1. Introduction

Development of a three-dimensional fair hull form is one of the main design requirements in the design of a marine vehicle. The final hull form must satisfy both the desired shape and performance characteristics. During the early stages of the design process there is little data available to produce a fair hull form and the designer often initiates the process with a rough sketch or few hull form parameters based on experience or empirical methods. In the past this was sufficient for simplified analytic approximations used for performance predictions. However, today designers require more accurate predictions on the performance characteristics of their designs and modern computing technology now available enables more extensive analysis to be carried out. Therefore the designer will require a precisely defined three-dimensional fair hull form even in the earlier stages of the design process.

Ship hull form surface is traditionally determined by a set of control points called offsets. Physical splines have long been used in drafting and design to construct fair curves that pass through specified offset points. The manual method requires the fairing of two-dimensional design curves on several planes in an iterative manner. However, this iterative process requires excessive time and experienced personnel, neither of which may be available in a tight-scheduled modern ship design process.

Fairing of ship hull forms has been one of the earlier applications of computers into shipbuilding (Berger et al., 1966; Hattori and Matida, 1977). These applications have generally been local and interactive where the shape is altered locally by an experienced designer (Fog, 1985; Horsham, 1988). However, the complex shape of the hulls has large number of offsets and it is not surprising to produce forms with wrinkles or any other flaws in parts of the surface. Furthermore, the interactive local fairing methods cannot ensure that three-dimensional fairness is achieved.

Parametric B-spline curves and surfaces are widely used to represent ship hull forms (Rogers, 1977). The main reason is that they have many superior geometric properties compared to other mathematical representation schemes (Rogers and Adams, 1989). B-splines have the capability to represent any complex shaped geometry such as ship hull forms, and therefore they have emerged as the de-facto industrial standard in the field of computer aided ship design (CASD). Fairness of a B-spline surface will depend on the quality of control point data. In many cases the designer must eliminate undesirable shape features in order to produce a smoother shape. While removing undesirable shape irregularities the designer must also preserve the shape and keep the form as close to the original as possible in order not to degrade specific performance characteristics. Although many ship hull form design software based on nonuniform rational B-splines (NURBS) are commercially available the designer still faces undesirable shape features in hull form surfaces that must be eliminated.

Since the 1970s there have been numerous attempts to produce automated fairing procedures for curves and surfaces. Pramila (1978) used linearized fairness functional which minimizes strain energy for ship hull surfaces. McCallum and Zhang (1986) described an automatic smoothing algorithm based on B-splines’ curvature behavior property and applied this to some curve forms used in ship design. Nowacki et al. (1989) described a surface approximation scheme based on minimization of the sum of the strain energy of mesh lines and the potential energy of springs attached to the data points. Rogers and Fog (1989) applied their constrained B-spline curve/surface-fitting algorithm to ship hull forms to generate defining polygons for curves and defining polygonal nets for surfaces. Sapidis and Farin (1990) proposed an automatic fairing algorithm for B-spline curves. The algorithm is based on removing and reinserting knots of the spline. Liu et al. (1991) introduced constrained smoothing B-spline curve fitting for mesh curves of ships by minimizing an energy functional as a fairness measure. Huanzong et al. (1991) proposed a fairing method by minimizing the elastic strain energy of mesh curves of hull surfaces. Moreton and Sequin (1992) applied nonlinear optimization techniques to minimize a fairness functional based on variation of curvature. Nowacki et al. (1992) generated an optimized rectangular network over the data of the ship hull and then constructed a curvature-continuous shape over the network. Nowacki and Lü (1994) proposed procedures for developing fair curves under constraints in which the fairness criterion is based on the linear combination of the square of the second- and the third-derivative norm and the constraints apply to approximation conditions, end conditions and an integral condition pertaining to the area under the curve. Pigounakis and Kaklis (1996) developed a two stage automatic algorithm for fairing cubic parametric B-splines under convexity, tolerance and end constraints. An iterative knot removal and reinsertion technique is employed which adopts the curvature-slope discontinuity as the fairness measure. Pigounakis et al. (1996) proposed three algorithms for fairing spatial B-spline curves: local fairing by knot removal and local/global fairing based on energy minimization; Hahmann (1998) proposed an automatic and local fairing algorithm for bi-cubic B-spline surfaces. In the proposed method a local fairness criterion selects the knot where the spline needs to be faired and the control net is modified by a constrained least-squares approximation. Poliakoff et al. (1999) presented an automated curve-fairing algorithm for cubic B-spline curves based on an extension of Kjellander's (1983) algorithm, which is based on finding and correcting the offending data point. The point to be faired is chosen by calculating for each point the distance to be moved and then choosing the one for which the distance is greatest. Kantorowitz et al. (2000) described a method for fairing ship hull lines which determines the suitable number of control points to produce the required shape of the body sections; Yang and Wang (2001) presented a method for planar curve fairing by minimal energy arc splines where as a first step the optimal tangents for curve interpolation are computed and the point positions are adjusted by smoothing discrete curvatures. Zhang et al. (2001) used strain energy minimization for fairing cubic spline curves and surfaces. Westgaard and Nowacki, 2001 G. Westgaard and H. Nowacki, A process for surface fairing in irregular meshes, Computer Aided Geometric Design 18 (2001) (7), pp. 619–638. Abstract | PDF (398 K) | View Record in Scopus | Cited By in Scopus (3)Westgaard and Nowacki (2001) suggested a stepwise automatic fairing process to construct a smooth surface by optimizing suitably chosen quantitative fairness measures. Kovibia and Pasadas (2004) presented an approximation method of surfaces by a new type of splines called fairness bicubic splines, and the surface is obtained by minimizing a quadratic fairness functional. Renka (2004) suggested a new method for constructing discrete approximation to fair curves and surfaces by directly minimizing an arbitrarily selected fairness functional subject to geometric constraints.

Generally, the outcome of the investigations in this area is the recommendation of using nonlinear optimization techniques that minimize a fairness functional based on the variation of curvature. Based on this outcome a numerical procedure using functional optimization to improve the fairness of ship hull forms is presented. This procedure is aimed at developing a practical design tool to create high-quality ship hull form geometry at the preliminary design stage prior to any refinements that are necessary at the production stage. An initial hull form based on the experience of the designer is assumed to be available. It is also assumed that the fairness quality of this form is not sufficient for extensive performance analysis. The hull form fairing problem is formulated as a nonlinear optimization problem in which the desired hull form will be the one that satisfies various geometric constraints while minimizing (or maximizing) a measure of form quality. A fairing functional based on geometric surface properties can be defined and fairness is measured by assigning every produced hull form a scalar value throughout the optimization process. Smaller values of fairness numerals indicate fairer surfaces. The desired shape will be the one that satisfies various geometric constraints while optimizing the measure of surface quality. The optimization variables of the procedure are the control points of a B-spline surface representing the initial hull form. These are obtained by applying a B-spline surface fitting procedure (Rogers and Adams, 1989).

In the first part of the paper, formulation of the optimization procedure is presented. It is shown that the hull form fairing process can be represented by a nonlinear optimization problem where the optimization variables are the defining control polygon of a B-spline surface representing the initial form and the objective function to be minimized is a fairness functional based on principal curvatures of the hull surface. Prior to the more complicated surface fairing problem the optimization-based fairing procedure is tested in Section 3 for a typical curve representing a ship section line. Section 4 presents typical examples of hull form fairing by using the optimization-based fairing procedure.