Study on adjoint variational principles in ship hydrofynamics

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A unified method of formulation for adjoint variational principles on the boundary value problem with unknown wetted surface is presented. This problem is reduced to solve a system of two integral equations. One is the equation on the hull boundary condition. The other is equality condition in height of the hull bottom to the water surface at the spray root line.Functional associated with the variational principles are formulated by integrating the weighted residuals of the kinematic condition on the water surface in addition to the one on the hull boundary. Both conditions are represented in the vortex line function set zero on the upstream water surface, and the adjoint generalized vortex line function, the one of the weight functions, is set zero on the downstream water surface. The other weight function is the integration factor found commonly in those two boundary conditions. Owing to the integration factor, the residual integration over the water surface is replaced with the bottom height of the hull at the spray root line, so that the equality condition in height is included smoothly in the functional.In 2D simple case, the adjoint variable is related to the natural one by coordinate reverse. As a result, the functional mentioned above is able to be reduced to the quadratic formula which does not include any more adjoint variable.