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Hull form generation using multi-objective optimisation techniques


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By: D Peacock, W F Smith & P K Pal


JUNE 1997




Hull-form Generation Using Multi-objective Optimisation Techniques


Research Student

Australian Maritime Engineering Cooperative Research Centre.

Sydney Core

The University of New South Wales, Sydney 2052 Australia


Senior Naval Architect

Directorate of Engineering Concepts, Capability and Naval Architecture

Department of Defence, Canberra 2600, Australia


Honoraiy Visiting Fellow

Australian Maritime Engineering Cooperative Research Centre,

Sydney Core

The University of New South Wales, Sydney 2052 Australia





Hull-form Generation Using Multi-objective Optimisation Techniques

DUGALD PEACOCK Research Student

Australian Maritime Engineering Cooperative Research Centre, Sydney Research Core

The University of New South Wales, Sydney, 2052, Australia


Senior Naval Architect

Directorate of Engineering Concepts, Capability and Naval Architecture Department of Defence, Canberra, 2600, Australia

PRABHAT KUMAR PAL Honorary Visiting Fellow

Australian Maritime Engineering Cooperative Research Centre, Sydney Research Core

The University of New South Wales, Sydney, 2052, Australia


During the preliminary ship design process, when only the principal design parameters are known, if

ship motions characteristics and stability particulars are required. then no accurate method to calculate

them is available unless a complete description of the hull-form is provided.

For this reason, a

methodology by which a ship's hull-form can be generated has been developed for non-interactive computer-based preliminary design, when only the principal design dimensions and form coefficients are known.

In this work, the problem of developing a displacement-type hull-form is formulated as a compromise decision support problem (DSP) and solutions are found using the decision support problem

technique (DSPT). The compromise DSP formulation is a multi-objective programming model

- a

hybrid, built on concepts from both traditional mathematical and goal progranmiing.

The hull-form matching a specified vector of design parameters is generated using B-spline curves

and surfaces. This approach models the hull-forms from first principles rather than from a basis ship. A computer program has been developed and validated using a set of design dimensions and form

coefficients for a published design.


AAP Area at transom (AP) D/T Depth-to-draft ratio

AFP Area at forward perpendicular (FP) DEPTH Minimum hull girder depth

AM Maximum sectional area DF/D Deck height forward to depth ratio

AP Aft perpendicular DKAFT Deck height aft-most point

AREA Area under curve DKB Deck beam maximum

B/T Beam to draft ratio D KB TR Deck beam at the transom

BEAM Maximum beam on DWL DKFWD Deck height forward-most point BTR Beam at the transom DOAP Deck overhang AP

Bi ith B-spline control vertex DOFP Deck overhang FP Bxi ith X component of control vertex FOREFT Forefoot location

Byi ith Y component of control vertex FP Forward perpendicular

Bzi ith Z component of control vertex HPOA Height of propeller opening aft

Cd Deck area coefficient HPOF Height of propeller opening forward

Cp Prismatic coefficient lE Half angle of entrance

Cwp Waterplane area coefficient ISTAS Counter for stations Cx Maximum section area coefficient L/B Length-to-beam ratio



The design of complex engineering shapes, such as ship hulls, requires a flexible but intelligent

method for representing these surfaces. In ship design, the traditional method of describing a hull-form is with a lines plan. (The lines plan is a two-dimensional orthographic drawing that gives a

representation of three-dimensional a hull-form.) However, the lines plan is not directly

mathematically related to the principal dimensions and form coefficients (subsequently known as design parameters). Therefore, there is a need to be able to establish a unique relationship between the design parameters and a lines plan. This relationship becomes very important when a preliminary design mathematical model is developed, which incorporates ship motions, resistance and stability calculations, all of which ase dependent on the lines being defined, as discussed by Peacock f 1]. Fundamentally, this relationship between the design parameters and the lines plan cannot have a unique solution, since there are many lines plans which can be described by the same set of design parameters. Fortunately, a set of design parameters with a fairness criterion can uniquely define a

lines plan.

Traditionally, three approaches to lines development have been used, these being systematic variation

of a parent hull-form, interactive modification of a parent hull-form, and generation of a hull-form

using first principles from design parameters. The first two of these methods involve the modification

of a parent hull-form. Thus, the subsequent hull-forms are limited to a certain design corridor related to the characteristics of the parent hull. The third of these methods offers the best opportunities for the preliminary design process, as the subsequent generated hull-forms are not directly limited to the

shape of parent hull-form and a wider range of hull shapes can be generated. Since no unique

mathematical transformation of the parameters exists, the best solution satisfying the required design

parameters with the best fairness characteristics will be the chosen hull-form.

The design problem is solved using the decision support problem technique (DSPT) suggested by

Mistree [2]. The DSPT was chosen primarily due to its ability to handle highly-constrained non-linear problems efficiently. In the current work a method is demonstrated that allows the rapid development

of ship hull-forms using a DSPT approach employing B-spline curves.

B-spline Curves

As B-spline curves are essential to this paper, a brief summary of the main properties of B-spline curves is detailed below. The underlying theory for B(asis)-Spline curves and surfaces is well established. A detailed explanation of B-spline curve and surfaces can be found in Rogers [3]. The

following key equations are written using Rogers' nomenclature.

A B-spline curve, shown in Figure 1, is a piece-wise continuous curve which is defined in the

following form: let F(t) be the position vector (coordinates of the point) along the ith segment of the

curve as a function of the parameter t.

LAFT Length of curve aft of amidships RISEK Rise of keel LBP Length between perpendiculars S TS L P Stem slope FP

LCB Longitudinal centre of buoyancy TAP Angle of section area curve at AP

LCB Calculated LCB TDPF Tangent at FP of main deck

LCF Longitudinal centre of flotation TFP Angle of section area curve at FP

LFBT Length flat bottom (propeller entrance) XBMAX Location of beam maximum

LFWD Length of curve forward of amidships XDM Location of minimum hull girder depth

Nir Basis function XDMAX Location of deck beam maximum


From all the different representations of the B-spline, the periodic description was chosen due to its

ability to be mathematically expressed in matrix form and its ability to allow control of the end conditions for the curve. This spline also allowed the vector area subtended under a B-spline curve to

be computed in matrix form using the formula suggested by MacCallum [6]. Figure 1: Single-Segment B-spline Curve


= Nr(t)H+r



where B. are the position vectors of the defining polygon vertices and


is the B-spline basis function. This basis function contains information on how the curve segments are joined together at the segment's ends, known as knots. This function, a blending function, also includes information on the behaviour of the curve derivatives at these knot points. The B-spline basis function is defined by a recursive relationship suggested independently by Cox [4] and de Boor [5]. There exist several types of spline basis functions, each differing by the way the knot vectors are defined. The B-spline curve chosen for use in this paper is the fourth-order periodic knot vector B-B-spline. This B-spline

A property exhibited by all non-trivial B-spline curves is that they do not necessarily pass through the control points of their defining polygon, except under special circumstances, such as multiple

coincident control vertices. Periodic B-spline curves also exhibit the extraproperty that the curves do not necessarily start nor end at the first or last control vertex. This is illustrated in Figure 2, where the

curve does not pass through the control vertex B9. However, the curve is forced to pass through control vertex B1 by the placement of multiple coincident control vertices at the start, B113. The curve is also forced to pass through control vertex B6, by the careful placement of vertices B and B7.

a = 1

B1 B




18 1



183 38



150 71



9 10 Bi_2 B_ B B1 {B2

is a piece-wise continuous cubic curve with second-order continuity.

spline can be expressed in matrix form as follows:

{b0(t) b1(t) b7(t) b3(r)] =


t2 t 6




1 The basis

3 3



3 0 0 3 0 4 1 0

function for this

B1 B2 B-Spline Curve


e Control Polygon


Figure 2: Example Periodic B-spline Curve

The end conditions of the periodic B-spline curve segments have been studied extensively by

Barsky [7]. In the study Barsky derived formulae to allow control of the curve's starting and ending points. The method he developed also allowed for the control of the curve's derivatives at the end points. The control of the curve's end points is accomplished by the placement of pseudo-vertices either side of the required end point. In the current research work all the curves are controlled by using pseudo-vertices as shown in Figure 3. The pseudo-vertices, labelled B1, B1, in Figure 3, are

placed a distanceARMapart with the curve end point vertex B located equally between the points.



Figure 3: Standard Arrangement for Controlling Curve End Conditions

The derivative conditions for the curves used in this paper are for natural spline ends with starting

points located at vertex and with the curve having an initial slope of F' at and ending at vertex

B with a slope of


at B. For other points on the curve for which this type of control is required, the curve is imagined to be broken into two separate curves at the point of interest. However, the

control vertices used are the same for the start and end of the curve, hence maintaining curve continuity. An example of this method of can be seen in Figure 2, where the curve has been forced to

pass through the vertex B6 with a specified slope.

This ability to control the end conditions, along with the curve's local control properties, made it the ideal choice for hull-generation. The local control property can be seen from the curve definition, where a point on the curve is only affected by the four local control vertices. The family of B-spline

curves also exhibit several other useful properties as described by Rogers: - The curve exhibits the variation diminishing property.

- The curve generally follows the shape of the defining polygon.

- Any transformation can be applied to the curve by applying it to the defining polygon vertices.

The most useful of these properties for this work is that the application of a transformation to the defining polygon vertices is the same as the application of that transformation to the curve. This property is used when mathematically fairing the curves.



Decision-based Design

Given that design activity relies on making decisions, Mistree has offered a method under the

Decision Based Design (DBD) umbrella known as the Decision Support Problem Tecimique (DSPT) to assist the designer in making better-informed choices. The underlying theory and a detailed explanation of the method was given by Mistree [2]. The DSPT consists of three principal components: a design philosophy rooted in systems thinking, an approach for identifying and

formulating decision support problems, and the supporting computer software. The DSPT has been

implemented in a program called DSIDES (Decision Support In the Design of Engineering Systems), documented by Reddy [8].

The specific DSPT method of interest in this paper is the compromise Decision Support Problem (DSP) which is a hybrid formulation of Mathematical Programming (MP) and Goal Programming

(GP). It has the ability of GP to incorporate the nature of real world problems. The compromise

DSP is similar to GP in that multiple objectives are formulated as system goals, and the deviation function is solely a function of the deviation variables. This is in contrast to traditional MP where

multiple objectives are modelled as weighted functions of the system variables. The compromise DSP

places special emphasis on the bounds of the system variables, unlike MP and GP. The compromise

DSP constraints and bounds are handled separately from the system goals which is contrary to the GP

method, where they are converted to and treated as goals. In the compromise formulation, the set of

system constraints and bounds define the feasible design space, and the set of system goals define the aspiration space.

The compromise decision support problem can be described in word form as follows:


An alternative to be improved through modification Assumptions used to model the domain of interest The system parameters

The constraints for the design The goals for the design

n p+q p q g,(X) fk(dI) Find:

number of system variables number of system constraints number of equality constraints number of inequality constraints system constraint function

function of deviation variables to be minimised in priority level k for the pre-emptive case

The values of the independent system variables and design variables, (which describe the

physical attributes of the problem)



The values of the deviation variables (which indicate the extent to which the goals are

achieved. dl is the under achievement of the ith goal, d7 is the over achievement of the ith




The system constraints that must be satisfied for the solution to be feasible (no restriction is

placed on linearity or convexity):





The system goals that must achieve a specified target value as far as possible G. (no restriction is placed on linearity or convexity):

A.(X)+d d



The upper and lower bounds on the system variables and deviation variables:


The deviation function, which is a measure of the deviation of the system performance from

that implied by the set of goals and their associated priority levels and relative weights. Pre-emptive' (Lexicographic) minimum

The DSPT has been employed to solve many complex engineering design problems. In the ship design arena the reader is referred to the publications of Mistree [9], Mistree [101, Pal [11}, and Smith [12].


Hull-Form Description

To design a hull-form to match a set of design parameters, a list of principal parameters was identified. These parameters were selected as the minimum number of parameters required to

adequately define a displacement-type hull-form. For the generation of a hull-form using the method

described in this paper it is required to develop a section area curve, a design waterline curve, a deck plan curve and a profile, and then assemble the curves in a three-dimensional method to form a hull. Details of the principal design parameters and form coefficients used to describe each of the principal curves follow. The symbols and their meanings are detailed in the nomenclature.

It was decided that the section area curve could be adequately represented by the principal design parameters shown in Figure 4. From these key variables, the tangent at the aft perpendicular (TAP)

was considered to be of secondary importance and it was decided not to use this parameter as an input design variable. This decision made the TAP a dependent variable. The value of TAP was computed

from the curve with the best fairness. A similar approach was taken for the longitudinal location of

maximum sectional area (XMSA).

The deck plan and design water line were described by a similar set of design variables. These are

detailed in Figure 5 for the design water line and Figure 6for the deck plan. In both of these curves it Pre-emptive means that if multiple levelsare considered, the deviation of the first level is minimised

before the second level is considered, and so on.


x1 x,


d, ,dt


and a,

(dl .d = O)


was decided not to include the longitudinal location of the centroid, as this inclusion tended to

over-specify the curve causing it to have ari irregular shape. Additionally it increased the number of design

parameters required to describe the hull-form. However, the location of the maximum beam and the maximum deck beam are required to be entered as input data. For the section area curve and the design water line curve the LAVE' and LFWD are equal to half the length between perpendiculars

(LBP). However, for the deck plan this is not necessarily correct. The use of this substitution of

LAFI and LFWD allowed the design waterline and deck plan curves to be generated using the same routines. H)REÍ



Figure 7: Stem and Keel Profiles Figure 8: Deck Profiles

It was decided to describe the stem and keel profiles using the variables displayed in Figure 7. The

deck profile is simply described using the variables indicated in Figure 8. The variables describing the keel profile were chosen such that they included a definition of the opening for the propulsion equipment. These variables were incorporated as the primary interest of developing the hull-form






Figure 4: Section-Area Curve





generation technique was to incorporate it into a hydrodynamic preliminary design package. With this in mind, a relatively simple curve has been used to describe the deck profile. The hull-forms can have skegs or other appendages. However, they have not been included in this explanation of the hull-form generation.

The development of the three principal curves: the section area curve, design water line and deck plan

curves, require a design optimisation problem to be solved for the best2 curve.


Solution of Curve Design

In this section, the solution for the design of a section area curve for a set of design parameters is

explained. The section area curve was chosen since it is the most complicated of ail the curves. Similar approaches are used for the design waterline and deck plan. The aim of this solution is to find the locus of points on the curve.

The curve is generated from three fixed points in space. These points are the section area at the aft-most point of the vessel, the forward-aft-most point of the vessel, and the maximum section area at an unknown longitudinal position. The curve also has derivative conditions at two of these key points which must be satisfied. These conditions are that the curve has a zero first derivative at the section with maximum area, and that the curve satisfies the required slope at the forward-most point of the


At each of these three points pseudo-vertices are inserted either side of the fixed points to allow for the control of the derivative and to force the curve to pass through the required point. The curve shown in

Figure 9 is defined using a total of ten control vertices and three arms. The ARMAFT and vertices B012 define the aft end point of the curve. The ARMMII) and vertices B456 define the longitudinal location of the position of maximum section area. The forward most point of the curve is specified using control vertices B8910 and ARMFWD. The arms ARMAFI, ARMMTD, ARMFWD are varied as design variables to allow the solution techniques to improve the curve fairness and to allow the

possible incorporation of a parallel middle body.

Figure 9: Control Vertices to Define the Section Area Curve.

The remaining two control vertices B37 are varied to satisfy the required LCB and displacement. The longitudinal position of the station of maximum section area is also allowed to vary, subject to the

condition that the vertices remain in the order defined.


'Best' here refers to the curve that satisfying the required input data and having the best fairness. The fairness criterion is discussed in Section 5.1. Should a feasible curve not be found, the curve with the geometry closest to satisfying all the design requirements is chosen independent of the curve fairness.

ARMMID - -- - Control Polygon

B456 B-Spline Curve


B - D


5. 1

Fairness of the Curve

Since many combinations for the locations of the vertices B3, B, B7 exist, the set of vertices which gives the curve the best fairness is chosen as the solution. Fairness is a subjective phenomenon and,

after investigation of combinations of curve derivatives [131 and other fairness criteria, itwas decided

to adopt a novel solution this being that the fairest curve is the one with the defining polygon of minimum length.

5 .2

Word Problem for Section Area Curve Design

The design of the section area curve in the standard compromise decision support form is as follows: Find:

Longitudinal location of control vertex 3 B3 Longitudinal location of control vertex 7 B37 Offset location of control vertex 3 B3

Longitudinal location of the section with the maximum area XMSA Length of the ARMAFT

Length of the ARMM[D Length of the ARMFWD

The tangent angle at aft perpendicular TAP

Sa tisfy.

Continuity of the longitudinal and transverse location of the control vertices.

B3, B3 B4 B3,2 B.3 B 6 Bi-, B8 B8 B7 B6 0.9xLCB LCB LCBC



The calculated LCBe should equal the required LCB.


+drd = 1.0

The length of the defining polygon should be as small as possible, for best fairness.


((B.1 i=o.9 LCB LBP


- B, )

+(B3 - B

The length of the arms should be as large as possible to increase fairness. This contradicts the previous goal. This contradiction of goals requires the problem to be formulated as a compromise DSP. ARMAFT +

+d d =1.0


+d d =1.0

LBP/2 4


50 45 40 35 30 Area 2 25


20 15 10 5 O -70 I I I I I I I I I I ARMFWD

+d d =1.0


To ensure that the problem is solved with each of the goals having the same emphasis in the objective

function, the goals were normalised as indicated above. The problem was formulated as a pre-emptive, lexicographic minimum problem with Archimedean sub-levels.

Z=((2d1+2d +d2),(d3 +d4 +d5))


Example of Design of Section Area Curves

As an example of designing a section area curve, several different section area curves were generated using the example ship data shown in Table 1. All of the data, apart from the LCB, remains constant for each curve. The LCB is varied from 46% LBP to 54% LBP forward of the AP. The resultant section area curves for each LCB value are shown in Figure 10. A summary of the requested and obtained LCB values, along with the resultant longitudinal location of the maximum section area and

the number of iterations required to obtain a convergent solution, are detailed in Table 2.

-60 -50 -40 -30 -20 -10 0 10 20 30 40 Length fwd of midships (m)

Figure 10: Section Area Curves for varying LCB

Table 1: Example Data for a Section Area Curve

LBP 124.0 m AFP 0.0 m2

LAFT 62.0 m AAP 0.0 m2

LFWD 62.0 m AM 45.744 m2

TFP 25.0 deg AREA 3359.6m2


Table 2: Resultant Data for an Example Section Area Curve


Constructing the Hull-form

After developing all the principal curves it is necessary to assemble them together to form a

three-dimensional hull-form. However by using a two-three-dimensional method of fitting sections to the section

area curve, design waterline and deck plan, it is hard to ensure that the longitudinal continuity of sections and the best overall longitudinal fairness is obtained. To overcome this problem the sections are fitted in a three-dimensional procedure to increase the hull-form's longitudinal fairness. This

fitting of the sections in a three-dimensional arrangement has also been formulated as a compromise decision support problem. The problem consists of finding the location of the B-spline control point

for each transverse section curve, whilst giving good longitudinal continuity and fairness.

At each station, vertices are inserted at known points on the station. Vertices B012 are inserted at the

keel line on the hull-form's centreline to control the end of the curve. Vertices B456 are inserted at the

design waterline, with the same slope as the side above the waterline to the set of coincident control vertices B789 detailing the deck edge. The remaining vertex B3 is adjusted in the two-dimensional

plane to find the equation of the line of points which satisfies the required sectionalarea. The location

of this imaginary line of points of the control vertex B3 can be seen in Figure 11.



Section curve Control polygon

o Imaginary Line


Figure 11: Location of Control Vertices For a Two-dimensional Section

The location of this imaginary line is found for each station along the vessel. Then, in a compromise

decision support problem, the following word problem is solved to locate the best location for the

vertex B3 at each station.

LCB requested %LBP LCB obtained %LBP XMSA %LBP Iterations 46.0 46.00 45.28 6 48.0 48.00 48.19 5 49.0 49.00 51.00 6 50.0 50.00 51.99 4 51.0 51.00 54.32 5 52.0 52.00 58.11 6 54.0 54.00 64.14 5




The first test for the method presented here was to see if it could generate an existing hull-form. This has been verified for the U.S. Navy's OLWER I-IAZZARD PERRY class (FFG7). The data was

scaled from the drawings published by Kehoe [14], and entered into the hull-form generation

program. The hull-form generated from the data summarised in Table 3 is shown in Figure 12a. The generated hull-form matched the displacement of the FFG7. The sectional shape of each of the stations appeared similar to the sectional shapes of the FFG7 hull-form. The hull-form generation

routine was also used to generate the remaining hull-forms shown in Figure 12. Each of the Hulls l2b to 12g has the same design parameters as that of the hull-form in Figure 12a with the design parameter listed under the figure changed.

Currently the method suggested in this paper is limited to round-bilge and hard-chine displacement monohulls. The method is valid for any practical hull-form with in these generic types. Whether the hull-form generated is a practical hull-form is not guaranteed, as it depends on the validity of the input. However, the hull-form will match as closely as possible the input data, subject to the list of goals and constraints described previously. The current version of the program generates files consistent with the input to the HYDROS [15] ship-motion package. The generated hull-form is

output as a rectangular topographical mesh.

This method of satisfying the design parameters for the hull-form using the first principles approach

allows two hull-forms to be generatedevery CPU second on a DEC Alphastation 500/266, depending on how difficult it is to satisfy the required hull-form for the given input design parameters.


((B31 i=O,8



2 +d dat = 1.0

) +(., _B,)

Match the polygon slope above and below the water line for each station istat = 0, nstats

B8 - B,




- B


nsfats±,siar nstats+zstar y8 y5 :4 :3

Minimise the longitudinal distance of the imaginary points LBP


((k, Bj2

B)2 +(B

)2) +d;*nstat+, U2*nstat+l = 1.0 Find:

The location of vertex B3 for each station


At each station



R.2 B., B4




a) Original Hull c) Cx = 0.875 e) Cp 0.50



b) L/B = 7.0 d)Cwp=0.810


f) LCB = 54.0% LBP

Figure 12: Example Generated Hull-forms



It has been demonstrated that it is possible to generate hull-forms matching the required naval architectural design parameters using the compromise Decision Support Problem Technique. The hull

forms are generated quickly and reliably for the round-bilge and hard-chine displacement monohull

domain. Using this method it is possible to generate new and existing hull-forms, using a

two-dimensional approach for the representation of the hull-form.

This method has shown that it is possible to generate hull-forms without the use of a parent hull-form. This non-dependence on a parent hull-form allows the technique described in the

paper to be

incorporated into a hydrodynarnic design program that searches for improved hull-form designs.

Future Work

The generation of the entire hull-form as a surface or as several surface patches has also been investigated. The method considered used the location of the control vertices as the design variables.

Each of these methods considered used a rectangular grid of control vertices. The longitudinal

positions of the control vertices were fixed at the design stations, as suggested by Rogers [161. Meshes of 8 x 8 control vertex gave over 100 unknown design variables, depending on the choice of fixed control points. This size of mesh was chosen as the minimum to adequately represent hull-forms in the displacement round-bilge domain. It was found that this method required considerable extra computational effort, and it was harder to ensure that the final hull-form could be adequately represented using the 25 key design parameters described previously in Section 4. This problem arose due to the concern of obtaining a good starting point for the solution to the problem. Noting that the full surface representation of the form is better for improving the fairness of the hull-form, it could be used following the method described in this paper to obtain a starting point for the location of control vertices. The required control vertices could then be obtained allowing for small

variations from those obtained using this first principles hull-form generation method.

The surface curvature of the surface could then be included in the designoptimisation problem. Work is currently in progress to incorporate both curves and surfaces for the constrained hull-form generation.

Table 3: Input Data for Original Hull-form


53.50 %BEAM

XBMAX 50.00 % LBP from AP lE 14.00 deg DKB

102.80 %BEAM


77.46 %BEAM

XDMAX 50.00 % LBP form AP LTFB

13.60 %LBP

HPOA 100.00 % DRAFT HPOF 51.11 %DRAE[ STSLP 35.00 deg FOREFT

44.35 % LBP


124.000 m

L/B 8.757 BIT 3.255 D/T 2.118 DF/D 1.350 LCB 49.770 % LBP from AP Cp 0.5923 Cm 0.7426 Cwp 0.8000 Cd 0.9080 AAP 0.00 TFPSA

25.00 deg




The first and third authors would like to acknowledge the support of the Australian Maritime

Engineering Cooperative Research Centre for their financial support and computing facilities. The

same authors wish to thank the Australian Department of Defence for allowing Dr Smith to participate

in this research work. The use of DSIDES is under contract licence with Professor Farrokh Mistree

of the Georgia Institute of Technology, USA.

1 1.


i. Peacock. D., Smith, W.F., and Pal, P.K., (1997), Minimal Ship Motion Hull-forn Design

Using Multi-Criterion Optimisation Techniques. Submitted for publication in FAST 97. Mistree, F., Hughes, OF., and Bras, B., (1992) Compromise Decision Support Problem and

Adaptive Linear Programming Algorithm. Progress in Astronautics and Aeronautics, Vol. 150, pp. 25 1-290.

Rogers, D.F. and Adams, J.A., Mathematical Elements fr Computer Graphics., Second

Edition. 1990. 611+xv.

Cox, M.G.,

The Numerical Evaluation of B-splines,

National Physical Laboratory DNAC4., 1991

de Boor, C., (1972), On Calculation with B-splines., Journal Approximation Theory, Vol. 6,

pp. 50-62.

MacCallum, K.J. and Zhang, J.M., (1986), Curve-smoothing Techniques Using B-splines.

The Computer Journal, Vol. 29, No. 6, pp. 564-57 1.

Barsky, B.A., (1982), End Conditions and Boundary Conditions for Uniform B-spline Curve and Surface Representations., Computers in Industry, Vol. 3, No. 1-2 , pp. 17-29.

Reddy, R., Smith, W.F., et al., (1992), DSIDES Decision Support In the Design of

Engineering Systems, SDL/REP.7. 1/92, Systems Design Laboratory, University of Houston, Houston, TX 77204-492.

Mistree, F., Smith, W.F., Kamal, S. and Bras, B., (1991), Designing Decisions: Axioms,

Models, and Marine Applications. IMSDC '91, pp. 1-24.

Mistree, F., Smith, W.F., Bras, B.A., Allen, J.K. and Muster, D., (1990), Decision-Based

Design: A Contemporary Paradigm fhr Ship Design. Transactions. Society of Naval

Architects and Marine Engineers, Vol. 98, pp. 565-597.

Pal, P.K., Gillies, D. and Peacock, D., (1985), Computer-aided Preliminary Design of

Hatchcoverless Container Ships. Practical Design of Ships and Mobile Units (PRADS'95),

Vol. 2, pp. 2.1475-2.1486.

Smith, W.F., Kamal, S.K. and Mistree, F., (1987), The Influence of Hierarchical Decisions on Ship Design. Marine Technology, Vol. 24, No. 2, pp. 13 1-142.

Nowacki, H.. Bloor, M.I.G. and Oleksiewicz, B.. Computational Geometry for Ships, World Scientific, Singapore., 1995, pp. 238+xii.

Kehoe, J.W., Brower, K.S., Meier, H.A. and Runnerston, E., (1983), U.S. and Foreign Hull

Form Machinery and Structural Design Practices. Naval Engineers Journal, Vol. 95,

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(1993), The HYDROS/3 Programs: Description and Documentation,

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in this investigation, Copper matrix Surface Composites (CmSCs) were reinforced with various ceramic particles like Alu- minum Nitrate (AlN), Titanium diboride (TiB 2 ), and rice

1) The GIS concept of the Atlas of Kraków province was chosen to enable use of data for analysis and decision support. Therefore, the Atlas can be regarded in extension as a

The levels of such parameters characterizing dynamic loads and overloads of examined movement structures as: maximal and average values of vertical ground reaction forces, total

The objective of the research study was to analyze the chemical composition, in- cluding amino acid composition, of the rapeseed protein-fibre concentrate (RPFC) as well as to

The scaling issue can be considered by performing calculation of the mentioned physical quantities for a different values of the bimorph geometrical dimensions, keeping the

The claim of the theorem concerned Galois module properties of class groups of towers of cyclotomic fields and was reformulated by Iwasawa in [I2] as a conjecture, later named the