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Automatic Hull Variation and Optimization

Stephen M. Hollister1

The process of design optimization is complicated by the infinite variability of the hull shape and the close interrelationship among all of the major design variables. As a vessel is stretched along its length, the prismatic coefficient, the longitudinal center of buoyancy, and the displacement all change. If a design evaluation program is used for optimization, it is difficult to isolate and examine the influence of each major hull shape parameter on the vessel. This paper describes a computer program that solves this problem by automatically varying the shape of a parent hull while maintaining constant values for the major design parameters. A series of hull shape variations can be automatically created and evaluated, with printed, plotted, or contoured results.

INTRODUCTION

Many design evaluation methods, such as resistance calculation techniques, use the major shape parameters of the hull to calculate their results. These parameters typically include length, beam draft, prismatic coefficient (Cp), longitudinal center of buoyancy (lcb), and midship coefficient (Cm). It is difficult to perform parametric or sensitivity studies using these techniques because all of the variables are interrelated and may change at the same time. For example, when a vessel's length is varied, it's Cp, lcb and displacement values all change at the same time. It would be desirable to be able to change any one of these major design variables, while maintaining constant values for the rest. Of course, this is impossible, because as one variable changes, at least one other variable must change to compensate.

Previous work in the area of hull shape creation approached the problem in three different ways: the series approach (such as Series 60), hull variation or distortion from a parent hull, and hull shape creation using form parameters. See references [1]2 - [13]. Many of these methods overlap, some are automatic, and some require user intervention. The program described in this paper uses specific hull variation or distortion techniques due to their generality and their fully automated approach.

This paper discusses a computer program which automatically varies the shape of a parent hull using any of the major design variables. A design constraint can be selected to control which variables are fixed and which variables are free to change. A search technique is

1 New Wave Systems, Inc., Jamestown, Rhode Island

2 Numbers in brackets designate References at end of paper.

Presented at the February 28, 1996 meeting of the New England Section of The Society of Naval Architects and Marine Engineers.

applied to create the derivative hull with the desired values of the major parameters. An inspection routine is then used to extract all hull shape dimensions required by the evaluation routine (resistance calculation). This process of automatic hull variation and evaluation collects all input, intermediate, and final results for printing, plotting, and contouring.



Fig. 1

The overall process flow is shown in the Fig. 1. The program starts with a parent hull and automatically varies the shape according to the designer's input. Each derivative hull shape is automatically evaluated, with the results printed, plotted, or contoured. Each box of the diagram is explained in the following sections.

The purpose of this program is to show the feasibility of carefully controlled hull variation for use by design optimization or evaluation methods. The overall design optimization is not fully automatic. It is under the control of the designer who can carefully vary one or two major shape variables at a time and evaluate the results. By graphing and contouring these results, the designer can quickly study tradeoffs for a wide range of hull shapes and progress toward an optimal design shape.


CREATE PARENT HULL SHAPE

There are many ways to define a hull shape for use by a computer program. The most common methods are:

o Station definition using offsets

Used for volumetric calculations

o B-spline (NURBS) surface definition

Used for fairing and construction

o Ruling line definition

Used to define developable plates

o Wireframe definition

Station definition plus control lines

o Polyhedron mesh definition

Used for flow and structural calculations

Although all of these hull definition methods could be applied to the techniques described in this paper, this example hull variation and evaluation program uses a station definition of the hull, since a perfectly fair hull surface is not required for the resistance evaluation.

This example program is part of the New Wave System's Nautilus System software for ship design and construction which allows you to create a station definition of a hull by typing in a table of offsets, by digitizing a body plan, by reading an existing offset file (SHCP-Ship Hull Characteristics Program, GHS-General Hydrostatics, etc.), or by converting a B-spline surface definition to a station definition. Once a station definition of a hull (parent hull) is created and stored in the vessel's database, the hull variation and evaluation program can be started.



INPUT DEFINITION

The program starts by reading the station definition of the parent hull from the vessel's database. Since the evaluation method being applied to the hull is a resistance calculation, the input values must either come from the shape of the boat or input by the designer. These inputs are listed below.

Basic input from the designer - This data includes things that are not dependent on the shape of the boat, such as boat velocity, water type, and drag options.

Input from the shape of the hull - All hull shape data required by the resistance evaluation program is obtained during the hull "Inspection" routine, discussed later in the paper.

Input of the hull variation constraints - This input tells the program which variables to vary to maintain the proper hull constraints. The two main constraints are constant draft (T) and constant displacement (Disp). This is discussed in the Hull Constraints section.

Input of the calculation and output choices - This input tells the program whether to perform a single calculation, a print or plot over a range of values, or a contour plot over a matrix of input values. The print or plot option allows the designer to vary one of the major design parameters while holding the rest of the variables constant. The contour plot allows the designer to vary two major design variables while holding the rest constant.

Once all user-defined input variables have been entered, the program begins its automatic hull variation and evaluation process.

HULL VARIATION

The process of hull variation produces a derivative hull from a parent hull by specifying new values for any of the following major parameters:

1. Length of the waterline (LWL)

2. Beam of the waterline (BWL)

3. Depth of the boat (Depth)

4. Draft (T)

5. Displacement (Disp)

6. Prismatic coefficient (Cp)

7. Longitudinal center of buoyancy (lcb)

8. Parallel middle body - forward (Pfwd)

9. Parallel middle body - aft (Paft)

10. Midship coefficient factor (Cmfact)

Because many of these parameters are interrelated, the designer must choose whether the calculations are done for a fixed draft or for a fixed displacement. These variables fall into one of the following hull variation routines: Stretch, Balance, Lackenby, or CmVary. Each of these will be discussed below.

STRETCH

This hull variation routine varies any or all of the three major dimensions (length, beam, and depth) by a scale factor. This is a very simple hull variation applied to all offsets of the station definition.

BALANCE

This step doesn't actually change the shape of the station definition, but it does modify some of the major parameters. If the user selects a constant displacement hull variation constraint, then this routine will raise or sink the hull to search for a new draft (T) which maintains a constant displacement. This is done with a searching technique which maintains zero trim while searching for the desired displacement. Note that if the draft changes, the LWL and the BWL will likely change. If a constant draft option is selected, this routine calculates the new displacement for the current hull shape

LACKENBY

This is a technique for hull variation developed by H. Lackenby [5] (see also [6]), which allows the designer to vary any of the following variables, without affecting the LWL, BWL, and the depth of the vessel.

o Prismatic coefficient (Cp)

o Longitudinal center of buoyancy (lcb)

o Parallel middle body - forward (Pfwd)

o Parallel middle body - aft (Paft)

The Lackenby method is a quadratic variation of the "one minus prismatic" approach whereby the lengths of the parallel middle body can be controlled independently of lcb and the prismatic coefficient. The sectional area curve is calculated for the boat and the curve is shifted to achieve the target values. Half-breadths and heights of the offsets in the station definition are not changed using this method; only the longitudinal locations of stations are shifted.

It can take a few iterations of this method to zero in on the desired values. The more stations in the definition, the more accurate it is and the less iterations it takes. The program developed for this paper repeats the Lackenby hull variation calculation until all four variables are within a small tolerance of the target values.

Although the Lackenby hull variation technique allows the designer to vary and set any one or all of these variables, the displacement does change for a constant draft.

An example of this variation can be seen in the following two figures. Fig. 2 shows a parent sailboat canoe-body hull shape with a prismatic coefficient of 0.534. Fig. 3 shows a derivative hull with a target prismatic coefficient of 0.60. Note that although the overall length may vary, the waterline length is fixed and the lcb position is fixed.


Fig 2. Sailboat Hull With a Prismatic Coefficient of 0.534


Fig 3. Sailboat Hull With a Prismatic Coefficient of 0.60

This example of the Lackenby approach was applied to a B-spline surface model of the sailboat to be able to show the smooth transition of buttock shapes. The hull variation program used for this paper, however, applies the same technique to a station definition of a hull.



CMVARY

This is a hull variation technique, developed by the author, that allows the midship shape of the vessel to change independently of the beam and depth of each station. It uses a value called Cmfact which varies the shape of each section diagonally in the direction of the bilge corner, defined as the intersection of the maximum beam and depth of the station. A Cmfact value of 1.0 means that each section shape is rectangular. (See Fig. 4)

Cmfact is related to the midship coefficient (Cm) of a vessel, except that it is based on the overall maximum beam and depth of the vessel, rather than the waterline beam and the draft. Since all of these hull variations affect the entire hull shape, a Cm-type factor was created that also affects the entire section, rather than just the area below the waterline. This makes the factor independent of the draft and displacement.

Fig 4. Station showing Cmfact diagonal

The Cmfact value for a particular station is the percent distance of the station curve from point P (0.0) towards point Q (1.0). The overall Cmfact for the hull is the largest Cmfact of all of the stations and is the value used by the CmVary routine to vary the shape of the hull.

CmVary Hull Variation Steps

1. Find the section with the largest Cmfact value. This is the one Cmfact value that is used for the entire boat.

2. Given a new, target Cmfact, determine the percent increase or decrease of the defining Cmfact section along the diagonal (P-Q) variation line.

3. Decrease this Cmfact change percentage parabolically to zero at each end of the boat. This means that this hull variation tapers off to zero change at the ends of the boat.

4. Apply the appropriately decreased Cmfact change percentage to each station. For each offset point (B), construct a line (R-S) parallel to the main station diagonal (P-Q) and determine the intersections with the defining beam-depth station box (points R and S). Stretch or shrink each offset point along this line by the appropriate Cmfact change factor. Point B is moved towards (or away from) point S by the same percentage as point A is moved towards (or away from) point Q. As the global value of Cmfact approaches 1.0, the shape of each section approaches the rectangular shape of the defining beam-depth box.

Fig 5. Parent hull shape with a Cmfact of 0.76

Fig 6. Derivative hull shape with a Cmfact of 0.80

Fig 7. Derivative hull shape with a Cmfact of 0.72

Figs. 5, 6, and 7 show the parent and two derivative hulls with larger and smaller values of Cmfact. Fig 5 shows the parent hull with a Cmfact of 0.76, Fig. 6 shows a derivative hull with a Cmfact of 0.80, and Fig. 7 shows another derivative hull with a Cmfact of 0.72.

There were difficulties with large changes in the Cmfact value due to its linear transformation of each station toward either of its two diagonal corners. As the change got larger, each section shape developed a corner knuckle or dimple, depending on the direction of variation from the parent shape. Rounding off the hard knuckle corner (at point Q) to use for a Cmfact value of 1 would improve this problem. It would give the vessel a small bilge radius for a Cmfact of 1.0, and would smooth the section shape changes over a greater range of Cmfact values.


OTHER VARIATIONS

There are many more potential avenues for hull variation, such as:

o Variation of the deadrise angle

o Variation of the profile rocker of the boat

o Change sections from U-shapes to V-shapes

o Change section shapes while maintaining constant LWL, BWL, draft, and section area.

Many of these hull variations are important in their own right, but for the resistance evaluation routine described in this program, they play a smaller role than varying the major hull shape parameters.


HULL CONSTRAINTS

The Stretch variation changes the length, beam, and depth of the boat, but the displacement, Cp, lcb, etc. all change. Balance determines the new draft (T) for a given displacement, but the length, beam, Cp, etc. all change. Lackenby maintains constant LWL, BWL, and draft, while varying Cp, lcb, Pfwd, and Paft, but displacement changes. CmVary maintains constant length, beam, and draft, but varies the displacement and prismatic coefficient. It is impossible to fix all of these major design variables to create one specific derivative hull shape. If one variable is set or changed by the program, then at least one other variable must change at the same time. The goal, however, is to allow the designer to change the major design variables in a controlled manner using constraints.

The following equations show the relationships among many of the major design variables.






where sa(x) is the sectional area at x

In an ideal case, the designer should be able to vary any parameter in the above equation and to select any other parameter to balance the equation so that the rest of the parameters remain fixed. This may be possible, but it might require a whole range of constraint maintenance routines. The program described in this paper only allows the designer to vary either the draft or the displacement to obtain target values of the major design parameters. For example, a designer might vary LWL, while maintaining fixed values for the rest of the parameters, except for draft, which would have to vary to maintain constant displacement.

Another way of looking at the problem is to fix all parameter values but one, and then solve (search) for a derivative hull that meets those set values. That is what the program in this paper does. Specifically, this program can generate a derivative hull shape with any values for the major variables, except for displacement and draft. If the draft constraint is set by the designer, the program will vary displacement to achieve all of the other target values. If the displacement constraint is set, the draft of vessel changes while achieving the rest of the target variables. This hull variation and constraint search loop is shown in Fig. 8.

Given target values for length, beam, depth, Cp, lcb, Pfwd, Paft, and Cmfact, this search loop will vary either draft or displacement to create a derivative hull with the desired values. This non-linear Hull Variation and Constraint search loop has gone through many variations to speed up the search process. It uses a form of a conjugate direction search technique to zero in on a solution. The balance step is performed after each hull variation step, since all parameters are very sensitive to draft and displacement.

This is just one approach to the constraint maintenance problem. One could derive a hull variation and constraint search loop based on varying any of the major design variables. Note that a new search technique would be needed by this program if both the displacement and the draft of the vessel are to be fixed. One could easily achieve this, however, by varying BWL, for example.

Fig. 8

There are also limits to the range of parameter values that can be achieved. Some shapes are physically not possible, but then again, those shapes are of no interest to the designer.


HULL INSPECTION ROUTINE

Once the hull variation and search loop is complete, this routine determines all of the hull shape information that is required by the Hull Evaluation routine. Exactly what is calculated here depends on the needs and sophistication of the evaluation routine. For many resistance calculation programs, this might mean calculating simple information, such as volume, wetted surface, waterplane area, and the half angle of the waterplane entry. Most of these values are determined automatically from the hydrostatic calculations in the Balance routine.

In addition, this inspection routine can also calculate the righting moments and righting arm areas of the vessel for any heel angle, with automatic balancing for trim. This allows the designer to study the trade-offs between resistance and stability.


HULL EVALUATION ROUTINE

In discussing the automatic hull variation and optimization process, little was said about the specific type of evaluation routine that would be used to analyze the shape of the hull. For this paper, a resistance technique based on the use of a "Geosim" coefficient approach is used. This technique estimates the overall resistance of a vessel by splitting the resistance into component parts, such as viscous resistance, residual resistance, and appendage resistance.

In general, however, this hull variation and constraint technique can be used with any hull shape evaluation technique. A sample of the major evaluation categories are given below. Following that is a more detailed description of the Geosim technique used in this program.

1. Geosim Coefficient Resistance Evaluation - These techniques break the overall resistance of the vessel into their component parts, such as viscous resistance (Cv), residual resistance (Cr), and appendage resistance. Residual resistances are often determined from a regression analysis of tank test data.

2. Planing Hull Resistance - This type of resistance calculation uses the same force breakdown techniques as the Geosim approach, but adds the moments of forces about the center of gravity of the boat. This defines a steady state free body diagram of the boat that can be solved for the resistance and trim of the boat.

3. Sailboat Velocity Prediction (VPP) -This type of resistance prediction is also based on a Geosim approach, but adds sail and hull lift and drag forces. A search technique is used to determine the velocity and heel angle of the boat for each wind speed and wind angle.

Note that for certain racing sailboats, the hull variation process is further complicated by the need to meet the shape constraints of any applicable rule, such as the America's Cup rule.

4. Computational Fluid Dynamics Methods (CFD) - These techniques typically use a polyhedron mesh generated from the shape of the hull to perform a full 3D analysis of the resistance of the hull.

5. Finite Analysis Methods (FEM) - These techniques use a polyhedron mesh generated from the shape of the hull to perform a 3D structural finite element analysis of the hull shell.


Geosim Coefficient Resistance Evaluation

The program described in this paper uses a Geosim approach to the evaluation of the vessel's resistance. This technique breaks the overall resistance into the following resistance components:

o Total Resistance (Rtot)


o Viscous Resistance (Rvisc)

Cf - Based on ATTC or ITTC57 friction lines

k - 3D viscous form factor

o Residual Resistance (Rresid)


o Correlation Allowance Resistance (Rcorr)

o Appendage Resistance (Rapp)

o Bulb Resistance (Rbulb)

o Transom Resistance (Rtrans)

o Air Resistance (Rair)


The program contains several different choices for each of the categories of resistance described above. The designer either selects an overall resistance approach from a menu or creates a customized approach by selecting one technique from each of the resistance categories.

Most of the results in this paper come from applying the Holtrop [14],[15] ship resistance methods for each of the resistance components.

The details of these calculations are not presented here, since the goal of this paper is to describe the overall hull variation, constraint, and evaluation process, rather than the particular resistance calculation methods used.


PRODUCTION OF RESULTS

The designer determines whether one condition is calculated and printed, a looping print is produced, a looping plot is produced, or a contour plot is produced. Each of these choices is described below. First, input and output variables and algebraic expressions will be briefly discussed.

Program Variables

All program input, intermediate, and final calculation variables are defined and labeled for access by the designer. Input variables are also broken into two categories: hull shape independent and hull shape dependent variables. Hull shape dependent input variables include LWL, BWL, Depth, T, Disp, Cp, lcb, Pfwd, Paft, and Cmfact. Hull shape independent variables include values such as boat velocity, water type and temperature, and vertical center of gravity (VCG).

Most intermediate and all final calculation values are also labeled for use by the designer. This is useful if there is a need to study the details of the resistance calculation technique itself. If some of the results seem questionable, the designer may evaluate all of the individual variables of the resistance calculation to see if some result values are abnormal. This is important for both program verification and resistance model validation. No calculation program should present itself to the designer as a "black box" evaluation.

Algebraic Expressions

For any of the output choices, the designer can also specify any algebraic combination of input and output variables to display or print. The program will parse the expression after the resistance calculation is done and evaluate the expression using the proper variables. For example, a designer might vary LWL over a range of values using a constant draft constraint and request that the program print the value for the expression (Rtot/Disp) as one of its output variables. In addition, all algebraic and trigonometric functions, such as SQRT, LOG, and TAN, can also be used.


Single Calculation

The designer selects the target values of all major hull shape parameters, along with a required constraint: constant draft or constant displacement. Input also consists of non-hull shape information, such as boat velocity, water type and temperature, and vertical center of gravity.

The calculation section first searches for the single target hull shape, then calculates the resistance information, and finally displays or prints all input, intermediate, and final results.


Looping Print Calculation

The user specifies any single input variable and the range of values it will take. The program automatically performs a complete hull variation for each value of the input variable, calculates the results, and displays or prints the designer-selected input and output variables and expressions.

The following output shows the results for varying BWL from 80 to 120 feet, with constant displacement and variable draft. Note that the other major design parameters are held constant, to within a one-half percent search tolerance.

BWL LWL Disp Cp LCB Rtot

80.02 307.51 15142 0.6794 0.0408 53.01

85.02 307.01 15147 0.6796 0.0409 51.51

90.02 306.56 15153 0.6797 0.0416 50.05

95.01 306.40 15158 0.6799 0.0402 48.82

99.97 305.79 15162 0.6801 0.0413 47.83

104.92 305.48 15162 0.6799 0.0415 47.17

109.88 305.08 15159 0.6799 0.0419 46.81

114.74 304.73 15159 0.6799 0.0419 46.74

119.93 305.76 15157 0.6799 0.0425 47.02

BWL T LWL/BWL LWL/T Rvisc Rresid

80.02 41.09 3.84 7.48 35.73 14.99

85.02 39.26 3.61 7.81 36.10 13.11

90.02 37.62 3.40 8.14 36.57 11.16

95.01 36.11 3.22 8.48 37.16 9.32

99.97 34.78 3.05 8.79 37.79 7.67

104.92 33.56 2.91 9.10 38.56 6.22

109.88 32.43 2.77 9.40 39.39 4.98

114.74 31.39 2.65 9.70 40.30 3.96

119.93 30.27 2.54 10.09 41.42 3.09

Also note that the purpose of this paper is to demonstrate the hull variation, constraint and evaluation techniques, and not to evaluate the actual numbers produced by the resistance part of the program.


Looping Plot Calculation

This function works the same as the Looping Print Calculation, except that the results are plotted. In this case, however, the designer specifies the independent input variable to vary and the data to be plotted.

The plotted data consists of one or more ordinate variables or expressions and one abscissa variable or expression. Note that the abscissa variable does not have to be the same as the independent input variable. The designer can plot any variable or expression versus any other variable or expression. This can be done for any input, intermediate, and final result variables. The designer can also specify the title, labels, tick marks, and grid options of the graph.

Plus - Total Resistance

Circle - Viscous Resistance

Triangle - Residual Resistance

Fig. 9 Example of the Looping Plot Output


Contour Plot Calculation

The contour plot function allows the designer to plot contour graphs of variables or expressions over a range of two input variables. For example, constant contour lines of Rtot can be plotted over a range of LWL and BWL values, using a constant displacement constraint. The program will vary the hull shape over a matrix of LWL and BWL values while maintaining constant values for the rest of the major design parameters (except for draft, which must vary to achieve a constant displacement).

Like the Looping Plot Calculation, this function allows you to specify multiple contour variables or expressions to be plotted at one time. The abscissa and ordinate values or expressions also do not have to be the same as the independent variables.

Fig. 10 Example of the Contour Plot Output

This graph shows constant resistance contour lines over a range of LWL and BWL values. This is drawn like a topographical map of total resistance, with the lowest and highest points of resistance identified and labeled (in tons of resistance). The lowest total resistance is located at the shortest length and the widest beam. Note that the draft is also less at this point because of the constant displacement constraint. Graphs like this might raise more questions about the resistance calculation than they answer, but that is preferable to using a resistance calculation program like a black box calculator. The designer can explore other variations in shape and compare the results with the equations used by the resistance model. In addition, this hull variation and evaluation approach can be invaluable to those developing new resistance techniques based on regression analysis techniques.


CONCLUSIONS

The computer program described in this paper demonstrates the ability to automatically vary the shape of a parent hull to obtain target values for the following major hull shape variables: LWL, BWL, depth, draft, displacement, Cp, lcb, parallel middle body-forward (Pfwd), parallel middle body-aft (Paft), and Cmfact. Target values are achieved using a constant displacement or constant draft constraint. If a constant draft constraint is selected, then all of the target values are obtained by varying the displacement of the vessel. If a constant displacement constraint is selected, then the target values are obtained by varying the draft of the vessel.

The program also demonstrates the process of automatically varying a parent hull shape over a range of target values, evaluating the hull shape using a "Geosim" resistance calculation routine, and printing, plotting, or contouring the results.



FUTURE WORK

This paper only scratches the surface in the area of automatic hull variation with constraints. As mentioned previously, the designer should be able to achieve the desired target values for all major design variables by varying just one selected parameter. In this paper, this was achieved by varying the draft or displacement. It should be a straight-forward change, however, to vary another variable, such as LWL, to achieve the desired target values.

There are also an infinite variety of other hull variations to investigate, such as variable bilge radius, variable deadrise angle, adjustable rocker shape, and adjustable section shape. This paper focused on variations of the major design parameters, since those variables had the greatest influence on the resistance calculation technique.

An associated area of development is the automatic creation of hull shapes from a set of standard form parameters. One could automatically create an approximate hull shape from form parameters and use the automatic hull variation process to achieve the exact target values.

Another area of development is to separate the hull variation routine, the hull inspection routine, the hull evaluation routine, and the output results routine into separate modules or programs, linked by a simple data and geometry file. This would allow designers to replace the existing evaluation routine or program with their own, but still use the standard hull variation and results evaluation framework.

Other versions of this program are under development; one for the planing boat model and another for the Velocity Prediction Program (VPP) model for sailboats. Each of these hull types have their own special hull variation and constraint requirements.

Another interesting area of development is in the automatic variation of CFD or FEM hull meshes. Rather than remeshing after each hull variation (which may not be automatic), the hull variation could be performed directly on a parent hull mesh. This would eliminate the need for remeshing after each step and would allow the analysis program to analyze many hull variations without human intervention.



REFERENCES

1. Todd, F., "Series 60, Methodized Experiments with Models of Single-Screw Merchant Ships" DTMB Report 1712, July, 1963

2. Gertler, M., "A Reanalysis of the Original Test Data for the Taylor Standard Series", DTMB Report 806, March, 1954

3. Moor, D., Parker, M., and Pattulo, R., "The BSRA Methodical Series - An Overall Presentation", Trans. RINA, vol 103, 1961

4. Lap, A., "Diagrams for Determining the Resistance of Single Screw Ships", International Shipbuilding Progress, vol 1, no. 4, 1954

5. Lackenby, H., "On the Systematic Geometrical Variation of Ship Forms", Trans. INA, Vol 92, 1950, pp 289-315

6. Botting, M., "Hull Form Derived From Parent", DTMB Report 2328, Feb, 1967

7. Volker, H., "Systematic Variation of Usual Ship Form", International Shipbuilding Progress, vol 1, no. 1, 1954

8. Reed, A. and Nowacki, H., "Interactive Creation of Fair Ship Lines", Journal of Ship Research, vol 18, 1974, pp 96-112

9. Kuiper, G. "Preliminary Design of Ship Lines by Mathematical Methods", Journal of Ship Research, vol 14, 1970, pp 52-66

10. Nowacki, H., Creutz, G., and Munchmeyer, F., "Ship Lines Creation by Computer - Objectives, Methods and Results", First International Symposium on Computer-Aided Hull Surface Definition (SCAHD77), pp 1-18, SNAME, Anapolis, MD, 1977

11. Soding, H. and Rabien, U., "Hull Surface Design by Modifying an Existing Hull", First International Symposium on Computer-Aided Hull Surface Definition (SCAHD77), pp 19-29, SNAME, Anapolis, MD, 1977

12. Nowacki, H. and Reese, D., "Design and Fairing of Ship Surfaces", Surfaces in Computer Aided Geometric Design, North Holland, Eds. Barnhill, R. and Boehm, W., 1983, pp 121-134

13. Standerski, N., "The Generation of Ship Hulls With Given Design Parameters Using Tensor Product Surfaces", Theory and Practice of Geometric Modeling, Eds. Strasser, W. and Seidel, H., Springer Verlag, 1989, pp 65-79

14. Holtrop, J. and Mennen, G., "An Approximate Power Prediction Method", International Shipbuilding Progress, Vol 29, July, 1982

15. Holtrop, J. "A Statistical Re-Analysis of Resistance and Propulsion Data", International Shipbuilding Progress, Vol 31, Nov, 1984