  This paper extends the Finite Elements with Switch Detection and Jumps
(FESD-J) [1] method to problems of rigid body dynamics involving patch
contacts. The FESD-J method is a high accuracy discretization scheme suitable
for use in direct optimal control of nonsmooth mechanical systems. It detects
dynamic switches exactly in time and, thereby, maintains the integration order
of the underlying Runge- Kutta (RK) method. This is in contrast to commonly
used time-stepping methods which only achieve first-order accuracy. Considering
rigid bodies with possible patch contacts results in nondifferentiable signed
distance functions (SDF), which introduces additional nonsmoothness into the
dynamical system. In this work, we utilize so-called equivalent contact points
(ECP), which parameterize force and impulse distributions on contact patches by
evaluation at single points. We embed a nondifferentiable SDF into a
complementarity Lagrangian system (CLS) and show that the determined ECP are
well-defined. We then extend the FESD-J discretization to the considered CLS
such that its integration accuracy is maintained. The functionality of the
method is illustrated for both a simulation and an optimal control example.
