A contact‐stabilized Newmark method for dynamical contact problems

The numerical integration of dynamical contact problems often leads to instabilities at contact boundaries caused by the non‐penetration condition between bodies in contact. Even an energy dissipative modification (see, e.g. (Comp. Meth. Appl. Mech. Eng. 1999; 180 :1–26)), which discretizes the non‐penetration constraints implicitly, is not able to circumvent artificial oscillations. For this reason, the present paper suggests a contact stabilization in function space, which avoids artificial oscillations at contact interfaces and is also energy dissipative. The key idea of this contact stabilization is an additional L²‐projection at contact interfaces, which can be easily added to any existing time integration scheme. In case of a lumped mass matrix, this projection can be carried out completely locally, thus creating only negligible additional numerical cost. For the new scheme, an elementary analysis is given, which is confirmed by numerical findings in an illustrative test example (Hertzian two‐body contact). Copyright © 2007 John Wiley & Sons, Ltd.     

Stress and fracture analysis in periodic contact problem for coated bodies

The periodic contact problem for the system of dies and two‐layered elastic foundation is considered. The imperfect adhesion between the layer and the substrate is taken into account by the boundary conditions at the interface. The solution of the problem is based on Hankel transform and the methods of localisation and superposition. The results are used to evaluate internal stresses in coated bodies which are in contact with rough surface and to analyse the influence of mechanical and geometrical characteristics of the coatings and the roughness parameters on the maximum stress value and its location, and on the types of coating fracture.     

Thermoelastic contact problem for a layer interacting with a rigid substrate under transient frictional heat generation conditions

A thermoelastic problem has been solved for the interaction between an elastic layer and a rigid substrate from which a body may come off under localized pressing load. The paper addresses the influence of an accompanying transient frictional heat generation and functional relation of a pressing load on the process of the layer displacement over the surface of a heat-insulated substrate. It is demonstrated that an increase in heat generation intensity reduces the contact area, where the time variation of the latter is governed by selection of a load function and a motion speed function: if the load is time dependent, the contact area remains unchanged; if the speed varies, the contact area exhibits a monotonic reduction. __________ Translated from Problemy Prochnosti, No. 3, pp. 132–151, May–June, 2008.     

Critical time step for DEM simulations of dynamic systems using a Hertzian contact model

The discrete element method (DEM) typically uses an explicit numerical integration scheme to solve the equations of motion. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time step. Currently, there are no comprehensive techniques for estimating appropriate DEM time steps when a nonlinear contact interaction is used. It is common practice to apply a large factor of safety to these estimates to ensure stability, which unnecessarily increases the computational cost of these simulations. This work introduces an alternative framework for selecting a stable time step for nonlinear contact laws, specifically for the Hertz-Mindlin contact law. This approach uses the fact that the discretised equations of motion take the form of a nonlinear map and can be analysed as such. Using this framework, we analyse the effects of both system damping and the initial relative velocity of collision on the critical time step for a Hertz-Mindlin contact event between spherical particles.     

A dual Lagrange method for contact problems with regularized contact conditions

This paper presents an algorithm for solving quasi‐static, non‐linear elasticity contact problems without friction in the context of rough surfaces. Here, we want to model the transition from soft to hard contact in case of rough surfaces on the micro‐scale. The popular dual mortar method is used to enforce the contact constraints in a variationally consistent way without increasing the algebraic system size. The algorithm is deduced from a perturbed Lagrange formulation and combined with mass‐lumping techniques to exploit the full advantages of the duality pairing. This leads to a regularized saddle point problem, for which a non‐linear complementary function and thus a semi‐smooth Newton method can be derived. Numerical examples demonstrate the applicability to industrial problems and show a good agreement to experimentally obtained results. Copyright © 2014 John Wiley & Sons, Ltd.     

A consistent approach for fluid-structure-contact interaction based on a porous flow model for rough surface contact

Simulation approaches for fluid-structure-contact interaction, especially if requested to be consistent even down to the real contact scenarios, belong to the most challenging and still unsolved problems in computational mechanics. The main challenges are 2-fold—one is to have a correct physical model for this scenario, and the other is to have a numerical method that is capable of working and being consistent down to a zero gap. Moreover, when analyzing such challenging setups of fluid-structure interaction, which include contact of submersed solid components, it gets obvious that the influence of surface roughness effects is essential for a physical consistent modeling of such configurations. To capture this system behavior, we present a continuum mechanical model that is able to include the effects of the surface microstructure in a fluid-structure-contact interaction framework. An averaged representation for the mixture of fluid and solid on the rough surfaces, which is of major interest for the macroscopic response of such a system, is introduced therein. The inherent coupling of the macroscopic fluid flow and the flow inside the rough surfaces, the stress exchange of all contacting solid bodies involved, and the interaction between fluid and solid are included in the construction of the model. Although the physical model is not restricted to finite element–based methods, a numerical approach with its core based on the cut finite element method, enabling topological changes of the fluid domain to solve the presented model numerically, is introduced. Such a cut finite element method–based approach is able to deal with the numerical challenges mentioned above. Different test cases give a perspective toward the potential capabilities of the presented physical model and numerical approach.     

A constrained‐optimization methodology for the detection phase in contact mechanics simulations

The detection phase in computational contact mechanics can be subdivided into a global search and a local detection. When potential contact is detected by the former, a rigorous local detection determines which surface elements come or may come in contact in the current increment. We first introduce a rigorous definition of the closest point for non‐differentiable lower‐dimensional manifolds. We then simplify the detection by formulating an optimization problem subject to inequality constraints. The formulation is then solved using different techniques from the field of mathematical optimization, for both linear and quadratic finite element meshes. The resulting general and robust detection scheme is tested on a set of problems and compared with other techniques commonly used in computational geometry. Copyright © 2013 John Wiley & Sons, Ltd.     

电连接器接触件应力场的数值分析与试验验证

利用ANSYS对某型号电连接器接触件应力场进行分析,以寻找应力变化规律.对接触件应力场分布特点、插孔形变、接触压力随温度升高的变化规律进行了仿真研究,并进行数据分析与接触压力的试验验证.结果表明:随着温度升高,插孔最大形变量增加,最大接触压力和接触压力区域都有所减少;尺寸较小的接触件插孔槽缝底部最大等效应力随温度升高增幅较大,交变载荷作用时易出现疲劳、破裂等,属产品失效薄弱点.通过分析得知最大等效应力值随温度上升而变化的趋势取决于温度软化效应和热应力增强作用的综合结果;仿真结果能较好地反映电连接器的工作应力状态,接触压力试验验证了有限元仿真方法的可行性.     

An augmented Lagrangian algorithm for contact mechanics based on linear regression

The penalty method for the solution of contact problems yields an approximate satisfaction of the contact constraints. Augmentation schemes, which can be adopted to improve the solution, either include the contact forces as additional unknowns or are strongly affected by the penalty parameter and display a poor convergence rate. In a previous investigation, an unconventional augmentation scheme was proposed, on the basis of estimating the ‘exact’ values of the contact forces through linear interpolation of a database extracted by the previous converged states. An enhanced version of this method is proposed herein. With respect to the original method, the enhanced one eliminates some numerical problems and improves the regularity of the convergence path by carrying out the estimate through linear regression methods. The resulting convergence rate is superlinear, and the method is quite insensitive to the penalty parameter. The main underlying concept is that, within the iterative solution of a non‐linear problem, linear regression techniques may be used as a tool to ‘shoot’ faster to the final solution, on the basis of a set of intermediate data. The generality of this concept makes it potentially applicable to contact problems in more general settings, as well as to other categories of non‐linear problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.     

Experimental observation and numerical modeling of formation of local plastic zones in hardened surface layers due to contact overloading

The problem of formation of plastic zones in case-hardened metallic bodies due to contact overloading is studied both experimentally and numerically. Metallic materials exposed to surface hardening demonstrate spatial variation of the material hardness and yield strength with a decreasing profile with depth and belong to the class of so-called plastically graded materials. The presented experimental program employs micro-Vickers hardness tests to map the variation in material hardness and corresponding yield strength for both virgin and loaded case-hardened specimens made of a chromium tool steel. It is shown that, depending on the profile of the yield strength in the near-surface zones and contact parameters, a plastic deformation can originate underneath the hardened layer. The distribution of the effective plastic strain extracted from the micro-hardness increment measurements are found in good agreement with the results of finite element simulations of a plastically graded material subjected to similar loading conditions. Numerical analysis reveals significant perturbations in the stress field distribution within the hardened layer due to formation of a closed-shaped plastic zone in the gradient layers, including development of a tensile stress on the boundary between the elastic and plastic zones as well as an overall increase in the effective stress intensity. It is shown that the hardened layer behaves similar to an elastic beam on a compliant foundation. These stress field perturbations in the hardened layers with low deformation capacity can greatly affect the durability and serviceability of surface treated mechanical parts.     

增强体对复合材料接触性能的影响

基于等效夹杂方法,引入一种数值建模方法用于求解Hertz接触载荷作用下复合材料次表面应力场。通过与有限元方法的对比验证本方法的有效性和优越性;讨论不同形状增强体深度、材料、尺寸、体积分数、相对位置等分布参数对基体应力场的影响。分析结果表明,双增强体接触模型中,次表面最大von Mises应力随增强体深度和半径的增大呈先增大后减小趋势,随增强体与基体之间材料差异的增大而单调递增;增强体体积分数及相对位置将对基体应力分布产生较大影响。通过钛基复合材料滚动接触疲劳实验验证了本文方法处理复合材料接触性能的能力。     

可控震源振动器平板-大地接触性质与能量传递研究

大地的表面形貌是影响可控震源振动器平板与大地之间接触性质的因素之一。为了掌握大地表面参数对可控震源振动器平板-大地接触系统接触性质及振动特性的影响规律，基于分形理论建立了三维粗糙大地表面形貌，构建了振动器平板-大地接触模型，得到了不同大地表面形貌及材料参数下振动器平板与大地之间的接触力-变形曲线；建立了振动器平板-大地接触振动动力学方程，并计算了振动器平板的位移响应以及能量传递。结果表明：振动器平板与大地之间接触力的非线性随大地表面粗糙度的增大而增大，振动器平板-大地接触系统的固有频率随大地表面粗糙度的增大而减小；粗糙表面的接触模型中振动器平板每一周期向大地传递的能量随时间的增大而减小。同时，大地表面材料参数的非线性也会影响平板-大地接触系统的振动响应和能量传递。由此可知，大地表面的粗糙度和材料的非线性是限制可控震源高频输出的重要原因。研究结果将为可控震源的优化和高频拓展提供参考。     

Hybrid hyper‐reduced modeling for contact mechanics problems

The model reduction of mechanical problems involving contact remains an important issue in computational solid mechanics. In this article, we propose an extension of the hyper‐reduction method based on a reduced integration domain to frictionless contact problems written by a mixed formulation. As the potential contact zone is naturally reduced through the reduced mesh involved in hyper‐reduced equations, the dual reduced basis is chosen as the restriction of the dual full‐order model basis. We then obtain a hybrid hyper‐reduced model combining empirical modes for primal variables with finite element approximation for dual variables. If necessary, the inf‐sup condition of this hybrid saddle‐point problem can be enforced by extending the hybrid approximation to the primal variables. This leads to a hybrid hyper‐reduced/full‐order model strategy. This way, a better approximation on the potential contact zone is further obtained. A posttreatment dedicated to the reconstruction of the contact forces on the whole domain is introduced. In order to optimize the offline construction of the primal reduced basis, an efficient error indicator is coupled to a greedy sampling algorithm. The proposed hybrid hyper‐reduction strategy is successfully applied to a 1‐dimensional static obstacle problem with a 2‐dimensional parameter space and to a 3‐dimensional contact problem between two linearly elastic bodies. The numerical results show the efficiency of the reduction technique, especially the good approximation of the contact forces compared with other methods.     

A reduced basis method for parametrized variational inequalities applied to contact mechanics

We investigate new developments of the reduced-basis method for parametrized optimization problems with nonlinear constraints. We propose a reduced-basis scheme in a saddle-point form combined with the Empirical Interpolation Method to deal with the nonlinear constraint. In this setting, a primal reduced-basis is needed for the primal solution and a dual one is needed for the Lagrange multipliers. We suggest to construct the latter using a cone-projected greedy algorithm that conserves the non-negativity of the dual basis vectors. The reduction strategy is applied to elastic frictionless contact problems including the possibility of using nonmatching meshes. The numerical examples confirm the efficiency of the reduction strategy.     

Algebraic multigrid methods for dual mortar finite element formulations in contact mechanics

This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so‐called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle‐point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact‐specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.     

A fracture mechanics based model for the analysis of seal effectiveness

The fluid containment in vessels, pipes, containers, etc. often requires the use of seals in order to assure the absence of leak in the junction zones. Sealing mechanism is typically achieved through the use of elastomeric elements that form contact with the surrounding rigid materials the containers are made of. A proper design and safety evaluation of the containment capacity of seals requires the careful evaluation of the contact pressure distribution between the soft (seal) and hard (vessel) elements. In the present paper such a problem is considered and solved through contact stress and strain evaluation based on fracture mechanics; numerical and experimental analyses on elastomeric elements are considered in order to verify the proposed modeling procedure. It is shown that the desired safety level against leakage can be ensured on the basis of the classical fracture mechanics parameters when the seal crack tip exists, or through contact strain assessment when the stress singularity vanishes. Such results can be useful in the design of seal shapes and for estimating the pressure to be applied to the sealed bodies in order to guarantee no leaks. Finally, some final relevant conclusions on the present study on leak containment are drawn.     

A solution method for frictional contact problems of a crack in FGMs under mechanical and thermal loads

This article provides a numerical treatment of a finite crack in an interfacial layer with spatially varying elastic properties under in-plane mechanical and thermal loading conditions. The variation of stress intensity factors and energy release rates with the functions which are governing the material properties of the interfacial layer is studied. Transient and steady-state response of a central crack in FGMs subjected to the mechanical and thermal loads are investigated. Unlike earlier studies which consider the cracks encountered as open, the current investigation studies cracks in an essentially compressive environment in which the crack faces are in contact and frictional effects play an important role. To solve this contact problem, a simple and efficient, iterative finite element method developed by authors is used. Numerical examples are provided to verify the technique and the results are compared with those of the published papers. This revised version was published online in July 2006 with corrections to the Cover Date.