Among the most useful applications of digital computers in naval architecture, giving direct geometrical answers, are:

• (a) Determining lines and offsets to suit arbitrary hull form characteristics derived from a prescribed parent form.
• (b) Final fairing and determination of closely spaced frame offsets for shipyard use based upon widely spaced preliminary design offsets.

The first of these applications provides the capability to carry out by computer and in a more general fashion what Adm. D. W. Taylor began prior to World War I, that is, to design ship lines mathematically. By the method of Taylor (1915) waterlines and sectional area curves took the form of a 5th order curve, separately for forebody and afterbody,

where t, a, b, c and d are constants. By suitable transformation, this equation was rewritten as,

where P is the waterplane area coefficient (for a water line), or prismatic coefficient (for a sectional area curve) of forebody or afterbody, t is the tangent of the curve at bow or stern, and a is a function of the second derivative of the curve at amidships (x = 1.0).A simple table was provided giving values of the coefficients, C, which were fixed for each body plan station. The resulting curves had at most one point of inflection. This method was used to draw the lines for Taylor's Standard Series. Taylor (1915) noted that, "practically all U.S. naval vessels designed during the last ten years have had mathematical lines,"

During the intervening 65 years, the use of mathematical ship lines appears to have declined until the advent of computers. A number of successful attempts have now been reported (Fuller, et al, 1977 and Soding, et al, 1977, for example) where ship lines in keeping with hull forms favored today have been produced and plotted with the aid of the computer. Polynomials of higher order than used by Taylor have been used for waterlines and sectional area curves, with particular attention taken to avoid unwanted points of inflection. However, unless some adjustment is done to the end profiles, the resulting hull forms are endowed with waterline endings or stern profiles that may not satisfy the user. Kuiper (1970) presented a method whereby the design waterline is expressed as two eight-term polynomials, one for forebody and one for afterbody, which are easily determined using the basic hull form characteristics. However, to define the hull form above and below the design waterline requires the use of seventeen form parameters which must be defined at all drafts, for forebody and afterbody.

Recently somewhat different computer techniques have been developed to assist in the early stage of lines development. For example, a Ship Hull Form Generator Program (HULGEN) was developed in the Ship Design Division of NAVSEC. (Fuller, et aI, 1977). The key to this program is. the use of polynomials in various combinations to build up a line-for-line definition of the hull form that is remarkably fair. The strength of the program is the user-oriented interactive-graphics method of data input, display and modification. Results of variations of parameters can be viewed instantly, or the hull form can be stretched and distorted into shapes to maintain those parameters.

The second application is that of final fairing of preliminary lines, which necessarily embodies judgment, in that the drafter's eye and opinion ultimately determine fairness. In this process, the drafter, or the mold loftsman, is faced with the problem of passing a curve through a set of points, usually equally spaced along a reference axis, and satisfying himself that the curve is smooth, with a minimum number of points of inflection and with curvature varying in a gradual way. In order to achieve fairness, the curve may have to miss some of the points by small amounts. Also, for consistency, interesecting curves in other views which contain these points must be checked and adjusted.

As previously noted, battens or splines are commonly used in drawing such curves, with batten weights (ducks) positioned to hold the batten at or near the given points. Therefore, computerized representations of ship lines often make use of the equations for spline curves. The bending induced in the batten by the ducks is describable by the theory of bending of a simple weightless beam with concentrated loads or supports at a series of discrete points, corresponding to the points of duck restraint. It is shown in Strength of Materials texts that the deflection of such a beam is given by polynomials no higher than the third order, that is, by cubic functions of the z-dimension parallel with the beam. Such cubics have continuity in first and second derivatives () at the points of application of the concentrated loads.

An assumption made in developing the deflection equation for a simple beam is that the deflections are small, and the beam assumes only small angles to the x-axis. Under these conditions the differential equation of bending is

This can be linearized by assuming that , in as much as y' is small; thus,

and y" becomes a linear function of x. A close approximation to y ", at point n for example, is given by the second difference , where

and h is the spacing between any pair of equally spaced points. Inasmuch as is quite sensitive to changes in curvature, it is apparent that by adopting values from a smooth curve, and by adjusting offsets to match the faired values, a curve through the adjusted offsets will usually be quite fair.

For illustrative purposes, Figure below

plots rough-and obviously unfair-points representing preliminary offsets of a rudder section. Also shown is a plot of from the given offsets, and a smooth curve interpreting the plot but missing some of the points.

The final faired rudder profile curve has been obtained from the smooth curve of , beginning at the nose of the section and working aft, but with the addition of two additional linear corrections, first to make the tail of the section sharp, and second, to make the average value of the faired offsets equal to the mean value of the given offsets.

The spline curve representation of ship lines by the differential equation of the deflection of a simple beam may become unrealistic when the slope y' of the line being represented becomes so large that it cannot be assumed equal to zero. Thus, most ships' waterlines can readily be represented by spline curve equations over most of the length of the ship. However, such is not the case for body plan stations, nor for many buttocks, especially near the ends of the ship, where steep slopes are often met. In order to overcome this problem, some early attempts to define ship lines with the aid of a computer required that the coordinate reference axes be rotated. More recently, lines have been expressed as parametric spline curves, by which the curves are defined by a parameter s, rather than directly by x,y coordinates. The parameter s is defined as the cumulative length of segments of the line from the start point up to the point in question (lIT Research Institute, 1980).

A computer program in which this representation is used is HULDEF, developed by the U.S. Navy for design use but now extended and made available to a number of shipyards in the U.S. for final hull form definition in the construction phase. HULDEF is said to be economical of computer time, and has been made compatible with other computer-based hull production programs. By HULDEF lines along the hull are developed from the given input waterlines and buttocks into iso-girth lines, formed by taking fixed percentages of the girthed length around each body plan station from centerline to deck edge (or chine) all along the hull from the tip of the bow to the stern. The lines are mathematized as parametric spline curves. The HULDEF system has been provided with interactive graphics capability so that the operator can readily display curves, first differences, and second differences, and can fair these on the scope to suit his own idea of fairness. This puts the fairing capability under the control and judgment of the operator just as it has been in the past under the control of the traditional drafter or mold loftsman. However, the previous time consuming operation of drawing the line-on a drafting table or on the mold loft floor-is no longer needed (Fuller, et al, 1977). Other similar systems are in use in some U.S. shipyards and abroad.

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