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Modern hull design often includes large bow flare and wide transom sterns. Optimization of these hulls when moving in a seaway puts new demands on the computational methods used. Nonlinear effects become important for wave loads, motions and added resistance in waves. Today, methods solving the unsteady viscous free surface flow around hulls undergoing large motions are available. Such simulations are very time consuming and require very powerful computers. Methods based on linear or partly nonlinear potential flow theory are commonly used, but most of these methods cannot handle the nonlinear effects for the modern hulls. The purpose of this paper is to present a method which aims to fill the gap between state-of-the-art RANSE methods and partly nonlinear panel methods. The method solves the fully nonlinear free surface time-domain potential flow problem including a hull undergoing rigid body motions. A mixed Euler-Lagrange (MEL) scheme is employed to evolve the free surface in time. Nonlinearities still within the hypothesis of potential flow are taken into account, i.e. higher and lower frequency components, hull shape above calm water line as well as interaction between incoming, radiated, diffracted, reflected and ship generated waves due to forward speed. The potential flow method alone cannot handle the roll motion since roll is dominated by viscous effects. A procedure to combine roll damping due to wave radiation as computed by the fully nonlinear potential flow method and viscous roll damping from roll decay experiments or from empirical estimations is presented. The method is applied to a container ship. In head waves, heave and pitch are compared with experimental data. For roll motion, roll decay and roll response are compared to experiments. A grid dependence and domain size study is also presented.