Design of Energy Conserving Algorithms for Frictionless Dynamic Contact Problems

Abstract

SUMMARYThis paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum, angular momentum, and energy in ÿnite element simulations. It is seen that a Lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties. A related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized. This penalty method sacriÿces the energy conservation property, but is dissipative under all conditions of changing contact so that the global algorithm remains stable. Notably, it is also shown that augmented Lagrangian iteration utilizing this penalty kernel reproduces the energy conserving (i.e. Lagrange multiplier) solution to any desired degree of accuracy. The result is a robust, stable method even in the context of large deformations, as is shown by some representative numerical examples. In particular, the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations.

Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles

“…Another way is to implicit the contact stress (see [2,3]) but the kinetic energy of the contacting nodes is lost at each impact. Looking for some energy conserving schemes is now a well-adressed problem, see for example [6,9,10,17]. Nevertheless, these schemes exhibit large oscillations on the contact stress.…”

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

“…Moreover, q (i) A denotes the standard variation of the nodal values in (14). The kinetic energy pertaining to each semi-discrete body assumes the form (21) Correspondingly, the total kinetic energy of the two-body system under consideration can be written as T = 1 2q ·Mq (22) where the system velocity vectorq follows from differentiating (16) with respect to time, and the system mass matrix M contains the elements (20), arranged consistently with the partitioning of q in (16).…”

“…To illustrate, the simulated motion snapshots of the two rings at successive points in time are depicted in Figure 7. After the initial free-flight phase contact takes place within the time interval of approximately [6,16].…”

Simulating granular materials by energy minimization

“…One can also look on the minimization approach as a kind of multi body contact problem [12][13][14] or a local-global solution strategy [15]. In these approaches, just as in the non-smooth contact dynamics method, contacts are generally modeled as infinitely stiff.…”