Towards an efficient numerical simulation of complex 3D knee joint motion

We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity together with advanced numerical algorithms needed for its simulation. The model comprises bones, cartilage and the major ligaments, while patella and menisci are still missing. Bones are modeled by linear elastic materials, cartilage by linear viscoelastic materials, and ligaments by onedimensional nonlinear Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity with strict contact inequalities. The spatiotemporal discretization follows a time layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts we use a new contactstabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet– Neumann method. The large algebraic systems of the bone– cartilage contact problems are solved efficiently by the truncated non-smooth Newton multigrid method. Communicated by : Gabriel Wittum. This work has been funded in part by the DFG research center Matheon. O. Sander (B) · J. Youett · R. Kornhuber Freie Universität Berlin Institut für Mathematik, Arnimallee 6, 14195 Berlin, Germany e-mail: [email protected] C. Klapproth · P. Deuflhard Zuse Institute Berlin (ZIB), Takustraße 7, 14195 Berlin, Germany