Study on adjoint variational principles in ship hydrofynamics

给出了未知湿表面边值问题伴随变分原理的统一表示方法。这个问题归结为求解两个积分方程组。一个是关于船体边界条件的方程。另一种是喷雾根线上船体底部到水面高度相等的条件，通过积分水面上的运动条件和船体边界上的运动条件的加权残值，得到了与变分原理相关的泛函。这两种情况均表示为上游水面上的涡线函数为零，下游水面上的伴随广义涡线函数为零。另一个权函数是这两个边界条件中常见的积分因子。由于引入了积分因子，用喷雾根线处船体底部高度代替水面上的残余积分，使得高度相等的条件顺利地包含在泛函中；在二维简单情况下，伴随变量通过坐标反转与自然变量联系起来。因此，上述泛函可以简化为不包含更多伴随变量的二次式。

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.