The potential flow velocity on the surface of the hull was obtained by 

 using the computer program of Chang, This program solves the double- 

 model Neumann problem by distributing a layer of doublets on the hull sur- 

 face and numerically solving the resulting integral equation by a panel 

 method. The main advantage of using a doublet distribution, rather than a 

 source distribution as in the Hess-Smith method, is that the surface 

 potential is obtained directly as the solution of the integral equation. 



The hull surface representation, Equation (23), is used to generate 

 the input for the potential flow program of Chang. A uniform rectangular 

 distribution of points in the ((}),6) plane determines a set of curved quadri- 

 lateral elements covering the hull surface. These curved quadrilateral 

 elements are approximated by plane quadrilateral elements and then used as 

 input into the Chang program. The results of the potential flow calcu- 

 lation are values of the surface velocity potential at the geometric mid- 

 points of the plane quadrilateral elements. These values of the surface 

 velocity potential are then assumed to be accurate values of the exact 

 surface velocity potential at the points of the hull surface that correspond 

 to the geometric centers of the rectangular elements in the (<t>,B) plane. 

 The value of the surface velocity at each point is obtained by numerical 

 differentiation of the surface velocity potential. The values of the 

 potential are first interpolated along (j) = constant curves by a periodic 



1 Q 



cubic spline. This interpolation yields accurate values of the surface 

 velocity potential at arbitrary locations on the 6 = constant curves. Then 

 the values of the surface velocity potential are interpolated along 

 = constant curves by another cubic spline. The surface velocities and 

 the derivatives of the surface velocities that are required in Equations 

 (3) and (7) are obtained, respectively, by differentiating the cubic 

 spline, evaluating the result, which gives the velocities, and then 



refitting the velocities with cubic splines and differentiating the second 



1 8 

 set of splines. This "spline-on-spline" procedure seems to be one of the 



best ways of obtaining two derivatives of a numerically defined function. 



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