Because of the complicated nature of the types of interactions between
the ship, the free surface and the flow, it is particularly important
to review the mathematical aspects of the numerical methods employed
(the boundary integral method with the time dependent Green's function
for the free surface). With respect to these, constraints on the
geometric characteristics of the hull surface have been found to be of
relevance to the properties of the computed solution. In 1950, Fritz
John [] wrote: "the proof of the uniqueness for the forced
motion is valid only under certain geometrical assumptions, namely
that the vertical line through any point of 
must not meet the
free surface ( is the ship's surface). Uniqueness is proved for a
free floating body only for frequencies
k of the primary waves... In the case where intersects the free
surface two complications arise. One is that for certain values of the
frequency k the integral equation has eigenfunctions. The other
difficulty is the presence of a strong singularity of the kernel in
the points of the curve of intersection of obstacle surface and free
surface".
The types of problems discovered by John have been encountered in
practice as years passed by different researchers. In 1965, W. D. Kim
[] indicates: "...there is a fundamental limitation to the
present numerical method. The kernels of the integral equations
oscillate rapidly as the parameter 
increases ( 
is wave frequency)''. J. V. Wehausen (1971)
[] reviewing John writes: "Existence of a solution to the
initial-value problem has apparently not been proved by anyone. For
the steady state time-harmonic problem John (1950) [] has
proved the existence of a solution for the case that the body plus its
reflection in the free surface is bounded by a convex surface of class
C''. Hence the body must intersect the free surface
perpendicularly".
More recently, the aspects of existence of solutions and appearance
of irregular frequencies can be found in textbooks. For example,
O. Faltinsen (1990) [] writes in page 114, citing John 1950:
``a solution may not exist for all frequencies... Ships with bulbous bow
do not satisfy conditions in John's analysis. Irregular frequencies
may cause the 3D technique to breakdown... Irregular frequencies in
the source technique represent eigenfrequencies for a fictious fluid
motion inside the body with the same free surface condition and the
body boundary condition 
. The determinant of the
coefficient matrix used to discretize the integral equation goes to
zero when the number of unknowns goes to infinity... Source methods
have been used on large volume offshore structures for about 20
years.
. Causes of
differences can be grid shape, size and distribution, geometry
approximation, singularity density distribution, Green function
calculation and how singularities are integrated over panels''.
Another recent review by J. N. Newman (1991) [] indicates:
"In the Neumann-Kelvin approach, an essential singularity occurs when
the source and field point are in the free surface, and no appropriate
algorithm exist which properly accounts for this singularity in panel
methods. ... for U ;SPMgt; 0 the essential singularity is an uncertain
source of errors. The problem is circumvented by using time-domain
analysis, with motions started from rest. For finite values of time
the essential singularity does not exist, although in principle waves
of monotonically decreasing wavelength will arise as time increases".
The considerations on the limitations of the numerical models employed in this project are of particular relevance since the design optimization studies will extensively explore different ranges of geometric parameters of the ship and sea states, which will usually push to the limit the ability of the codes to provide accurate results. In particular, it must be noted that a bulbous bow may be a source of problems as stated explicitly by Faltinsen. The optimization studies can be severely limited by a numerical model that involves inverting ill conditioned matrices as the previous considerations indicate. One of the examples proposed by SAIC for the design studies (the Truman geometry) presents a bulbous bow. The accuracy of the model employed in the design will also be a fundamental factor in finding optimized designs that are realistic in practice.