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In this paper, multigrid methods are tested on unilateral problems with friction. An optimal strategy is presented and efficiency of the solver is discussed on several examples.

Multigrid methods have been widely used in fluids mechanics when large numbers of degrees of freedom are involved. Usually the geometries are sufficiently simple to enable the generation of multiple overlapped meshes in an easy way (essentially in the context of finite difference methods). In nonlinear structure mechanics, the computational costs increase because of the treatment of nonlinearities and finite elements methods are dominant because of the complexity of the geometries. The present work investigates the ability of multigrid methods to reduce the computational times and analyzes the specific problems of formulation and implementation related to the treatment of nonlinearities in the context of finite element methods. This work is conducted on contact problems involving unilateral contact and friction between an elastic body and a rigid obstacle. The nonlinearities are stiff because the contact behavior laws are nonsmooth (the nonpenetration is characterized by the nonregularized Signorini conditions) and nondifferentiable because of the use of the nonregularized Coulomb law.

Key words: Unilateral contact, friction, multigrid.

The efficient modeling of dynamical contact problems with friction is still a callenge in non-linear implicit structural analysis. We employ a mixed formulation in space with the displacement as primal variable and the contact stress as dual variable. For the discretization of the latter we use a discrete Lagrange multiplier space with biorthogonal basis functions. For the treatment of the nonlinear frictional contact conditions semi-smooth Newton methods are applied. To avoid oscillations in the Lagrange multiplier during the solution of dynamical contact problems with mass, we locally under-integrate the mass matrix. We also show the applicability of the mixed formulation to a velocity driven rigid-plastic problem.

Key words: Coulomb friction, semi-smooth Newton methods, non-oscillating Lagrange multiplier, energy conservating time integration.

This paper reviews some advances and applications in mathematical programming method for numerical modeling of elastic-plastic contact problems. Emphases are on the parametric variational principle and quadratic programming method used for analysis of elastic-plastic contact problems with isotropic/orthotropic friction law. The contact problem with friction between two elastic-plastic Cosserat bodies is treated in the same way as that in the conventional plastic analysis. There is no available rule for choosing a reasonable value of the penalty factors for simulation of the contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Two numerical examples are presented to show the efficiency of the model and algorithm presented.

Key words: Computational contact mechanics, mathematical programming method, parametric variational principle.

A nonsmooth contact class of algorithms were introduced by Kane et al. [1] and extended to the case of friction by Pandolfi et. al [2]. The formulation specifically addresses geometries for which neither normals nor gap functions can be properly defined, e.g. bodies with corners. The formulation provides the incremental displacements in variational form, following from a minimum principle. Selected numerical examples of application of the algorithm are presented here.

Key words: Frictional contact, non-smooth contact, finite elements, variational formulations.

Dynamic contact problems in elasticity are treated within a finite element framework by employing the well-established node-to-segment method. A new formulation of the algorithmic forces of contact is proposed which makes possible the design of energymomentum conserving integrators. The numerical example presented herein indicates that the present approach provides enhanced numerical stability.

Key words: Large deformation contact problems, conserving time integration, node-tosegment method.

The standard implementation of the classical Coulomb frictionmodel together with the Newton iterative method for the finite element method leads to non-symmetric tangent matrices for sliding zones of contact surfaces. This fact is known in literature as consequence of the non-associativity of the friction law. Considering anisotropic models for friction, especially including coupling of adhesion and friction, leads to additional non-symmetries due to anisotropy. Since, non-symmetry of matrices is a non-desirable feature of most engineering problems, various proposals for symmetrization are known in computational mechanics. A further suggestion is made in this contribution. The covariant approach for both isotropic and anisotropic frictional contact problems leads to a very simple structure of the tangent matrices. This allows to obtain very robust tangent matrices within the symmetrized Augmented Lagrangian method. In the current contribution, the nested Uzawa algorithm is applied for symmetrization within the Augmented Lagrangian approach for an anisotropic friction model including adhesion and friction. The numerical examples show the good convergence behavior for various problems such as small and large sliding problems.

Key words: Anisotropic friction and adhesion, covariant description, Augmented Lagrangian method, symmetrization.

We describe two methods for the numerical solution of two-body frictional contact problems based on pure boundary integral formulations, and corresponding automatic mesh refinement procedures. For the -version of the penalty BEM we use local reliable and efficient a posteriori error estimators of the residual type [1]. For the -version of the mortar BEM we use heuristic local a posteriori residual error indicators and a three-step refinement procedure [1].

Key words: A posteriori error estimate, adaptive mesh refinement, contact, Tresca friction, Steklov-Poincaré operator.

Mortar element methods have recently been successfully applied to large deformation dry-frictional contact problems. In this contribution, we summarize two recent extensions of the mortar based contact approach: solution of lubricated contact problems between deformable solid bodies, and implementation of mortar methods in parallel equation solving frameworks.

Lubricated contact problems, in which thin fluid films exist between solid contact surfaces, are widely found in engineering and sciences. Based on Reynolds equations, different numerical methods have been developed for solving lubricated contact problems between rigid or linearly elastic solid bodies (see, for example, Christopherson [2] and Hamrock et al. [6]). In our approach, the fluid film thickness is computed from a least squares projection based on dual basis functions [12]. The free boundary problem for the fluid phase is regularized with a penalty method. The solid and the fluid phase unknowns are solved in a fully coupled system of equations, based on the linearization of the weak forms of the governing equations for the solid and the fluid phases. The performance and potential of the proposed method is demonstrated here in a two dimensional example of elastohydrodynamic lubrication.

Another important issue for mortar based contact formulations lies in determining their effectiveness within parallel computing architectures. In general, contact problems result in potentially ill-conditioned and highly unsymmetric stiffness matrices, so that direct methods have been preferred (particularly in highly nonlinear large deformation applications). However, direct methods require memory and computation time up to () and (), respectively, so that their applicability to very large scale contact simulations appears to be limited. Iterative methods require memory and computation time of (n) and (), making their use in large-scale numerical simulations tempting. We consider here a conceptually simple implementation which uses iterative methods to solve the linear systems generated by consistent linearization of the mortar contact formulation, on distributed memory machines, and examine performance in a representative numerical example.

Key words: Lubrication, friction, contact, mortar formulation, parallel computation.

The problem of interface decohesion in laminated beams is addressed with reference to the double cantilever beam (DCB) geometry. The paper deals with the analysis of the influence of non-uniform bonding properties or interfacial defects on the crack propagation process and its stability. In spite of the relative simplicity of the considered case study, which is under pure Mode I deformation, a contact problem takes place due to the decohesion. The occurrence of contact makes the problem highly non-linear, since the extension of the contact area is unknown. The finite element method with a generalized interface constitutive law is used to enforce the contact constraint along the interface and to model the progress of the delamination process.

Key words: Contact, decohesion, cohesive models, laminated beams, finite elements.

In this paper a smoothing procedure for the 3D beam-to-beam contact is presented. A smooth segment is based on current position vectors of three nodes for two adjacent finite elements. The approximated fragment of a 3D curve modeling a beam axis spans between the centre points of these elements. The curve is described parametrically using three Hermite polynomials. The four boundary conditions used to determine the coefficients for each of these polynomials involve co-ordinates and slopes at the curve ends. The slopes are defined in terms of the element nodal co-ordinates, too, so there is no dependence on nodal rotations and this formulation can be embedded in a beam analysis using any type of beam finite element. This geometric representation of the curve is incorporated into the 3D beam-to-beam frictional contact model with the penalty method used to enforce contact constraints. The residual vector and the corresponding tangent stiffness matrix are determined for the normal part of contact and for the stick or slip state of friction. A numerical example is presented to show the performance of the suggested smoothing procedure in a case of large frictional sliding.

Key words: 3D curve smoothing, Hermite polynomials, frictional contact, consistent linearisation.