A contact formulation based on localized Lagrange multipliers: formulation and application to two‐dimensional problems

The non‐penetration condition in contact problems is traditionally based on the classical Lagrange multiplier method. This method makes extensive use of modelling details of the contacting bodies for contact enforcement as the contact surface meshes are in general non‐matching. To deal with this problem we introduce a novel element in the Lagrange multiplier approach of contact modelling, namely, a contact frame placed in between contacting bodies. It acts as a medium through which contact forces are transferred without violating equilibrium in the contact domain for discrete contact models. Only nodal information of the contacting bodies is required which makes the proposed contact enforcement generic. The contact frame has its own independent freedoms, which allows the formulation to pass contact patch tests by design. Copyright © 2002 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.     

A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method

This paper introduces a new algorithm to define a stable Lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method. In contrast to earlier approaches, we do not work with an interior penalty formulation as, e.g. for Nitsche techniques, but impose the constraints weakly in terms of Lagrange multipliers. Roughly speaking a stable and optimal discrete Lagrange multiplier space has to satisfy two criteria: a best approximation property and a uniform inf–sup condition. Owing to the fact that the interface does not match the edges of the mesh, the choice of a good discrete Lagrange multiplier space is not trivial. Here we propose a new algorithm for the local construction of the Lagrange multiplier space and show that a uniform inf–sup condition is satisfied. A counterexample is also presented, i.e. the inf–sup constant depends on the mesh‐size and degenerates as it tends to zero. Numerical results in two‐dimensional confirm the theoretical ones. Copyright © 2008 John Wiley & Sons, Ltd.     

Toeplitz矩阵压缩恢复的两种中值修正的增广Lagrange乘子算法

增广Lagrange乘子算法是求解矩阵压缩恢复的一种有效迭代方法．为了有效求解Toeplitz矩阵压缩恢复模型,本文提出了两种中值修正的增广Lagrange乘子算法．在新算法中,对增广Lagrange乘子算法每步产生的迭代矩阵进行中值修正并保证其Toeplitz结构．新算法不仅减少了奇异值分解所用的时间和CPU时间,而且获得更精确的迭代矩阵．同时,本中还详细给出了两种新算法的收敛性分析．最后通过数值例子验证了新算法的可行性和有效性,并展示了新算法在计算时间和精度方面比增广Lagrange乘子算法更有优势．     

巴西圆盘中心裂纹摩擦接触问题的逐点Lagrange乘子法

采用逐点Lagrange乘子法求解巴西圆盘中心裂纹在压剪荷载作用下裂纹面可能发生的摩擦接触问题。为了避免传统的Lagrange乘子法中总刚度阵求逆的困难,将Lagrange乘子逐点转到局部坐标系下,采用Gauss-Seidel迭代法求解法向和切向乘子,同时注意在求解的过程中对切向乘子约束修正,待所有点乘子求解完成后再变换到整体坐标系下迭代求解位移。与传统接触算法相比,该算法无需对总刚度阵求逆,降低了求解规模,提高了计算效率。通过该方法计算了巴西圆盘中心裂纹两种典型情况下的应力强度因子,计算结果与文献比较,吻合良好。考虑不同荷载角和裂纹长度对位移,应力强度因子和接触区的影响,并对不同摩擦系数下应力强度因子的影响进行了分析。结果表明:忽略裂纹接触摩擦作用,应力强度因子可能被高估。     

Microcrack nucleation in thermal barrier coating systems

Crack nucleation in thermal-barrier coating (TBC) systems subjected to a monotonic cooling process is studied. The TBC system is modeled using the finite element method, where cracks are represented as discrete discontinuities across continuum elements using the partition-of-unity method. The numerical implementation used for crack nucleation is based on an algorithm where, at insertion of a discontinuity, the traction response is derived from a cohesive zone model that has been modified to (i) behave like an initially rigid cohesive model, and to (ii) ensure smoothness of the traction-separation law at zero crack opening. Accordingly, an adequate convergence behavior of the numerical formulation can be warranted in boundary value problems of systems with relatively complex geometries. In the present numerical study, a comparison is made between TBC systems composed of different constitutive models. The fracture patterns and evolutions of the overall crack growth resulting from the simulations clearly illustrate the importance of accounting for the effects of plasticity in the bond coating and anisotropy in the top coating. The computed fracture profile is in good correspondence with experimental observations reported in the literature.     

Interface problems with quadratic X‐FEM: design of a stable multiplier space and error analysis

The aim of this paper is to propose a procedure to accurately compute curved interfaces problems within the extended finite element method and with quadratic elements. It is dedicated to gradient discontinuous problems, which cover the case of bimaterials as the main application. We focus on the use of Lagrange multipliers to enforce adherence at the interface, which makes this strategy applicable to cohesive laws or unilateral contact. Convergence then occurs under the condition that a discrete inf‐sup condition is passed. A dedicated P1 multiplier space intended for use with P2 displacements is introduced. Analytical proof that it passes the inf‐sup condition is presented in the two‐dimensional case. Under the assumption that this inf‐sup condition holds, a priori error estimates are derived for linear or quadratic elements as functions of the curved interface resolution and of the interpolation properties of the discrete Lagrange multipliers space. The estimates are successfully checked against several numerical experiments: disparities, when they occur, are explained in the literature. Besides, the new multiplier space is able to produce quadratic convergence from P2 displacements and quadratic geometry resolution. Copyright © 2014 John Wiley & Sons, Ltd.     

无单元Galerkin方法中周期边界条件的处理

本文研究了无单元Galerkin方法中周期边界条件的处理技术,将Lagrange乘子法用于周期边界条件的处理.数值计算结果表明,该方法具有较高的计算精度.另外,它与无单元Galerkin方法中本质边界条件处理的Lagrange乘子法具有统一性,对于周期、本质混合型边界条件的处理尤为方便.     

A yield‐limited Lagrange multiplier formulation for frictional contact

An analogy with rigid plasticity is used to develop a constitutive framework for quasi‐static frictional contact between finitely deforming solids. Within this setting, a Lagrange multiplier method is used to impose a sharp distinction between stick and slip. The scope of the multipliers is limited by a constitutively defined ‘yield’ function and a finite element‐based predictor–corrector scheme is employed to efficiently determine the regions of stick and slip and the associated tractions. Selected simulations of planar quasi‐static problems are presented to validate the method and illustrate its capabilities. Copyright © 2000 John Wiley & Sons, Ltd.     

Delamination and matrix cracking of cross-ply laminates due to a spherical indenter

An investigation was performed to study delamination growth induced by matrix cracks in cross-ply composites resulting from a spherical indenter. The major focus of the study was to understand fundamentally the damage mechanics in terms of matrix cracking and delamination growth, and interaction between them. A nonlinear three-dimensional finite element model based on an updated Lagrange formulation was developed during the investigation. An augmented Lagrangian method was utilized to model the delamination interface condition. A general contact node search algorithm was proposed which can handle complex contact conditions, such as arbitrary slippage and discontinuous curvature. The indentation resulting from the spherical rigid indenter was also modeled. Fracture mechanics was applied to determine the delamination propagation in three dimensions. The strain energy release rates were calculated by a crack-closure technique. The model was verified analytically and experimentally.     

A consistent tangent stiffness matrix for three-dimensional non-linear contact analysis

A consistent tangent stiffness matrix for the analysis of non-linear contact problems is presented. The associated element has three or four nodes and establishes contact between three-dimensional structures like solids and shells. It accounts for the non-linear kinematics of large deformation analysis and guarantees a quadratic convergence rate. Two formulations, the penalty method and the Lagrange multiplier method, are investigated.     

A stable energy‐conserving approach for frictional contact problems based on quadrature formulas

A common approach for the numerical simulation of non‐linear multi‐body contact problems is the use of Lagrange multipliers to model the contact conditions. The stability of standard algorithms is improved by introducing a modified mass matrix which assigns no mass to the potential contact nodes. By this, the spurious algorithmic oscillations in the multiplier do not occur any more, which facilitates the application of the primal–dual active set strategy to dynamical contact problems. The new mass matrix is calculated via a modified quadrature formula that needs no extra computational cost. In addition the conservation properties of the underlying algorithm are transferred to the modified mass version. Different numerical examples for frictional two‐body contact problems illustrate the improvement in the results for the contact stresses. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic and contact analysis of a bimodal ultrasonic motor

A bimodal ultrasonic motor, which operates with only one power amplifier, uses two simultaneously excited modes to drive the rotor; a longitudinal mode and a flexural mode. The equations of motion describing the vibrations and contact behavior are derived by Hamilton's principle and the geometry constraint. The Lagrange multiplier method is used to treat the frictional contact problem. The finite element method and numerical integration scheme are used to simulate the dynamic responses of this system with and without contact. Some important factors are studied for the bimodal ultrasonic motor design. The factors include structure design, amplitude of input voltage, phase displacement, exciting frequency, and contact behavior.     

Extended meshfree methods without branch enrichment for cohesive cracks

An extended meshless method for both static and dynamic cohesive cracks is proposed. This new method does not need any crack tip enrichment to guarantee that the crack closes at the tip. All cracked domains of influence are enriched by only the sign function. The domain of influence which includes a crack tip is modified so that the crack tip is always positioned at its edge. The modification is only applied for the discontinuous displacement field and the continuous field is kept unchanged. In addition to the new method, the use of Lagrange multiplier is explored to achieve the same goal. The crack is extended beyond the actual crack tip so that the domains of influence containing the crack tip are completely cut. It is enforced that the crack opening displacement vanishes along the extension of the crack. These methods are successfully applied to several well-known static and dynamic problems.     

Frictionless 2D Contact formulations for finite deformations based on the mortar method

In this paper two different finite element formulations for frictionless large deformation contact problems with non-matching meshes are presented. Both are based on the mortar method. The first formulation introduces the contact constraints via Lagrange multipliers, the other employs the penalty method. Both formulations differ in size and the way of fulfilling the contact constraints, thus different strategies to determine the permanently changing contact area are required. Starting from the contact potential energy, the variational formulation, the linearization and finally the matrix formulation of both methods are derived. In combination with different contact detection methods the global solution algorithm is applied to different two-dimensional examples.     

DESIGN OF ENERGY CONSERVING ALGORITHMS FOR FRICTIONLESS DYNAMIC CONTACT PROBLEMS

This paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum, angular momentum, and energy in finite element simulations. It is seen that a Lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties. A related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized. This penalty method sacrifices the energy conservation property, but is dissipative under all conditions of changing contact so that the global algorithm remains stable. Notably, it is also shown that augmented Lagrangian iteration utilizing this penalty kernel reproduces the energy conserving (i.e. Lagrange multiplier) solution to any desired degree of accuracy. The result is a robust, stable method even in the context of large deformations, as is shown by some representative numerical examples. In particular, the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations. © 1997 by John Wiley & Sons, Ltd.     

广义多体系统动力学的速度变换矩阵综合法

文中提出了广义多体系统和速度变换矩阵的概念,提出了一种新的加速度变换关系,以带不定乘子的拉格朗日方程为基础推导得到了求解复杂系统动力学问题的一种新方法,即广义多体系统的速度变换矩阵综合法。利用该方法,可根据无耦合广义体的动力学参数和系统的速度变换矩阵直接获得广义多体系统的动力学方程,其中不含拉格朗日不定乘子和约束反力,且方程中逆矩阵求解的维数等于系统的自由度数,因而有利于提高计算效率。该方法主要面向计算机实现程式化的算法,系统的动力方程可以由计算机自动完成运算,从而避免了繁琐的解析推导工作。     

Fast BEM–FEM mortar coupling for acoustic–structure interaction

A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure–acoustic field interaction. Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain. The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non‐matching grids. The resulting saddle‐point problem is solved by an approximate Uzawa‐type scheme in which the matrix–vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method. The algorithm is demonstrated on the example of a cavity‐backed elastic panel. Copyright © 2005 John Wiley & Sons, Ltd.     

一种适用于求解大规模结构的分步接触算法及其工程应用

大规模结构接触非线性问题的求解是当前工程界研究的热点和难点。该文基于传统的Lagrange乘子法提出了一种新的分步接触算法。该算法的基本原理是将接触问题分两步求解,第一步求解由整体系统力系平衡方程构成的控制方程,第二步求解接触局部区域的约束方程。该算法利用Lagrange乘子来精确模拟接触约束条件,同时对传统的Lagrange乘子法进行了解耦降维处理,所需存储量小、易于实现并行化,且通过引入缩放因子进一步提高了其求解效率,故非常适合高效求解大规模结构的接触问题。经典Hertz接触算例和平面双缝坝算例的结果验证了该算法的正确性,考虑内外衬接触非线性的穿黄隧洞整体模型工程应用算例说明了该算法的有效性。     

Level set topology optimization of structural problems with interface cohesion

This paper presents a finite element topology optimization framework for the design of two‐phase structural systems considering contact and cohesion phenomena along the interface. The geometry of the material interface is described by an explicit level set method, and the structural response is predicted by the extended finite element method. In this work, the interface condition is described by a bilinear cohesive zone model on the basis of the traction‐separation constitutive relation. The non‐penetration condition in the presence of compressive interface forces is enforced by a stabilized Lagrange multiplier method. The mechanical model assumes a linear elastic isotropic material, infinitesimal strain theory, and a quasi‐static response. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are computed by the adjoint method. The performance of the presented method is evaluated by 2D and 3D numerical examples. The results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered. In addition, 3D examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality. The optimization framework presented is limited to two‐phase structural systems where the material interface is coincident in the undeformed configuration, and to structural responses that remain valid considering small strain kinematics. Copyright © 2017 John Wiley & Sons, Ltd.