An augmented Lagrangian method for discrete large-slip contact problems

An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.     

An augmented Lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems

A finite element formulation for three dimensional (3D) contact mechanics using a mortar algorithm combined with a mixed penalty–duality formulation from an augmented Lagrangian approach is presented. In this method, no penalty parameter is introduced for the regularisation of the contact problem. The contact approach, based on the mortar method, gives a smooth representation of the contact forces across the bodies interface, and can be used in arbitrarily curved 3D configurations. The projection surface used for integrating the equations is built using a local Cartesian basis defined in each contact element. A unit normal to the contact surface is defined locally at each element, simplifying the implementation and linearisation of the equations. The displayed examples show that the algorithm verifies the contact patch tests exactly, and is applicable to large displacements problems with large sliding motions.Copyright © 2012 John Wiley & Sons, Ltd.     

A computational method for frictional contact problem using finite element method

Using the finite element method a numerical procedure is developed for the solution of the two-dimensional frictional contact problems with Coulomb's law of friction. The formulation for this procedure is reduced to a complementarity problem. The contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem. Several examples are given to demonstrate the validity of the present formulation.     

On augmented Lagrangian algorithms for thermomechanical contact problems with friction

The detailed discretization of contact zones with contact stiffness based on real physical characteristics of contact surfaces can produce stiffness terms which induce ill-conditioning of the global stiffness matrix. Moreover the consistent treatment of frictional behaviour generates non-symmetric tangent stiffness matrices due to the non-associativity of the slip phase. Other non-symmetries are due to the coupling terms and to the dependencies on various parameters that can be involved. To overcome these difficulties almost consistent techniques based on two-step algorithms have been proposed in the past. Here an augmentation technique is proposed which takes into account micro-mechanical effects, and permits the symmetrization of the tangent stiffness during frictional slip phase.     

Numerical analysis of a quasistatic thermoviscoelastic frictional contact problem

We study a quasistatic frictional contact problem between a viscoelastic body and a deformable obstacle. A thin lubricant layer is assumed on the contact surface and, then, a normal damped response contact condition considered. Thermal and frictional effects are also taken into account. A fully discrete scheme is proposed, using the finite element method for the spatial approximation and the Euler scheme for discretizing the time derivatives. Error estimates on the solutions are derived and the linear convergence, under suitable regularity hypotheses, is obtained. The scheme was implemented and some numerical examples are included to show the performance of the method.Acknowledgments This work was partially supported by MCYT-Spain (Project BFM2003-05357) and it is also part of the project New Materials, Adaptive systems and their Nonlinearities: Modelling, Control and Numerical Simulation carried out in the framework of the european community program Improving the Human Research Potential and the Socio-Economic Knowledge Base (Contract n° HPRN-CT-2002-00284).     

Combined sticking: a new approach for finite-amplitude Coulomb frictional contact

International Journal of Mechanics and Materials in Design - Engineering-level accuracy of discretization methods for frictional contact originates from precise representation of discontinuous...     

Analysis of frictional contact problem using boundary element method and domain decomposition method

In this paper, we study the bilateral or unilateral contact with Coulomb friction between two elastic solids, using a domain decomposition method coupled with the boundary element method. The decomposition method we have selected is the Schur complement method, a non‐overlapping technique. It enables to reduce the solution of the global problem to the solution of a problem defined only on the contact surface. Moreover, its principal advantage is that computing is done separately on each solid. We have chosen to associate it with the boundary element method. Indeed, it only requires the discretization of the boundaries of solids. This technique of coupling reduces the number of unknowns and the time of computing. We have applied it to the study of indentation of an elastic foundation by an elastic flat punch and a sphere. In this last case, our results are in conformity with the Hertz theory and the analytical solution of Spence. Moreover, we have shown the influence of friction on the size of the contact radius and on the normal pressure at centre. Copyright © 1999 John Wiley & Sons, Ltd.     

An augmented Lagrangian algorithm for contact mechanics based on linear regression

The penalty method for the solution of contact problems yields an approximate satisfaction of the contact constraints. Augmentation schemes, which can be adopted to improve the solution, either include the contact forces as additional unknowns or are strongly affected by the penalty parameter and display a poor convergence rate. In a previous investigation, an unconventional augmentation scheme was proposed, on the basis of estimating the ‘exact’ values of the contact forces through linear interpolation of a database extracted by the previous converged states. An enhanced version of this method is proposed herein. With respect to the original method, the enhanced one eliminates some numerical problems and improves the regularity of the convergence path by carrying out the estimate through linear regression methods. The resulting convergence rate is superlinear, and the method is quite insensitive to the penalty parameter. The main underlying concept is that, within the iterative solution of a non‐linear problem, linear regression techniques may be used as a tool to ‘shoot’ faster to the final solution, on the basis of a set of intermediate data. The generality of this concept makes it potentially applicable to contact problems in more general settings, as well as to other categories of non‐linear problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method

NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.     

A solution method for frictional contact problems of a crack in FGMs under mechanical and thermal loads

This article provides a numerical treatment of a finite crack in an interfacial layer with spatially varying elastic properties under in-plane mechanical and thermal loading conditions. The variation of stress intensity factors and energy release rates with the functions which are governing the material properties of the interfacial layer is studied. Transient and steady-state response of a central crack in FGMs subjected to the mechanical and thermal loads are investigated. Unlike earlier studies which consider the cracks encountered as open, the current investigation studies cracks in an essentially compressive environment in which the crack faces are in contact and frictional effects play an important role. To solve this contact problem, a simple and efficient, iterative finite element method developed by authors is used. Numerical examples are provided to verify the technique and the results are compared with those of the published papers. This revised version was published online in July 2006 with corrections to the Cover Date.     

A new strategy for the solution of frictional contact problems

This article is devoted to the development of a new heuristic algorithm for the solution of the general variational inequality arising in frictional contact problems. The existing algorithms devised for the treatment of the variational inequality representing frictional contact rely on the decomposition of the physical problem into two sub-problems which are then solved iteratively. In addition, the penalty function method and/or the regularization techniques are typically used in the solution of these reduced sub-problems. These techniques introduce user-defined parameters which could influence the convergence and accuracy of the solution. The new method presented in this article overcomes these difficulties by providing a solution for the general variational inequality without decomposition into sub-problems. This is accomplished using a new heuristic algorithm which utilizes mathematical programming techniques, and thus avoids the use of penalty or regularization methods. The versatility and reliability of the developed algorithm were demonstrated through implementation to the case of frictional contact of an elastic hollow cylinder with a rigid foundation. © 1998 John Wiley & Sons, Ltd.     

A phase-field method for modeling cracks with frictional contact

We introduce a phase-field method for continuous modeling of cracks with frictional contacts. Compared with standard discrete methods for frictional contacts, the phase-field method has two attractive features: (i) it can represent arbitrary crack geometry without an explicit function or basis enrichment, and (ii) it does not require an algorithm for imposing contact constraints. The first feature, which is common in phase-field models of fracture, is attained by regularizing a sharp interface geometry using a surface density functional. The second feature, which is a unique advantage for contact problems, is achieved by a new approach that calculates the stress tensor in the regularized interface region depending on the contact condition of the interface. Particularly, under a slip condition, this approach updates stress components in the slip direction using a standard contact constitutive law, while making other stress components compatible with stress in the bulk region to ensure nonpenetrating deformation in other directions. We verify the proposed phase-field method using stationary interface problems simulated by discrete methods in the literature. Subsequently, by allowing the phase field to evolve according to brittle fracture theory, we demonstrate the proposed method's capability for modeling crack growth with frictional contact.     

Lagrangian relaxation method for price-based unit commitment problem

The unit commitment problem consists of determining the schedules for power generating units and the generating level of each unit. The decisions concern which units to commit during each time period and at what level to generate power to meet the electricity demand. The problem is a typical scheduling problem in an electric power system. The electric power industry is undergoing restructuring and deregulation. This article developes a stochastic programming model which incorporates power trading. The uncertainty of electric power demand or electricity price are incorporated into the unit commitment problem. It is assumed that demand and price uncertainty can be represented by a scenario tree. A stochastic integer programming model is proposed in which the objective is to maximize expected profits. In this model, on/off decisions for each generator are made in the first stage. The approach to solving the problem is based on Lagrangian relaxation and dynamic programming.     

A new non-smooth model for three dimensional frictional contact problems

This paper presents a new non-smooth model for three dimensional contact problems with Coulomb friction. The problem is formulated exactly as a system of non-smooth equations without employing any external variables or approximation. As compared with the existing models, the present model does not utilize the slip angle as a variable. Therefore, transformation of variables is not required and the formulation is simpler. For solving a three dimensional contact problem, the nodus is to determine the slip direction at the contact nodes because the relative slipping of the contact may occur in any direction on the contact interface. The proposed model solves this problem in a simple manner by formulating it as an equivalent non-smooth equation. Based on the theory of non-smooth analysis, a generalized derivative is introduced to solve the non-smooth equations. Thus, the non-smooth damped Newton method can be implemented directly. The proposed method has been tested using a number of numerical examples. Received 23 March 2000     

Improvement of a frictional contact algorithm for strongly curved contact problems

One of the challenges in contact problems is the prediction of the actual contact surface and the kind of contact that is established in each region. In numerical simulation of deep drawing problems the contact conditions change continuously during the forming process, increasing the importance of a correct evaluation of these parameters at each load step. In this work a new contact search algorithm devoted to contact between a deformable and a rigid body is presented. The rigid body is modelled by parametric Bézier surfaces, whereas the deformable body is discretized with finite elements. The numerical schemes followed rely on a frictional contact algorithm that operates directly on the parametric Bézier surfaces. The algorithm is implemented in the deep drawing implicit finite element code DD3IMP. This code uses a mechanical model that takes into account the large elastoplastic strains and rotations. The Coulomb classical law models the frictional contact problem, which is treated with an augmented Lagrangian approach. A fully implicit algorithm of Newton–Raphson type is used to solve within a single iterative loop the non‐linearities related with the frictional contact problem and the elastoplastic behaviour of the deformable body. The numerical simulations presented demonstrate the performance of the contact search algorithm in an example with complex tools geometry. Copyright © 2003 John Wiley & Sons, Ltd.     

Smoothing Newton method for solving two- and three-dimensional frictional contact problems

Two- and three-dimensional frictional contact problems are uniformly formulated as a system of non-differentiable equations based on variational inequality theory. Through constructing a simple continuously differentiable approximation function to the non-differentiable one, the smoothing Newton method is directly implemented as an exact method. Both the global convergence and the local quadratic convergent rate of the method are guaranteed. None of the additional variables and linear approximations on Coulomb friction law is introduced and hence the formulation exactly describes the frictional contact phenomenon in both two- and three-dimensional cases. Numerical experiments suggest that the method is very efficient and promising. © 1998 John Wiley & Sons, Ltd.     

Thermoelastic contact problem for a layer interacting with a rigid substrate under transient frictional heat generation conditions

A thermoelastic problem has been solved for the interaction between an elastic layer and a rigid substrate from which a body may come off under localized pressing load. The paper addresses the influence of an accompanying transient frictional heat generation and functional relation of a pressing load on the process of the layer displacement over the surface of a heat-insulated substrate. It is demonstrated that an increase in heat generation intensity reduces the contact area, where the time variation of the latter is governed by selection of a load function and a motion speed function: if the load is time dependent, the contact area remains unchanged; if the speed varies, the contact area exhibits a monotonic reduction. __________ Translated from Problemy Prochnosti, No. 3, pp. 132–151, May–June, 2008.     

A parallel noninvasive multiscale strategy for a mixed domain decomposition method with frictional contact

A multiscale extension for a parallel noninvasive mixed domain decomposition method is presented. After briefly exposing our noninvasive implementation of the Latin method, we present how the scalability of the algorithm is obtained by the partial verification of the constitutive law of the interfaces. We propose a new interpretation of the classical macrostrategy that imposes the overall balance of interface's forces. We also propose a new study of the macrostrategy that consists in enforcing the coarse continuity of the interfaces' displacement.