A formulation of arbitrarily shaped surface elements for three-dimensional large deformation contact with friction

The paper introduces a general theory for the numerical simulation of large deformation contact problems. The contacting bodies under consideration may be of two- or three-dimensional shape modelled by finite elements. A contact finite element which can be applied to handle multi-body contact as well as contact with rigid bodies is developed. The element is universal in the sense that it can be used as a surface element for any known finite element model and includes friction. The frictional behaviour of the model obeys Coulomb's law of friction distinguishing between sticking and sliding contact. The algorithmic treatment is based on a penalty formulation for the normal and sticking contact. The corresponding consistent tangential stiffness matrices are derived, leading to an overall quadratic convergence behaviour for the method. This feature is demonstrated in a number of representative examples. © 1997 by John Wiley & Sons, Ltd.     

Tangent operators,sensitivity expressions,and optimal design of non-linear elastica in contact with applications to beams

The tangent operator and design sensitivity expressions for non-linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method. The sensitivity computations are then combined with a numerical programming package to create an optimal design environment. To exemplify the optimal design environment, a beam's contact surface is contoured to minimize bending stress. This work combines the research on the sensitivity analysis of non-linear elastic bodies subject to constraints, finite strain non-linear elastic beam analysis and contact analyses. The analysis is valid for any smooth contact surface; and specialized for the case in which the surface is represented by a cubic spline. The direct differentiation method is utilized to perform the sensitivity analysis. In an example problem, the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve.     

具有曲面接触斑弹性体滚动接触理论及其数值方法

该文基于了Kalker三维弹性体非Hertz滚动接触理论模型,考虑滚动接触物体具有曲面接触斑,利用有限元法,推导出物体柔度系数,即力与位移之间的关系,将理论模型转化为数学上的非线性规划问题。结合拉格朗日乘子法,求解非线性方程组,从而得到接触斑力学行为。该模型是考虑曲面接触斑三维弹性体滚动接触理论模型,考虑了滚动接触物体的真实几何尺寸和接触区外边界因素对滚动接触行为的影响。为解决任何几何形状弹性体滚动接触问题提供了方法。该文主要对二维问题的数值模拟,所得数据结果较为合理。并结合商业有限元软件计算结果对静态问题进行对比验证,两种模型的分析结果吻合的较好。该文的模型和数值方法进一步完善后将适用于任意曲面接触斑滚动接触问题的求解。     

Finite element method used in contact problems with dry friction

In this paper, we present a formulation of numerical approximations of the frictional quasi-contact problem with dry friction between a deformable body and a foundation with possibility to consider the case of two deformable bodies. We consider numerical approximations of 3D static contact problem with dry friction, using finite contact elements. Saddle-point algorithm, Lagrange incremental multipliers method and penalty functions are used to enforce finite element surface contact constrains for incremental formulation of the frictional quasi-static problem. Some typical examples in the elastic contact problems with dry friction are presented.     

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Large‐scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.     

Three‐dimensional numerical solution for wear prediction using a mortar contact algorithm

A finite element formulation to compute the wear between three‐dimensional flexible bodies that are in contact with each other is presented. The contact pressure and the bodies displacements are calculated using an augmented Lagrangian approach in combination with a mortar method, which defines the contact kinematics. The objective of this study is to characterize the wear rate coefficients for bimetallic pairs and to numerically predict the wear depths in new component designs. The proposed method is first validated with the classical pin‐on‐disc problem. Then, experimental results of wear for the metallic pairs used in internal combustion engine valves and inserts are presented and are taken as a reference solution. An example is provided that shows agreement of the numerical and experimental solution. Finally, the proposed algorithm is used to predict the wear in an application example: the wear in an internal combustion engine valve. Copyright © 2013 John Wiley & Sons, Ltd.     

A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact

The complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods. A new technique for the finite element modelling of elastic–plastic fatigue crack growth with unilateral contact on the crack faces is presented. The extended finite element method (X-FEM) is used to discretize the equations, allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh. This paper presents an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic–plastic crack growth with treatment of contact. An original formulation, which takes advantages of two powerful numerical methods, is presented. Next the numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element model for contact analysis of multiple Cosserat bodies

The objective of this paper is to develop a finite element model for multi-body contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed for finite element analysis of contact problems. The contact problem with friction between two Cosserat bodies is treated in the same way as in plastic analysis. The penalty factors, that are normally introduced into the algorithm for contact analysis, have a direct influence on accuracy of solution. There is no available rule for choosing a reasonable value of these factors for simulation of contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Compared with the conventional work on Cosserat elasticity, the newly developed model is on the contact analysis of the Cosserat materials and is seldom found in the existing literatures. Four examples are computed to illustrate the validity and importance of the model developed.     

On stability and reflection‐transmission analysis of the bipenalty method in contact‐impact problems: A one‐dimensional,homogeneous case study

The stability and reflection‐transmission properties of the bipenalty method are studied in application to explicit finite element analysis of one‐dimensional contact‐impact problems. It is known that the standard penalty method, where an additional stiffness term corresponding to contact boundary conditions is applied, attacks the stability limit of finite element model. Generally, the critical time step size rapidly decreases with increasing penalty stiffness. Recent comprehensive studies have shown that the so‐called bipenalty technique, using mass penalty together with standard stiffness penalty, preserves the critical time step size associated to contact‐free bodies. In this paper, the influence of the penalty ratio (ratio of stiffness and mass penalty parameters) on stability and reflection‐transmission properties in one‐dimensional contact‐impact problems using the same material and mesh size for both domains is studied. The paper closes with numerical examples, which demonstrate the stability and reflection‐transmission behavior of the bipenalty method in one‐dimensional contact‐impact and wave propagation problems of homogeneous materials.     

Cn continuous modelling of smooth contact surfaces using NURBS and application to 2D problems

This paper presents a strategy for the finite element implementation of Cⁿ continuous contact surfaces for deformable bodies undergoing finite deformations, whereby n represents an arbitrary level of continuity. The proposed novel approach avoids the non‐physical oscillations of contact forces which are induced by the traditional enforcement of kinematic contact constraints via faceted surfaces discretizing the interacting boundaries. In particular, for certain problems, the level of continuity may influence the rate of convergence significantly within a non‐linear solution scheme. A hierarchical tree data structure is proposed for an efficient search algorithm to find the neighbour elements on adaptively refined meshes, which are involved in the smoothing process of a particular finite element. The same data structure is used for the automatic detection of the contact surfaces of a body. Three representative numerical examples demonstrate the increased rate of convergence, the ability to trace the actual surface more accurately and the prevention of pressure jumps of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

Contact forces of polyhedral particles in discrete element method

A general contact force law for arbitrarily shaped bodies is presented. At first an advanced contact force law is derived from the well know Hertz contact law. The obtained formulation of the Hertz contact law can be applied to the contact of arbitrarily shaped bodies. In a second step this contact model is applied to the contacts among polyhedral particles. The results are compared to finite element simulations. The model is extended by terms for damping and friction. The behaviour of the damping and friction model are demonstrated with simple examples. The force law is then implemented in the discrete element method (DEM). The application of this DEM is demonstrated by a simulation of the particle movement in a mixer.     

A simple finite element‐based framework for the analysis of elastic pseudo‐rigid bodies

This article advocates a general procedure for the numerical investigation of pseudo‐rigid bodies. The equations of motion for pseudo‐rigid bodies are shown to be mathematically equivalent to those corresponding to certain constant‐strain finite element approximations for general deformable continua. A straightforward algorithmic implementation is achieved in a classical finite element framework. Also, a penalty formulation is suggested for modelling contact between pseudo‐rigid bodies. Representative planar simulations using a non‐linear elastic model demonstrate the predictive capacity of the pseudo‐rigid theory, as well as the robustness of the proposed computational procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

The bi-potential method applied to the modeling of dynamic problems with friction

The bi-potential method has been successfully applied to the modeling of frictional contact problems in static cases. This paper presents an extension of this method for dynamic analysis of impact problems with deformable bodies. A first order algorithm is applied to the numerical integration of the time-discretized equation of motion. Using the Object-Oriented Programming (OOP) techniques in C++ and OpenGL graphical support, a finite element code including pre/postprocessor FER/Impact is developed. The numerical results show that, at the present stage of development, this approach is robust and efficient in terms of numerical stability and precision compared with the penalty method.     

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.     

Numerical solution of elastic bodies in contact by FEM utilising equilibrium displacement fields

In this paper a new effective formulation of the computation of the statics of elastic bodies in contact is described and demonstrated. Line contact, plane strain, and negligible friction are assumed. The formulation is an extension of the standard finite element method (FEM). With the aim to utilise the Hertz theory directly, we use the exact solution of the elastic 2-dimensional halfspace loaded by Hertzian pressure distribution and enforce the contact condition by the method of Lagrange multipliers. In numerical examples we have focussed on the demonstration and evaluation of the accuracy of the new formulation for selected applications compared with the state of the art node-to-segment contact algorithm implemented in the software system ADINA. Proposed formulation is more accurate for problems where Hertz contact dominates the strain state, especially for small number of elements, whereas we obtained a fairly good agreement with ADINA for a more general bending problem.     

Simulation of two dimensional unilateral contact using a coupled FE/EFG method

In this paper, simulation of two dimensional unilateral contact problems using a coupled finite element/element free Galerkin method is proposed. For the analysis, the element free Galerkin method and Galerkin formulation for two dimensional elasticity problems are considered. Then, the penalty method for imposition of contact constraint is proposed. The finite element shape functions are used in the penalty term of contact constraint. Finally, the accuracy of the presented method is verified through some examples. The numerical results have demonstrated that the presented approach is simple and accurate for frictionless contact analysis of 2D solids.     

Semi-Lagrangian reproducing kernel particle method for fragment-impact problems

Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes.     

Impact analysis of laminated composite plates and shells by super finite elements

A super finite element method that exhibits coarse-mesh accuracy is used to predict the transient response of laminated composite plates and cylindrical shells subjected to non-penetrating impact by projectiles. The governing equations are based on the classical theories of thin laminated plates and shells taking into account the von Karman kinematics assumptions for moderately large deflections. A non-linear Hertzian-type contact law accounting for curvatures of the colliding bodies is adopted to calculate the impact force . The theoretical basis of the present finite element model is verified by analysing impact-loaded laminated composite plate and shell structures that have previously been studied through analytical or other numerical procedures. The predictive capability of the present numerical approach is successfully demonstrated through comparisons between experimentally-measured and computed force-time histories for impact of carbon fibre-reinforced plastic (CFRP) plates. The current computational model offers a relatively simple and efficient means of predicting the structural impact response of laminated composite plates and shells.     

Dynamic contact analysis technique for rapidly sliding elastic bodies with node-to-segment contact and differentiated constraints

For a stabilized Newmark time integration of dynamic contact problems of the rapidly sliding bodies, considering the equality and inequality contact constraints and a high-speed contact point motion sliding on the deforming contact surface, the velocity and acceleration contact constraints are derived. Also, to suppress the numerical oscillations accompanied by the node-to-segment contact of the finite element models, a pseudo-node-to-node contact technique is suggested with the linear shape function elements having the almost equal segment lengths on the contact surface. The numerical simulations are performed with a high-speed punch moving on the beam and the high-speed rotating disks to check the stability and accuracy of the solution.     

The Yeoh model applied to the modeling of large deformation contact/impact problems

The present paper is devoted to the modeling of finite deformations of hyperelastic bodies described by the Yeoh model under contact/impact conditions. A total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. A first order algorithm is applied to integrate the equations of motion. For the finite element implementation, an explicit expression of the tangent operator is derived. Two numerical examples are presented to show the applicability of the developed approach.