The Method of Substationary Point Search

Abstract

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.