Keywords: nonconvex optimization, neural networks, frictional contact.

Electronic circuits can be designed nowadays to emulate the dynamical behaviour of a given, maybe nonlinear, dynamical system. Therefore, following the dynamical system approach to optimization and data processing the potential of applying electronic devices for on-line solution of large scale problems seems straightforward. In fact, one of the aspects of neural network computing is exactly this application. An iterative algorithm is best suited for neural network implementation, if it can be resolved into a number of relatively simple steps which can be executed by separate elements (distributed), which are interconnected by appropriate connection lines. The high parallelization of this scheme makes it also appropriate for a parallel computer implementation, if a hardware implementation is not available or at the development stage.

In the present paper, a neural network approach is proposed dealing with the solution of frictional contact problems. The friction process along the interfaces is described by means of nonmonotone, possibly multivalued, stress-strain or reaction-displacement laws that include complete jumps or softening branches corresponding to the evolution of the stick-slip process. The character of these material and/or boundary laws is nonclassic due to the softening branches and the vertical jumps of the stress-strain or reaction-displacement diagrams. Therefore, the analysis of such problems cannot be performed numerically by means of the classic analysis methods. It is known that if the arising material and/or boundary laws are of monotone nature, the problem can be formulated as a variational inequality problem that leads to a convex minimization problem [1]. In addition, since the theory of the aforementioned variational inequalities is related to the notion of convexity, the treatment of these problems can be numerically carried out by solving equivalent optimization problems that express the principle of minimum potential and/or complementary energy at the position of equilibrium [1]. Monotonicity in the arising stress-strain relations and/or boundary conditions, which is the common feature of these problems, leads to convex, generally non - differentiable energy potentials. The corresponding energy functions have been called convex superpotentials.

Cases lacking monotonicity correspond to nonconvex potentials and cannot be formulated by the same mathematical tools as the previous ones. For that, a new variational theory was introduced into Mechanics by Panagiotopoulos [2] by applying the mathematical notion of generalized gradient of Clarke [3]. Within this framework, the so-called hemivariational inequalities have been obtained, which constitute generalizations of the classic variational inequalities and lead to substationarity principles for the potential and complementary energy. The latter constitute extensions of the propositions of minimum potential and complementary energy in the case of nonconvexity and nonsmoothness of the energy potential [2].

The present work contributes to the numerical treatment of nonconvex and nonsmooth optimization problems. The proposed approach permits the rational treatment of the aforementioned limit states. In particular, discretizing the structure by means of a suitable finite element scheme, the structural behaviour is described by a discrete hemivariational inequality. An effective algorithm based on [4] equivalently transforms the initial nonmonotone problem into a sequence of monotone, Coulomb friction problems. Then, a neural network computing system is applied in order to solve efficiently the arizing optimization problems.

1

Panagiotopoulos, P.D., "Inequality problems in mechanics and applications. Convex and nonconvex energy functions", Birkhäuser, Basel - Boston - Stuttgart, 1985, russian translation, MIR Publ., Moscow 1988.

2

Panagiotopoulos, P.D., "Hemivariational inequalities. Applications in mechanics and engineering", Springer, Berlin - Heidelberg - New York, 1993.

3

Clarke, F.H., "Optimization and nonsmooth analysis", J. Wiley, New York, 1983.

4

Mistakidis, E.S., Stavroulakis, G.E., "Nonconvex optimization in Mechanics. Algorithms, heuristics and engineering application by the F.E.M.", Kluwer, Boston, 1997.

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