Normally Admissible Stratifications and Calculation of Normal Cones to a Finite Union of Polyhedral Sets

Abstract

This paper considers computation of Fréchet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to those already known in the literature. Finally, we apply this approach to a class of time dependent problems and provide an illustration on a special structure arising in delamination modeling.

The intermediate set and limiting superdifferential for coalitional games: between the core and the Weber set

“…Even though the computation of these objects may be rather a challenging task, see e.g. [1,9], the presented framework allows for a significant simplification. For the general approach based on upper semicontinuous functions, we refer the reader to [17].…”

Identification of some nonsmooth evolution systems with illustration on adhesive contacts at small strains

“…However, when computing N gph Q on the basis of N gphQ and the chain rule from [14, Theorem 4.1] one obtains only an upper estimate and not equality. That is why we make use of [2] where formulas for Fréchet and limiting normal cones to a finite union of convex polyhedra have been derived and then applied to a special structure arising in time dependent problems. For simplicity, we will show the result only for the simplest case of M = 1.…”

“…As we have mentioned in the introduction, it would be possible to use standard calculus rules to obtain a formula for N gph Q based on multiple computations of N gph Q. The graph of Q can easily be visualized in Figure 1 That is why we make use of [2] where formulas for Fréchet and limiting normal cones to a finite union of convex polyhedra have been derived and then applied to a special structure arising in time dependent problems. For simplicity, we will show the result only for the simplest case of M = 1.…”

“…The state variable q may involve displacements and various internal parameters (but also various concentrations of some constituents subjected to diffusive processes). In this paper, we focus on a subclass of such problems where the state has the structure q = (u, z) (2) for each time instance t in a Banach space U × Z. In this way, a quasistatic plasticity, or damage, or various phase transformations at small strains can be modelled, and also various problems in contact mechanics like friction or adhesion, together with various combinations of these phenomena.…”

“…The application of standard tools of generalized differential calculus provides us in this case only with an upper estimate of the coderivative of the normal cone mapping to the overall constraint set, which could be a substantial drawback both in the optimality conditions as well as in the used numerical approach. To overcome this hurdle, we have employed a new result from [2] which enables us to compute the limiting coderivative of the mentioned normal cone mapping exactly.…”