Solution Of Frictionless Contact Problems Research Articles

On the finite element solution of frictionless contact problems using an exact penalty approach

In this article, an exact penalty method is introduced for the enforcement of the impenetrability constraint in problems of frictionless two-body contact solved by the finite element method. A complete algorithmic implementation is presented, including an automated scheme for the selection of the penalty parameter. Numerical examples are employed to assess the accuracy and robustness of the exact penalty method in comparison to classical penalty and Lagrange multiplier alternatives.

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.

A numerically scalable domain decomposition method for the solution of frictionless contact problems

AbstractWe present a domain decomposition method with Lagrange multipliers for solving iteratively frictionless contact problems. This method, which is based on the FETI method and therefore is named here the FETI‐C method, incorporates a coarse contact system that guides the iterative prediction of the active zone of contact. We demonstrate numerically that this method is numerically scalable with respect to both the problem size and the number of subdomains. Copyright © 2001 John Wiley & Sons, Ltd.

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.

A Lagrange multiplier method for the finite element solution of frictionless contact problems

This article proposes a novel Lagrange multiplier-based formulation for the finite element solution of the quasi-static two-body contact problem in the presence of finite motions and deformations. The main idea rests in the interpretation of the two-body contact as a composition of two simultaneous Signorini-like problems, which naturally yield geometrically unbiased approximations of the kinematics and kinetics of frictionless contact. A two-dimensional finite element is introduced that exactly satisfies the impenetrability constraint and allows for the direct computation of consistent pressure distributions on the interacting surfaces.

Frictionless Contact of Layered Half-Planes, Part II: Numerical Results

In Part I of this paper, analytical development of a method was presented for the solution of frictionless contact problems of multilayered half-planes consisting of an arbitrary number of isotropic, orthotropic, or monoclinic layers arranged in any sequence. The local/global stiffness matrix approach similar to the one proposed by Bufler (1971) was employed in formulating the surface mixed boundary condition for the unknown stress in the contact region. This approach naturally facilitates decomposition of the integral equation for the contact stress distribution on the top surface of an arbitrarily laminated half-plane into singular and regular parts that, in turn, can be solved using a numerical collocation technique. In Part II of this paper, a number of numerical examples is presented addressing the effect of off-axis plies on contact stress distribution and load versus contact length in layered half-planes laminated with unidirectionally reinforced composite plies. The results indicate that for the considered unidirectional composite, the load versus contact length response is significantly influenced by the orientation of the surface layer and the underlying half-plane, while the corresponding contact stress profiles are considerably less affected.

Frictionless Contact of Layered Half-Planes, Part I: Analysis

A method is presented for the solution of frictionless contact problems on multilayered half-planes consisting of an arbitrary number of isotropic, orthotropic, or monoclinic layers arranged in any sequence. A displacement formulation is employed and the resulting Navier equations that govern the distribution of displacements in the individual layers are solved using Fourier transforms. A local stiffness matrix in the transform domain is formulated for each layer which is then assembled into a global stiffness matrix for the entire multilayered half-plane by enforcing continuity conditions along the interfaces. Application of the mixed boundary condition on the top surface of the medium subjected to the force of the indenter results in an integral equation for the unknown pressure in the contact region. The integral possesses a divergent kernel which is decomposed into Cauchy type and regular parts using the asymptotic properties of the local stiffness matrix and the ensuing relation between Fourier and finite Hilbert transform of the contact pressure. For homogeneous half-planes, the kernel consists only of the Cauchy-type singularity which results in a closed-form solution for the contact stress. For multilayered half-planes, the solution of the resulting singular integral equation is obtained using a collocation technique based on the properties of orthogonal polynomials. Part I of this paper outlines the analytical development of the technique. In Part II a number of numerical examples is presented addressing the effect of off-axis plies on contact stress distribution and load versus contact length in layered composite half-planes.

The solution of large displacement frictionless contact problems using a sequence of linear complementarity problems

AbstractA Newton method for solution of frictionless contact problems is presented. A finite element discretization is performed and the contact constraints are given as complementarity conditions. The resulting equations, which represent the equilibrium of the system, are formulated as a generalized equation. Generalized equations, from the discipline of Mathematical Programming, are a way of writing multi‐valued relations, such as complementarity conditions, in a way that is similar to ordinary equations. Newton's method is then used, in a straightforward way, to solve the present non‐linear generalized equation, resulting in a sequence of Linear Complementarity Problems (LCP's).

Mixed finite element formulation for frictionless contact problems

Simple mixed finite element models and a computational procedure are presented for the solution of frictionless contact problems. The analytical formulation is based on a form of Reissner's large-rotation theory with the effects of transverse shear deformation included. The contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the internal forces (stress-resultants), the generalized displacements, and the Lagrange multipliers associated with the contact conditions. The elemental arrays are obtained by using a modified form of the two-field, Hellinger-Reissner mixed variational principle. The internal forces and the Lagrange multipliers are allowed to be discontinuous at interelement boundaries. The Newton-Raphson iterative scheme is used for the solution of the nonlinear algebraic equations, and for the determination of the contact region and the contact pressures. Two numerical examples, axisymmetric deformations of a hemispherical shell and planar deformations of a circular ring, are presented. Both structures are pressed against a rigid plate. Detailed information about the response of both structures is presented. These examples demonstrate the high accuracy of the mixed models and the effectiveness of the computational procedure developed.

The Frictionless Contact of Cracked Elastic Bodies

AbstractIn this paper a method was developed for solving the general mixed boundary‐value problem in static elasticity by numerically integrating a system of singular integral equations describing the boundary conditions of the problem, as well as the single‐valuedness of the displacements. The method was adapted to the solution of frictionless contact problems, in the case when one of the contacted bodies contains a crack. The solution of the problem yields not only the stress distribution along the contact zone but also the stress intensity factors at the tips of the crack, as well as the unknown contact length in cases of the praxis, where the contact zone is not assumed infinitesimal. The method was applied with success to the inclusion problem of an infinite cracked plate.