Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems

In this paper, we use parametric quintic splines to derive some consistency relations which are then used to develop a numerical method for computing the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Numerical evidence is presented to show the applicability and superiority of the new method over other collocation, finite difference, and spline methods.     

Numerical solutions for system of second order boundary value problems

We investigate some numerical methods for computing approximate solutions of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that cubic spline method gives approximations which are better than that computed by higher order spline and finite difference techniques.     

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.     

A discretization theory for a class of semi-coercive unilateral problems

Summary. In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a Signorini-Fichera unilateral contact model and some obstacle problem with frictions are studied. The theoretical conditions proved are in good agreement with the numerical ones. Received January 14, 1999 / Revised version received June 24, 1999 / Published online July 12, 2000     

On the Parametric Cubic Spline Approach for Solving Second Order Boundary Value Problems

In this article some comments on the paper “parametric cubic spline approach to the solution of a system of second order boundary value problems” in (Khan and Aziz, J. Optim. Theory Appl. 118:45–54, 2003) are given. This paper concerns with a numerical method for solving a second order boundary value problem associated with obstacle, unilateral and contact problems. Corrections are given for the convergence analysis of the numerical method and the computational experiments.     

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.     

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.     

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 INITIAL DIRICHLET BOUNDARY VALUE PROBLEM FOR GENERAL SECOND ORDER PARABOLIC SYSTEMS IN NONSMOOTH MANIFOLDS

Abstract

We consider some unilateral boundary value problems in polygonal and polyhedral domains with unilateral transmission conditions. Regularity results in terms of weighted Sobolev spaces are obtained using a penalization technique, similar regularity results for the penalized problems and by showing uniform estimates with respect to the penalization parameter.     

Two-sided approximations for unilateral variational inequalities by multi-grid methods

For the numerical solution of unilateral variational inequalities two iterative schemes are developed which provide approximations from below resp. from above. Both schemes are based on some kind of active set strategy and require the solution of an algebraic system of equations at each iteration step which is done by means of multigrid techniques. Convergence results are established and illustrated by some numerical results for the elastic-plastic torsion problem     

三次多项式样条解一类四阶边值问题

应用三次多项式样条函数解一组四阶单侧、障碍、接触边值问题,取半结点为网格点,并增加了边界方程,应用方法解文献中的数值例子,说明了方法的高效性,数值结果也显示了方法的优越性.     

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.

     

Augmented Lagrangian Active Set Methods for Obstacle Problems

Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.     

Optimal Control of a Variational Inequality with Application to Equilibrium Problem of an Elastic Nonhomogeneous and Anisotropic Plate Resting on Unilateral Elastic Foundation

Several optimal control problems with the same state problem—a variational inequality with a monotone operator—are considered. The inequality represents bending of an elastic, nonhomogeneous, anisotropic Kirchhoff plate resting on some unilateral elasto-rigid foundation and point supports. Both the thickness of the plate and the coefficient of the unilateral elastic foundation play the role of design variables. Cost functionals include the work of external forces (compliance), total reaction forces of the foundation or the weight of the plate. The solvability of all the problems is proved. Moreover, approximate methods for the optimal control and weight minimization problems are proposed, making use of finite elements. The design variables are approximated by piecewise affine functions. The solvability of the approximate problems is proved and some convergence analysis is presented.     

Some Remarks on a Vibro-Impact Scheme

We consider a mechanical system subject to a unilateral constraint for which an impact law is defined. The impact process is considered to be instantaneous and the behavior at the impact is described through a restitution coefficient e. We solve this problem using an algorithm developed and used by Paoli and Schatzman. We show that this algorithm develops numerical instabilities related to the restitution coefficient used in the impact law. Finally, we propose a new scheme which introduces some numerical damping in order to avoid the numerical instabilities.     

Free material optimization via mathematical programming

This paper deals with a central question of structural optimization which is formulated as the problem of finding the stiffest structure which can be made when both the distribution of material as well as the material itself can be freely varied. We consider a general multi-load formulation and include the possibility of unilateral contact. The emphasis of the presentation is on numerical procedures for this type of problem, and we show that the problems after discretization can be rewritten as mathematical programming problems of special form. We propose iterative optimization algorithms based on penalty-barrier methods and interior-point methods and show a broad range of numerical examples that demonstrates the efficiency of our approach. Supported by the project 03ZO7BAY of BMBF (Germany) and the GIF-contract 10455-214.06/95.     

Finite element approximation to a contact problem in linear thermoelasticity

A finite element approximation to the solution of a one-dimensional linear thermoelastic problem with unilateral contact of the Signorini type and heat flux is proposed. An error bound is derived and some numerical experiments are performed.

     

On some class of nonlinear parabolic inequalities in Orlicz spaces

The paper deals with the existence of solutions of some unilateral problems in the framework of Orlicz spaces.     

Verification for existence of solutions for some generalized obstacle problems

We describe some numerical methods to automatically prove the existence of solutions for some generalized obstacle problems. In this paper, our goal is to establish a new procedure for numerical verification of generalized obstacle problems.     

Formulation and analysis of two energy-consistent methods for nonlinear elastodynamic frictional contact problems

Energy-conserving algorithms are necessary to solve nonlinear elastodynamic problems in order to recover long term time integration accuracy and stability. Furthermore, some physical phenomena (such as friction) can generate dissipation; then in this work, we present and analyse two energy-consistent algorithms for hyperelastodynamic frictional contact problems which are characterised by a conserving behaviour for frictionless impacts but also by an admissible frictional dissipation phenomenon. The first approach permits one to enforce, respectively, the Kuhn–Tucker and persistency conditions during each time step by combining an adapted continuation of the Newton method and a Lagrangean formulation. In addition the second method which is based on the work in [P. Hauret, P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916] represents a specific penalisation of the unilateral contact conditions. Some numerical simulations are presented to underscore the conservative or dissipative behaviour of the proposed methods.