 4Ž .a

On DD*, the conjugate function of P is then obtained as

P* e* s J* L*e*, e* s F* L*e* y W * e* . 26Ž . Ž . Ž . Ž . Ž .

From the theory of convex analysis, P*: DD* ª R] is always concave, u.s.c.Ž .So the dual ¨ariational problem DVP can be proposed as: To find

e* g DD* such that

DVP P* e* s sup P e* ;e* g DD*. 27Ž . Ž . Ž . Ž .

The associated dual ¨ariational inequality problem is to find e* g DD* suchthat

² : W * e* , e* y e* G F* L*e* y F* L*e* ;e g DD*. 28Ž . Ž . Ž . Ž .

If P* is Gateaux differentiable on DD*, thenˆ

P* e* q te* y P* e*Ž . Ž .² :dP* e*; e* s lim s DP* e* , e* ,Ž . Ž .

q ttª0

DAVID YANG GAO682

where the Gateaux derivative DP* is given asˆ

DP* e* s L DF* L*e* y DW * e* .Ž . Ž . Ž .Ž .Then the DVI can be written as

² :DVI9 DP* e* , e* y e* F 0 ;e* g DD*. 29Ž . Ž . Ž .If the inverse operator Ly1 of L exists, we can let

DD [ e g EE N Ly1e, B* L*e* G 0 ;e* g DD* . 30Ž . Ž .� 4Ž .Ž .The dual complementarity problem DCP then can be formulated as

DCPŽ .y1e* g DD*, P* e* g DD, L P* e* , B* L*e* s 0.Ž . Ž . Ž .Ž .

31Ž .

Furthermore, if the problem has only the external complementarity condi-Ž . y1tion, W: EE ª R is convex, differentiable such that e s W * e* s C e*,

Ž .then the dual variational inequality DVI9 can be reduced into the form

A9u*, ¨* y u* G F* ¨* y F* u* ;¨* g L*( DD*. 32Ž . Ž . Ž . Ž .w xThis is the well-known Mosco dual variational inequality 29 , where

A9 s yAy1 y : ¨* ª yAy1 y¨*Ž .is a formal inverse of A s L*CL. Unfortunately, in infinite-dimensionalproblems, where L is a gradient-like operator, to find the inverse Ay1 sLy1 Cy1L*y1 is almost impossible. This is the reason why the applicationsof Mosco’s dual variational inequality are limited. But in dual variational

Ž . Ž .inequalities DVI and DVI9 , instead of ¨*, we use e* g EE* as thevariational argument, so these dual problems are very easy to formulate.

If the system has the associated internal complementarity conditiononly, the dual complementarity problem can be simply given as

UDCP9 e* g EE , 0 g P* e* . 33Ž . Ž . Ž .a

In many engineering problems, this dual problem is much easier than theprimal problem.

THEOREM 1. If P: CC ª R[ is con¨ex and Gateaux differentiable, thenˆŽ . Ž . Ž . Ž .the primal problems PVP , PVI , PCP and the dual problems DVP ,

Ž . Ž .DVI , DVI9 are equi alent in the sense that they ha¨e same solutions setand

inf P u s sup P* e* . 34Ž . Ž . Ž .ugCC e*gDD*


Ž . Ž . [Proof. Let us prove DVI m DVP first. Since W *: EE* ª R is aconvex, l.s.c., one has

² :W * e* y W * e* G W * e* , e* y e* ;e*, e* g EE*.Ž . Ž . Ž .Ž .If e* g DD* is a solution of DVI , then

² :P* e* y P* e* G W * e* , e* y e* y F* L*e* q F* L*e* G 0Ž . Ž . Ž . Ž . Ž .;e* g DD*.

Ž . Ž .This shows that DVI « DVP . Since DD* is convex, for any givenw xu g 0, 1 , one has

e* g DD*, e* g DD* « u e* q 1 y u e* g DD*Ž .From the DVP, we should have

P* e* G P* ue* q 1 y u e* s P* e* q u e* y e* ;u G 0,Ž . Ž . Ž .Ž . Ž .i.e.,

1P* e* q u e* y e* y P* e* F 0 ;u G 0.Ž . Ž .Ž .

u

Taking u ª 0q, we obtain

1² :0 G lim P* e* q u e* y e* y P* e* s DP* e* , e* y e* .Ž . Ž . Ž .Ž .

q uuª0

Ž . Ž . ]This shows that DVP « DVI9 . Since F*: DD* ª R is concave, u.s.c.,and Gateaux differentiable, one hasˆ

F* L*e* y F* L*e* F F* L*e* , L*e* y L*eŽ . Ž . Ž .Ž .² :s L F* L*e* , e* y e* ;e* g DD*.Ž .

Ž . Ž . ]So it is easy to find that DVI9 m DVI . Actually, if P*: DD* ª R isstrictly concave and DD* is a closed subset of a reflexive Banach space,then each dual problem has a unique solution.

Ž . Ž .Now we are going to prove that the dual solution of DVP solves PCP .Introducing the Lagrange multiplier u g VV to relax the constraint inaŽ . Ž . U ]DVP , the Lagrangian L u, e* : VV = EE ª R associated with DVP cana abe given by

L u , e* s P* e* y B u , B* L*e* . 35Ž . Ž . Ž . Ž . Ž .Ž .So the DVP is equivalent to the saddle-point problem

inf sup L ¨ , e* . 36Ž . Ž .U¨gVVa e*gEEa

DAVID YANG GAO684

The Kuhn]Tucker optimality conditions for this saddle-point problem are

0 g L u , e* , B u G 0, B* L*e* F 0,Ž . Ž . Ž .e* 37Ž .B u , B* L*e* s 0.Ž . Ž .Ž .

Since W: EE ª R[ is convex, l.s.c., the partial superdifferential inclusion0 g L gives the subdifferential constitutive relationse*

Lu g W * e* m e* g W Lu .Ž . Ž .

Ž .This shows that the solution of the saddle-point problem 36 solves theŽ .complementarity problem PCP .

For any given ¨ g CC and e* g DD*, using the Legendre transformationŽ . Ž . Ž . Ž .F* L*e* s L*e*, ¨ y F ¨ to replace F* in P*, the Lagrangian L ¨ , e*

Ž .associated with the dual variational problem DVP can be written as

² :L ¨ , e* s L¨ , e* y W * e* y F ¨ . 38Ž . Ž . Ž . Ž .

Ž . Ž .Since W e is convex, W ** s W, it is easy to find that sup L e*, ¨ se*Ž . Ž . Ž .P ¨ . On the other hand, if e* g DD*, inf L e*, ¨ s P* e* . Since L:¨

CC = DD* ª R is a saddle-point functional, we have

inf P ¨ s inf sup L ¨ , e* s sup inf L ¨ , e* s sup P* e*Ž . Ž . Ž . Ž .¨ ¨ ¨e* e * e *

; ¨ , e* g CC = DD*.Ž .

Ž .So Eq. 34 is proved.

Let us now demonstrate how the above scheme fits in with the finite-dimensional nonlinear programming and NCP. Let VV s VV * s Rn, EE sEE* s Rm, with the standard coordinatewise partial ordering. Consider thegeneral global optimization problem:

P min f u s.t. u g CC , g Lu G 0,Ž . Ž . Ž .

where CC ; Rn is a nonempty convex cone, f : CC ª R is l.s.c., L g Rm= n isa matrix, and g : EE ª Rm is componentwise l.s.c. on EE. To reformulate

Ž .this nonlinear constrained optimization problem in the model form PVP ,Ž . Ž . Ž .i.e., P u s W Lu y F u , we need only set

F u s yf x y C u ,Ž . Ž . Ž .CC

0 if g e G 0,Ž .W e sŽ . ½q` otherwise.


Ž .Then the finite-dimensional optimization problem P can be written as aunconstrained nonsmooth optimization problem:

P u s W Lu y F u ª min ;u g Rn .Ž . Ž . Ž .Consider the linear programming case, where

� n 4CC s u g R N u G 0 ,

f u s u , u* , g Lu s Lu y b.Ž . Ž . Ž .n m Ž .For fixed u* g R and b g R , P is a linear programming

PVP min u , u* s.t. u G 0, g Lu s Lu y b G 0.Ž . Ž . Ž .lin

The conjugate functions in this elementary case may be calculated at onceŽ . Ž .from 9 and 25 as

² :b , e* , if e* F 0,W * e* sŽ . ½q` otherwise,

0 if u* q u* G 0,F* u* sŽ . ½y` otherwise.

So the dual problem is

DVPŽ .lin

² :max y b , e* s.t. e* F 0, g* e* s yL*e* y u* F 0.Ž .Ž .The Lagrangian 38 in this case is

² :L ¨ , e* s ¨ , u* q L¨ y b , e* ;¨ G 0, e* F 0,Ž . Ž .i.e., the Lagrangian multiplier of the primal problem should be thesolution of the dual variational problem. The Kuhn]Tucker conditions forthis problem are

u G 0, L*e* q u* G 0, u , L*e* q u* s 0,Ž .² :e* F 0, Lu y b G 0, Lu y b , e* s 0.

So this is a bi-complementarity problem.

4. CONTACT PROBLEM OF ELASTOPLASTIC BEAM

As a typical example, we will use the extended elastic]perfectly-plasticw xbeam theory developed recently 13, 14 to illustrate the primal and dual

bi-complementarity problems. Let us consider a elastic]perfectly-plasticbeam possibly in contact with a rigid obstacle, which is described by a

Ž . Ž .strictly concave function c x see Fig. 2 .

DAVID YANG GAO686

FIG. 2. Frictional contact problem of elastoplastic beam.

�Ž .Suppose that the beam in the x-y plane is a rectangle V s x, y g2 4R N 0 F x F L, yh F y F h . The beam is subjected to a given distributed

q TŽ Ž . Ž .. Ž .load p s q x , p x on the top surface y s h see Fig. 2 . On theyŽ .bottom surface, the beam is subjected to a frictional shear force q x ,

which is unknown until the problem is solved. Displacement of materialŽ .point x, y in the beam is described by the vector

u x , yŽ .u x , y s , x , y g V .Ž . Ž .ž /w xŽ .

The first component describes the horizontal displacement of the materialpoint, while the second describes the displacement of the middle axis,which coincides with the x axis in the equilibrium state.

The general strain vector is given by the geometrical equation

u 0

u x , y x x Ž .ee s s s s Lu. 39Ž .gž / u w ž /w xŽ .q� 0 � 0 y x y x

In this problem, the geometrical operator is a linear differential operator:

0

xL [ . 40Ž . � 0

y x


Let VV be the general displacement space,

u x , yŽ . 1 1w xVV [ u s N u x , y g HH V , w x g HH 0, L ,Ž . Ž . Ž .½ 5ž /w xŽ .where HH 1 s WW 2, 1 is the standard Sobolev space. The source space VV * s FF

is a general force space:

q xŽ . 2 w xFF [ p s N q x , p x g LL 0, L .Ž . Ž .½ 5ž /p xŽ .Ž .The bilinear form ), ) : FF = UU ª R can be written as

L Lq yp, u s q x u x , h dx q q x u x , yh dxŽ . Ž . Ž . Ž . Ž .H H0 0

Lq p x w x dx.Ž . Ž .H

0

The general strain space EE and its dual space EE* s SS , the general stressspace, in this problem are

e x , yŽ . 2EE [ e s e x , y , g x , y g LL V ,Ž . Ž . Ž .½ 5ž /g x , yŽ .

s x , yŽ . 2SS [ s s s x , y , t x , y g LL V .Ž . Ž . Ž .½ 5ž /t x , yŽ .² :The bilinear form , : SS = EE ª R, i.e.,

² : ² : ² :s, e s s , e q t , g s se q tg dV ,Ž .HV

encompasses the duality relationship between SS and EE.Ž .TUsing the Gauss]Green theorem, for any given u s u, w such that

Ž .e s Lu, the duality relation 1 in this problem is² :s, Lu s L*s, u , 41Ž . Ž .

² : Ž .where L*: SS ª FF is the operator adjoint to L relative to ), ) , ), )defined by

t s tL*s, u s y w dV y q u dVŽ . H H ž / x x yV V

h hq s n u dy N q t n w dy NH Hx xs0, L x xs0, L

yh yh

Lq t n u dx N .H y ys" h

0

DAVID YANG GAO688

Thus L* involves the ‘‘interior’’ operator in the domain V defined by

y y

x yhUL s dy. 42Ž .Hint yh � 00 y x

and the ‘‘boundary’’ operator on V given by

s n at x s 0, L in x direction,¡ x~t n at x s 0, L in y direction,L*s s 43Ž .x¢t n at y s "h.y

The feasible set VV ; VV is a subspace incorporating the essential bound-aary conditions. For example, if the beam is clamped at both ends, then thisspace can be given as

TVV s u , w g VV N u s w s 0 at x s 0, L .Ž .� 4a

² : Ž . Ž .Thus for any given u g VV , the duality relation Lu, s s u, L*s s u, pagives equilibrium conditions for this extended beam theory:

¡s q t s 0 ; x , y g V ,Ž .

x y~ tL*s s p « 44h Ž .w xy dy s p x ; x g 0, L ,Ž .H

xyh¢ " w xt x , " h s "q x ; x g 0, L .Ž . Ž .

Ž .Assuming that the shape of obstacle c x is a strictly concave function, forthe given external load p, the external complementarity condition for thiscontact problem is the well-known Signorini condition:

w x y c x G 0, p x y p x F 0,Ž . Ž . Ž . Ž .45Ž .w xw x y c x p x y p x s 0 ; x g 0, L .Ž . Ž . Ž . Ž .Ž . Ž .

Ž . Ž Ž . Ž ..T Ž .For any solution u x , y s u x , y , w x and s x , y sŽ Ž . Ž ..T w xs x, y , t x, y , the contact region ZZ ; 0, L and its boundary can bedefined by

w x w xZZ [ x g 0, L N w x s c x ; x g 0, L ,� 4Ž . Ž .thw x ZZ [ x g 0, L N w x s c x , dy q p x s 0 .Ž . Ž . Ž .H½ 5 xyh


This boundary is unknown until the problem is solved. It is called the freeboundary.

yŽ .For a lubricated contact problem, the lubrication implies that q x s 0.In the case of Coulomb’s law of dry friction, we have

y< <q x - yn p « u x , yh s 0 ; x g ZZ ,Ž . Ž .y y< <q x s yn p « u x , yh s ymq xŽ . Ž . Ž .

; x g ZZ for some m G 0.

Here n is the coefficient of friction. For more general frictional contactŽ . Ž Ž ..problems, we can define a convex, l.s.c. functional j u s j u x, yh such

Ž w x.that the frictional law can be given as cf., e.g., 31, 32 :

y w xyq x s t x , yh g j u ; x g 0, L .Ž . Ž . Ž .

This subdifferential relation is understood pointwise. It can also be writtenin the inverse form

w xu x , yh g j* s ; x g 0, L ,Ž . Ž .Ž . Ž Ž ..where j* s s j* t x, yh is the conjugate function of j.

When a certain combination of stresses in the beam results in strainsexceeding the limit of elastic behavior, plastic deformation begins to takeplace. Let V ; V be the elastic zone and V ; V be the plastic zonee psuch that V j V s V, V l V s B. In elastic zone V ; V, thee p e p econstitutive equation is given by Hooke’s law,

E 0 ess s s s Ce, 46Ž .ž / gž /ž /t 0 G

where E, G are positive elastic constants. However, in the plastic zoneV ; V, the strain vector e can be split into two part: e s ee q ep, i.e., thepelastic strain ee, which is given by Hooke’s law, and the plastic strain ep,which is given by Hencky’s plastic constituti e law:

g* sŽ .pe s l in V ,p s

47Ž .s.t. l G 0, g* s F 0, lg* s s 0 a.e. in V .Ž . Ž .Ž .Here l is a plastic flow factor; g* s is the plastic yield function,

2 2'5 5g* s s s y s s s q at y s , 48Ž . Ž .a b b

s is a material constant, and a ) 0 is a parameter. For the von MisesbŽ w x. 5 5material, a s 3; for the Tresca material, a s 4 cf., e.g., 3 . s sa

DAVID YANG GAO690

2 2's q at is the effective stress, which is a norm of the stress vector s.For a proportional loading problem, i.e., during plastic deformation, thestrain vectors ee and ep stay in the same direction, it is easy to find that

1y1 2 2 y15 5l s g e s e y E s s e q g y E s , 49Ž . Ž .(1r a b ba

2 25 5 'where e s e q 1ra g is the norm of the strain vector. By usingŽ .1r a

Ž . Žthe step function d g* , the constitutive relation can be written as cf., e.g.,w x.18 :

¡ g* sŽ .y1e s E s q g d g* s ,Ž .Ž . g* sŽ . sy1 ~e s C s q g d g* s orŽ .Ž . g* s s Ž .y1g s G t q g d g* s ,Ž .Ž .¢

t

50Ž .subject to the internal complementarity conditions

g e G 0, g* s F 0, g e g* s s 0. 51Ž . Ž . Ž . Ž . Ž .The elastic region V and the plastic region V can be given by thee pinternal complementarity conditions

V [ x , y g V N g e x , y - 0 or g* s x , y - 0 ; x , y g V ,� 4Ž . Ž . Ž . Ž .Ž . Ž .e

V [ x , y g V N g* s x , y s 0 or g e x , y G 0 ; x , y g V ,� 4Ž . Ž . Ž . Ž .Ž . Ž .p

respectively. The interface of elastoplastic region G s V l V is de-ep e pfined by

G [ x , y g V N g e x , y g* s x , y s 0 .� 4Ž . Ž . Ž .Ž . Ž .ep

Ž .For the given external force p, the obstacle c x and the boundarycondition set VV , the contact problem of this extended beam theory can beagiven as:

Ž .Problem 4. For the given obstacle function c x and the external loadq TŽ . Ž . Ž . Ž Ž . Ž ..p x , q x , find the displacement field u x, y s u x, y , w x g VVa

Ž Ž . Ž ..Tand the stress s field s s s x, y , t x, y such that the followingequations are satisfied:

1. Geometrical Equation:

0

u x , y x Ž .ee s Lu or sgž / ž /w xŽ .� 0 y x


2. Constituti e Equation:

¡ g* sŽ .y1e s E s q g d g* s ,Ž .Ž . g* sŽ . sy1 ~e s C s q g d g* s orŽ .Ž . g* s s Ž .y1g s G t q g d g* s .Ž .Ž .¢

t

3. Equilibrium Condition:

t¡ h w xy dy s p x ; x g 0, L ,Ž .H xyh~L*s s p or

s q t s 0 ; x , y g V ;Ž .¢ x y

q w xt x , h s q x , t x , yh g j u x , yh ; x g 0, L .Ž . Ž . Ž . Ž .Ž .

4. External Complementarity Condition:

w x y c x G 0, p x y p x F 0, w y c p y p s 0.Ž . Ž . Ž . Ž . Ž . Ž .

5. Internal Complementarity Condition:

g e G 0, g* s F 0, g e g* s s 0.Ž . Ž . Ž . Ž .

This is a strong nonlinear bicomplementarity problem. Its framework isshown in Fig. 3.

Let KK denote a convex subset of stresses

KK s s g SS N g* s x , y F 0 ; x , y g V , 52� 4Ž . Ž . Ž .Ž .U Ž .the plastic superpotential W s can be defined byp

0, if s g KK,UW s s 53Ž . Ž .p ½q` otherwise,

which is the indicator function of the convex set KK. Its subdifferential is apoint to set map

¡ g* sŽ .g , if g* s s 0, g G 0,Ž .

sU ~ W s s 54Ž . Ž .p � 40 , if g* s - 0,Ž .¢B, if g* s ) 0.Ž .

DAVID YANG GAO692

FIG. 3. Framework for contact problem of elastoplastic beam theory.

Ž .So by this subdifferentiation notation, the plastic flow law 47 can bewritten in the form

ep g W U s . 55Ž . Ž .p

Using the Legendre]Fenchel transformation, the conjugate function ofW U can be given asp

0, in V ,¡ ep ~² :W e s sup s, e sŽ .p 2 2 y1's e q 1ra g y E s dV in V ,Ž .H¢ ž /b b psgKK

V


which is called the support function of the convex set KK. The elasticpotential is a quadratic functional:

1T c y1 2W e s e Ced g e q E s d g e dV . 56Ž . Ž . Ž . Ž .Ž . Ž .He b2 V

The superpotential in this problem is

W e s W e q W e .Ž . Ž . Ž .e p

So the constitutive equation and the internal complementarity conditioncan be written in the subdifferential form

s g W e . 57Ž . Ž .The external potential energy F: VV ª R] can be defined as

F u s u, p y C u y j uŽ . Ž . Ž . Ž .VV a

L¡ qpw q q u x , h dx y j u x , yh ,Ž . Ž .Ž .H0~s 58Ž .Tif u s u , w g VV ,Ž . a¢y`, otherwise.

Ž . Ž . Ž .So the total potential energy P u s W Lu y F u is a convex, l.s.c.functional. The convex cone CC is

T w xCC s u s u , w g VV N w x G c x ; x g 0, L . 59Ž . Ž . Ž . Ž .� 4By the Lemma 1, the bi-complementarity problem for this extended beamtheory is equivalent to the following primal variational problems: FindŽ .Tu, w g CC such that

TPVP P u , w s inf P ¨ , z ; ¨ , z g CC . 60Ž . Ž . Ž . Ž . Ž .Ž .The associated variational inequality problem PVI has a very compli-

cated form:

L* W Lu , v y u q j vŽ . Ž .Ž .L qG p z y w q q ¨ x , h y u x , h dx q j u x , yhŽ . Ž . Ž . Ž .Ž Ž .H

0

T;v s ¨ , z g VV l CC . 61Ž . Ž .a

Since the superpotential W is a nonsmooth function, the primal variationalapproaches for solving this bi-complementarity problem are very difficult.Now let us consider the dual approaches.

DAVID YANG GAO694

The dual feasible set SS s EEU in this problem isa a

s tTSS s s s s , t g SS q s 0 ; x , y g V ,Ž . Ž .a ½ x y

q w xt x , h s q x ; x g 0, L .Ž . Ž . 5DD* ; SS in this problem is a so-called statically admissible space:a

thT w xDD* [ s s s , t g SS dy y p x F 0 ; x g 0, L . 62Ž . Ž . Ž .Ha½ 5 xyh

By the Legendre]Fenchel transformation, the external complementaryenergy F*: SS ª R] can be obtained as

t¡ L hy c x dy q p x dx y j* t ,Ž . Ž . Ž .H H

x0 yh~F* L*s sŽ .Tif s s s , t g DD*,Ž .¢y`, otherwise.

63Ž .

The internal complementary energy W *: SS ª R[ is a convex, l.s.c.function

1¡ T y1s C s dV , if s g KK,U U H~W * s s W s q W s s 64Ž . Ž . Ž . Ž .2e p V¢q`, otherwise.

Ž .On the space DD* l KK, the total complementary energy functional P* scan be simply written as

tL hP* s , t s y c x p x q dy dxŽ . Ž . Ž .H H

x0 yh

1y1 2 y1 2w xy E s q G t dV y j* t . 65Ž . Ž .H 2V

Thus the dual variational problem for this nonlinear bi-complementarityTŽ .problem can be given as the following: Find s s s , t such that

TDVP P* s , t G P* s , t ; s , t g DD* l KK. 66Ž . Ž . Ž . Ž . Ž .


The dual variational inequality for this problem has a very simple form:

DVIŽ .y1 y1E s s y s q G t t y t dV q j* tŽ . Ž . Ž .H

V

TG c x t y t dV q j* t ; s , t g DD* l KK.Ž . Ž . Ž . Ž .H xV

67Ž .

Ž . Ž .For the lubricated contact problem, the dual problems DVP and DVIare quadratic programmings on the closed convex subset DD* l KK. SinceW *: SS ª R is a strictly convex functional, the solution of the dualproblems is unique. By Theorem 1, we know that both dual problems areequivalent to the primal problems and BCP. However, compared with

Ž . Ž . Ž . Ž .primal problems 60 and 61 , the dual problems 66 or 67 are mucheasier to solve. In numerical analysis, the dual approaches will provide thelower bound solutions. Moreover, using the linear equilibrium constraintin the statically admissible space DD*, the degrees of freedom in nonlinear

w xiteration can be reduced via the mixed finite element method given in 11 .The complementary finite element method for fully nonlinear, nonsmooth

w xvariational problems was discussed in 13, 14 .

ACKNOWLEDGMENT

The reviewer’s valuable suggestions are gratefully acknowledged.

REFERENCES

1. G. Allen, Variational inequalities, complementarity problems, and duality theorems,Ž .J. Math. Anal. Appl. 58 1977 , 1]10.

2. J. P. Aubin, ‘‘Applied Functional Analysis,’’ Wiley, New York, 1979.3. J. Chakrabarty, ‘‘Theory of Plasticity,’’ McGraw-Hill, New York, 1987.4. R. W. Cottle, ‘‘Nonlinear Programs with Positively Bounded Jacobians,’’ Ph.D. thesis,

Department of Mathematics, University of California, Berkeley, 1964.5. R. W. Cottle, J. S. Pang, and R. E. Stone, ‘‘The Linear Complementarity Problems,’’

Academic Press, New York, 1992.6. V. F. Dem’yanov, G. E. Stavroulakis, L. N. Polyakova, and P. D. Panagiotopoulos,

‘‘Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Eco-nomics,’’ Kluwer, Dordrecht, 1996.

7. G. Duvaut and J. L. Lions, ‘‘Inequalities in Mechanics and Physics,’’ Springer-Verlag,Berlin, 1976.

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

DAVID YANG GAO696

9. M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarityŽ .problems, SIAM Re¨ . 39 1997 , 669]713.

10. A. Friedman, ‘‘Variational and Free Boundary Problems,’’ Springer-Verlag, New York,1993.

11. D. Y. Gao, Panpenalty finite element programming for limit analysis, Comput. Struct. 28Ž .1988 , 749]755.

12. D. Y. Gao, Duality theory in nonlinear buckling analysis for von Karman equations, Stud.Ž .Appl. Math. 94 1995 , 423]444.

13. D. Y. Gao, Nonlinear elastic beam theory with application in contact problems andŽ .variational approaches, Mech. Res. Comm. 23 1996 , 11]17.

14. D. Y. Gao, Complementary finite-element method for finite deformation nonsmoothŽ .mechanics, J. Eng. Math. 30 1996 , 339]353.

15. D. Y. Gao, Dual extremum principles in finite deformation theory with applications inŽ .post-buckling analysis of nonlinear beam model, Appl. Mech. Re¨ . ASME 50 1997 ,

S64]S71.16. D. Y. Gao and D. L. Russell, An extended beam theory for smart materials applications,

Ž .J. Appl. Math. Optim. 34 1996 , 279]298.17. D. Y. Gao and G. Strang, Geometric nonlinearity: Potential energy, complementary

Ž .energy, and the gap function, Quart. Appl. Math. XLVII 1989 , 487]504.18. D. Y. Gao and G. Strang, Dual extremum principles in finite deformation elastoplastic

Ž .analysis, Acta Appl. Math. 17 1989 , 257]267.19. R. Glowinski, J. L. Lions, and R. Tremolieres, ‘‘Numerical Analysis of Variational´ `

Inequalities,’’ North-Holland, Amsterdam, 1981.20. J. Haslinger and P. D. Panagiotopoulos, The reciprocal variational approach to the

Signorini problem with friction, approximation results, Proc. Roy Soc. Edinburgh Sect. AŽ .98 1984 , 365]383.

21. I. Hlavacek, J. Haslinger, J. Necas, and J. Lovisek, ‘‘Solution of Variational Inequalitiesin Mechanics,’’ Springer-Verlag, New York, 1982.

22. G. Isac, ‘‘Complementarity Problems,’’ Lecture Notes in Mathematics, Springer-Verlag,New York, 1992.

Ž .23. S. Karamardian, Generalized complementarity problems, J. Optim. Theory Appl. 6 1971 ,161]168.

24. N. Kikuchi and J. T. Oden, ‘‘Contact Problems in Elasticity: A Study of VariationalInequalities and Finite Element Methods,’’ SIAM, Philadelphia, 1988.

25. D. Kinderlehrer and G. Stampacchia, ‘‘An Introduction to Variational Inequalities andTheir Applications,’’ Academic Press, New York, 1980.

26. A. Klarbring, A. Mikelic, and M. Shillor, Duality applied to contact problems withŽ .friction, Appl. Math. Optim. 22 1990 , 211]226.

27. J. Lovısek, Duality in the obstacle and unilateral problem for the biharmonic operator,´Ž .Aplikace Matematiky 26 1981 , 291]303.

28. Z. Q. Luo, J. S. Pang, and D. Ralph, ‘‘Mathematical Programs with Equilibrium Con-straints,’’ Cambridge University Press, Cambridge, UK, 1996.

Ž .29. U. Mosco, Dual variational inequalities, J. Math. Anal. Appl. 40 1972 , 202]206.30. J. T. Oden and J. N. Reddy, ‘‘Variational Methods in Theoretical Mechanics,’’ Springer-

Verlag, Berlin, 1976.31. P. D. Panagiotopoulos, ‘‘Inequality Problems in Mechanics and Applications,’’ Birkhauser,¨

Boston, 1985.32. P. D. Panagiotopoulos, ‘‘Hemivariational Inequalities, Applications in Mechanics and

Engineering,’’ Springer-Verlag, Berlin, 1993.Ž33. J. S. Pang, Complementarity problems, in ‘‘Handbook in Global Optimization’’ R. Horst

.and P. Padalos, Eds. , Kluwer, Boston, 1994.


34. B. D. Reddy, Mixed variational inequalities arising in elastoplasticity, Nonlinear Anal. 19Ž .1992 , 1071]1089.

35. R. T. Rockafellar, Duality and stability in extremum problems involving convex functions,Ž .Pacific J. Math. 21 1967 , 167]187.

36. J. F. Rodrigues, ‘‘Obstacle Problems in Mathematical Physics,’’ Elsevier, Amsterdam,1987.

37. G. Strang, ‘‘Introduction to Applied Mathematics,’’ Wellesley-Cambridge Press, Welles-ley, MA, 1986.

Ž .38. E. Tonti, 1972 , A mathematical model for physical theories, I, II, Rend. Accad. LinceiŽ .LII 1972 , 133]139, 350]356.

Ž .39. S. T. Yau and D. Y. Gao, 1992 , Obstacle problem for von Karman equations, Ad¨ .Ž .Appl. Math. 13 1992 , 123]141.

Bi-Complementarity and Duality: A Framework in Nonlinear … · 2017-02-15 · the well-known books...

Documents

Transcript of Bi-Complementarity and Duality: A Framework in Nonlinear … · 2017-02-15 · the well-known books...