The given paper represents an attempt to generalize some practical principles of simulating vessel surfaces with the use of NURBS.

A.Aleksanov.
Alex@orkinus.com
 

NURBS-based representation of curves and surfaces has already become a standard of computer simulation for many industries. Since 2000 the leading producers of CAD systems for shipbuilding also use this standard for introducing a vessel surface.

NURBS-based representation gives such wide opportunities for simulation as no other means does and, at the same time, NURBS has a series of features creating difficulties for understanding by the user.
It is a mechanism which controls the shape of curves and surfaces that a user faces at first simulating with the help of NURBS. Traditionally, in all previously existing mold-loft methods, control of the shape of curves and surfaces was carried out by defining points through which there passed a required curve or a surface. In a case with NURBS the curve passes only through finite points of a reference polygon. The shape of a curve will be defined by the shape of the reference polygon but to make a curve or a surface pass through the given point auxiliary conditions are required.

During a long time the basic method of simulating a vessel surface was a method of defining a grid of lines on this surface. Various mathematical models were used as the mathematical representation of curves. The most widespread model was a spline, which is analogous to a flexible spilling batten. It was a grid of intersecting lines that determined a vessel surface. Frames, water-lines and buttock-lines, generally, were used as the lines. If during the operation with the surface there arose a necessity of deriving coordinates of whatever point of the surface, the algorithm for calculation of the given point was started depending on the shape of adjacent lines. Advantages of the given method include operation of the user with lines which are natural for a ship constructor. Disadvantage of this method is the absence of an analytically continuous (on tangents and curvature) surface. Notwithstanding the fact that NURBS-based representation of a vessel surface meets the requirements of analyticity and smoothness, the absence of possibility of controlling the shape of the surface by means of direct modification of frames, water-lines and buttock-lines may become a major problem for a novice user. Besides, as a rule, a NURBS surface is visualized as a set of lines of equal parameter which hardly explain the shape of the given surface to the user. Therefore, many NURBS modelers include a method of shaping a surface which passes through a set of cross sections (Cross sectional design). Unfortunately, this method is not always applicable in case of a vessel surface since it is impossible to describe intricate contours, as for example, a bow bulb. One of solutions of this problem can be the possibility to dynamically map the deformation of sections while editing breakpoints of the surface. The user will control the shape of the surface by moving the breakpoints of a polyhedron and thus the user can watch interactive deformation of sections.

As distinct from aviation or automotive industry where the shape of the hull is developed and optimized during a long time, the terms of developing a surface in shipbuilding are very tight and in this case the optimum division of the surface into sections becomes of high importance for construction of a NURBS-based vessel surface. While dividing the surface into sections, it is necessary to take into account a series of mathematical features of curves and surfaces. Based on a long-term experience of the NURBS-based simulation of vessel surfaces it is possible to offer the following principal requirements for dividing a surface into sections:

• usage of degree of NURBS surfaces is not higher than the third degree. The higher degree gives an additional smoothness of a surface and at the same time augments a range of variation of the surface at correction of one breakpoint. Thus, the property of localization of the surface modification is lost and there occurs a necessity of increasing the amount of breakpoints.

• obligatory segmentation into separate sections of the surface between lines of fractures. Usage of mathematical NURBS properties for creation of fractures inside the surface section is possible, but labor-intensive for control and is practically not supported by many systems if it is necessary to transmit data from one system to another.

• obligatory segmentation of flat side sections, flat bottom sections and sections of ruled surfaces allows to effectively control the shape of these lines. Without segmentation into separate sections of the surface it is practically impossible to obtain a correct line of a flat side or bottom only by means of breakpoints of the surface.

• attributable to NURBS limitation of the amount of margin lines of the surface sections (usually three or four) can be bypassed by cutting-off the surfaces. Thus, the modeling surface is extended beyond the boundaries of the modeled area and is cut off along these boundaries.

• it is necessary to avoid segmentation of smooth surfaces into separate sections. Joining of such sections, as a rule, is performed only by the first derivative and does not result in required smoothness. Sections of the surface in this region will not look smooth enough even visually. Only in case of radial conjugation of the surfaces the joining is admissible and looks quite natural, in this case it is not necessary to maintain the condition of continuity of flexions.

• difficulties in simulating the shape of the surface section angles can be avoided only by changing the shape of the margin lines of the surface section. Breakpoints inside the surface cannot practically affect the entrance angles of the surface sections in the environ of the angle point.

Observing the above said requirements will allow avoiding many problems at simulation, to decrease time and increase quality of the surface.

The quality of simulating of the vessel surface is one of the main factors for more accurate simulation of vessel structures, reduction the time periods and improvement of the hull assembly and welding quality that, in the long run, results in significant decrease of construction expenses. Thus, for example, the bent shell plates are among the most labor-consuming components of the hull both with regard to the manufacture and installation. The quality of simulating the shell plates directly depends on the quality of simulating of the vessel surface. In case of poor-quality simulation of the vessel surface and plate developments the errors, as a rule, are revealed only at assembly, therefore, the production workers have to add rough tolerances to edges of the shell plates and to trim them on-site during assembly. This process requires a large amount of time and completely degrades the mounting and welding quality. In the long run, the expenditures on elimination of defect can considerably exceed the expenditures on simulation. Simulation of hull plates without rough tolerances is considered to be optimal, but in this case the quality of the vessel surface should be faultless.

Thus, control of the surface quality is one of the most important constituents in any vessel surface simulation system. The quality of obtained surface directly depends on implementation degree of control tools in the system. Until recently the quality of a vessel surface was examined visually on the hull sections traced on a large scale and this it was a labor-intensive process. It was necessary to trace, make modifications and trace once and again. Thus, operation speed was determined only by the speed of the plotter. In many NURBS-based state-of-the-art systems there exist the tools for control of the shape of the modeled surface on the basis of visualization of a Gaussian curvature of the surface. It is not always applicable in case of a vessel surface as the specialist can hardly make any judgments concerning the acceptability of the shape of the vessel hull sections. Sometimes, the vessel surface, quite acceptable by the Gaussian curvature, leads to inadmissible bends of structural sections of the hull. Thus, while modeling the surface the probability of errors is augmented and the quality is degraded. The same refers to visualization of filled surfaces. The filled surface always looks much better than it actually is.

It is possible to enumerate some of the principal tools for control of the vessel surface quality which make it unnecessary to constantly print the drawing on paper:

• visualization of a Gaussian curvature. Despite of ambiguity and non-demonstrativeness of the vessel surface representation it is possible to reveal problem zones.

• visualization of radii of curvature of the curves, sections and surfaces. Presently, it is one of the principal control tools for many ship-building systems. Visualization of graphs of the radii of curvature (instead of curvatures) allows to control the most problematic areas of the curves, i.e. regions with a curvature which is close to zero.

• visualization of inflection points of curves, knuckle lines of surfaces and sections.

This allows to more graphically imagine the shape of the surface, the law of variation of the shape of the surface sections. It shows the regions of cambers and excessive undulation even between the surface sections.

• a dynamic variation of the shape of the knuckle lines and curvatures at variation of the surface shape. It is most convenient for planarization in the manual mode, while repositioning the breakpoints. The user can see not only variation of the shape of the surfaces sections, but also the radii of curvature and knuckle lines.

• visualization of lines and sections in a compressed form in one of the coordinates. It is a very important property for simulation of surfaces, which are significantly elongated in one of the coordinates as, for example, surfaces of the wings, keels of the yachts or a rudder blade, particularly, if the system allows editing breakpoints of the surface in this mode. It is also a very useful property for control of butting the lines of the sections to a flat side, flat bottom, as well as for control of joining the bow and stern sections of surfaces in the midship. Compressing a model in one of the coordinates is similar to visual inspection of the shape of the curves in the drawing if viewed along the curve and, thus, the viewer eye level is hardly above the level of the table. If in the compressed form the model looks smooth, it will look even better in the normal form.

• an automatic control of deviations from the source data. In some cases when it is necessary to achieve a high degree of approximation of the modeled surface to the initial one, a constant control of deviations from the initial points is required. It is especially important if it is necessary to restore the surface of an existing vessel for reconditioning or remodeling. In such cases the criterion of the surface quality is minimization of deviations from the source data.

Application of the surface quality control tools mentioned above allows to completely abandon printing the drawings on paper, to greatly reduce the time of simulation and considerably increase the quality of a modeled surface.

The rules and requirements described in this paper are founded on a long-term experience of using the Sea Solution system. In view of similarity of the mathematical representation of curves and surfaces this is applicable for most of other NURBS-based systems.

Summarizing the aforesaid it is possible to distinguish some the principal stages of designing a vessel surface:

• the analysis of the shape of a modeled surface and selection of segmentation into sections. It is a very important factor while creating a surface. The labor intensity of simulation and quality of a surface depends on optimality of segmentation of the surface into sections.

• the definition of the surface sections with a minimum set of breakpoints, approximation of the shape of a surface to the source data and allocation of the breakpoints. Your idea of the shape of a surface, if the hull is designed “from scratch” or the line of a rough lines drawing, or the offset table points can be taken as the source data.

• increase in amount of the breakpoints, definition of knuckle lines on frames, water-lines and buttock-lines, more exact approximation to the source data. At this stage it is necessary to make efforts for minimizing the deviation from the source data and at the same time defining correct arrangement of the knuckle lines, i.e. eliminating errors, which exist in the initial source data.

• minimization of the curvature of water-lines, frames and buttock-lines. At this final stage it is necessary to achieve the acceptable smoothness of lines of the sections. It is a repetitive process and, generally, a small displacement of breakpoints is enough for meeting the smoothness requirements. During the smoothing process it is also necessary to control position of the knuckle lines.

Following this technique and using the Sea Solution system during the last five years only the “Steelcad Consultants AS” company has simulated more than 200 surfaces of the hulls of different types of the vessels. The most of them were used at construction of the vessels and have proved to be the high quality surfaces and shell plate developments. This paper is only a small part of accumulated practical experience, but it also can be useful both for novice users and specialists in simulating with the use of NURBS.

A.Aleksanov.
Alex@orkinus.com
 

Fig.1. The example of segmentation of a fore end of a supply vessel into surface sections. The basic surface is carried out for one section and formed by a line of a flat bottom, a line of bilge radius, a line of a flat side transforming into a deck-line and a line of a diametric buttock-line (blue lines). In the area of the upper deck the surface is elongated beyond the fracture line of the side and is cut off by a vertical surface of the bulwark (a red line).
 

Fig.2. The same after cutting off the surfaces.
 

Fig.3. The final variant of the surface divided into sections 32х32 breakpoints. A degree of the surface is 3.
 

Fig.4. The example of a singular point in an angle of the surface section. The line of a flat bottom and a line of a flat bottom (Линия плоского днища и линия плоского днища образуют) form a local plane within the area of the angle point of the surface. The entrance angle of a frame within the area of the angle point is determined as the angle of the given local plane and frame plane intersection line. Thus, you can manipulate the entrance angle of the frame only by changing the configuration of the margin lines of the surface. In this case it is the abutment angle of the flat side line and diametric buttock-line.

Fig.5. The example of radial mating of a stern transom of a trawler ship.
 

Fig.6. The example of radial cross-structure of a twin-hulled vessel.
 

Fig.7. The example of a surface section without segmentation of the flat side line. It is practically impossible to affect the shape of the buttock-lines within the area of the flat side.
 

Fig.8. The example of a surface with a flat side line. The shape of the buttock-lines, which are close to the flat side, iterates the shape of the flat side line.
 

Fig.9. The example of unsuccessful segmentation of the bow surface. The vertical light green line is the boundary of two sections. Thus, it was possible to achieve smoothness only on tangents, but not on the curvature. In the long run the lines still looked insufficiently smooth.
 

Fig.10. The example of allocation of the breakpoints on the projected hull and visualization of knuckle lines of the frames. Since the third degree NURBS is used the line of bends is not smooth.
 

Fig.11. Visualization of the radii of curvature of the frames within the area of transition of the bilge to the dead flat. Local flattening of the frames is visible.
 

Fig.12. Visualization of a compressed model in coordinate X. It is easy to control the joining of two sections of the surface within midship.
 

Fig.13. Visualization of a compressed model in coordinate X. Compression factor is 0.1. Visual control of smooth conjugation of two surface sections within midship.
 

Fig.14. The same without compressing. It is very difficult to control the shape of the lines.