To obtain a mesh that wraps around the body and conforms to the other 



boundaries, the physical fluid region is divided into several subregions (Fig. 



4). Each subregion is mapped onto a rectangle of computational space. In 



each rectangle of computational space, the inverted Laplace equation is solved 



subject to Dirichlet boundary conditions, transformed Neumann boundary 



conditions, or matching boundary conditions where rectangles overlap. 



The Laplace equation for the velocity potential (j) transforms exactly as 



Equations (26a, b). The transformed Laplace equation is given by 



a<t) - 23(t) + Yi}) =0 (27f) 



55 5n nn 



where a, 6, Y, are defined by Equations (27c-e). 



Wherever possible, central differencing is used to discretize Equations 

 (27a-f). At the boundaries of the fluid region where a Neumann boundary 

 condition is specified, second-order one-sided finite differences replace some 

 of the derivatives in Equations (27a-f). (See Coleman and Haussling [16] for 

 more details.) The resulting system of quasilinear equations for x, y, and <l> 

 at the grid points is solved by successive overrelaxation. 



Lin et al. [1] cite the works of various researchers who show that a 

 logarithmic singularity exists in the velocity potential of free-surface 

 potential flow near the intersection of the free surface with a vertical 

 waveraaker in horizontal motion. But, since the body contour for the problem 

 considered is nearly vertical at the intersection with the free surface and 

 since the horizontal velocity component of the body is zero, the logarithmic 

 singularity in the velocity potential may be relatively weak. Thus special 

 numerical treatment of the singularity may not be critical. In fact, nothing 

 special has been included in the numerical method to accommodate such a 

 singularity if it should arise. 



17 



