A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems

In the present work the mortar method is applied to planar large deformation contact problems without friction. In particular, the proposed form of the mortar contact constraints is invariant under translations and rotations. These invariance properties lay the foundation for the design of energy‐momentum time‐stepping schemes for contact–impact problems. The iterative solution procedure is embedded into an active set algorithm. Lagrange multipliers are used to enforce the mortar contact constraints. The solution of generalized saddle point systems is circumvented by applying the discrete null space method. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy‐momentum scheme. Copyright © 2008 John Wiley & Sons, Ltd.     

A Nitsche embedded mesh method

A new technique for treating the mechanical interactions of overlapping finite element meshes is proposed. Numerous names have been applied to related approaches, here we refer to such techniques as embedded mesh methods. Such methods are useful for numerous applications e.g., fluid-solid interaction with a superposed meshed solid on an Eulerian background fluid grid or solid-solid interaction with a superposed meshed particle on a matrix background mesh etc. In this work we consider the interaction of two elastic domains: one mesh is the foreground and defines the surface of interaction, the other is a background mesh and is often a grid. We first employ a classical mortar type approach [see Baaijens (Int J Numer Methods Eng 35:743–761, 2001)] to impose constraints on the interface. It turns out that this approach will work well except in special cases. In fact, many related approaches can exhibit mesh locking under certain conditions. This motivates the proposed version of Nitsche’s method which is shown to eliminate the locking phenomenon in example problems.     

Transient 3d contact problems—NTS method: mixed methods and conserving integration

The present work deals with a new formulation for transient large deformation contact problems. It is well known, that one-step implicit time integration schemes for highly non-linear systems fail to conserve the total energy of the system. To deal with this drawback, a mixed method is newly proposed in conjunction with the concept of a discrete gradient. In particular, we reformulate the well known and widely-used node-to-segment methods and establish an energy-momentum scheme. The advocated approach ensures robustness and enhanced numerical stability, demonstrated in several three-dimensional applications of the proposed algorithm.     

A mortared finite element method for frictional contact on arbitrary interfaces

This work concerns finite-element algorithms for imposing frictional contact constraints on intra-element, or embedded surfaces. Existing techniques typically rely on the underlying bulk mesh to implicitly partition the surface, a strategy that can give rise to overconstraint. In the present work, we first apply a mortaring algorithm to the modeling of frictional contact conditions on arbitrary interfaces. The algorithm is based upon a projection of the bulk and surface fields onto independent mortar fields at the interface. We examine the advantages of this approach when combined with extended finite-element approximations to the bulk fields. In particular, the method allows for bulk and surface domains to be partitioned separately, as well as enforce nonlinear contact constraints on surfaces that are not explicitly “fitted” to the bulk mesh. Results from several benchmark problems in frictional contact are provided to demonstrate the accuracy and efficacy of the method, as well as the improvement in robustness compared to existing techniques. We also provide an example that illustrates the effectiveness of the approach in high-speed machining simulation.     

A finite deformation mortar contact formulation using a primal–dual active set strategy

In recent years, nonconforming domain decomposition techniques and, in particular, the mortar method have become popular in developing new contact algorithms. Here, we present an approach for 2D frictionless multibody contact based on a mortar formulation and using a primal–dual active set strategy for contact constraint enforcement. We consider linear and higher‐order (quadratic) interpolations throughout this work. So‐called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non‐penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. The necessity of such a linearization in order to obtain a consistent Newton scheme is demonstrated. By further interpreting the active set search as a semi‐smooth Newton method, contact nonlinearity and geometrical and material nonlinearity can be resolved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation contact problems. Numerical examples illustrate the efficiency of our method and the high quality of results. Copyright © 2009 John Wiley & Sons, Ltd.     

A dual mortar approach for 3D finite deformation contact with consistent linearization

In this paper, an approach for three‐dimensional frictionless contact based on a dual mortar formulation and using a primal–dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two‐dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi‐smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Toward robust and accurate contact solvers for large deformation applications: a remapping/adaptivity framework for mortar-based methods

In recent years, the mortar method has proven to be an effective spatial discretization strategy for large deformation contact problems, particularly when such problems feature deformable-to-deformable contact. The mortar approach has been shown to greatly enhance both the spatial accuracy and the robustness with which such problems can be solved in many circumstances. In this work, we concern ourselves with problems that arise in the context of many practical applications, both in manufacturing and in other areas. Specifically, it is frequently necessary to remesh a problem in the midst of an ongoing incremental loading strategy, either because of adaptive mesh refinement being used to improve resolution, or excessive mesh distortion which necessitates an overall remeshing. This work focuses on a particularly important issue associated with contact remeshing; i.e., the remapping of contact variables after the remesh so that a simulation can successfully continue. We develop and demonstrate our algorithm in the context of a mortar-discretized approach to contact. The approach is applicable to either two or three dimensional analysis, and is demonstrated by a number of three dimensional numerical examples.     

Transient three‐dimensional domain decomposition problems: Frame‐indifferent mortar constraints and conserving integration

The present work deals with transient large‐deformation domain decomposition problems. The tying of dissimilar meshed grids is performed by applying the mortar method. In this connection, a reformulation of the original linear mortar constraints is proposed, which retains frame‐indifference for arbitrary discretizations of the interface. Furthermore, a specific coordinate augmentation technique is proposed to make possible the design of an energy–momentum scheme. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energy–momentum scheme for three‐dimensional problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

A two‐dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non‐smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi‐smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher‐order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples. Copyright © 2010 John Wiley & Sons, Ltd.     

The energy-momentum method for the stability of non-holonomic systems

In this paper, we analyze the stability of relative equilibria of non-holonomic systems (that is, mechanical systems with non-integrable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit both neutrally stable and asymptotically stable, as well as linearly unstable relative equilibria. To carry out the stability analysis, we use a generalization of the energy-momentum method combined with the Lyapunov-Malkin theorem and the center manifold theorem. While this approach is consistent with the energy-momentum method for holonomic systems, it extends it in substantial ways. The theory is illustrated with several examples, including the rolling disk, the roller racer and the rattleback top     

Crack face contact in X‐FEM using a segment‐to‐segment approach

In this study, we propose a segment‐to‐segment contact formulation (mortar‐based) that uses Lagrange's multipliers to establish the contact between crack faces when modeled with the extended finite element method (X‐FEM) in 2D problems. It is shown that, in general, inaccuracies arise when the contact is formulated following a point‐to‐point approach. This is due to the non‐linear character of the X‐FEM interpolation along the crack faces that leads to crack face interpenetration. However, the segment‐to‐segment approach optimizes the fulfilment of the contact constraints along the whole crack segment, and in practice the contact is modeled precisely. Convergence studies for mesh sequences have been performed, showing the advantages of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

This paper focuses on the application of NURBS‐based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar‐based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C⁰‐continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.     

Frictional mortar contact for finite deformation problems with synthetic contact kinematics

In this paper we present a mortar based method, for frictional two dimensional contact problems. It is based on the work by Tur et al. (Comput Methods Appl Mech Eng 198(37–40):2860–2873, 2009) and uses the same concentrated integration scheme as well as a non regularized tangential contact formulation based on Lagrange multipliers only. We abstract the contact kinematics to a rather synthetic formulation. Therefore we are able to use two different methods of defining the normal field on the discretized surface normal: The popular method of averaged non-mortar side normal and the rather simple non continuous mortar side normal field. The problem is solved with a fixed point Newton–Raphson procedure and for both normal fields the full linearizations are derived. With numerical examples we show the performance of the more concise formulation of the non averaged non continuous mortar side normal field.     

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.     

Isogeometric contact analysis using mortar method

In the present work, an isogeometric contact analysis scheme using mortar method is proposed. Because the isogeometric analysis is employed for contact analysis, the geometric exactness of the contact region is maintained without any loss of geometric data because of geometry approximation. Thus, the proposed method can overcome underlying shortcomings that result from the geometric approximation of contact surfaces in the conventional finite element (FE)‐based contact analysis. For an isogeometric contact analysis, the schemes for treating the contact conditions and detecting the real contact surfaces are essentially required. In the proposed method, the mortar method is adopted as a nonconforming contact treatment scheme because it is expected to be in good harmony with the useful characteristics of nonuniform rational B‐spline A new matching algorithm is proposed to combine the mortar method with the isogeometric analysis to guarantee consistent contact surface information with the nonuniform rational B‐spline curve. The present scheme is verified by patch test and the well‐known problems which have theoretical solutions such as interference fit and the Hertzian contact problem. It is shown that the problems with curved contact surfaces which are difficult to treat by conventional approaches can be easily dealt with. Copyright © 2011 John Wiley & Sons, Ltd.     

An implicit finite wear contact formulation based on dual mortar methods

Finite deformation contact problems with frictional effects and finite shape changes due to wear are investigated. To capture the finite shape changes, a third configuration besides the well‐known reference and spatial configurations is introduced, which represents the time‐dependent worn state. Consistent interconnections between these states are realized by an arbitrary Lagrangean–Eulerian formulation. The newly developed partitioned and fully implicit algorithm is based on a Lagrangean step and a shape evolution step. Within the Lagrangean step, contact constraints as well as the wear equations are weakly enforced following the well‐established mortar framework. Additional unknowns due to the employed Lagrange multiplier method for contact constraint enforcement and due to wear itself are eliminated by condensation procedures based on the concept of biorthogonality and the so‐called dual shape functions. Several numerical examples in both 2D and 3D are provided to demonstrate the performance and accuracy of the proposed numerical algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

Weak Cn coupling for multipatch isogeometric analysis in solid mechanics

The objective of this contribution is to design a novel methodology to enforce interface conditions preserving higher-order continuity across the interface. In recent years, isogeometric methods using NURBS as basis functions have gained increasing attention, especially in the context of higher-order partial differential equations. They require, in general, higher continuity across interfaces, and thus, new methodologies for domain decomposition constraints capable to deal with those requirements have to be developed. In this contribution, we introduce, in a first step, the coupling constraints using a Euclidean norm on the interface and construct new basis functions. A reformulation as saddle point system allows for a comparison with classical mortar approaches and leads finally to an extended mortar method to enforce Cⁿ continuity.