An asynchronous, variational method for simulating elastica in complex contact and impact scenarios is developed. Asynchronous Variational Integrators [1] (AVIs) are extended to handle contact forces by associating dierent time steps to forces instead of to spatial elements. By discretizing a barrier potential by an innite sum of nested quadratic potentials, these extended AVIs are used to resolve contact while obeying momentum- and energy-conservation laws. A series of two- and three-dimensional examples illustrate the robustness and good energy behavior of the method.
The simplest contact models for finite element simulation follow the early analytical work of Hertz @cite_21 in assuming frictionless contact of planar (or nearly planar) surfaces with small strain. In this regime, several approaches have been explored to arrive at a weak formulation of contact; for a high-level survey of these approaches, see for example the overview by Belytschko al @cite_25 or Wriggers @cite_12 . The first of these are the use of penalty forces, described for instance by Oden @cite_13 and Kikuchi and Oden @cite_17 . The penalty approach results in a contact force proportional to an arbitrary parameter and to the rate of interpenetration, or in more general formulations to an arbitrary function of rate of interpenetration and interpenetration depth; Belytschko and Neal @cite_0 discuss the choosing of this parameter in Section 8. Recent work by Belytschko al @cite_39 uses moving least squares to construct an implicit smooth contact surface, from which the interpenetration distance is evaluated. Peric and Owen @cite_30 describe how to equip penalty forces with a Coulomb friction model.
@cite_30 @cite_21 @cite_39 @cite_0 @cite_13 @cite_25 @cite_12 @cite_17
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Making use of a perturbed Lagrangian formulation, a finite element procedure for contact problems is developed for the general case in which node-to-node contact no longer holds. The proposed procedure leads naturally to a discretization of the contact interface into contact segments. Within the context of a bilinear interpolation for the displacement field, a mixed finite element approximation is introduced by assuming discont naous contact pressure, constant on the contact segment. Because of this piece-wise constant approximation, the gap function enters into the formulation in an ‘average’ sense instead of through a point-wise definition. Numerical examples are presented that illustrate the performance of the proposed procedure. Current finite element formulations for contact problems based on either the classical Lagrange parameter procedure [l-3, 12, 203 or the penalty-function method [4-6,11], are characterized by a point-wise enforcement of the contact-constraint condition, in the sense that penetration of the bodies is established on a nodal basis. Moreover, in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure. Within the framework of classical Lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables (Lagrange multipliers). These extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms. Penalty methods, on the other hand, enable one to transform the constrained problem into an unconstrained one without introducing additional variables. The constraint condition is now satisfied only approximately for finite values of the penalty parameter. The main difhculty associated with these methods, however, lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition. This is a well-understood phenomenon, particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics (e.g. see [l&22,25] for a review). Recently, augmented Lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and ‘regularize’ the penalty formulation (e.g. see the survey in [7,8]). Within the framework of Zinearited e u c , it is possible to restrict the finite element formulation of contact problems by assuming that node-to-node contact occurs. This is in fact the case often considered in the literature [l, 2, 4, 6, 12, 20, 211. In the general context of fully We propose a new explicit contact algorithm for finite element discretized solids and shells with smooth and non-smooth geometries. The equations of motion are integrated in time with a predictor-corrector-type algorithm. After each predictor step, the impenetrability constraints and the exchange of momenta between the impacting bodies are considered and enforced independently. The geometrically inadmissible penetrations are removed using closest point projections or similar updates. Penetration is measured using the signed volume of intersection described by the contacting surface elements, which is well-defined for both smooth and non-smooth geometries. For computing the instantaneous velocity changes that occur during the impact event, we introduce the decomposition contact response method. This enables the closed-form solution of the jump equations at impact, and applies to non-frictional as well as frictional contact, as exemplified by the Coulomb frictional model. The overall algorithm has excellent momentum and energy conservation characteristics, as several numerical examples demonstrate. Copyright © 2005 John Wiley & Sons, Ltd. Abstract The nature of the constitutive contact force law utilized to describe contact–impact events in solid contact interfaces plays a key role in predicting the response of multibody mechanical systems and in the simulation of engineering applications. The goal of this work is to present a comparative study on the most relevant existing viscoelastic contact force models. In the sequel of this process, their fundamental characteristics are examined and their performances evaluated. Models developed based on the Hertz contact theory and augmented with a damping term to accommodate the dissipation of energy during the impact process, which typically is a function of the coefficient of restitution between the contacting solids, are considered in this study. In particular, the identified contact force models are compared in the present study for simple solid impact problems with the sole purpose of comparing the performance of the various models and examining the corresponding system behavior. The outcomes indicate that the prediction of the dynamic behavior of contacting solids strongly depends on the selection of the contact force model. The use of Coulomb's friction law with the principles of classical rigid-body dynamics introduces mathematical inconsistencies. Specifically, the forward dynamics problem can have no solutions or multiple solutions. In these situations, compliant contact models, while increasing the dimensionality of the state vector, can resolve these problems. The simplicity and efficiency of rigid-body models, however, provide strong motivation for their use during those portions of a simulation when the rigid-body solution is unique and stable. In this paper, we use singular perturbation analysis in conjunction with linear complementarity theory to establish conditions under which the solution predicted by the rigid-body dynamic model is stable. We employ a general model of contact compliance to derive stability criteria for planar mechanical systems. In particular, we show that for cases with one sliding contact, there is always at most one stable solution. Our approach is not directly applicable to transitions between rolling and sliding where the Coulomb friction law is discontinuous. To overcome this difficulty, we introduce a smooth nonlinear friction law, which approximates Coulomb friction. Such a friction model can also increase the efficiency of both rigid-body and compliant contact simulation. Numerical simulations for the different models and comparison with experimental results are also presented.