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

Abstract

A class of evolution quasi-static systems which leads, after a suitable time discretization, to recursive non-linear programs, is considered and optimal control or identification problems governed by such systems are investigated. The resulting problem is an evolutionary Mathematical Programs with Equilibrium Constraints. A subgradient information of the (in general nonsmooth) composite objective function is evaluated and the problem is solved by the implicit programming approach. The abstract theory is illustrated on an identification problem governed by delamination of a unilateral adhesive contact of elastic bodies discretized by finite-element method in space and a semiimplicit formula in time. Being motivated by practical tasks, an identification problem of the fracture toughness and of the elasticity moduli of the adhesive is computationally implemented and tested numerically on a two-dimensional example. Other applications including frictional contacts or bulk damage, plasticity or phase transformations are outlined.

Keywords:

• rate-independent systems
• optimal control
• identification
• fractional-step time discretization
• quadratic programming
• gradient evaluation
• variational analysis
• implicit programming approach
• limiting subdifferential
• coderivative
• nonsmooth contact mechanics
• delamination

AMS Subject Classifications:

• 35Q90
• 49N10
• 65K15
• 65M32
• 74M15
• 74P10
• 90C20

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research has been supported by GA ČR through the projects 201/12/0671 ‘Variational and numerical analysis in nonsmooth continuum mechanics’, 201/10/0357 ‘Modern mathematical and computational models for inelastic processes in solids’, 13-18652S ‘Computational modeling of damage and transport processes in quasi-brittle materials’, 14-15264S ‘Experimentally justified multiscale modelling of shape memory alloys’ with the institutional support RVO:61388998 (ČR) and SVV 2015 No. 2602255.