On the Numerical Treatment of Hemivariational Inequalities

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.