We present a simulation strategy, beyond-equilibrium molecular dynamics (BEMD), which demonstrates the application of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and the calculation of the friction matrix. The method is based on - and restricted to - the regime, where a generalized canonical ensemble provides a sufficiently rigorous description in terms of microscopic expressions for nonequilibrium variables. Multiplostatted equations of motion (Nose-Hoover variants of Hamilton's classical equations) are employed to maintain this ensemble. The friction matrix appearing in the dynamical equation is iteratively obtained employing a Green-Kubo type expression. Since the remaining 'building blocks' for the GENERIC equation are readily accessible via static Monte Carlo simulation (for the present application they are available analytically), BEMD provides the desired information to perform multiscale simulations. We demonstrate the efficiency of BEMD for the calculation of theological properties of a rarefied gas in the weak shear flow regime and discuss its limitations. Here, our choice of variables is inspired by Grad's moment method for gases. (C) 2004 Elsevier B.V. All rights reserved.