Generalized penalty method for history-dependent variational–hemivariational inequalities
Introduction
A large number of processes which arise in Mechanics, Physics and Engineering Sciences are described by boundary value problems which, in a weak formulation, lead to mathematical models expressed in terms of variational or hemivariational inequalities. Variational inequalities refer to those inequality problems which have a convex structure. They have been studied extensively for over half a century since 1960s, both theoretically and numerically, by using arguments of convex analysis. Representative references in the field include [1], [2], [3] and, more recently, [4], [5], [6], [7], [8], [9], [10]. The notion of hemivariational inequality was introduced in [11] in the study of engineering problems involving non-smooth, non-monotone and possibly multivalued relations for deformable bodies. Since then, the theory of hemivariational inequalities grew rapidly. It uses as main ingredient the properties of the subdifferential in the sense of Clarke, defined for locally Lipschitz functions which may be nonconvex. Comprehensive references in the area include [12], [13], [14] and, more recently, [15], [16], [17], [18], [19], [20]. Finally, variational–hemivariational inequalities represent a special class of inequalities, driven by both convex and nonconvex functions. They represent a powerful tool in the study of a wide range of nonlinear boundary value problems with or without unilateral constraints, as shown in [21], [22], [23], [24], [25].
History-dependent operators represent a class of nonlinear operators defined on spaces of vector-valued continuous or Lebesgue integrable functions. They arise in Contact Mechanics and describe various memory effects which appear either in the material’s behaviour or in the contact conditions. Variational–hemivariational inequalities involving in their structure a history-dependent operator are called history-dependent variational–hemivariational inequalities. They have been intensively studied in the recent literature. General existence and uniqueness results can be found in [26], [27], together with various applications in Contact Mechanics. A convergence result which shows the continuous dependence of the solution with respect to the data was obtained in [28]. The numerical analysis of history-dependent variational–hemivariational inequalities can be found in [29], [30]. There, numerical schemes have been considered and error estimates have been derived. Additional results on inequality problems with history-dependent operators can be found in [26], [31], [32], [33].
The current paper represents a continuation of [34], [35]. Indeed, the paper [34] was devoted to the study of a penalty method for history-dependent variational–hemivariational inequalities. There, the corresponding unconstrained problems have been constructed with a given penalty operator and a convergence result was proved. In [35] we studied a generalized penalty method in the study of elliptic variational–hemivariational inequalities, i.e., time-independent variational–hemivariational inequalities. There, in contrast to [34], the unconstrained problems have been constructed with a sequence of penalty operators, denoted .
The aim of the current paper is twofold. The first one is to use the generalized penalty method in the study of history-dependent variational–hemivariational inequalities. Thus, our main convergence result, Theorem 16, extends our previous work in [34], [35], since these results can be obtained in the particular cases when for each and when the time is removed, respectively. Recall also that the proof of Theorem 16 is based on assumptions on the locally Lipschitz function which are less restrictive than that used in [34], [35]. This ingredient represents one of the traits of novelty of this contribution, as we mention in Remark 20. The second aim of the current paper is to illustrate the use of Theorem 16 in the study of a new mathematical model of contact and to provide the corresponding mechanical interpretations.
The outline of the paper is as follows. Basic notation and preliminary material needed in the rest of the paper are recalled in Section 2. In Section 3 we state the original inequality problem and the penalty problems, together with their unique solvability. Then, in Section 4 we state and prove our main result, Theorem 16, which states that the sequence of solutions of the generalized penalty problems converges to the solution of the original problem. The proof is carried out in several steps, based on arguments of compactness, pseudomonotonicity and the properties of the Clarke subdifferential. Finally, in Section 5 we illustrate the use of this abstract convergence result in the study of a nonlinear boundary value problem which describes the quasistatic evolution of a rod–spring system with unilateral constraints.
Section snippets
Background material
In this section we shortly recall some notation and preliminaries which are needed in the rest of the paper. For more details on the material presented below we refer the reader to [10], [12], [36], [37], [38].
Everywhere in this paper represents a reflexive Banach space with dual and denotes the duality between and . We use the notations and for the norm on the spaces and , respectively, and for the zero element of . Throughout the paper all the limits, upper
Problem statement and well-posedness results
We now turn to the inequality problem we consider in this paper. The functional framework is the following. Let be a subset of and let be a normed space. Given two operators and , a function , a locally Lipschitz function and a function , we consider the following history-dependent variational–hemivariational inequality.
Problem 11 Find such that, for all ,
A convergence result
In the section we move to the convergence of solution to generalized penalty problem. To this end, besides the assumptions introduced in the previous section, we consider the following assumptions.
: as .
: For each , there exists a sequence such that
: There exists an operator such that:
- (a)
for any sequence satisfying in and we have
- (b)
implies
A spring-rod system with unilateral constraints
The abstract results presented in Sections 3 Problem statement and well-posedness results, 4 A convergence result in this paper are useful in the study of various mathematical models which describe the equilibrium of viscoelastic or viscoplastic bodies in unilateral contact with a foundation. In this section we present a one-dimensional example which illustrates the applicability of these results.
Consider a viscoelastic rod which occupies the interval on the axis with . The rod is
Acknowledgements
This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 — CONMECH. It is supported by the National Science Centre of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, National Science Centre of Poland under Preludium Project No. 2017/25/N/ST1/00611, and International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No.
References (41)
- et al.
Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces
J. Differential Equations
(2017) Existence results for quasilinear parabolic hemivariational inequalities
J. Differential Equations
(2008)- et al.
Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model
Nonlinear Anal.: RWA
(2018) - et al.
Optimization problems for a viscoelastic frictional contact problem with unilateral constraints
Nonlinear Anal.: RWA
(2019) - et al.
On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity
Appl. Math. Lett.
(2019) - et al.
On the optimal control of variational–hemivariational inequalities
J. Math. Appl. Anal.
(2019) - et al.
Analysis of a general dynamic history-dependent variational–hemivariational inequality
Nonlinear Anal. RWA
(2017) - et al.
History-dependent variational-hemivariational inequalities in contact mechanics
Nonlinear Anal.: RWA
(2015) - et al.
A penalty method for history-dependent variational-hemivariational inequalities
Comput. Math. Appl.
(2018) - et al.
Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems
(1984)
Numerical Analysis of Variational Inequalities
Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods
Golden ratio algorithms with new stepsize rules for variational inequalities
Math. Methods Appl. Sci.
Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities
J. Optim. Theory Appl.
Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of -Laplacian type
Inverse Problems
Tykhonov well-posedness of a viscoplastic contact problem
Evol. Equ. Control Theory
Optimal control for a class of mixed variational problems
Z. Angew. Math. Phys.
Nonconvex energy functions, hemivariational inequalities and substationary principles
Acta Mech.
Optimization and Nonsmooth Analysis
Cited by (7)
Penalty method for solving a class of stochastic differential variational inequalities with an application
2023, Nonlinear Analysis: Real World ApplicationsTwo adaptive modified subgradient extragradient methods for bilevel pseudomonotone variational inequalities with applications
2022, Communications in Nonlinear Science and Numerical SimulationCoupled variational–hemivariational inequalities with constraints and history-dependent operators
2024, Mathematical Methods in the Applied SciencesOPTIMAL CONTROL OF A QUASISTATIC FRICTIONAL CONTACT PROBLEM WITH HISTORY-DEPENDENT OPERATORS
2023, International Journal of Numerical Analysis and Modeling