Abstract

The author and his research group in computational medicine have come across dynamical contact problems in the context of a collaboration in orthopaedic surgery. Special first attention has focussed on the motion of the patient-specific knee.

As it turned out, the numerical integration of time dependent contact problems has stayed unsatisfactory for decades. The classical Newmark method, which is quite popular in the engineering world, is a real ’’perpetuum mobile’’ in that in generates energy! A rather recent improvement due the Caltech group around Marsden and Ortiz is energy dissipative, but still unsatisfactory, since it produces artificial oscillations (untolerable in the collaboration with surgeons!). For this reason, the author and his coworkers have suggested a further modification called ’’contact-stabilized Newmark method’’. This scheme now is energy dissipative and avoids artificial oscillations. However, the new scheme escapes the usual domain of consistency theory for numerical integration. After a long investigative period, we meanwhile found the theoretical key to this kind of integrators. The new theoretical characterization requires bounded variation in terms of a physical energy functional that includes kinetic energy, elastic energy, and visco-elastic energy. First numerical findings (a few days old) for an adaptive stepsize control in a Hertzian contact problem will be presented.