解可分离结构变分不等式的一种新的交替方向法

交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性.     

A class of implicit variational inequalities and applications to frictional contact

This paper deals with the mathematical and numerical analysis of a class of abstract implicit evolution variational inequalities. The results obtained here can be applied to a large variety of quasistatic contact problems in linear elasticity, including unilateral contact or normal compliance conditions with friction. In particular, a quasistatic unilateral contact problem with nonlocal friction is considered. An algorithm is derived and some numerical examples are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of a time discretization for an implicit variational inequality modelling dynamic contact problems with friction

Some dynamic contact problems with friction can be formulated as an implicit variational inequality. A time discretization of such an inequality is given here, thus giving rise to a so‐called incremental solution. The convergence of the incremental solution is established, and then the limit is shown to be the unique solution of the variational inequality. This paper contains therefore not only some new results concerning the numerical aspect of some models of contact and friction but also a constructive existence result. Copyright © 2001 John Wiley & Sons, Ltd.     

Fractional variational integrators for fractional variational problems

In this paper, the fractional variational integrators for a class of fractional variational problems are developed. The fractional discrete Euler-Lagrange equation is obtained. Based on the Grünwald-Letnikov method and Diethelm’s fractional backward differences, some fractional variational integrators are presented and the fractional variational errors are discussed. Some numerical examples are presented to illustrate these results.     

Numerical solution of a system of fourth order boundary value problems using variational iteration method

In this paper, the variational iteration method is used to solve a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. Numerical solution obtained by the method is of high accuracy. Moreover, the higher-order derivatives of numerical solution can also approximate the higher-order derivatives of exact solution well. Five examples compared with those considered by Siddiqi and Akram [S.S. Siddiqi, G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Applied Mathematics and Computation 190 (2007) 652–661] show that the method is more efficient.     

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.     

Parametric Cubic Spline Approach to the Solution of a System of Second-Order Boundary-Value Problems

We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.     

The solution of three-dimensional friction contact problems

A variational method is developed for solving friction contact problems, in which the friction obeys Coulomb's of friction law in velocities, and numerical solutions of three-dimensional problems of the contact of a sphere, a cylinder of finite length and a cube with an elastic half-space are constructed. It is established that the maximum frictional forces correspond to a boundary point of the regions of adhesion and slippage. When the number of steps,increase this maximum decreases, and the distribution of the frictional forces becomes smoother. Certain undesirable effects that can arise during numerical implementation of the method – numerical artefacts – are described. These effects can occur in the numerical solution of problems with a different physical content, the mathematical structure of which is similar to the structure of the contact problems investigated, as the artefacts are caused by the presence of unilateral constraints and by the dependence on external effects of the region in which unilateral constraints with an equally sign occur. This problem is solved by an appropriate choice of the load-step zero approximations.     

A comparison of solvers for linear complementarity problems arising from large-scale masonry structures

We compare the numerical performance of several methods for solving the discrete contact problem arising from the finite element discretisation of elastic systems with numerous contact points. The problem is formulated as a variational inequality and discretised using piecewise quadratic finite elements on a triangulation of the domain. At the discrete level, the variational inequality is reformulated as a classical linear complementarity system. We compare several state-of-art algorithms that have been advocated for such problems. Computational tests illustrate the use of these methods for a large collection of elastic bodies, such as a simplified bidimensional wall made of bricks or stone blocks, deformed under volume and surface forces. This work was supported by the Engineering and Physical Science Research Council of Great Britain under grant GR/S35101, and the first author was supported by a fellowship from the Royal Society of Edinburgh.     

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.     

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.     

A dynamic unilateral contact problem with adhesion and friction in viscoelasticity

The aim of this paper is to study an interaction law coupling recoverable adhesion, friction and unilateral contact between two viscoelastic bodies of Kelvin–Voigt type. A dynamic contact problem with adhesion and nonlocal friction is considered and its variational formulation is written as the coupling between an implicit variational inequality and a parabolic variational inequality describing the evolution of the intensity of adhesion. The existence and approximation of variational solutions are analysed, based on a penalty method, some abstract results and compactness properties. Finally, some numerical examples are presented.     

Numerical analysis of a quasi-static contact problem for a thermoviscoelastic beam

In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in [1]. The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.     

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

     

Sextic spline method for the solution of a system of obstacle problems

We use sextic spline function to develop numerical method for the solution of system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the approximate solutions obtained by the present method are better than those produced by other collocation, finite difference and spline methods. A numerical example is given to illustrate practical usefulness of our method.     

Some projection methods with the BB step sizes for variational inequalities

Since the appearance of the Barzilai-Borwein (BB) step sizes strategy for unconstrained optimization problems, it received more and more attention of the researchers. It was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints. In this paper, we further extend the BB step sizes to more general variational inequality (VI) problems, i.e., we adopt them in projection methods. Under the condition that the underlying mapping of the VI problem is strongly monotone and Lipschitz continuous and the modulus of strong monotonicity and the Lipschitz constant satisfy some further conditions, we establish the global convergence of the projection methods with BB step sizes. A series of numerical examples are presented, which demonstrate that the proposed methods are convergent under mild conditions, and are more efficient than some classical projection-like methods.     

A Modified Alternating Direction Method for Variational Inequality Problems

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method. Accepted 4 May 2001. Online publication 19 October, 2001.     

Goldstein, Levitin-Polyak投影法中的一个可行步长准则

１引　言设Ω是Ｒｎ空间的一个非空的凸闭紧子集,Ｆ是Ｒｎ→Ｒｎ的算子．我们考虑变分不等式问题： 变分不等式问题在数学规划中起着很重要的作用,因此,长期以来一直受到广泛的重视．求解变分不等式问题的方法中,有一类投影迭代方法,例如［１,４,６,９］．在所有的投影迭代方法中,Ｇｏｌｄｓｔｅｉｎ［６］,Ｌｅｖｉｔｉｎ－Ｐｏｌｙａｋ［９］所提出的方法;是最简单的．这里,ＰΩ（ｘ）是ｘ在　上的投影,即　的唯一解． 我们称算子Ｆ在集合Ω上是单调的,若在用Ｇｏｌｄｓｔｅｉｎ,Ｌｅｖｉｔｉｎ－Ｐｏｌｙａｋ方法（２）求…     

From solvability and approximation of variational inequalities to solution of nondifferentiable optimization problems in contact mechanics

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law.     

Direct Numerical Algorithm for Constrained Variational Problems

We propose a direct treatment for the numerical simulation of optimal solutions for vector, one-dimensional variational problems under pointwise constraints in the form of several inequalities. It is an iterative procedure to approximate the optimal solutions of such variational problems that rely on our ability to e?ciently approximate the optimal solutions of variational problems without restrictions, except possibly for end point constraints. One main advantage is that there is no need to control the free boundary, or the contact set, during the iterative process where constraints are active. In addition to proving some convergence results, the scheme is illustrated through several typical situations.