Multi-field Modeling of Nonsmooth Problems of Continuum Mechanics, Differential Mixed Variational Inequalities and Their Stability

Abstract

This paper surveys various nonsmooth problems in continuum mechanics, presents multi-field variational models for these problems in the form of mixed variational inequalities and differential mixed variational inequalities, and exhibits stability results for differential variational inequalities with respect to perturbations of the data.

Asymptotically periodic solutions for fractional differential variational inequalities

“…For various motivations, differential variational inequalities (DVIs) in Euclidean spaces and general Banach spaces have been studied extensively recently; e.g. Pang and Stewart [28,31], Avgerinos and Papageorgiou [7], Gwinner [15,16], Liu Zhenhai et al [24,23,22,21], Anh [2,3], Ke et al [5,6,20] and references cited therein. Built on dynamical systems associated with variational inequalities, differential variational inequalities open up a broad paradigm for the enhanced modeling of complex real-world engineering systems, such as mechanical impact problems, electrical circuits with ideal diodes, Coulomb friction problems for contacting bodies, economical dynamics and hybrid engineering systems with variable structures.…”

On the use of elliptic regularity theory for the numerical solution of variational problems

“…Often in applications they have a clear physical meaning and are more of interest than the primal variables; speaking in the language of continuum mechanics, the engineer is often more interested in the stresses and strains than in the displacements. This motivates multifield variational formulations and multiple saddle point problem formulations, see [19,26,27]. While for linear elliptic boundary value problems the passage from the primal variational formulation to a dual mixed formulation or a saddle point problem form involving a Lagrange multiplier is a standard procedure and while there are the well-established mixed finite element methods [11,4] for their numerical treatment, such a procedure for non-smoothly constrained problems has to overcome several difficulties.…”