Radek Kučera

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.

Convergence Rate of an Optimization Algorithm for Minimizing Quadratic Functions with Separable Convex Constraints

A new active set algorithm for minimizing quadratic functions with separable convex constraints is proposed by combining the conjugate gradient method with the projected gradient. It generalizes recently developed algorithms of quadratic programming constrained by simple bounds. A linear convergence rate in terms of the Hessian spectral condition number is proven. Numerical experiments, including the frictional three-dimensional (3D) contact problems of linear elasticity, illustrate the computational performance.

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.

An Optimal Algorithm for Minimization of Quadratic Functions with Bounded Spectrum Subject to Separable Convex Inequality and Linear Equality Constraints

An, in a sense, optimal algorithm for minimization of quadratic functions subject to separable convex inequality and linear equality constraints is presented. Its unique feature is an error bound in terms of bounds on the spectrum of the Hessian of the cost function. If applied to a class of problems with the spectrum of the Hessians in a given positive interval, the algorithm can find approximate solutions in a uniformly bounded number of simple iterations, such as matrix-vector multiplications. Moreover, if the class of problems admits a sparse representation of the Hessian, it simply follows that the cost of the solution is proportional to the number of unknowns. Theoretical results are illustrated by numerical experiments.

Minimizing quadratic functions with separable quadratic constraints

This article deals with minimizing quadratic functions with a special form of quadratic constraints that arise in 3D contact problems of linear elasticity with isotropic friction [Haslinger, J., Kučera, R. and Dostál, Z., 2004, An algorithm for the numerical realization of 3D contact problems with Coulomb friction. Journal of Computational and Applied Mathematics, 164/165, 387–408.]. The proposed algorithm combines the active set method with the conjugate gradient method. Its general scheme is similar to the algorithms of Polyak’s type that solve the quadratic programming problems with simple bounds. As the algorithm does not terminate in a finite number of steps, the convergence is proved. The implementation uses an adaptive precision control of the conjugate gradient loops. Numerical experiments demonstrate the computational efficiency of the method.

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.

An interior-point algorithm for the minimization arising from 3D contact problems with friction

The paper deals with a variant of the interior-point method for the minimization of strictly quadratic objective function subject to simple bounds and separable quadratic inequality constraints. Such minimizations arise from the finite element approximation of contact problems of linear elasticity with friction in three space dimensions. The main goal of the paper is the convergence analysis of the algorithm and its implementation. The optimal preconditioners for solving ill-conditioned inner linear systems are proposed. Numerical experiments illustrate the computational efficiency for large-scale problems.

Shape Optimization in Three-Dimensional Contact Problems with Coulomb Friction

We study the discretized problem of the shape optimization of three-dimensional (3D) elastic bodies in unilateral contact. The aim is to extend existing results to the case of contact problems obeying the Coulomb friction law. Mathematical modeling of the Coulomb friction problem leads to an implicit 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 2D case of this problem was studied by the authors in [P. Beremlijski, J. Haslinger, M. Kocvara, and J. V. Outrata, SIAM J. Optim., 13 (2002), pp. 561-587]; there we used the so-called implicit programming approach combined with the generalized differential calculus of Clarke. The extension of this technique to the 3D situation is by no means straightforward. The main source of difficulties is the nonpolyhedral character of the second-order (Lorentz) cone, arising in the 3D model. To facilitate the computation of the subgradient information, needed in the used numerical method, we exploit the substantially richer generalized differential calculus of Mordukhovich. Numerical examples illustrate the efficiency and reliability of the suggested approach.

On the Moore–Penrose inverse in solving saddle-point systems with singular diagonal blocks‡

SUMMARY This paper deals with the role of the generalized inverses in solving saddle-point systems arising naturally in the solution of many scientific and engineering problems when finite-element tearing and interconnecting based domain decomposition methods are used to the numerical solution. It was shown that the Moore–Penrose inverse may be obtained in this case at negligible cost by projecting an arbitrary generalized inverse using orthogonal projectors. Applying an eigenvalue analysis based on the Moore–Penrose inverse, we proved that for simple model problems, the number of conjugate gradient iterations required for the solution of associate dual systems does not depend on discretization norms. The theoretical results were confirmed by numerical experiments with linear elasticity problems. Copyright

A New FETI-based Algorithm for Solving 3D Contact Problems with Coulomb Friction

The paper deals with solving of contact problems with Coulomb friction for a system of 3D elastic bodies. The iterative method of successive approximations is used in order to find a fixed point of certain mapping that defines the solution. In each iterative step, an auxiliary problem with given friction is solved that is discretized by the FETI method. Then the duality theory of convex optimization is used in order to obtain the constrained quadratic programming problem that, in contrast to 2D case, is subject to quadratic inequality constraints. The solution is computed (among others) by a novelly developed algorithm of constrained quadratic programming. Numerical experiments demonstrate the performance of the whole computational process.