A mortared finite element method for frictional contact on arbitrary interfaces

This work concerns finite-element algorithms for imposing frictional contact constraints on intra-element, or embedded surfaces. Existing techniques typically rely on the underlying bulk mesh to implicitly partition the surface, a strategy that can give rise to overconstraint. In the present work, we first apply a mortaring algorithm to the modeling of frictional contact conditions on arbitrary interfaces. The algorithm is based upon a projection of the bulk and surface fields onto independent mortar fields at the interface. We examine the advantages of this approach when combined with extended finite-element approximations to the bulk fields. In particular, the method allows for bulk and surface domains to be partitioned separately, as well as enforce nonlinear contact constraints on surfaces that are not explicitly “fitted” to the bulk mesh. Results from several benchmark problems in frictional contact are provided to demonstrate the accuracy and efficacy of the method, as well as the improvement in robustness compared to existing techniques. We also provide an example that illustrates the effectiveness of the approach in high-speed machining simulation.     

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.     

A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

A finite element method for contact using a third medium

The numerical simulation of contact problems is nowadays a standard procedure in many engineering applications. The contact constraints are usually formulated using either the Lagrange multiplier, the penalty approach or variants of both methodologies. The aim of this paper is to introduce a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. To be able to account for the contact constraints, the property of the medium, that imbeds the bodies coming into contact, has to change with respect to the movements of the bodies. Within this approach the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions. In this paper we will derive a new finite element formulation that is based on the above mentioned ideas. The formulation is presented for large deformation analysis and frictionless contact.     

A mortar‐finite element formulation for frictional contact problems

In this paper a finite element formulation is developed for the solution of frictional contact problems. The novelty of the proposed formulation involves discretizing the contact interface with mortar elements, originally proposed for domain decomposition problems. The mortar element method provides a linear transformation of the displacement field for each boundary of the contacting continua to an intermediate mortar surface. On the mortar surface, contact kinematics are easily evaluated on a single discretized space. The procedure provides variationally consistent contact pressures and assures the contact surface integrals can be evaluated exactly. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical solution of the point contact problem using the finite element method

The elastohydrodynamic lubrication problem, in which the lubricant pressure and film thickness are sensitive to surface deformation, is solved by using a finite element procedure and the Newton method. The numerical procedure is applied to the point contact problem, in which a thin lubricant film is maintained between two balls loaded together by a high load under conditions of pure rolling. The present analysis shows that pressure spikes are formed near the outlet region, a result which has been found in the line contact problem and which has been conjectured in the present problem.     

A mixed finite element model for the elastic contact problem

The static elastic contact problem is approached using Lagrange multipliers, leading to a mixed finite element problem. A non-linear friction law is introduced explicitly and the non-local character of the friction phenomena is implicitly assumed. In order to avoid stress oscillations near singular points, a perturbed Lagrangian functional is considered. The algorithms herein proposed do not impose nodal dependencies over the contact surfaces, allowing for the independent discretization of both bodies. The method is able to model simultaneous contact over different regions of any geometrical shape. Computer code, examples and results presented here are restricted to axisymmetrical and bidimensional cases.     

Reducing computational costs using a multi-region finite element method for electromagnetic scattering

Electromagnetic scattering problems involving multiple scatterers can be solved by the finite element method using a single domain truncated with an absorbing boundary condition, but often it is more efficient to separate the single domain into several subdomains, separated by free-space, and to solve the set of subdomains iteratively. This multi-region method has been reported in the literature. Its relative computational cost is investigated and formulas for determining are provided when it is advantageous to use a multi-region against a single-region method.     

A computational procedure for flexible beams with frictional contact constraints

This paper deals with the application of a parametric quadratic programming (PQP) method to the numerical solution of large-deflection beams involving frictional contact constraints. The flexibility of the structure is modelled by an intrinsic spatial beam theory which is approximated by transverse-shear deformable linear beam elements. The linear complementary problem (LCP) without the penalty function resulting from PQP is made part of a Newton-Raphson search. The tool for solving the complementary equations is Lemke's algorithm, in which frictional contact conditions are enforced and new contact surfaces are updated during iteration. Applying the resulting contact element, a more accurate approximation of the contact point can be guaranteed, and the contact force can be directly computed by the adjacent beam elements. Three numerical examples are analysed to show the effectiveness and validity of the method.     

A finite element solution method for contact problems with friction

A new finite element solution method for the analysis of frictional contact problems is presented. The contact problem is solved by imposing geometric constraints on the pseudo equilibrium configuration, defined as a configuration at which the compatibility conditions are violated. The algorithm does not require any a priori knowledge of the pairs of contactor nodes or segments. The contact condition of sticking, slipping, rolling or tension release is determined from the relative magnitudes of the normal and tangential global nodal forces. Contact iterations are in general found to converge within one or two iterations. The analysis method is applied to selected problems to illustrate the applicability of the solution procedure.     

A finite element method for the computational modelling of cohesive cracks

The present contribution is concerned with the computational modelling of cohesive cracks in quasi‐brittle materials, whereby the discontinuity is not limited to interelement boundaries, but is allowed to propagate freely through the elements. In the elements, which are intersected by the discontinuity, additional displacement degrees of freedom are introduced at the existing nodes. Therefore, two independent copies of the standard basis functions are used. One set is put to zero on one side of the discontinuity, while it takes its usual values on the opposite side, and vice versa for the other set. To model inelastic material behaviour, a discrete damage‐type constitutive model is applied, formulated in terms of displacements and tractions at the surface. Some details on the numerical implementation are given, concerning the failure criterion, the determination of the direction of the discontinuity and the integration scheme. Finally, numerical examples show the performance of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

Symmetric Galerkin boundary element method for quasi-brittle-fracture and frictional contact problems

The analysis of elastic quasi-brittle structures containing cohesive cracks and contacts with friction is given a unitary formulation in the framework of incremental plasticity. Integral equations for displacements and tractions are enforced by a weighted-residual Galerkin approach so that symmetry is preserved in the key operators (in contrast to collocation BE approaches) and cracks (either internal or edge cracks) can be dealt with by a single-domain BE formulation. The space-discrete problem in rates is expressed as a linear complementarity problem centered on a symmetric matrix or, equivalently, as a quadratic programming problem in variables pertaining to the displacement discontinuity locus only. Criteria for overall instabilities and bifurcations are derived from this formulation. The BE approach proposed and implemented by a suitable time-stepping technique, is comparatively tested by numerical solutions of cohesive-crack propagation problems.     

A finite element method for contact problems related to fracture mechanics

A finite element method for contact problems in crack mechanics is developed on the basis of the penalty function method. The method is successfully applied to three important problems in fracture mechanics: a crack propagated from a pin hole, a two-point supported specimen with an edge crack loaded by a stamp, and a thick plate with a through-wall crack under bending force.     

A combined surface integral and finite element solution for a three-dimensional contact problem

A method is described to determine contact stresses and deformation using a combination of the finite element method and a surface integral form of the Bousinesq solution. Numerical examples of contacting hypoid gears are presented.     

A second order finite element method for the one-dimensional Stefan problem

We describe a finite element method for the one-dimensional Stefan problem. The elements are quadrilaterals of the space-time plane which are determined at each time-step in relation with the position of the free boundary. The method appears as a generalization of the classical Crank-Nicolson scheme, since it is identical to this scheme in the case of rectangular elements; it has the advantage of providing a simple and accurate determination of the free boundary. Numerical experiments show that the order of accuracy is equal to 2.     

A contact algorithm for cohesive cracks in the extended finite element method

Cracks with quasibrittle behavior are extremely common in engineering structures. The modeling of cohesive cracks involves strong nonlinearity in the contact, material, and complex transition between contact and cohesive forces. In this article, we propose a novel contact algorithm for cohesive cracks in the framework of the extended finite element method. A cohesive-contact constitutive model is introduced to characterize the complex mechanical behavior of the fracture process zone. To avoid the stress oscillations and ill-conditioned system matrix that often occur in the conventional contact approach, the proposed algorithm employs a special dual Lagrange multiplier to impose the contact constraint. This Lagrange multiplier is constructed by means of the area-weighted average and biorthogonality conditions at the element level. The system matrix can be condensed into a positive definite matrix with an unchanged size at a very low computational cost. In addition, we illustrate solving the cohesive crack contact problem using a novel iteration strategy. Several numerical experiments are performed to illustrate the efficiency and high-quality results of our method in contact analysis of cohesive cracks.     

Singularity enrichment for complete sliding contact using the partition of unity finite element method

In this paper, the numerical modelling of complete sliding contact and its associated singularity is carried out using the partition of unity finite element method. Sliding interfaces in engineering components lead to crack nucleation and growth in the vicinity of the contact zone. To accurately capture the singular stress field at the contact corner, we use the partition of unity framework to enrich the standard displacement‐based finite element approximation by additional (enriched) functions. These enriched functions are derived from the analytical expression of the asymptotic displacement field in the vicinity of the contact corner. To characterize the intensity of the singularity, a domain integral formulation is adopted to compute the generalized stress intensity factor (GSIF). Numerical results on benchmark problems are presented to demonstrate the improved accuracy and benefits of this technique. We conduct an investigation on issues pertaining to the extent of enrichment, accurate numerical integration of weak‐form integrals and the rate of convergence in energy. The use of partition of unity enrichment leads to accurate estimations of the GSIFs on relatively coarse meshes, which is particularly beneficial for modelling non‐linear sliding contacts. Copyright © 2008 John Wiley & Sons, Ltd.     

齿轮副动态啮合特性的接触有限元分析

齿轮副动态啮合特性对齿轮系统振动机理研究及动态设计都具有重要意义,而它又与齿轮副啮合位置变化、受载弹性变形及滑动摩擦等因素密切相关。本文首先建立了精确的啮合齿轮副有限元分析模型,并在此基础上提出了一种可综合考虑齿轮副连续弹性啮合过程中多种影响因素的接触有限元分析方法。然后,利用本文提出的方法分别研究了考虑滑动摩擦、齿廓修形及时变刚度等因素的齿轮副低速和高速工况下连续弹性啮合过程的动态啮合特性。研究表明：本文提出的分析方法不但可以有效研究由滑动摩擦引起的节点冲击激励,以及齿廓修形设计对齿轮副啮入、啮出冲击激励的影响,而且还能有效分析具有时变刚度激励的齿轮副参数振动响应特性,可为齿轮副动态啮合特性分析提供有效的分析工具。     

A computational investigation and smooth-shaped defect synthesis for eddy current testing problems using the subregion finite element method

Eddy Current Testing (ECT) plays a key role in detecting cracks and defects in conductors. The present study examines for the first time how the subregion method as an effective mathematical and computational technique can be admixed with Finite Element Method (FEM) to study multiple defects parameters for ECT issues. Separating a defect region from the entire domain in any computational technique will save both time and storage space. Examples of different types of defects are presented in this article . A tangible result of processing time reduction by 90% has been achieved which has led us to consider the subregion FEM method as an effective method in solving different Nondestructive Evaluation (NDE) problems. An agreement between our results and others using classical FEM has been achieved which could lead to using this technique in online and field testing problems. The presented subregion FEM algorithm was verified experimentally with good agreement by testing Aluminum (T6061-T6) samples with defects. A Tunneling Magnetoresistive (TMR) sensor was used to measure the component of the magnetic field from normal to the sample top surface. A major component of minimizing processing time was achieved, which could lead to using this technique in online and field testing problems.     

Finite deformation formulation for embedded frictional crack with the extended finite element method

We reformulate an extended finite element (FE) framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation, including finite stretching and rotation. For the FE representation, we consider a Galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration. Contact and frictional constraints employ two Kuhn–Tucker conditions, a contact/separation constraint nesting over a stick/slip constraint for the case when the crack faces are in frictional sliding mode. We integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics. We then present plane strain simulations demonstrating various aspects of the extended FE solutions. The mechanisms considered include combined opening and frictional sliding in initially straight, curved, and S‐shaped cracks, with and without bulk plasticity. To gain further insight into the extended FE solutions, we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks, including the distribution of the normal component of traction on the crack faces. Copyright © 2009 John Wiley & Sons, Ltd.