Penalization of history-dependent variational inequalities

2013 ◽  
Vol 25 (2) ◽  
pp. 155-176 ◽  
Author(s):  
M. SOFONEA ◽  
F. PĂTRULESCU
Keyword(s):  
Variational Inequalities ◽  
Viscoelastic Body ◽  
Regularity Result ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Quasivariational Inequalities ◽  
Penalization Method ◽  
New Class ◽  
Mechanical Interpretation ◽  
Appl Math

The present paper represents a continuation of Sofonea and Matei's paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material's behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

Numerical analysis of history-dependent variational–hemivariational inequalities with applications to contact problems

2015 ◽  
Vol 26 (4) ◽  
pp. 427-452 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
WEIMIN HAN ◽  
STANISŁAW MIGÓRSKI
Keyword(s):  
Numerical Analysis ◽  
Real World ◽  
Viscoelastic Body ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Hemivariational Inequalities ◽  
New Class ◽  
Nonlinear Anal ◽  
Dependence Result

A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law.

History-dependent quasi-variational inequalities arising in contact mechanics

2011 ◽  
Vol 22 (5) ◽  
pp. 471-491 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
ANDALUZIA MATEI
Keyword(s):  
Variational Inequalities ◽  
Frictional Contact ◽  
Deformable Body ◽  
Regularity Result ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Signorini Condition ◽  
Normal Damped Response ◽  
Weak Solvability

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

scholarly journals A HISTORY-DEPENDENT FRICTIONAL CONTACT PROBLEM WITH WEAR FOR THERMOVISCOELASTIC MATERIALS

2019 ◽  
Vol 24 (3) ◽  
pp. 351-371
Author(s):  
Lamia Chouchane ◽  
Lynda Selmani
Keyword(s):  
Contact Problem ◽  
Thermal Effects ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Viscoelastic Body ◽  
Perturbation Parameter ◽  
Convergence Result ◽  
Uniqueness Result ◽  
Quasivariational Inequalities ◽  
Wear And Friction

In this manuscript we study a contact problem between a deformable viscoelastic body and a rigid foundation. Thermal effects, wear and friction between surfaces are taken into account. A variational formulation of the problem is supplied and an existence and uniqueness result is proved. The idea of the proof rested on a recent result on history-dependent quasivariational inequalities. Finally, a perturbation of the data is initiated and a convergence result is demonstrated when the perturbation parameter converges to zero.

scholarly journals A VISCOELASTIC CONTACT PROBLEM WITH ADHESION AND SURFACE MEMORY EFFECTS

2014 ◽  
Vol 19 (5) ◽  
pp. 607-626 ◽  
Author(s):  
Mircea Sofonea ◽  
Flavius Patrulescu
Keyword(s):  
Long Memory ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Viscoelastic Body ◽  
Uniqueness Result ◽  
Unilateral Constraint ◽  
Memory Effects ◽  
Constitutive Law ◽  
Classical Formulation ◽  
Viscoelastic Contact

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The material's behavior is modelled with a constitutive law with long memory. The contact is with normal compliance, unilateral constraint, memory effects and adhesion. We present the classical formulation of the problem, then we derive its variational formulation and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities and fixed point.

Convergence results for primal and dual history-dependent quasivariational inequalities

2018 ◽  
Vol 149 (2) ◽  
pp. 471-494
Author(s):  
Mircea Sofonea ◽  
Ahlem Benraouda
Keyword(s):  
Fixed Point ◽  
Variational Formulation ◽  
Continuous Dependence ◽  
Viscoelastic Body ◽  
Rigid Foundation ◽  
Quasivariational Inequalities ◽  
Class A ◽  
Convergence Results ◽  
Mechanical Interpretation ◽  
Primal And Dual

AbstractWe consider a class of history-dependent quasivariational inequalities for which we prove the continuous dependence of the solution with respect to the set of constraints. Then, under additional assumptions, we associate with each inequality in the class a new inequality, the so-called dual variational inequality, for which we state and prove existence, uniqueness, equivalence and convergence results. The proofs are based on various estimates, monotonicity and fixed-point arguments for history-dependent operators. Our abstract results are useful in the study of various mathematical models of contact. To provide an example, we consider a boundary value problem which describes the equilibrium of a viscoelastic body in contact with an elastic-rigid foundation. We list the assumptions on the data and derive both the primal and the dual variational formulation of the problem. Then, we state and prove existence, uniqueness and convergence results. We also provide the link between the two formulations, together with their mechanical interpretation.

A regularization method for a viscoelastic contact problem

2016 ◽  
Vol 23 (2) ◽  
pp. 181-194
Author(s):  
Flavius Pătrulescu
Keyword(s):  
Mathematical Model ◽  
Contact Problem ◽  
Variational Inequalities ◽  
Long Memory ◽  
Regularization Method ◽  
Viscoelastic Body ◽  
Constitutive Law ◽  
Normal Compliance ◽  
Viscoelastic Contact

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a deformable obstacle, the so-called foundation. The material’s behaviour is modelled with a viscoelastic constitutive law with long memory. The contact is frictionless and is defined using a multivalued normal compliance condition. We present a regularization method in the study of a class of variational inequalities involving history-dependent operators. Finally, we apply the abstract results to analyse the contact problem.

scholarly journals On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 266 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Optimal Control Theory ◽  
Infinite Dimensional ◽  
Nash Games ◽  
New Class ◽  
Set Valued Mappings ◽  
Differential Variational Inequalities ◽  
Theoretical Developments

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

scholarly journals Mixed quasi invex equilibrium problems

2004 ◽  
Vol 2004 (57) ◽  
pp. 3057-3067 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Strong Monotonicity ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases ◽  
Weaker Conditions

We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.

AN ANALOGUE OF EULER’S IDENTITY AND SPLIT PERFECT PARTITIONS

2018 ◽  
Vol 99 (03) ◽  
pp. 353-361
Author(s):  
MEGHA GOYAL
Keyword(s):  
Generating Function ◽  
New Class ◽  
Appl Math ◽  
Euler’S Identity

We give the generating function of split$(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split$n$-colour partitions. We derive a combinatorial relation between the number of restricted split$n$-colour partitions and the function$\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with$d(a)$copies of each part$a$and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’,Indian J. Pure Appl. Math 22(9) (1991), 737–743].

scholarly journals Quasi cl-supercontinuous functions and their function spaces

2012 ◽  
Vol 45 (3) ◽  
Author(s):  
J. K. Kohli ◽  
Jeetendra Aggarwal
Keyword(s):  
Continuous Functions ◽  
Function Spaces ◽  
Strong Form ◽  
General Topology ◽  
Regular Space ◽  
Basic Properties ◽  
New Class ◽  
Mathematical Literature ◽  
Class Of Functions ◽  
Appl Math

AbstractA new class of functions called ‘quasi cl-supercontinuous functions’ is introduced. Basic properties of quasi cl-supercontinuous functions are studied and their place in the hierarchy of variants of continuity that already exist in the mathematical literature is elaborated. The notion of quasi cl-supercontinuity, in general, is independent of continuity but coincides with cl-supercontinuity (≡ clopen continuity) (Applied General Topology 8(2) (2007), 293–300; Indian J. Pure Appl. Math. 14(6) (1983), 767–772), a significantly strong form of continuity, if range is a regular space. The class of quasi cl-supercontinuous functions properly contains each of the classes of (i) quasi perfectly continuous functions and (ii) almost cl-supercontinuous functions; and is strictly contained in the class of quasi

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