Conserving algorithms for frictionless and full stick friction dynamic contact problems using the direct elimination method

In this paper, conserving time‐stepping algorithms for frictionless and full stick friction dynamic contact problems are presented. Time integration algorithms for frictionless and full stick friction dynamic contact problems have been designed to preserve the conservation of key discrete properties satisfied at the continuum level. Energy and energy‐momentum–preserving algorithms for frictionless and full stick friction dynamic contact problems, respectively, have been designed and implemented within the framework of the direct elimination method, avoiding the drawbacks linked to the use of penalty‐based or Lagrange multipliers methods. An assessment of the performance of the resulting formulation is shown in a number of selected and representative numerical examples, under full stick friction and slip frictionless contact conditions. Conservation of key discrete properties exhibited by the time‐stepping algorithm is shown.     

Time integrators based on approximate discontinuous Hamiltonians

We introduce a class of time integration algorithms for finite dimensional mechanical systems whose Hamiltonians are separable. By partitioning the system's configuration space to construct an approximate potential energy, we define an approximate discontinuous Hamiltonian (ADH) whose resulting equations of motion can be solved exactly. The resulting integrators are symplectic and precisely conserve the approximate energy, which by design is always close to the exact one. We then propose two ADH algorithms for finite element discretizations of nonlinear elastic bodies. These result in two classes of explicit asynchronous time integrators that are scalable and, because they conserve the approximate Hamiltonian, could be considered to be unconditionally stable in some circumstances. In addition, these integrators can naturally incorporate frictionless contact conditions. We discuss the momentum conservation properties of the resulting methods and demonstrate their performance with several problems, such as rotating bodies and multiple collisions of bodies with rigid boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Improved implicit integrators for transient impact problems—geometric admissibility within the conserving framework

The value of energy and momentum conserving algorithms has been well established for the analysis of highly non‐linear systems, including those characterized by the nonsmooth non‐linearities of an impact event. This work proposes an improved integration scheme for frictionless dynamic contact, seeking to preserve the stability properties of exact energy and momentum conservation without the heretofore unavoidable compromise of violating geometric admissibility as established by the contact constraints. The physically motivated introduction of a discrete contact velocity provides an algorithmic framework that ensures exact conservation locally while remaining independent of the choice of constraint treatment, thus making full conservation equally possible in conjunction with a penalty regularization as with an exact Lagrange multiplier enforcement. The discrete velocity effects are incorporated as a post‐convergence update to the system velocities, and thus have no direct effect on the non‐linear solution of the displacement equilibrium equation. The result is a robust implicit algorithmic treatment of dynamic frictionless impact, appropriate for large deformations and fully conservative for a range of geometric constraints. Copyright © 2001 John Wiley & Sons, Ltd.     

A formulation of conserving impact system based on localized Lagrange multipliers

This paper proposes a new finite element method of frictional impact of elastic bodies. The formulation introduces a contact frame that is placed in between contacting bodies and represents the contact surface. The nonpenetration condition and the slip–stick condition are defined between the contacting body and the contact frame with the aid of the independent localized Lagrange multipliers representing the contact forces. The position of the contact frame and the local coordinate of the contacting node along the contact frame are also treated as the independent variable, which enables the exact satisfaction of the constraint conditions without the deficiency or redundant constraint. The energy and momentum conservation algorithm is applied to the proposed impact system. For the case of frictional impact, the linear momentum is exactly conserved and the angular momentum is approximately conserved with negligible error. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite difference energy techniques for arbitrary meshes applied to linear plate problems

A new energy based finite difference analytical technique is introduced. The method incorporates certain energy concepts and the ability to use arbitrary, irregular meshes within the framework of the Finite Difference Method. This formulation reduces any governing partial differential equations to a set of difference equations containing partial derivatives up to and including the second order. Further, certain strong similarities with the popular Finite Element Method are shown and the ability to solve problems with irregular boundaries is discussed. To demonstrate the Finite Difference Energy Method several plate bending problems are solved.     

A direct automated procedure for frictionless contact problems

The solution of elastostatic bodies in frictionless contact is obtained by an automated direct method which exploits the theory of linear elasticity and circumvents the need for the inclusion of artificial interface elements, mathematical programming techniques or computation of contact pressure. The method is simple and economical to use and can be easily appended to existing numerical schemes such as the finite element method. The formulation and numerical algorithm are presented for body combinations which are independent of relative tangential displacements along the contact surface. The method is illustrated through an elementary example amenable to hand calculation. Numerical results for more realistic problems are given and compared to known solutions. It is concluded that the method provides a powerful means for both the analysis and design of contacting bodies when used in conjunction with a finite element computer program.     

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. Copyright © 2004 John Wiley & Sons, Ltd.     

An energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems

Numerical solution of dynamic problems requires accurate temporal discretization schemes. So far, to the best of the authors’ knowledge, none have been proposed for adhesive contact problems. In this work, an energy‐momentum‐conserving temporal discretization scheme for adhesive contact problems is proposed. A contact criterion is also proposed to distinguish between adhesion‐dominated and impact‐dominated contact behaviors. An adhesion formulation is considered, which is suitable to describe a large class of interaction mechanisms including van der Waals adhesion and cohesive zone modeling. The current formulation is frictionless, and no dissipation is considered. Performance of the proposed scheme is compared with other schemes. The proposed scheme involves very little extra computational overhead. It is shown that the proposed new temporal discretization scheme leads to major accuracy gains both for single‐degree‐of‐freedom and multi‐degree‐of‐freedom systems. The single‐degree‐of‐freedom system is critically analyzed for various parameters affecting the response. For the multi‐degree‐of‐freedom system, the effect of the time step and mesh discretization on the solution is also studied using the proposed scheme. It is further shown that a temporal discretization scheme based on the principle of energy conservation is not sufficient to obtain a convergent solution. Results with higher order contact finite elements for discretizing the contact area are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

An augmented Lagrangian method for discrete large-slip contact problems

An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.     

A finite element method for contact using a third medium

The numerical simulation of contact problems is nowadays a standard procedure in many engineering applications. The contact constraints are usually formulated using either the Lagrange multiplier, the penalty approach or variants of both methodologies. The aim of this paper is to introduce a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. To be able to account for the contact constraints, the property of the medium, that imbeds the bodies coming into contact, has to change with respect to the movements of the bodies. Within this approach the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions. In this paper we will derive a new finite element formulation that is based on the above mentioned ideas. The formulation is presented for large deformation analysis and frictionless contact.     

A continuum-based finite element formulation for the implicit solution of multibody,large deformation-frictional contact problems

In this paper, a formulation is presented for the finite element treatment of multibody, large deformation frictional contact problems. The term multibody is used to mean that when two bodies mechanically contact, both may be deformable. A novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first, before deriving the corresponding finite element equations This feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general. In particular, the approach yields a characterization of the frictional constraint (assuming a Coulomb law) suitable for arbitrary discretizations in either two or three dimensions. A geometric framework is constructed within which both frictionless and frictional response are naturally described, making subsequent finite element discretization a straightforward substitution of finite-dimensional solution spaces for their continuum counterparts. To our knowledge, this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature. The development includes exact linearization of the statement of virtual work, which enables optimal convergence properties for Newton-Raphson solution strategies, and which appears to be highly desirable (if not essential) for the general robustness of implicit finite element techniques. Since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur, the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non-linear behaviour. This assertion is substantiated through presentation of a variety of examples in both two and three dimensions.     

Frictional contact of flexible and rigid bodies

 This paper is about planar frictional contact problems of both flexible and rigid bodies. For the flexible case a nonlinear finite element formulation is presented, which is based on a modified Coulomb friction law. Stick-slip motion is incorporated into the formulation through a radial return mapping scheme. Linearly interpolating four node elements and three node contact elements are utilized for the finite element discretization. The corresponding tangent stiffness matrices and residual vectors of the equations of motion are presented. In the rigid body case the contact problem is divided into impact and continual contact, which are mathematically described by linear complementarity problems. The impact in normal direction is modeled by a modified Poisson hypothesis, which is adapted to allow multiple impacts. The formulation of the tangential impact is grounded on Coulombs law of friction. The normal contact forces of the continual contact are such that colliding bodies are prevented from penetration and the corresponding tangential forces are expressed by Coulombs law of friction. Examples and comparisions between the different methods are presented. Received: 10 January 2001     

Solution of frictional contact problems by an EBE preconditioner

We present an Element by Element (EBE) procedure to solve non symmetric linear systems arising from the solution of contact with friction problems. Hybrid formulation is introduced and different types of contact elements are reviewed. To deal with large scale problems the EBE method has proved to be a strongly parallel algorithm. Numerical experiments described in this paper confirmed the efficiency of this specific solver.     

The solution of large displacement frictionless contact problems using a sequence of linear complementarity problems

A Newton method for solution of frictionless contact problems is presented. A finite element discretization is performed and the contact constraints are given as complementarity conditions. The resulting equations, which represent the equilibrium of the system, are formulated as a generalized equation. Generalized equations, from the discipline of Mathematical Programming, are a way of writing multi-valued relations, such as complementarity conditions, in a way that is similar to ordinary equations. Newton's method is then used, in a straightforward way, to solve the present non-linear generalized equation, resulting in a sequence of Linear Complementarity Problems (LCP's).     

Crack identification in two-dimensional unilateral contact mechanics with the boundary element method

The identification of a unilateral frictionless crack is performed in nonlinear elastostatics by using boundary measurements for given static loadings. The procedure proposed takes into account the possibility of a partial or total closure of the crack during the identification process; that makes the present formulation more complex than others referred to permanently open cracks. The Linear Complementarity Problem (LCP), which provides at each step contact tractions and relative displacements along the crack, is discretised by means of the Dual Boundary Element Method (DBEM) and solved explicitly by Lemke's algorithm. The identification procedure is based on a first-order nonlinear optimisation technique in which the gradients of the cost function are obtained by solving again a LCP with a considerably reduced number of variables. Some numerical examples show the applicability of the method. Received 23 November 1998     

Stochastic second-order BEM perturbation formulation

This paper presents the stochastic second order moment perturbation approach to the classical deterministic Boundary Element Method (BEM) formulation. Numerous applications of such a formulation in different problems of stochastic mechanics, especially in the field of computational modeling of structural defects in homogeneous and composite materials occurring randomly in solids and engineering structures, were the main reasons to introduce the proposed model. The stochastic boundary element method (SBEM) formulation of the general linear elasticity boundary value has been provided together with an appropriate discretization. The equations describing the expected values and the covariances of stress and strain tensors for points lying on the boundary and inside the region are considered. This set of equations constitutes a formal mathematical statement of the problem and is suitable for computational implementation.     

A general boundary element method approach to the solution of three-dimensional frictionless contact problems

A direct boundary element method is presented for three-dimensional stress analysis of frictionless contact problems. The isoparametric formulation of the boundary element method is implemented for the general case of contact in the absence of friction, which is limited to linear elastic homogeneous and isotropic materials. An iterative procedure is employed to determine the correct size of the contact zone by finding a boundary solution compatible with the contact condition. The applicability of the procedure is tested by application to three problems of advancing and conforming contact. The computed results are compared with numerical and analytical solutions where possible.     

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.