Hemivariational Inequalities

The aim of Chapter 1 is to provide some notions and propositions of Nonsmooth Analysis that will be used in the next Chapters for the study of engineering problems leading to hemivariational inequalities. The propositions are given here without proofs. In this Chapter we primarily rely on the books and monographs by Moreau [Mor69], Rockfellar [Rock60,68,70,79,80], Göpfert [Göp], Eke-land and Temam [Eke], Aubin [Aub77,79a,84], Aubin and Francowska [Aub90], Clarke [Clar83] and Panagiotopoulos [Pan85]. The reader is referred there for the proofs of the propositions.

In this Chapter we explain the origins of Nonsmooth Mechanics and of the Inequality Problems. To do this we use the two notions of convex and of nonconvex superpotentials. We consider boundary conditions and material laws resulting from convex or nonconvex, nonsmooth energy functions using the concept of sub differential or of generalized gradient. For additional information on these subjects the reader is referred to the monographs and books of Duvaut and Lions [Duv72], Panagiotopoulos [Pan85], Hlavacek et al. [H188], Moreau, Pana-giotopoulos, Strang [Mor88a,b], Antes, Panagiotopoulos [Ant92], as well as to [Mor68,86,88c] and [Ger74].

This chapter constitutes a continuation of the previous chapter. First we deal with material laws derived by convex superpotentials and then with material laws derived by nonconvex superpotentials. Special attention is paid to the loading and unloading problem. After a short illustration of the advantages of the use of superpotentials we introduce material laws and boundary conditions expressed by means of the new notions of fans, quasi differentials and co differentials. For further reading on these subjects the reader may consult the following books and papers: [Pan85,87b,88,92], [Ant92,] [Mor88a,b], [Hal74,75], [Léné73,74], [Stav91,93a,b].

In the present chapter we present the method for the formulation of hemivariational inequalities and related variational expressions. For static problems the classical minimum propositions for the potential and the complementary energy are extended to analogous substationarity propositions.

Multivalued nonmonotone boundary conditions lead to hemivariational formulations on the boundary of the body which are equivalent to multivalued integral boundary equations. Analogous results hold for interface problems with the difference that, in this case the multivalued integral equations are extended over the interface. As the present chapter is rather brief, we refer the reader to the monograph by Antes and the author [Ant92] and to the monograph by the author [Pan85] p. 160–162 for further information on the material.

In the present chapter we study static hemivariational inequalities concerning the existence of their solutions. Some approximation results are also given. We distinguish the coercive and the more difficult semicoercive case where the rigid body displacements play an important role. After the study of hemivariational inequalities we deal with variational hemivariational inequalities and we derive some existence and approximation results. Finally the mathematical results obtained are applied to concrete engineering problems. The present chapter is mainly based on [Pan88a,89a,90,91,92c]. For other types of existence results we refer to Naniewicz [Nan88,89a,89b].

The present chapter is devoted to the study of the eigenvalue problem for hemi-variational inequalities and to the study of dynamic hemivariational inequalities. In Chapt. 4 we described certain mechanical problems leading to the eigenvalue problem for hemivariational inequalities and to dynamic hemivariational inequalities. Here we shall give some existence results. For the eigenvalue problem the minimax technique is applied [Mot93] and for the dynamic hemivariational inequalities the regularization method. The corresponding results for dynamic hemivariational inequalities appear here for a first time.

The present chapter addresses the optimal control and the parameter inden-tification problem of systems governed by hemivariational inequalities. This is a nonclassical mathematical problem, because the state of the problem is connected with the control function through a hemi variational inequality. We recall here that optimal control problems governed by state variational inequalities have been already studied (cf. e.g. [Yvon], [Lio71], [Panag77], [Mign76, 84], [Shi], [Hasli86a,b] [Barb]). However the present problem is more complicated, because we have state hemivariational inequalities. Here due to the lack of convexity of the superpotentials involved, compactness arguments will be applied. The mathematical framework is quite general to cover most of the usual engineering structures, as e.g. beams, plates in stretching and bending etc. The chapter is based mainly on [Panag92] and [Hasli93]. We refer also to [Panag84,89,90,91], [Hasli89]. Some application of the theory to engineering problems close the present chapter and illustrate the prospects for applications of the developed theory.

This chapter is the first one of a series of chapters concerning the numerical solution of hemivariational inequalities. Since we are interested in engineering applications we deal with realistic problems, which have a large number of unknowns. After a short description of the first attempts to find the numerical solution of a hemivariational inequalities we deal mainly with four methods which seem today to be quite efficient for nonmonotone possibly multivalued engineering problems leading to hemivariational inequalities. The two first methods are, the “Microspring Approximation of the Decreasing Branches” and the “Decreasing Branch Approximation by Monotone Laws” which are described in this chapter. The second method is generalized in the next chapter for any hemivariational inequality. The two other methods are, the method of the “Sub-stationarity Point Search” (Chapt. 11 and the “Decomposition into Two Convex Problems” Chapt. 12. It should be noted that the numerical methods presented in this part of the book do not treat all types of hemivariational inequalities. We prefer to sacrifice this in favour of the efficiency of the numerical treatment of more restricted classes of nevertheless, interesting problems. On the contrary the initial numerical efforts aimed to the maximum of generality; this was very soon abandoned. The chapters of the present part of the book do not treat the typical questions of convergence, stability etc. with respect to the developed numerical methods.

The present chapter deals with a generalization of the method presented in Sect. 9.3, i.e. the approximation of a decreasing branch by monotone laws. After a general description of the method based on [Mis92a,b,93] we give some numerical applications. Moreover a comparison with the path following method [Cris91] is attempted. The material of the present chapter permits us to consider zigzag nonmonotone multivalued material laws and boundary conditions for twodimensional and threedimensional bodies.

The present chapter is devoted to the presentation of another numerical method for the treatment of hemivariational inequalities. This method deals with the search for the substationarity points of the potential or the complementary energy of the structure. We recall (cf. Sect. 4.3) that besides all local minima, substationarity points are also the classical stationary points and some local maxima (cf. Sect. 1.2). Note that a hemivariational inequality is generally not equivalent to the corresponding substationarity problem. However, Prop. 6.3.1 holds for most practical applications and we have a complete equivalence between the hemivariational inequality and the substationarity problem. The method presented here makes use of efficient algorithms of numerical optimization and is an extension of a method presented in [Tzaf91,91a,93]. The hemivariational inequality is decomposed into a finite number of variational inequalities (monotone problems). This is achieved if the epigraph(s) of the su-perpotential(s) involved can be split into convex parts. This is possible, e.g. in the onedimensional case, if the superpotential is the minimum of a finite number of convex functionals. After a description of the algorithm and a discussion of its numerical properties we give a number of numerical applications.

This chapter presents a numerical method related to the quasidifferentiability concept which has been studied in Sect. 1.4, 3.5 and 4.5. We deal especially with problems for which the nonconvex nonsmooth superpotential can be expressed as the difference of two convex functions. In this case the problem can be decomposed into two variational inequalities or equivalently into two convex minimization problems. The method presented here leads to an efficient algorithm for large classes of hemivariational inequalities arising in engineering problems. We base our presentation mainly on [Stav91,93a,b] and on some results of Auchmuty [Auch83,89].

In this chapter we present certain numerical applications concerning the numerical treatment of dynamic hemivariational inequalites and of hemivariational inequalities arising in crack problems. In this area the theory of hemivariational inequalities permits the formulation and study of new and very interesting engineering problems, as is the analysis of cracks which are repaired by an adhesive material. The dynamic hemivariational inequalities are “transformed” through time discretization into static hemivariational inequalities which can be treated by one of the methods developed in the previous chapters.

This chapter deals with certain applications of the theory of hemivariational inequalities to the grasping problem of multifingered grippers of robots. We assume that the action of the support on each finger is described by a non-monotone possibly multivalued relationship, or more generally by a nonconvex superpotential law. A special case of the general problem studied is the grasping of a robot gripper with adhesive action, or the grasping of a robot gripper on the assumption of nonmonotone frictional forces. The final aim is the solution of the corresponding optimal control problem. We base our method on the formulation of the classical grasping problem of a robot gripper as a non-symmetric linear complementarity problem (L.C.P.). Under the term “classical” we mean that the grasping action is subjected to the classical Signorini condition (i.e nonpenetration inequality with the object assumed to be rigid) and to Coulomb frictional forces depending on the normal reaction between fingertip and object. The formulation of this problem as a L.C.P. was achieved in [Al-Fah91a,b,92a,b,93] where we have tried to introduce into the area of robotics certain methods of the Inequality Problems of Mechanics. Here we derive the corresponding hemivariational inequality by “superimposing” to the Signorini law and to the Coulomb friction law the necessary nonmonotonicities. The numerical solution proposed for both the arising hemivariatonal inequalities and the corresponding optimal control problem is achieved by a repeated solution of the L.C.P. according to a scheme analogous to the algorithm of Sect. 10.3.

This last chapter of the book gives some information on some recent developments in the theory of hemivariational inequalities. We deal first with fractal interfaces in the theory of hemivariatonal inequalities and secondly with the neurocomputing approach to the numerical treatment of hemivariational inequalities. The first part is important from the theoretical point of view, due to the arising noninteger dimension, and from the practical point of view as the real geometry of most physical objects is fractal. The second part is also very important because of the trend of evolution existing in the computer science towards neurocomputing. We base this chapter mainly on [Pang92a,b], [Pdp90a,b,91,92a,b,c,d], [Art], [Mis93b], [The93a] for the fractals and on [Kor], [Avd], [The91a,93b] for the neural networks.