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.
Non-smooth contact requires special consideration, since in the non-smooth regime there is no straightforward way of defining a contact normal or penetration distance. Simo al @cite_3 discretize the contact surface into segments over which they assume constant contact pressure; this formulation allows them to handle non-node-to-node contact using a perturbed Lagrangian. Kane al @cite_10 apply non-smooth analysis to resolve contact constraints between sharp objects. Pandolfi al @cite_15 extend the work of Kane al ,by describing a variational model for non-smooth contact with friction. Cirak and West @cite_43 decompose contact resolution into an impenetrability-enforcement and momentum-transfer step, thereby exactly enforcing non-interpenetration while nearly conserving momentum and energy.
@cite_43 @cite_15 @cite_10 @cite_3
2075921599 2099876657 2067309337 2030010167
The present work extends the non-smooth contact class of algorithms introduced by to the case of friction. The formulation specifically addresses contact geometries, e.g. involving multiple collisions between tightly packed non-smooth bodies, for which neither normals nor gap functions can be properly defined. A key aspect of the approach is that the incremental displacements follow from a minimum principle. The objective function comprises terms which account for inertia, strain energy, contact, friction and external forcing. The Euler–Lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact. In addition to its value as a basis for formulating numerical algorithms, the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non-uniqueness. We present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms, as well as the ability of the approach to account for stick and slip conditions. 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. 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. 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