Jaroslav Haslinger

Solution of variational inequalities in mechanics

Etude des problemes unilateraux pour les fonctions scalaires. Contact unilateral de corps elastiques. Problemes de la theorie de la plasticite

Shape Optimization and Fictitious Domain Approach for Solving Free Boundary Problems of Bernoulli Type

This contribution deals with an efficient method for the numerical realization of the exterior and interior Bernoulli free boundary problems. It is based on a shape optimization approach. The state problems are solved by a fictitious domain solver using boundary Lagrange multipliers.

Approximation of the Signorini problem with friction by a mixed finite element method

In many technical and physical situations one meets problems when one deformable body comes into contact with another. A mathematical formulation of these problems leads to the use of variational inequalities. A detailed analysis of contact problems between two elastic bodies has been given in [5], assuming that no friction occurs on the contact surface (for related results see also [9]). It is clear that the frictionless problem gives an approximation of real situation only, so that an involvement of the friction is desirable. A mathematical formulation of the contact problem between an elastic body and a perfectly rigid support (known as the Signorini problem), involving the friction governed by Coulomb’s law, has been introduced in [ 21. The existence of its solution for simple geometrical situations has been proven in [8] for the first time. From the existence proof an algorithm (method of successive approximations) for the numerical approximation of the problem follows. Unfortunately, the convergence of the algorithm is an open problem up to this time. The present paper deals with the approximation of one iterative step, which is defined by the Signorini problem with prescribed normal forces on the contact surface. A mixed formulation of this problem is derived, making use of the duality approach. This formulation allows us to approximate independently the displacement field in the body and the normal and 99 0022-247x/82/030099-24eo2.oq/0

On a splitting type algorithm for the numerical realization of contact problems with Coulomb friction

This paper presents and analyses an iterative process for the numerical realization of contact problems with Coulomb friction which is based on the method of successive approximations combined with a splitting type approach. Numerical examples illustrate the efficiency of this method.

Multidisciplinary Free Material Optimization

We present a mathematical framework for the so-called multidisciplinary free material optimization (MDFMO) problems, a branch of structural optimization in which the full material tensor is considered as a design variable. We extend the original problem statement by a class of generic constraints depending either on the design or on the state variables. Among the examples are local stress or displacement constraints. We show the existence of optimal solutions for this generalized free material optimization (FMO) problem and discuss convergent approximation schemes based on the finite element method.

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss-Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.

Shape Optimization in Contact Problems with Coulomb Friction

The paper deals with a discretized problem of the shape optimization of elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems following the Coulomb friction law. Mathematical modelling of the Coulomb friction problem leads to a quasi-variational inequality. It is shown that for small coefficients of friction the discretized problem with Coulomb friction has a unique solution and that this solution is Lipschitzian as a function of a control variable describing the shape of the elastic body. The shape optimization problem belongs to a class of so-called mathematical programs with equilibrium constraints (MPECs). The uniqueness of the equilibria for fixed controls enables us to apply the so-called implicit programming approach. Its main idea consists of minimizing a nonsmooth composite function generated by the objective and the (single-valued) control-state mapping. In this paper, the control-state mapping is much more complicated than in most MPECs solved in the literature so far, and the generalization of the relevant results is by no means straightforward. Numerical examples illustrate the efficiency and reliability of the suggested approach.

On numerical solution of hemivariational inequalities by nonsmooth optimization methods

In this paper we consider numerical solution of hemivariational inequalities (HVI) by using nonsmooth, nonconvex optimization methods. First we introduce a finite element approximation of (HVI) and show that it can be transformed to a problem of finding a substationary point of the corresponding potential function. Then we introduce a proximal budle method for nonsmooth nonconvex and constrained optimization. Numerical results of a nonmonotone contact problem obtained by the developed methods are also presented.

Optimal control of variational inequalities. Approximation theory and numerical realization

An optimal-control problem of a variational inequality of the elliptic type is investigated. The problem is approximated by a family of finite-dimensional problems and the convergence of the approximated optimal controls is shown. The finite-dimensional problems, being nonsmooth, are to be optimized by a bundle algorithm, which requires an element of Clarkes generalized gradient of the minimized function. A simple algorithm which yields this element is proposed. Some numerical experiments with a simple model problem have also been carried out.

Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach

SUMMARY The paper deals with a fast method for solving large scale algebraic saddle-point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside of the original domain. This approach has a significantly higher convergence rate, however the algebraic systems resulting from finite element discretizations are typically non-symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle-point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved by a projected Krylov subspace method for non-symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non-projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright c � 2007 John Wiley & Sons, Ltd.