Recent Progress in the Calculation of Potential Flows 



elements in the neighborhood of p, to limit the solid angles to small values. 

 But the elements can easily be divided around the point calculated (maintaining 

 a- for the subelements), if the analytical definition is known. 



Finally, I want to congratulate Dr. Smith for his achievements in this study, 

 which opens many possibilities. 



REPLY TO DISCUSSION 



A. M. O. Smith 



I wish to thank Dr. Weinblum for his complimentary remarks. I am glad to 

 be reminded of the work by Karl Maruhn on flow fields about ellipsoids. About 

 the time of World War II, I was aware of his work, but it had since gradually 

 faded from my memory. The report (Ref. 1) mentioned in my paper, covering 

 velocity fields, includes the ellipsoid family, but of course, it also covers many 

 more shapes. 



With regard to Dr. Landweber's comments, I wish to say that the present 

 paper is restricted to a very brief discussion of various aspects of the method. 

 A more complete analysis of iterative solutions of this problem is contained in 

 section 5.4 of Ref. 1. 



The analytical procedure mentioned by Dr. Landweber is numerically ap- 

 proximated by the point- Jacobi iterative matrix method. Indeed, for exterior 

 flows this method has a negative convergence factor slightly less than unity in 

 absolute value. This means that once the procedure has steadied out, succes- 

 sive iterates oscillate about the true solution with slow convergence. Clearly, 

 in this circumstance the averaging of two successive iterates produces a much 

 improved result. The only problem is how many iterations are required before 

 the iterative procedure becomes steady. For single smooth bodies only a few 

 iterations are required, but multiple-body problems require a large number. 

 Averaging is not effective for interior flows. For these the convergence factor 

 is positive and thus the iterates form a monotonic sequence. 



Reference 1 states that the iterative procedure actually used is the Gauss- 

 Seidel, which is always superior to the point- Jacobi. For typical exterior flows 

 this procedure requires only four iterations per decimal place of accuracy - a 

 very fast convergence. For interior flows it requires about half as many itera- 

 tions as the point- Jacobi method. The convergence factor is always positive. 

 Tables summarizing the detailed results may be found on pages 78 and 80 of 

 Ref. 1. 



Recent experience has shown that direct-matrix solution is efficient at 

 higher element numbers than had been supposed. Eventually a direct solution, 

 whose computing effort varies as the cube of the matrix order, must be slower 



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