3D Contact Problems Research Articles

A numerical dynamic behaviour model for 3D contact problems with friction

This paper proposes a novel algorithm for the condition detection in which the slip state transitions in a stick–slip motion, or the solution breaks down and also to study the state transition of nodes belonging to the contact area: stick, slip or open contact state. We designed a Matlab Simulink program to simulate the occurrence conditions for the slip–stick transition analysing three types of contact surface materials, with respectively 0.5, 0.75 and 0.9 friction coefficients, using finite element contact. The proposed method is able to detect the stick–slip motion and implicit the numerical instability of the model. By applying this method to control walking robots on uncertain, unknown and unstructured surfaces, the occurrence conditions for the slip–stick transition depending on the friction coefficient of contact material were determined. The presented simulations demonstrates through a numeric modelling of the dynamic behaviour of 3D contact problems with friction we can detect the slip/stick phenomenon for a walking robot motion on a uneven terrain, so it can improve the real time control in order to predict and avoid robot overthrow.

Shape Optimization in Contact Problems with Coulomb Friction and a Solution-Dependent Friction Coefficient

The present paper deals with shape optimization in discretized two-dimensional (2D) contact problems with Coulomb friction, where the coefficient of friction is assumed to depend on the unknown sol...

Applying Nagata patches to smooth discretized surfaces used in 3D frictional contact problems

The accurate solution of large deformation frictional contact problems using the finite element method is still a challenging task due to the strong nonlinearities involved. This paper presents a smoothing method applicable to 3D contact surfaces discretized with an arbitrary mesh topology. The quadratic Nagata patch interpolation is adopted to define the smooth surface. The resulting contact surface passes through all nodes of the mesh while providing a smooth description, with at least G1 continuity at the nodes and quasi-G1 continuity between the patches. Thus, the proposed method avoids the non-physical oscillations in the contact force, which are induced by the traditionally used faceted contact surfaces description, when slave nodes slide over several master segments. Moreover, it allows the accurate evaluation of kinematic variables, leading to important improvements in terms of convergence rate within the Newton–Raphson iteration loop. The developed global and local contact search algorithms, designed for contact surfaces described by Nagata patches, are described in detail. Three numerical examples were selected to illustrate the advantages of the proposed smoothing method, including a complex industrial example of sheet metal forming process. The results show the significant improvements attained with the proposed approach, in terms of efficiency, robustness and accuracy, when compared with the traditional faceted contact surfaces description.

Semi-Smooth Newton Method for Solving 2D Contact Problems with Tresca and Coulomb Friction

The contribution deals with contact problems for two elastic bodies with friction. After the description of the problem we present its discretization based on linear or bilinear finite elements. The semi--smooth Newton method is used to find the solution, from which we derive active sets algorithms. Finally, we arrive at the globally convergent dual implementation of the algorithms in terms of the Langrange multipliers for the Tresca problem. Numerical experiments conclude the paper.

An approximate model of cylindrical and spherical contact on composite laminates of finite thickness

An empirical force–contact area relationship for the two-dimensional (2D) plane strain line contact between a stiff cylindrical indenter and a composite laminate of finite thickness supported on a rigid substrate is presented. The validity of a 2D Hertzian line contact force–contact length relation originally meant for isotropic contact problems on laminates of finite thickness is first investigated. The same relationship can be applied to orthotropic composite plates provided that an equivalent indentation modulus is known. The indentation modulus can then be corrected for the finite thickness of the plates. The results correlate well with experimental observation. Similar procedures can be performed to find the corresponding correction for a 3D axisymmetric contact problem.

Fast FE–BEM algorithms for orthotropic frictional contact

SUMMARYThis paper presents a new and efficient methodology for solving 3D frictional contact problems considering an orthotropic friction law. The contact methodology is based on a proposed augmented Lagrangian formulation for orthotropic frictional contact problems, and a new discrete contact operator, which allows to reduce the number of unknowns in a Newton‐like algorithm that accelerates the attainment of the solution. A fast Uzawa scheme is also proposed on the basis of the Steffensen's method. Both algorithms prove to be very robust and efficient to solve orthotropic frictional contact problems. The proposed formalism has the advantage of being very compact and valid for both the FEM and the BEM. Numerical results are given to demonstrate the validity of the formulation and algorithms proposed. Copyright © 2013 John Wiley & Sons, Ltd.

3D frictionless contact problems with large load-steps based on the covariant description for higher order approximation

Based on the consideration of contact from the surface geometry point of view using a covariant contact description in the local surface coordinate system a 3D extension of the so-called large penetration algorithm is presented. Within the Node-To-Surface (NTS) approach this scheme allows the computation with larger load increments than within a usual consistent linearization, see [2]. The required specific structure of the tangent matrix for this 3D scheme is found in a straightforward fashion using the covariant contact description. In order to show the efficiency of the scheme for further approaches like e.g. the Surface-To-Analytical-Surface (STAS) examples with contact, bending and sliding of shells modeled with elements of quadratic approximation, are presented. Furthermore a comparison regarding the efficiency for both considered approaches, NTS and STAS, is presented.

Adaptive finite elements with thin obstacles

This work describes concepts for a posteriori error estimation and adaptive mesh design for dual-mixed finite element models where the solution is subjected to inequality constraints. These methods are developed here especially for variational problems with a so-called thin obstacle, i.e. the restrictions are imposed on the boundary of the domain under consideration. The studies presented in this work are partly motivated by our memberships in research groups , where one focus is on the numerical FE-analysis of 3D-contact problems. Eventually, numerical tests confirm our theoretical results.

Asymptotics for thin elastic fibers in unilateral contact

AbstractWhat is the contact condition in a 1D beam‐model and is it possible to obtain the frictional moments and forces from the 3D traction? If it is possible does the cross‐section of the beams influence these values? These questions motivate to study the dimension reduction of a 3D contact problem for beams. This paper is a continuation of [1]. In [1] the asymptotic dimension reduction of a Robin‐type elasticity boundary value problem was presented. In this work the explicit relation between a 3D contact problem and a 3D Robin‐type elasticity boundary value problem are established and the 1D equations derived in [1] are interpreted as 1D contact conditions, further some numerical examples are shown. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scalable TFETI with optional preconditioning by conjugate projector for transient frictionless contact problems of elasticity

The FETI based domain decomposition method is adapted to implement the time step of the Newmark scheme for the solution of dynamic contact problems without friction. If the ratio of the decomposition and discretization parameters is kept uniformly bounded, then the cost of the time step is proved to be proportional to the number of nodal variables. The algorithm uses our in a sense optimal MPRGP algorithm for the solution of strictly convex bound constrained quadratic programming problems with optional preconditioning by the conjugate projector to the subspace defined by the trace of the rigid body motions on the artificial subdomain interfaces. The proof of optimality combines the convergence theory of our MPRGP algorithm, the classical bounds on the spectrum of the mass and stiffness matrices, and our theory of the preconditioning by a conjugate projector for nonlinear problems. The results are confirmed by numerical solution of 2D and 3D dynamic contact problems.

A comparative study between two smoothing strategies for the simulation of contact with large sliding

The numerical simulation of contact problems is still a delicate matter especially when large transformations are involved. In that case, relative large slidings can occur between contact surfaces and the discretization error induced by usual finite elements may not be satisfactory. In particular, usual elements lead to a facetization of the contact surface, meaning an unavoidable discontinuity of the normal vector to this surface. Uncertainty over the precision of the results, irregularity of the displacement of the contact nodes and even numerical oscillations of contact reaction force may result of such discontinuity. Among the existing methods for tackling such issue, one may consider mortar elements (Fischer and Wriggers, Comput Methods Appl Mech Eng 195:5020---5036, 2006; McDevitt and Laursen, Int J Numer Methods Eng 48:1525---1547, 2000; Puso and Laursen, Comput Methods Appl Mech Eng 93:601---629, 2004), smoothing of the contact surfaces with additional geometrical entity (B-splines or NURBS) (Belytschko et al., Int J Numer Methods Eng 55:101---125, 2002; Kikuchi, Penalty/finite element approximations of a class of unilateral contact problems. Penalty method and finite element method, ASME, New York, 1982; Legrand, Modeles de prediction de l'interaction rotor/stator dans un moteur d'avion These de doctorat. PhD thesis, Ecole Centrale de Nantes, Nantes, 2005; Munoz, Comput Methods Appl Mech Eng 197:979---993, 2008; Wriggers and Krstulovic-Opara, J Appl Math Mech (ZAMM) 80:77---80, 2000) and, the use of isogeometric analysis (Temizer et al., Comput Methods Appl Mech Eng 200:1100---1112, 2011; Hughes et al., Comput Methods Appl Mech Eng 194:4135---4195, 2005; de Lorenzis et al., Int J Numer Meth Eng, in press, 2011). In the present paper, we focus on these last two methods which are combined with a finite element code using the bi-potential method for contact management (Feng et al., Comput Mech 36:375---383, 2005). A comparative study focusing on the pros and cons of each method regarding geometrical precision and numerical stability for contact solution is proposed. The scope of this study is limited to 2D contact problems for which we consider several types of finite elements. Test cases are given in order to illustrate this comparative study.

Anisotropic wear framework for 3D contact and rolling problems

This work presents a new methodology to compute anisotropic wear on 3D frictional contact and/or rolling problems. The damage on the solids’ geometries and the contact pressures evolution under anisotropic tribological properties can be predicted using this contact framework. The present formulation uses the Boundary Element Method and/or the Finite Element Method for computing the elastic influence coefficients, and enforces contact constraints using a new augmented scheme. An anisotropic wear model is considered, together with an orthotropic friction law. The methodology and the proposed algorithm are illustrated with some examples, in which different studies about the influence of anisotropy on wear are presented.

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.

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.

Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction

The paper deals with shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems, where the coefficient of friction depends on the solution. We consider the two- dimensional Signorini problem, coupled with the physically less accurate model of given friction, but assume a solution-dependent coefficient of friction. First, we investigate the shape optimization problem in the continuous, infinite-dimensional setting, followed by a suitable finite-dimensional approximation based on the finite- element method. Convergence analysis is presented as well. Next, an algebraic form of the state problem is studied, which is obtained from the discretized problem by further approximating the frictional term by a quadrature rule. It is shown that if the coefficient of friction is Lipschitz continuous with a sufficiently small modulus, then the algebraic state problem is uniquely solvable and its solution is a Lipschitz continuous function of the control variable, describing the shape of the elastic body. For the purpose of numerical solution of the shape optimization problem via the so-called implicit programming approach we perform sensitivity analysis by using the tools from the generalized differential calculus of Mordukhovich. The paper is concluded first order optimality conditions.

Qualitative analysis of 3D elastostatic contact problems with orthotropic Coulomb friction and solution-dependent coefficients of friction

This paper analyzes a discrete form of 3D contact problems with local orthotropic Coulomb friction and coefficients of friction which may depend on the solution itself. The analysis is based on the fixed-point reformulation of the original problem. Conditions guaranteeing the existence and uniqueness of discrete solutions are established. Finally, numerical results of a model example are presented.

A theoretically supported scalable TFETI algorithm for the solution of multibody 3D contact problems with friction

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3D and proved to be scalable for Tresca friction. The analysis admits “floating” bodies. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded. The analysis is based on the classical results by Farhat et al. on scalability of FETI for linear problems and on our development of optimal quadratic programming algorithms. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate that our algorithm is effective. The effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with Coulomb’s friction is illustrated on the analysis of a mine support.

On the Solution of the 3D Crack Surface Contact Problem Using the Boundary Element Method

The efficient solution of the 3D crack surface contact problem utilizing the boundary element method (BEM) is presented. The dual discontinuity method (DDM), a special formulation of the BEM, is applied. This method deals directly with the relative displacements and the discontinuities of the tractions at the crack. For the normal behavior a unilateral contact is assumed and for the description of the tangential behavior Coulomb’s frictional law is utilized. The hard contact formulation is regularized by the application of the penalty method. An incremental iterative procedure based on a radial return mapping algorithm is applied for the solution of this non-linear problem. Based on the stress field the fracture mechanical parameters are determined by an extrapolation method for all increments of a characteristic load cycle. By the analysis of this load cycle the cyclic fracture mechanics values are obtained. Due to the non-linear nature of crack growth the simulation is implemented in the framework of a predictor-corrector scheme. For the investigation of the influence of the crack surface roughness on the behavior of cracks two numerical examples are presented.

An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements

An adaptive contact analysis approach is presented for 2D solid mechanics problems using only triangular elements and the subdomain parametric variational principle (SPVP). The present approach is implemented for the node-based smoothed FEM (or NS-FEM), the edge-based smoothed FEM (ES-FEM) and the standard FEM models with automatically adaptive refinement scheme. A modified Coulomb frictional contact model and its corresponding discrete equations are introduced. The global discretized system equations are then formulated in an incremental form with the aid of the basic boundary value equations for friction contact and the subdomain parametric variational principle. A simple adaptive refining scheme is presented, and the Voronoi vertices are taken as candidate points to become new nodes because of duality property between the Voronoi diagrams and Delaunay triangulation. The present adaptive approach can properly simulate variable behaviors of a contact interface such as bonding/debonding, contacting/departing, and sticking/slipping. Several examples are presented to numerically validate the proposed approach via the comparison with reference solutions obtained by ABAQUS ®, and to investigate the effects of the various parameters used in the computations on the response of the contact system. The numerical results have demonstrated that the present adaptive contact analysis approach using the ES-FEM has higher accuracy and convergence rate in the strain energy than that using FEM and NS-FEM. However, the latter two methods can provide the lower and upper bound solution for the system strain energy, respectively.

3D frictional contact of anisotropic solids using BEM

A numerical method to study three-dimensional (3D) contact problems in solids with anisotropic elastic behavior is developed in this work. This formulation is based on the Boundary Element Method (BEM) for computing the elastic influence coefficients and on projection functions over the augmented Lagrangian for contact restrictions fulfillment. The constitutive equations of the potential contact zone are Signorini’s contact conditions and Coulomb’s law of friction. The formulation uses a recently introduced explicit approach for fundamental solutions evaluation, which are valid for general anisotropic behavior meanwhile mathematical degeneracies are allowed. The accuracy and robustness of the proposed method is illustrated by solving some examples previously presented in the literature. This approach is further applied to study the influence of solids anisotropy on the contact problem.