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In this paper we consider travelling surface waves on a layer of water of decreasing depth. A numerical scheme based on the boundary element method is used to present calculations for the run-up of a solitary wave. The numerical results are compared with an analytical approximation based on a modified Korteweg-de Vries equation.

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In this paper we present a generic optimization algorithm for the allocation of dynamic positioning actuators, such as azimuthing thrusters and fixed thrusters. The algorithm is based on the well-known Lagrange multipliers method. In the present approach the Lagrangian functional represents not only the cost function (the total power delivered by all actuators), but also all constraints related to thruster saturation and forbidden zones for azimuthing thrusters.In the presented approach the application of the Lagrange multipliers method leads to a nonlinear set of equations, because an exact expression for the total power is applied and the actuator limitations are accounted for in an implicit manner, by means of nonlinear constraints. It is solved iteratively with the Newton-Raphson method and a step by step implementation of the constraints related to the actuator limitations.In addition, the results from the non-linear solution method were compared with the results from a simplified set of linear equations, based on an approximate (quadratic) expression for the thruster power. The non-linear solution was more accurate, while requiring only a slightly higher computational effort.An example is shown for a thruster configuration with 8 azimuthing thrusters, typical for a DP semi-submersible. The results show that the optimization algorithm is very stable and efficient.Finally, some options for improvements and future enhancements -such as including thruster-thruster and thrusterhull interactions and the effects of current -are discussed.

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