LAGRANGE MULTIPLIER METHOD FOR SOLVING VARIATIONAL INEQUALITY IN MECHANICS

Abstract

Abstract. Lagrange multiplier method for solving the contact problemin elasticity is considered. Based on lower semicontinuity of sensitivityfunctional we prove the convergence of modified dual scheme to corre-sponding saddle point. IntroductionDuality methods based on classical schemes for constructing Lagrangianfunctionals are widely used for solving variational inequalities in mechanics. Ingeneral it is not able to prove their convergence to the corresponding saddlepoint. For coercive variational inequalities the convergence with respect to theprimal variable can be shown only. It is provided under assumption that thestep size of dual variable is sufficiently small. For semicoercive variational in-equalities it is not able to use classical Lagrangian functional because of thequadratic form of the functional to be minimized has a nontrivial null space.To remedy this situation, a modified Lagrangian functionals are examined. Inorder to prove that duality method based on modified Lagrangian functionalconverges to a saddle point we show that the corresponding sensitivity func-tional is a weakly lower semicontinuous in the original Hilbert space.1. Semicoercive contact problem of elasticityConsider a two-dimensional contact problem between an trapezoid elasticbody Ω and an absolutely rigid support (Figure 1).The boundary Γ of domain Ω is equal to Γ