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.

scholarly journals Tykhonov triples and convergence results for history-dependent variational inequalities

2020 ◽  
Vol 34 ◽  
pp. 01006
Author(s):  
Mircea Sofonea
Keyword(s):  
Mathematical Model ◽  
Variational Inequality ◽  
Continuous Dependence ◽  
Viscoelastic Body ◽  
Time Dependent ◽  
Well Posedness ◽  
Rigid Foundation ◽  
Frictionless Contact ◽  
Convergence Results ◽  
Engineering Sciences

We deal with the Tykhonov well-posedness of a time-dependent variational inequality defined on the unbounded interval of time ℝ+ = [0, +∞ ), governed by a history-dependent operator. To this end we introduce the concept of Tykhonov triple, provide three relevant examples, then we state and prove the corresponding well-posedness results. This allows us to deduce various corollaries which illustrate the continuous dependence of the solution with respect to the data. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in Mechanics, Physics and Engineering Sciences. To give an example, we consider a mathematical model which describes the equilibrium of a viscoelastic body in frictionless contact with a rigid foundation.

scholarly journals Quasistatic Elastic Contact with Adhesion

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Boudjemaa Teniou ◽  
Sabrina Benferdi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Elastic Contact ◽  
Rigid Foundation ◽  
Mechanical Problem ◽  
Quasistatic Process ◽  
Variational Study ◽  
Nonlinear Elastic

The aim of this paper is the variational study of the contact with adhesion between an elastic material and a rigid foundation in the quasistatic process where the deformations are supposed to be small. The behavior of this material is modelled by a nonlinear elastic law and the contact is modelled with Signorini's conditions and adhesion. The evolution of bonding field is described by a nonlinear differential equation. We derive a variational formulation of the mechanical problem, and we prove the existence and uniqueness of the weak solution using a theorem on variational inequalities, the theorem of Cauchy-Lipschitz, a lemma of Gronwall, as well as the fixed point of Banach.

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.

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.

Verification of Iterative Methods for the Linear Complementarity Problem

2016 ◽  
pp. 545-580 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Linear Complementarity Problem ◽  
Numerical Experiments ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Convergence Results ◽  
Fixed Point Principle ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

In the present chapter, we give an overview of iterative methods for linear complementarity problems (abbreviated as LCPs). We also introduce these iterative methods for the problems based on fixed-point principle. Next, we present some new properties of preconditioned iterative methods for solving the LCPs. Convergence results of the sequence generated by these methods and also the comparison analysis between classic Gauss-Seidel method and preconditioned Gauss-Seidel (PGS) method for LCPs are established under certain conditions. Finally, the efficiency of these methods is demonstrated by numerical experiments. These results show that the mentioned models are effective in actual implementation and competitive with each other.

Tykhonov well-posedness of a frictionless unilateral contact problem

2019 ◽  
Vol 25 (6) ◽  
pp. 1294-1311 ◽  
Author(s):  
Zhenhai Liu ◽  
Mircea Sofonea ◽  
Yi-bin Xiao
Keyword(s):  
Contact Problem ◽  
Variational Formulation ◽  
Unilateral Contact ◽  
Well Posedness ◽  
Unilateral Constraints ◽  
The Family ◽  
Contact Models ◽  
Mechanical Interpretation ◽  
Well Posed ◽  
Natural Way

We consider a frictionless contact problem, Problem [Formula: see text], for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem [Formula: see text]. Then we consider a perturbation of Problem [Formula: see text], which could be frictional, governed by a small parameter [Formula: see text]. This perturbation leads in a natural way to a family of sets [Formula: see text]. We prove that Problem [Formula: see text] is well-posed in the sense of Tykhonov with respect to the family [Formula: see text]. The proof is based on arguments of monotonicity, pseudomonotonicity and various estimates. We extend these results to a time-dependent version of Problem [Formula: see text]. Finally, we provide examples and mechanical interpretation of our well-posedness results, which, in particular, allow us to establish the link between the weak solutions of different contact models.

scholarly journals Linear program fixed-point representation

MATEMATIKA ◽  
2017 ◽  
Vol 33 (1) ◽  
pp. 55
Author(s):  
Jalaluddin Morris Abdullah
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Symmetric Matrix ◽  
Fixed Point Problem ◽  
Linear Program ◽  
Point Problem ◽  
Central Elements ◽  
Point Representation ◽  
Primal And Dual Problems ◽  
Primal And Dual

From a linear program and its asymmetric dual, invariant primal and dual problems are constructed. Regular mappings are defined between the solution spaces of the original and invariant problems. The notion of centrality is introduced and subsets of regular mappings are shown to be inversely related surjections of central elements, thus representing the original problems as invariant problems. A fixed-point problem involving an idempotent symmetric matrix is constructed from the invariant problems and the notion of centrality carried over to it; the non-negative central fixed-points are shown to map one-to-one to the central solutions to the invariant problems, thus representing the invariant problems as a fixed-point problem and, by transitivity, the original problems as a fixed-point problem.

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