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.     

Development of a finite element contact analysis algorithm for charged‐hydrated soft tissues with large sliding

This paper focuses on a finite element analysis of contact phenomena with large sliding between charged‐hydrated biological soft tissues, such as articular cartilages, based on the triphasic theory. The impenetrability constraint between the contacting bodies and the continuity of the interstitial fluid and ion phases at the contact surfaces are imposed by applying a Lagrange multiplier approach with the contact pressure, chemical potential of the fluid and electrochemical potentials of ions as Lagrange multipliers. A node‐to‐segment one‐pass approach is adopted to cope with large deformations and sliding between the contact surfaces. To pass the contact patch test, contact boundary integrations are performed on both the master and slave contact surfaces. On the other hand, the degrees of freedom of the multipliers at the master nodes are eliminated by projecting the master nodes onto the slave surface to avoid overconstraint. The effectiveness of the proposed algorithm is verified by a couple of numerical examples, in which continuous distributions of displacement, fluid flow, ionic molar flow and Lagrange multipliers on or across the contact surface are confirmed. Copyright © 2008 John Wiley & Sons, Ltd.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

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.     

An X‐FEM approach for large sliding contact along discontinuities

The extended finite element method (X‐FEM) has been developed to minimize requirements on the mesh design in a problem with a displacement discontinuity. This advantage, however, still remains limited to the small deformation hypothesis when considering sliding discontinuities. The approach presented in this paper proposes to couple X‐FEM with a Lagrangian large sliding frictionless contact algorithm. A new hybrid X‐FEM contact element was developed with a contact search algorithm allowing for an update of contacting surfaces pairing. The stability of the contact formulation is ensured by an algorithm for fulfilling Ladyzhenskaya‐Babuska‐Brezzi (LBB) condition. Several 2D simple examples are presented in this paper in order to prove its efficiency and stability. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

3D contact modelling of large plastic deformation in powder forming processes

In this paper, the three-dimensional large frictional contact deformation of powder forming process is modeled using a node-to-surface contact algorithm based on the penalty and augmented-Lagrange approaches. The technique is applied by imposing the normal and tangential contact constraints and modifying the contact properties of frictional slip. The Coulomb friction law is employed to simulate the friction between the rigid punch and the work piece. It is shown that the augmented-Lagrange technique significantly improves imposing of the constraints on contact surfaces. In order to predict the non-uniform relative density and stress distributions during the large deformation of powder die-pressing, the nonlinear contact friction algorithm is employed within the framework of large finite element deformation, in which a double-surface cap plasticity model is used for highly nonlinear behavior of powder. Finally, the numerical schemes are examined for accuracy and efficiency in modeling of a set of powder components.     

移动式架车机伸缩托头-托架接触的有限元分析

采用基于接触对的非线性接触分析处理接触问题时,其分析结果更接近工程应用实际，但计算成本比较高且存在不收敛的情况。而自由度节点耦合分析不但能降低计算成本，而且能够保证一定精度。为了探究在一定误差范围内选用自由度节点耦合分析来替代基于接触对的非线性接触分析的可行性，以举升重量为15 t的移动式架车机为研究对象，在ANSYS软件中建立伸缩托头-托架的接触模型与节点耦合模型，在举升高度最大且伸缩托头完全伸出的工况下进行有限元分析，考虑接触刚度系数与摩擦因数对有限元分析结果的影响，分析了2种模型各接触区域等效应力的区别。通过比较2种模型的有限元分析结果可知，2种模型各接触区域的等效应力值存在一定的相对误差，且接触区域内2个接触面的相对滑动程度决定了接触分析的精度。对于精度要求不高的机械结构，可以考虑用自由度节点耦合分析来替代基于接触对的非线性接触分析。     

基于时变载荷的齿轮摩擦功率损失计算研究

应用齿轮啮合原理、齿轮接触分析和摩擦学理论,提出考虑齿轮瞬态啮合过程中传动效率随齿面接触载荷时变的齿轮摩擦功率损失计算方法;建立了啮合齿面间的相对运动速度、接触压力、滑动摩擦系数及摩擦功率损失的计算模型,并编制了计算程序;分析了齿轮几何参数、运行工况及润滑油温度等对齿轮摩擦功率损失的影响规律.研究结果为齿轮传动及其润滑系统的合理设计提供理论依据.     

无轨伸缩式门式起重机接触部位的有限元分析

非线性接触分析在处理接触问题时可以更加接近工程实际,但在实际应用中,接触参数设置复杂、计算效率低以及计算收敛性难以保证等因素制约了它的实用性。节点耦合分析是不存在收敛问题的线性分析,计算成本比接触分析低。为了探讨2种模型在应用中的差异,研究节点耦合线性分析代替非线性接触分析的可行性,以起重量为12 t的无轨伸缩式门式起重机为例,在有限元分析软件ANSYS中分别构建其接触模型和节点耦合模型,考虑接触刚度和摩擦系数对计算结果的影响,在伸缩支腿全伸的工况下比较2种模型下无轨伸缩式门式起重机各接触部位等效应力值的差异。通过对比计算结果发现：2种模型中,最大等效应力均出现在车架施加起升载荷处,计算值相差1.3%,结果差异不大。由于节点耦合分析忽略了摩擦系数,所以2种模型中各接触部位等效应力值存在差异,根据接触部位的相对滑动,在接触模型中选择合适的摩擦系数可减小此差异。研究结果表明：考虑接触区域相对滑动对计算结果的影响,节点耦合分析能较好地模拟接触分析,对于精度要求不高或接触区域基本无相对滑动的模型,可采用节点耦合分析这种简化方式来代替接触分析,以提高计算效率。     

A contact algorithm for 3D discrete and finite element contact problems based on penalty function method

A contact algorithm in the context of the combined discrete element (DE) and finite element (FE) method is proposed. The algorithm, which is based on the node-to-surface method used in finite element method, treats each spherical discrete element as a slave node and the surfaces of the finite element domain as the master surfaces. The contact force on the contact interface is processed by using a penalty function method. Afterward, a modification of the combined DE/FE method is proposed. Following that, the corresponding numerical code is implemented into the in-house developed code. To test the accuracy of the proposed algorithm, the impact between two identical bars and the vibration process of a laminated glass plate under impact of elastic sphere are simulated in elastic range. By comparing the results with the analytical solution and/or that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the algorithm proposed in this paper is proved.     

基于数值仿真的冲压成形界面接触压力

为探究冲压成形过程中板料-凹模圆角区界面接触压力,采用有限元静力算法建立了U形件小圆角半径弯曲成形过程的数值仿真模型,完成了板料和模具界面接触压力数值模拟,并参数分析了钢板强度、相对圆角半径、压边力和摩擦系数对板料界面接触压力分布的影响.研究表明:与压边力和摩擦系数相比,钢板强度和相对圆角半径更明显地影响着板料界面接触压力,并随着材料强度增加和相对模具圆角半径减小界面接触压力明显增加;随着压边力和摩擦系数的增加,界面接触压力宽度也随之增大.     

Investigation into relative slip during fretting fatigue under partial slip contact condition

A finite element analysis based methodology was developed to compute local relative slip on contact surface from the measured global relative slip away from contact surface. A set of springs was included in finite element model to simulate fretting fatigue test system. Compliance of springs was calibrated by comparing experimental and computed global relative slips. This methodology was then used to investigate local relative slip during fretting fatigue in cylinder‐on‐flat contact configuration under partial slip contact condition for unpeened and shot‐peened titanium alloy, Ti–6Al–4V. Relative slip on contact surface is significantly smaller (about one order) than the measured global relative slip by using a conventional extensometer near the contact surface. Effects of coefficient of friction, rigidity of fretting fatigue system and applied stress to specimen on the global and local relative slips were characterized. Coefficient of friction and contact load have considerable effect on local relative slip, and practically no effect on global relative slip. Gross slip condition can develop at some locations on contact surface in spite of overall partial slip condition. Increase in rigidity of fretting fatigue system increases local relative slip but decreases global relative slip. Finally, fatigue life diagrams based on relative slip on contact surface are established for both unpeened and shot‐peened titanium alloy. These show the same characteristics as of the conventional S–N diagram where fatigue life decreases with increase of relative slip.     

Simulation of surface pitting due to contact loading

A computational model for simulation of surface pitting of mechanical elements subjected to rolling and sliding contact conditions is presented. The two-dimensional computational model is restricted to modelling of high-precision mechanical components with fine surface finishing and good lubrication, where the cracks leading to pitting are initiated in the area of largest contact stresses at certain depth under the contacting surface. Hertz contact conditions with addition of friction forces are assumed and the position and magnitude of the maximum equivalent stress is determined by the finite element method. When the maximum equivalent stress exceeds the local material strength, it is assumed that the initial crack develops along the slip line in a single-crystal grain. The Virtual Crack Extension method in the framework of finite element analysis is then used for two-dimensional simulation of the fatigue crack propagation under contact loading from the initial crack up to the formation of the surface pit. The pit shapes and relationships between the stress intensity factor and crack length are determined for various combinations of contacting surface curvatures and loadings. The model is applied to simulation of surface pitting of two meshing gear teeth. Numerically predicted pit shapes in the face of gear teeth show a good agreement with the experimental observations. © 1998 John Wiley & Sons, Ltd.     

Frictional contact analysis of multiple cracks by incremental displacement and resultant traction boundary integral equations

Based on the merits of the dual boundary element technique, a modified dual boundary element technique is extended to deal with the frictional contact of a finite plate with arbitrarily distributed multiple cracks. Besides establishing the incremental displacement boundary integral equation on the outer boundary, the resultant traction boundary integral equation on one of the crack surfaces is also developed. Since the resultant traction instead of incremental traction on the crack surface is introduced, the computed resultant contact tractions under sliding condition satisfy the Coulomb's friction law directly. Hence, as compared with the authors' previous work, only very few computation iterations are required by this method to accurately describe the contact situations of crack surfaces. As a result, not only the linear cracks, but also other types of multiple cracks, for example, curved and kinked cracks, can be tackled. The effects of friction and interaction among cracks on the computation of stress intensity factors are also displayed.     

New benchmark problems for verification of the curve-to-surface contact algorithm based on the generalized Euler–Eytelwein problem

Development of the numerical contact algorithms for finite element method usually concerns convergence, mesh dependency, etc. Verification of the numerical contact algorithm usually includes only a few cases due to a limited number of available analytic solutions (e.g., the Hertz solution for cylindrical surfaces). The solution of the generalized Euler–Eytelwein, or the belt friction problem is a stand alone task, recently formulated for a rope laying in sliding equilibrium on an arbitrary surface, opens up to a new set of benchmark problems for the verification of rope/beam to surface/solid contact algorithms. Not only a pulling forces ratio ${\displaystyle \frac{T}{T_0}}$, but also the position of a curve on a arbitrary rigid surface withstanding the motion in dragging direction should be verified. Particular situations possessing a closed form solution for ropes and rigid surfaces are analyzed. The verification study is performed employing the specially developed Solid-Beam finite element with both linear and $C1$-continuous approximations together with the Curve-to-Solid Beam (CTSB) contact algorithm and exemplary employing commercial finite element software. A crucial problem of "contact locking" in contact elements showing stiff behavior despite the good convergence is identified. This problem is resolved within the developed CTSB contact element.     

Simple Common Plane contact algorithm

The common‐plane (CP) algorithm is widely used in the discrete element method to model contact forces between interacting particles or blocks of rock. A new simple contact algorithm, similar to the CP algorithm, is proposed to model discontinuities such as joints, faults and material interfaces in an explicit finite difference code. The CP is defined as a plane separating interacting faces of grid cells, instead of blocks or particles used in the original CP method. The new method does not require iterations even for very stiff contacts. It is very robust and easy to implement, both in 2D and 3D parallel codes. Copyright © 2011 John Wiley & Sons, Ltd.     

Direct numerical simulation of the dynamics of sliding rough surfaces

The noise generated by the friction of two rough surfaces under weak contact pressure is usually called roughness noise. The underlying vibration which produces the noise stems from numerous instantaneous shocks (in the microsecond range) between surface micro-asperities. The numerical simulation of this problem using classical mechanics requires a fine discretization in both space and time. This is why the finite element method takes much CPU time. In this study, we propose an alternative numerical approach which is based on a truncated modal decomposition of the vibration, a central difference integration scheme and two algorithms for contact: The penalty algorithm and the Lagrange multiplier algorithm. Not only does it reproduce the empirical laws of vibration level versus roughness and sliding speed found experimentally but it also provides the statistical properties of local events which are not accessible by experiment. The CPU time reduction is typically a factor of 10.