A Parallel Finite Element Method for 3D Two-Phase Moving Contact Line Problems in Complex Domains

Moving contact line problem plays an important role in fluid-fluid interface motion on solid surfaces. The problem can be described by a phase-field model consisting of the coupled Cahn–Hilliard and Navier–Stokes equations with the generalized Navier boundary condition (GNBC). Accurate simulation of the interface and contact line motion requires very fine meshes, and the computation in 3D is even more challenging. Thus, the use of high performance computers and scalable parallel algorithms are indispensable. In this paper, we generalize the GNBC to surfaces with complex geometry and introduce a finite element method on unstructured 3D meshes with a semi-implicit time integration scheme. A highly parallel solution strategy using different solvers for different components of the discretization is presented. More precisely, we apply a restricted additive Schwarz preconditioned GMRES method to solve the systems arising from implicit discretization of the Cahn–Hilliard equation and the velocity equation, and an algebraic multigrid preconditioned CG method to solve the pressure Poisson system. Numerical experiments show that the strategy is efficient and scalable for 3D problems with complex geometry and on a supercomputer with a large number of processors.     

A contact friction algorithm including nonlinear viscoelasticity and a singular yield surface provision

A finite element solution method for two-dimensional boundary value problems involving nonlinear viscoelasticity and contact friction is presented. The simulation of ceramic composites at elevated temperatures is the motivation of coupling viscoelasticity and contact friction. Three major topics are discussed; (i) the time-integration scheme developed for coupling the interface contact friction with nonlinear viscoelasticity in the surrounding continuum, (ii) the spatial discretization of a generalized two-dimensional deformation field using finite elements, and (iii) two methodologies for treating the nonlinearities introduced by the contact friction. The implementation of the contact friction utilizes a penalty method coupled with an incremental plasticity formulation. This formulation results in a highly nonlinear problem, and many solution techniques have difficulty with convergence due to a directional sensitivity arising from the unknown slip direction in the case of three-dimensional contact friction (or the unknown slip direction for hardening or dilatant two-dimensional friction problems). This directional sensitivity is illustrated along with an algorithm which alleviates this difficulty. Also, an algorithm based upon proportional stressing is developed to eliminate problems created by a singular yield surface for idealized Coulomb friction.     

A mixed interface finite element for cohesive zone models

The phenomena of crack initiation, propagation and ultimate fracture are studied here under the following assumptions: (i) the crack law is modelled by means of a cohesive zone model and (ii) the crack paths are postulated a priori. In this context, a variational formulation is proposed which relies on an augmented Lagrangian. A mixed interface finite element is introduced to discretise the crack paths, the degrees of freedom of which consist in the displacement on both crack lips and the density of cohesive forces. This enables an exact treatment of multi-valued cohesive laws (e.g. initial adhesion, contact conditions, possible rigid unloading, etc.), without penalty regularisation.A special attention is paid to the convergence with mesh-refinement, i.e. the well-posedness of the problem, on the basis of theoretical results of contact mechanics and some complementary numerical investigations. Fulfilment of the LBB condition is the key factor to gain the desired properties. Moreover, it is shown that the integration of the constitutive law admits a unique solution as soon as some condition on the augmented Lagrangian is enforced. Finally, a 3D simulation shows the applicability to practical engineer problems, including in particular the robustness of the formulation and its compatibility with classical solution algorithms (Newton method, line-search, path-following techniques).     

A Dirichlet–Neumann type algorithm for contact problems with friction

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We introduce a new algorithm for the numerical solution of a nonlinear contact problem with Coulomb friction between linear elastic bodies. The discretization of the nonlinear problem is based on mortar techniques. We use a dual basis Lagrange multiplier space for the coupling of the different bodies. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from the mortar discretization on non-matching triangulations. We apply a nonlinear block Gauss–Seidel method as iterative solver which can be interpreted as a Dirichlet–Neumann algorithm for the nonlinear problem. In each iteration step, we have to solve a linear Neumann problem and a nonlinear Signorini problem. The solution of the Signorini problem is realized in terms of monotone multigrid methods. Numerical results illustrate the performance of our approach in 2D and 3D. Received: 20 March 2001 / Accepted: 1 February 2002 Communicated by P. Deuflhard     

A FEM–BEM coupling method for a nonlinear transmission problem modelling Coulomb friction contact

A nonlinear transmission problem modelling elastoplastic interface problems with contact and Coulomb friction is reduced to a boundary/domain variational inequality. We present a corresponding FEM/BEM coupling procedure and derive for its Galerkin solution an a priori error estimate. Furthermore we reformulate the problem as an equivalent saddle point problem whose discretization can be solved by the Uzawa algorithm. Convergence of the FEM/BEM coupling method is proved and numerical results are given.     

Surface Smoothing Procedures in Computational Contact Mechanics

This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.     

滚动接触疲劳载荷对铁轨表面接触疲劳裂纹应力强度因子的影响

为研究轮轨滚动接触疲劳（Rolling Contact Fatigue,RCF）载荷对铁轨表面裂纹应力强度因子的影响,以UIC60铁轨轮廓尺寸为依据建立轮轨接触的三维有限元模型,通过改变RCF载荷大小、轮轨表面摩擦因数和接触中心位置等轮轨接触的输入参数,计算铁轨表面接触裂纹尖端的应力强度因子,分析RCF载荷对铁轨表面接触疲劳裂纹的影响. 结果表明RCF载荷作为控制铁轨表面接触裂纹的重要因素,其变化直接导致裂纹尖端应力强度因子的变化,从而改变裂纹的扩展状况;为减缓铁轨表面裂纹的扩展,可以针对载荷采取均匀分布载重量、使用润滑剂降低轮轨摩擦因数等相应措施.     

XFEM在高压管道裂纹扩展计算中的应用

介绍XFEM的基本理论,针对常规有限元计算困难的三维裂纹,运用XFEM完成三维高压管道中的裂纹扩展计算.根据管道裂纹扩展的计算结果,总结得到管道裂纹扩展的规律,得到径向裂纹比轴向裂纹更严重的结论.     

A finite element technique for determining mode I stress intensity factors: Application to no-slip bimaterial crack problems

A comparative finite element technique, using conventional finite elements, is presented for the determination of mode I stress intensity factors in two-dimensional crack problems. Given a crack problem to be solved and an auxiliary crack problem for which the mode I stress intensity factor K_I is readily available, it is argued that the ratio of K_Is for these two problems can be approximated by the ratio of corresponding crack opening displacements near the crack tips, as obtained from finite element solutions. The geometry and loading of the auxiliary problem need not be related to those of the problem to be solved; however, it is essential that the mesh configurations around the crack tips be identical so that numerical errors inherent to the finite element discretization process be the same for the two problems. The validity of this technique is checked for several two-dimensional problems for cracks in homogeneous material whose solutions are available in the literature. Then, it is verified that the method applies to problems of no-slip cracks at a bimaterial interface, in which the no-slip condition is enforced by including no-slip blocks along the crack faces. Finally, this technique is used to predict the stress intensity factors for a four-point bending specimen with an edge no-slip crack at the bimaterial interface.     

New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem

We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed.     

Variational formulation and nonsmooth optimization algorithms in elastodynamic contact problems for cracked body

A mathematical model of an elastodynamic contact problem for a body with a crack with unilateral restrictions and friction on the crack faces is presented in classical and weak forms. Different variational formulations of unilateral contact problems with friction based on the principles of Hamilton–Ostrogradskii and Tupin, and boundary variational principles are considered. In particular, boundary variational functionals that are used with boundary integral equations are established. Nonsmooth optimization algorithms of Udzawa type for the solution of these unilateral contact problems with friction are developed. The convergence of the proposed algorithms is studied numerically.     

Finite element algorithms for contact problems

Summary The numerical treatment of contact problems involves the formulation of the geometry, the statement of interface laws, the variational formulation and the development of algorithms. In this paper we give an overview with regard to the different topics which are involved when contact problems have to be simulated. To be most general we will derive a geometrical model for contact which is valid for large deformations. Furthermore interface laws will be discussed for the normal and tangential stress components in the contact area. Different variational formulations can be applied to treat the variational inequalities due to contact. Several of these different techniques will be presented. Furthermore the discretization of a contact problem in time and space is of great importance and has to be chosen with regard to the nature of the contact problem. Thus the standard discretization schemes will be discussed as well as techiques to search for contact in case of large deformations.     

聚氨酯橡胶摩擦轮有限元及疲劳寿命分析

针对汽车滑板输送机聚氨酯橡胶摩擦轮的开裂问题,对摩擦轮与滑板之间的相互作用进行预压紧力和接触摩擦驱动有限元分析,获得摩擦轮的应力和变形分布等结果. 在获得摩擦轮周向应力分布的基础上,根据疲劳裂纹扩展公式对聚氨酯橡胶部分进行疲劳寿命分析,验证是否满足设计要求. 分析过程和结果能为聚氨酯橡胶摩擦轮的详细设计提供参考.     

聚氨酯橡胶摩擦轮有限元及疲劳寿命分析

针对汽车滑板输送机聚氨酯橡胶摩擦轮的开裂问题,对摩擦轮与滑板之间的相互作用进行预压紧力和接触摩擦驱动有限元分析,获得摩擦轮的应力和变形分布等结果.在获得摩擦轮周向应力分布的基础上,根据疲劳裂纹扩展公式对聚氨酯橡胶部分进行疲劳寿命分析,验证是否满足设计要求.分析过程和结果能为聚氨酯橡胶摩擦轮的详细设计提供参考.     

Influence of micro-cracking and contact on the effective properties of composite materials

In the present work a novel micro-mechanical approach to analyze the influence of micro-crack evolution and contact on the effective properties of elastic composite materials is proposed, based on homogenization techniques, interface models and fracture mechanics concepts. By means of the finite element method, enhanced non-linear macroscopic constitutive laws are developed by taking into account changes in micro-structural configuration associated with the growth of micro-cracks and with contact between crack faces. Numerical simulations are carried out for the cases of a porous composite with edge cracks and of a debonded fibre reinforced composite, loaded along extension/compression uniaxial macro-strain paths. Micro-crack propagation is modelled by using an original methodology based on the J-integral technique in conjunction with an interface model taking into account the unilateral contact of crack faces. In the context of a micro-to-macro transition obtained by controlling the macro-deformation of the micro-structure, the effects of adopting three types of boundary conditions on the macroscopic constitutive law, namely linear deformation, uniform tractions and periodic deformations and anti-periodic tractions, are studied. As a consequence, the proposed method can be applied to a large class of problems including periodic, locally periodic and irregular composite materials. Micro-crack and contact evolution result in a progressive loss of stiffness and can lead to failure for homogeneous macro-deformations associated with unstable crack propagation.     

An extended finite element method for modeling crack growth with frictional contact

A new technique for the finite element modeling of crack growth with frictional contact on the crack faces is presented. The eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry are independent of the finite element mesh. This method greatly facilitates the simulation of a growing crack, as no remeshing of the domain is required. The conditions which describe frictional contact are formulated as a non-smooth constitutive law on the interface formed by the crack faces, and the iterative scheme implemented in the LATIN method [Nonlinear Computational Structural Mechanics, Springer, New York, 1998] is applied to resolve the nonlinear boundary value problem. The essential features of the iterative strategy and the X-FEM are reviewed, and the modifications necessary to integrate the constitutive law on the interface are presented. Several benchmark problems are solved to illustrate the robustness of the method and to examine convergence. The method is then applied to simulate crack growth when there is frictional contact on the crack faces, and the results are compared to both analytical and experimental results.     

Asymptotic numerical method for problems coupling several nonlinearities

The objective of this paper is to assess the efficiency of the asymptotic numerical method to solve problems coupling various nonlinearities. The 3D hemispherical stretching of a circular sheet, that involves geometrical, material and red unilateral contact nonlinearities is chosen as an example. An elastoplastic model based on the plasticity deformation theory is adopted. The structural discretization is performed by a shell finite element well adapted for problems involving large displacements and large rotations. The unilateral contact problem is identified to boundary conditions which are replaced by force–displacement relations and solved using a special algorithm. Comparisons with results obtained by the help of an industrial code establish the interest and the performance of the present method.     

基于群体智能的三维碎片全局最优匹配方法

针对传统三维碎片整体匹配过程中误差积累的问题,提出了一种基于群体智能的全局最优匹配方法。该方法对破碎物体的三维多碎片全局匹配建立全局整体碎片匹配的数学模型,将碎片的整体最优匹配求解问题转换为求满足一定约束条件的最优匹配矩阵的组合优化问题,通过将自然社会认知优化算法进行离散化来求解该NP问题。典型实例分析验证了所提方法全局优化能力强,与初始位置无关,有较强的鲁棒性,为三维碎片整体匹配提供一个有效的方法。     

A family of space–time connecting discretization schemes with local impact detection for elastodynamic contact problems

We derive, formulate and analyze a new family of discretization schemes for elastodynamic contact problems which implicitly resolve the individual impact times for each node on the contact interface. Within our approach, information from the space discrete setting is incorporated into the time discretization by means of pointwise chosen parameters for the time discretization scheme. The members of this family can be interpreted as modified Newmark schemes, thus making them easily understandable and implementable. We prove that for certain parameter choices the algorithms are dissipative methods. Further, as our analysis and our numerical experiments show, a special solution dependent choice of parameters leads to a new space–time connecting discretization with a highly stable behavior of displacements, velocities and boundary stresses at the contact interface.