Improved implicit integrators for transient impact problems––dynamic frictional dissipation within an admissible conserving framework

Abstract

This work presents a frictional extension of an improved conservative integration framework for dynamic contact, extending the details of the frictionless scheme to encompass Coulomb stick/slip friction and the associated conservation/dissipation. It addresses algorithmic details that do not appear in the frictionless context, establishing an appropriate and objective treatment of relative tangential motion that is necessary to rigorously ensure conservation of angular momentum in the fully discrete setting. It also further extends the functionality of a discrete contact velocity term proposed in the frictionless implementation, using it to preserve conservation or dissipation locally as warranted by the frictional model and effectively enabling an enforcement of the contact constraints independent of energy considerations. The result is a robust implicit algorithmic framework for dynamic frictional impact, viable for large deformation analysis, appropriately conservative or dissipative for both stick and slip phenomena on a local scale, and readily extensible to more complex frictional models.

Introduction

There exists currently a growing body of computational work in non-linear dynamics that utilizes a class of conservative integration methods, i.e. methods that are designed to better emulate a given physical system by algorithmically preserving a particular set of physically motivated quantities by construction. One such method, introduced for elastodynamics by Simo and Tarnow [16] and generalized by Gonzalez [8], establishes a means of preserving algorithmic versions of the system energy as well as the linear and angular momenta over a discretized bulk continuum. The method is well conceived, as energy is often the defining metric for determining numerical stability in a system, and a number of sources [6], [8] have demonstrated the possibility of ‘instability’, in the form of boundless energy growth, that can result from an ad hoc application or extension of a traditional linear integration method to a system with even relatively simple non-linearities.

The non-linearities associated with dynamic impact are relatively complex, even given the simplest mathematical description. In the most basic form a contact problem must define or determine the time of initial contact and release for each of a potentially changing but common set of surface points on a pair of colliding bodies, and also consider the magnitudes and directions of any associated contact effects, all of which are generally unknown and characterized by both spatial and temporal discontinuities. The contact entities themselves are also heavily influenced by the associated bulk media, and successful description of the former necessitates a careful consideration of the latter. A few contact descriptions, notably Armero and Petocz [1], Laursen and Chawla [10], Demkowicz and Bajer [7], and Laursen and Love [11], have chosen to consider frictionless dynamic impact under the auspices of a conservative system, extending the framework to incorporate the conserving mathematical idealization of a frictionless surface interaction.

It might seem a contradiction of sorts to extend such algorithms to the realm of physically dissipative phenomena, whose presence tends to ‘naturally’ stabilize their respective systems and appears to obviate the need for a conserving application. Developments in the interest of controlling numerical dissipation (e.g. [3], [4]) lead to the conclusion that, given suitable a priori energy estimates for the effects of a given dissipative phenomena, it is possible to accurately render these effects by capturing the dissipative values within a conservative context, free from the parameter-dependent numerical dissipation inherited from traditional temporal integration schemes. This approach has been recently applied to bulk elastoplasticity in such works as Meng and Laursen [14] and has been advocated for friction by both Armero and Petocz [2], and Chawla and Laursen [5] as extensions of their respective frictionless works.

These frictional extensions, however, make the same concessions as their respective frictionless foundations in the treatment of the contact interface. As discussed in Laursen and Love [11], the method of Laursen and Chawla [10] partially compromises the normal contact constraints in the interest of making a consistent accounting of system energy. In retaining this discrete overlap through the frictional extension, Chawla and Laursen [5] are forced to concede algorithmic conservation of angular momentum for frictional traction, in large part due to the unfortunate selection of non-invariant algorithmic description for the relative tangential motion. Armero and Petocz [2] do maintain conservation of momenta through a more judicious frame-indifferent discretization of the tangential rate terms, but retain the algorithmic artifacts of their frictionless case, namely a localized storage of non-physical surface energy in the regularization potential and the associated caveat that complete energy balance is only achieved upon full release of the contact constraints. In addition, their treatment of frictional energy change only adheres to the continuum estimate of energy dissipation in a qualitative sense, ensuring a consistently dissipative treatment over the course of a contact event, but not quantifying the dissipation in relation to the established a priori estimates.

In Laursen and Love [11], we treated the frictionless impact problem by using a discrete contact velocity term as a means of resolving conservation of energy without placing undue constraint on the normal contact forces. In this extension to frictional problems, we will show that with careful discretization of a frictional system, it is possible to apply the discrete contact velocity framework so that the discrete system not only conserves both linear and angular momentum, but respectively conserves or dissipates energy appropriate to the local frictional phenomenon. The energy dissipation is quantitatively established by a direct discretization of the continuum dissipation estimates, thus correctly capturing physical dissipation in the regularization limit.

Section 2 provides the notational foundation for the subsequent algorithmic development, outlining the virtual work description and contact kinematics for generalized dynamic contact with Coulomb friction. The continuum system is then systematically discretized in 3.1 System discretization within an energy–momentum framework, 3.2 Contact surface discretization, culminating with the addition of the discrete velocity update in Section 3.3. The remaining portions of Section 3 describe the algorithmic details necessary to ensure momentum conservation and consistent energy dissipation (3.4 Momentum conservation, 3.5 Ensuring energy consistency respectively), outline a regularization of the discrete constraints (Section 3.6), and provide a summary of the process. The numerical examples of Section 4 demonstrate the situationally dependent conservative/dissipative behavior captured by the algorithm for a sampling of dynamic frictional impact problems.