Extremum property of constraint reactions

The problem of finding constraint reactions in mechanical systems is discussed. It is shown that in a number of cases its solution reduces to finding the minimum of the generalized potential energy, whose expression depends on the active forces and the inertial forces and includes the strain energy. The condition of a minimum value provides a means, in particular, to single out “the most realistic” solution of statically indeterminate problems. In systems with Coulomb static friction, verification of the equilibrium conditions can be reduced to a convex minimization problem. Typical expressions for the generalized potential energy in systems with kinetic friction with one bilateral or unilateral constraint, which enable the dynamics of the system to be determined in cases of indeterminancy, are presented.     

A Decomposition Method for a Unilateral Contact Problem with Tresca Friction Arising in Electro-elastostatics

This article is concerned with the numerical modeling of unilateral contact problems in an electro-elastic material with Tresca friction law and electrical conductivity condition. First, we prove the existence and uniqueness of the weak solution of the model. Rather than deriving a solution method for the full coupled problem, we present and study a successive iterative (decomposition) method. The idea is to solve successively a displacement subproblem and an electric potential subproblem in block Gauss-Seidel fashion. The displacement subproblem leads to a constraint non-differentiable (convex) minimization problem for which we propose an augmented Lagrangian algorithm. The electric potential unknown is computed explicitly using the Riesz's representation theorem. The convergence of the iterative decomposition method is proved. Some numerical experiments are carried out to illustrate the performances of the proposed algorithm.     

A dynamic viscoelastic contact problem with normal compliance,finite penetration and nonmonotone slip rate dependent friction

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb’s law of dry friction. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modelled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solution of the regularized problem is obtained. Next, we prove the convergence of the weak solution of the regularized problem to the weak solution of the initial nonregularized problem. Then, we introduce a fully discrete approximation of the variational problem based on a finite element method and on a second order time integration scheme. The solution of the resulting nonsmooth and nonconvex frictional contact problems is presented, based on approximation by a sequence of nonsmooth convex programming problems. Finally, some numerical simulations are provided in order to illustrate both the behaviour of the solution related to the frictional contact conditions and the convergence result.     

Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction

The paper deals with shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems, where the coefficient of friction depends on the solution. We consider the two-dimensional Signorini problem, coupled with the physically less accurate model of given friction, but assume a solution-dependent coefficient of friction. First, we investigate the shape optimization problem in the continuous, infinite-dimensional setting, followed by a suitable finite-dimensional approximation based on the finite-element method. Convergence analysis is presented as well. Next, an algebraic form of the state problem is studied, which is obtained from the discretized problem by further approximating the frictional term by a quadrature rule. It is shown that if the coefficient of friction is Lipschitz continuous with a sufficiently small modulus, then the algebraic state problem is uniquely solvable and its solution is a Lipschitz continuous function of the control variable, describing the shape of the elastic body. For the purpose of numerical solution of the shape optimization problem via the so-called implicit programming approach we perform sensitivity analysis by using the tools from the generalized differential calculus of Mordukhovich. The paper is concluded first order optimality conditions.     

Existence of a solution for a quasistatic problem of unilateral contact with local friction

The existence result in linear elasticity obtained for the quasistatic problem of unilateral contact with regularized Coulomb friction is extented to a local friction problem. After discretizing the implicit variational inequality with respect to time, we have to solve a sequence of variational inequalities similar to the one of the static problem. If the friction coefficient is small enough, we show the existence of the incremental solution. We construct a suitable sequence of functions converging towards a quasistatic solution of the problem.     

An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.     

Rolling Contact Problem with the Generalized Coulomb Friction

This paper deals with the numerical solution of the wheel - rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element analysis of a static contact problem with Coulomb friction

A unilateral contact problem with a variable coefficient of friction is solved by a simplest variant of the finite element technique. The coefficient of friction may depend on the magnitude of the tangential displacement. The existence of an approximate solution and some a priori estimates are proved.     

A VARIATIONAL-HEMIVARIATIONAL INEQUALITY IN CONTACT PROBLEM FOR LOCKING MATERIALS AND NONMONOTONE SLIP DEPENDENT FRICTION

We study a new class of elliptic variational-hemivariational inequalities arising in the modelling of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modelled by the nonmonotone multivalued subdifferential condition which depends on the slip. The problem is governed by a nonlinear elasticity operator,the subdifferential of the indicator function of a convex set which describes the locking constraints and a nonconvex locally Lipschitz friction potential. The result on existence and uniqueness of solution to the inequality is shown. The proof is based on a surjectivity result for maximal monotone and pseudomonotone operators combined with the application of the Banach contraction principle.     

A newton-penalty method for a simplified liquid crystal model

In this paper we are concerned with the computation of a liquid crystal model defined by a simplified Oseen-Frank energy functional and a (sphere) nonlinear constraint. A particular case of this model defines the well known harmonic maps. We design a new iterative method for solving such a minimization problem with the nonlinear constraint. The main ideas are to linearize the nonlinear constraint by Newton’s method and to define a suitable penalty functional associated with the original minimization problem. It is shown that the solution sequence of the new minimization problems with the linear constraints converges to the desired solutions provided that the penalty parameters are chosen by a suitable rule. Numerical results confirm the efficiency of the new method.     

Heat transfer in sliding of a plane-parallel layer over a base

An analytic solution of the thermal problem of friction for a plane-parallel layer–base tribosystem under conditions of incomplete thermal contact between contacting bodies is obtained. Asymptotics of the obtained solution for small and large values of time are determined. For the materials of a cermet layer–iron base friction pair, we investigate the influence of the thermal conductivity coefficient of the contact on the temperature distribution and intensity of a heat fluxes.     

DD‐type Domain Decomposition Method for Frictional Contact Problem

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method. The global problem is transformed to a smaller problem on the contact surface. The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure.     

A Domain Decomposition Method for Frictional Contact Problem

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method. The global problem is transformed to a smaller problem on the contact surface. The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure.     

The Signorini problem with Coulomb friction for a hyperelastic body under finite deformations

The quasi-static three-dimensional problem of elasticity theory for a hyperelastic body under finite deformations, loading by bulk and surface forces, partial fastening and unilateral contact with a rigid punch and in the presence of time-dependent anisotropic Coulomb friction is considered. The equivalent variational formulation contains a quasi-variational inequality. After time discretization and application of the iteration method, the problem arising with “specified” friction is reduced to a non-convex miniumum functional problem, which is studied by Ball's scheme. The operator in contact stress space is determined. It is shown that a threshold level of the coefficient of friction corresponds to each level of loading, below which there is at least one fixed point of the operator. If the solution at a certain instant of time is known, the iteration process converges to the solution of the problem at the next, fairly close instant of time.     

Le problème de Signorini dans la théorie des plaques minces de Kirchhoff–LoveSignorini's problem in the Kirchhoff–Love theory of plates

In the framework of the Kirchhoff–Love asymptotic theory of elastic thin plates we consider the unilateral contact problem with friction for a plate on a rigid foundation (Signorini problem with friction). First, we notice, when the thickness vanishes, that the order of the friction force must be lower than that of the contact pressure. These two measures are connected by Coulomb law. Consequently, at least formally, the friction force must be vanishing when the thickness goes to zero. We actually prove that any sequence of solution of the sequence of three-dimensional scaled Signorini problems with friction strongly converges to the unique solution of a two-dimensional Signorini plate problem without friction. To cite this article: J.-C. Paumier, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 567–570.     

Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

In this paper, we prove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.     

A majorization–minimization algorithm for split feasibility problems

The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range constraints. Split feasibility generalizes important inverse problems including convex feasibility, linear complementarity, and regression with constraint sets. When a feasible point does not exist, solution methods that proceed by minimizing a proximity function can be used to obtain optimal approximate solutions to the problem. We present an extension of the proximity function approach that generalizes the linear split feasibility problem to allow for non-linear mappings. Our algorithm is based on the principle of majorization–minimization, is amenable to quasi-Newton acceleration, and comes complete with convergence guarantees under mild assumptions. Furthermore, we show that the Euclidean norm appearing in the proximity function of the non-linear split feasibility problem can be replaced by arbitrary Bregman divergences. We explore several examples illustrating the merits of non-linear formulations over the linear case, with a focus on optimization for intensity-modulated radiation therapy.     

Exact penalty property for a class of inequality-constrained minimization problems

We use the penalty approach in order to study constrained minimization problems. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish the exact penalty property for a large class of inequality-constrained minimization problems.     

Convergence of Rothe scheme for a class of dynamic variational inequalities involving Clarke subdifferential

In the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational–hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we apply obtained existence result to provide the weak solvability of contact problem.