'-A0.AIIO 77 DAVID V TAYLOR NAVAL. SHIP RESEARCH AND DEVELOPMENT CE--ETC F/6 80/'.DOCUMENTATION FC'. SWATH SHIP RESISTANCE OPTIMIZATION PROMAN (S-ETC (U)SEP I T J NA8LU. AN04REEDUNLSSFE flflflflflflflf0 llff

I1.81111 .2 5 111111.6______111 11111 I .

MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAUI OfI STANTFIRDI I1

DAVID W. TAYLOR NAVAL SHIPU RESEARCH AND DEVELOPMENT CENTER

Bethesda, Maryland 20084z

DOCUMENTATION FOR SWATH SHIP RESISTANCE

OPTIMIZATION PROGRAM (SWATHO) USER'S AND

MAINTENANCE MANUAL

-.4

.9 Toby J. Nagle r-o* 7

and

0 Arthur M. Reed FEB 11198 2

0-4 E

-€ .APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

EEP-4

EnH SHIP PERFORMANCE DEPARTMENT

September 1981 DTNSRDC/SPD-0927-03

N1W\TNSRDC %602/3012800180)1 6lspwn3M/046)82 2 1 06

MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS

DTNSRDC

COMMANDER 00TECHNICAL DIRECTOR

01

OFFICE R-I N-CHARGE I____OFFICER-IN-CHARGECARDEROCK IANNAPOLIS

SYSTEMSDEVELOPMENT

DEPARTMENT

SHIP PERFORMANCE AVIATION AND

DEPARTMENT SURFACE EFFECTSDEPARTMENT

15 16

STRUCTURES COMPUTATION,

DEPARTMENT MATHEMATICS ANDLOGISTICS DEPARTMENT

17 18

SHIP ACOUSTICS PROPULSION ANDDEPARTMENT AUXILIARY SYSTEMS

DEPARTMENT19 27

SHIP MATERIALS CENTRALENGINEERING INSTRUMENTATIONDEPARTMENT DEPARTMENT

28 29

GPO 866 993 NOW-DTNSRDC 30/43 (Rev 2-90)

UNCLASSIFIED_______* j SECURITY CLASSIFICATION OF THIS PAGE (When Doe Entered)

• REPORT DOCUMENTATION PAGE READ INSTRUCTIONSREPORT DOCUMENTATION BEFORE COMPLETING FORM

I REPORT NUMBER 2. GOVT ACCESSION NO. 3 RECIPIENT'S CATALOG NUMBER

DTNSRDC/SPD-0927-03 ' - L f" -C TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

DOCUMENTATION FOR SWATH SHIP RESISTANCEOPTIMIZATION PROGRAM (SWATHO) USER'S ANDMAINTENANCE MANUAL - 6. PERFORMING ORG. REPORT NUMmER

7. AUTHOR(s) 1. CONTRACT OR GRANT NUMBERs)

by Toby J. Nagle & Arthur M. ReedS., / " /-

9. PERFORMING ORGANIZATION NAME AND ApDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WO")I.jNIT NUMBERS

David W. Taylor Naval Ship R&D Center 6 254 N,'ZF43-421-01Bethesda, Md. 20084 1500-104-70

Ii. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

David W. Taylor Naval Ship R&D Center September 1981

Code 1506 13. NUMBER OF PAGESBethesda. MD 20084" .

14. MONITORING AGENCY NAME & ADORESS(Il different from Controlling Office) IS. SECURITY CLASS. (of thie report)

UNCLASSIFIED

15a. DECL ASSI FIC ATION" DOWNGRADINGSCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

17. DISTRIBUTION STATEMENT (of the abstrect entered In Block 20, It different frog" Report)

IS. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue on reverse wide If neceeary and Identify by block number)

Small-waterplane-area Twin-hull (SWATH) ShitsResistanceResistance Minimization

20 ABSTRACT (Continue on reverse aide if necessary and Identify by block number)This report documents a computer program, SWATHO, which optimizes SWATH

ship geometry, resulting in a minimum resistance and EHP for a specified speedof the ship. Contained in the report is a detailed description of the proce-dure for assembling the input deck for use on the CDC-6000 computer system at

*, DTNSRDC, running the program on this system and understanding the output fromthe program. Also included are main program and subroutine descriptions ase].l as the theory behind the computations being performed.

DD ,JA 73 1473 EDITION OF I OV65ISOBSOLETE UNCLASSIFIED . 7S7 ' 1J"S N 0102- LF- 014- 6601 SECURITY CLASSIFICATION OF THIS PAGE (Who Date 81ne01)

SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)

SErIJRITY CLASSIFICATION OF THIS PAGE(Whn Dorm Enfered)

TABLE OF CONTENTSPage

LIST OF FIGURES........................................................... iii

NOMENCLATURE.............................................................. iv

ABSTRACT..................................... ................... ......... 1

ADMINISTRATIVE INFORMATION-....... o........................... .......... 1

INTRODUCTION,...................................... ....................... 1I

USER MANUAL

PROGRAM DESCRIPTION................. ........... ............ .......... 3

INPUT DESCRIPTION ..................... ..................... ........... 4

INPUT EXAMPLE.-................... ...................... ............ 9

OUTPUT DESCRIPTION ........ ............................. ........ .... 10

OUTPUT EXAMPLE..................................... .................. 12

JOB CONTROL SETUP FOR SWATHO.................. .......... ............ 34

MAINTENANCE MANUAL

INTRODUCTION.......................... ............................... 35

FUNCTIONAL LISTING OF FUNCTIONS AND SUBROUTINES..................... 40

COMMON BLOCK DESCRIPTIONS................................... ........ 43

SUBROUTINE AND FUNCTION DESCRIPTIONS................................. 59

REFERENCES............................................... ................ 100

APPENDIX A Objective and Penalty Functions........... .................. 101

APPENDIX B Computational Procedure for SWATH Resistance Prediction ...... 106APPENDIX C Relationships Between Chebychev Series and SWATH Huilform.

Coefficients .............................. ... ...............119

ACCeSicn

'For

NT7IS1J"r

1JIw _

AvN

ile

LIST OF FIGURES

Page

I - SWATH Ship Body and Strut Profiles (SWATHO).......................... 8

2 - 'Free Diagram of SWATH Program........................................ 39

3 - Form Drag Coefficient................................................ 67

4 - Body-Strut Configuration............................................. 95

fit

NOMENCLATURE

Smo Description

A(x) Body sectional area curve

A Maximum section area of body

A Bm Symmetric coefficients of Chebychev

series for body

A Symmetric coefficients of ChebychevSm series for strut

AW Area of the waterplane

b Half of the separation distance of

the hulls

BBm Anti-symmetric coefficients of Chebychev

series for body

B Sm Anti-symmetric coefficients of Chebychev

series for strut

CA Correlation allowance

CF Frictional resistance coefficient

C Fm Form drag coefficient

Cp PBody prismatic coefficient

CWP Waterplane coefficient

C Waterplane longitudinal inertiaCIW coefficient

g Acceleration due to gravity

hB Maximum depth of submergence of axis of

body

hs Maximum draft of strut

n(a) Bessel functions

iv

NOMENCLATURE

(Continued)

Symbol Description

LB Maximum length of body

LS Maximum length of strut

PE Effective power

R Reynolds numbern

R Total ship resistanceT

R WB Wave resistance due to one main body

R Frictional resistance

RFm Form drag

RW SWave resistance due to one strut

Wave resistance due to the interactionwSB of strut and main body

RW Wave resistance

S Wetted surface

T (T also used) Thickness at mid length of strutmax

t(x) Strut half thickness function

TB Auxiliary wave resistance functionmn used in calculation of RW

TS Auxiliary wave resistance function

mn used in calculation of RWs

V

NOMENCLATURE

(Continued)

Symbol Desc r ipt ion

T TSB Auxiliary wave resistance function usedmn in calculation of RWS B

Un (x) Chebychev cosine series term (symmetric)

V Velocity of ship

V (x Chebychev sine series term (asymmetric)m

WE Auxiliary wave resistance function usedran in calculation of RW

B

WS Auxiliary wave resistance function usedmn in calculation of RW

WSB Auxiliary wave resistance function usedmn in calculation of RuS

Variable of integration

P Density of water

Y' Dimensionless wave number related to body¥obob length and ship speed

- Dimensionless wave number related to strutos length and ship speed

4Variable of integration

vi

L I"" M nn. . .. .. ,,. ..._ . .

ENGLISH/SI EQUIVALENTS

I degree (angle) = 0.01745 rad (radians)

1 foot = 0.3048 m (meters)

1 foot per second (fps) = 0.3048 m/sec (meters per second)

I inch = 25.40 mm (millimeters)

1 knot = 0.5144 m/s (meters per second)

1 lb (force) = 4.448 N (Newtons)

I inch-lb - (Inch lbs) = 0.1130 N'm (Newton-meter)

I long ton (2240 pounds) = 1.016 metric tons; or 1016 kilograms

I horsepower = 0.746 kW (kilowatts)

vii

ABSTRACT

This report documents a computer program,

SWATHO, which optimizes SWATH ship geometry,resulting in a minimum resistance and EHP for

a specified speed of the ship. Contained inthe report is a detailed , escription of theprocedure for assembling the input deck for use

on the CDC-6000 computer system at DTNSRDC, runningthe program on this system and understanding theoutput from the program. Also included are mainprogram and subroutine descriptions as well as thetheory behind the computations being performed.

ADMINISTRATIVE INFORMATION

This investigation was authorized under a direct funded block from the

Naval Material Command (NAVMAT 08T2) under Program Element 62543N, Task

Area ZF43-421-001, and administered bv the David W. Taylor Naval Ship

Research and Development Center (DTNSRDC), Work Unit 1500-102-30, and

1-1500-104-70.

INTRODUCTION

The computer program SWATHO, developed at DTNSRDC, optimizes SWATH ship

geometry to arrive at minimum values of resistance and effective horsepower for

a predetermined speed. The minimum values arrived at by the optimization are

a result of initial ship geometry and constraints on the geometry as specified

by the program user.

The calculations use standard EHP prediction methods accounting for

frictional resistance, form drag, and wave resistance. Wave making predictions1

are made using the work of Lin and Day on twin hull ships.

Contained in this report is a user manual which explains how to assemble

the input deck as well as how to run the program on the CDC-6000 computer

1References are listed on page 100

~1

system at DTNSRDC. A description of the output and how to interpret it is also

included along with an overview of how the main program works. Following the

user's manual, a maintenance manual is presented, which contains a detailed

description o1 the common blocks, functions and subrcutines, and a trt-e

(i ag ruam. The appendices contain a description of the objective and penaltv fnc-

tions used for minimization of effective power, a presentation of the theor,'

behind the computational procedures for the resistance predictions, and art explana-

tion of the relationships between Chebychev series and SWATH hullform ctefficients.

This program will assist naval architects in finding a minimum resistance

hull design which satisfies the given geometric constraints. The User's Manual

portion of this document is intended to provide the user with the information

needed to prepare input for the program, and with a description ef the resulting

output. The Maintenance Manual provides the documentation needed to understand

the functioning of the program and its various subprograms, at a detailed enough

level that changes could be made in the program if necessary.

2

USER'S MANUAL

PROGRAM DESCRIPTION

The Program SWATHO optimizes a SWATH ship geometry to obtain a minimum

value of resistance and EHP for a specified speed of the ship. The program

can be better understood if it is divided into four parts, each part having

a specific purpose and result.

The first part of the program reads the input data that the user has

prepared on cards; thus initializing the ship description as well as setting

up the constraints. The array of scaling factors, which is used as the initial

geometry, is optimized and initially set to 1 in this part. After each run,

the values in the scaling array which correspond to the value of minimum EMP

at that point are input to the next run of the program, keeping the rest of

the INPUT cards the same. The actual description and format of the INPUT

cards are found in the INPUT DESCRIPTION and INPUT EXAMPLE of this USER'S MANUAL.

The second part of the program actually performs the optimization of

the EHP function. A detailed description of the method used and the actual

subroutines which perform the optimization can be found in the MAINTENAN'CE MANUAL.

The third part of the program evaluates the objective function, thus

finding intermediate values and a final value of EHP. A penalty function

is also calculated in this part, based on any violation of constraints

by the ship geometry at that time. Then more optimization is performed.

This optimization cycle is repeated until the computational time is exceeded

or sufficient convergence is reached.

The last part of the program outputs the initial ship description,

intermediate curves and a design description, as well as resistance calculations

over a range of speeds and other results of calculations. A detailed

description of the output is found in the OUTPUT DESCRIPTION and OUTPUT

EXAMPLE in the USER'S MANUAL.

3

INPUT DESCRIPTION

Diagrams showing the orientation of the geometry represented by the variablesinvolved in this input description are found in Figure 1.

INPUT CARDS (Total number of input cards is 8)

I. The first card contains the alphanumeric title of the model.

Variable: TITLE (8)

Format: 8A10

1 11 21 31 41 51 61 71[ITLE (1) ITITLE (2)ITITLE (3) TITLE (4)ITITLE (5) TITLE (6) TITLE (7)ITITLE (8)1

2. Data on the second card are:

XLSI - Length of strut in ft

HSI - Draft of strut in ft

TSMAXI - Strut thickness in ft at XLSI/2

CWPI - Waterplane area coefficient

CLCFI - Waterplane moment coefficient

KG - Height of CG above baseline

Variable: XLSI, HSI, TSMAXI, CWPI, CLCFI, KG

Format: 6FI0.6

1 1.1 21 31 41 51XLSI ]HSI TSMAXI CWPI CLCFI KG

4

3. Data on the third card are:

XLBI - Length of body in ft

BDIAI - Body diameter in ft at XLBI/2

BSI - Separation of hull centerlines

CSTRTI - Djst. of strut CL fwd of body CL

CPI - Body prismatic coefficient

CLCBI - Body moment coefficient

Variable: XLBI, BDIAI, BSI, CSTRTI, CPI, CLCBI

Format : 6FI0.6

1 ____11 21 31 41 51fiLEBI I BDIAI I BSI ICSTRTI ICPI ICL-CBI J1

4-5 Data on the fourth and the fifth card are:

S - Array of scale factors

Variable: S

Format :8 F10.6

1 11 21 31 141 51 61 715S(1) IS(2) S(3) IS(4) IS(5) S(6) S() S(8)

1 11 21 31IS(9) IS(10) I (11) I (12)

5

6. Datt oni the sixth card is:

OPTSPID - Optimization speed in knots

Vairiable: OPTSPD

Format: FIO.6

I

7. Data on the seventh card are:

DISPMN - Minimum ship displacement in long tons

DFAC - Minimum draft/body diameter ratio

DRFTMX - Maximum draft of ship

WMAX - Maximum width of ship

XFWD - Minimum distance from body leading edgeto strut leading edge

XAFT - Minimum distance from body trailing edgeto strut trailing edge

Variable: DISPMN, DFAC, DRFTMAX, WMAX, XFWD, XAFT

Format: 6F10.4

1 11 21 31 41 51DISPMN DFAC DFTMAX [WMAX ]XFWD XAFT

6

8. Data on the eighth card are:

MINGMT - Minimum TRANSVERSE GM

MINGML - Minimum LONGITUDINAL GM

LODMAX - Maximum body length over diameter ratio

LODMIN - Minimum body length over diameter ratio

TSMIN - Minimum strut thickness at 50 percent chord

Variable: MINGMT, MINGML, LODMAX, LODMIN, TSMIN

Format: 5FIO.l

MINGMT MINGML 2 LODMAX LODMIN 4 TSMIN

7

cnn

I E-4

E-44

K cn

rA4

INPUT EXAMPLE

Example of Input Data (Optimization):

1 11 21 31 41 51 61 71

ITLE (1) TITLE (2) TITLE (3) TITLE (4) TITLE (5) TITLE (6) TITLE (7) TITL(8

SWATH OPTIMIZATION PROGRAM

1 11 21 31 41 51

XLSI HSI TSMAXI CWPI CLCFI KG

210. 10.725 6.500 .8511 .014286 20.

1 11 21 31 41 51

XLBI EDIAI BSI CSTRTI CPI CLI

260. 14.30 68.50 15. .849 .01538

1 11 21 31 41 51 61 71

(1) S(2) S(3) S(4) S(5) S(6) S(7) S(8)

1.0154 .9900 0.9900 0.9783 -1.0 .9032 -1.0000 1.005

1 11 21 31

S(9 I- S(10) S(11) I5(12

.9900 1.2500 1.0000 .5000

1

20.

1 11 21 31 41 51

DISPNN DFAC DRFTMX WMAXI XFWD L AF

2675.0 1.67 30.0 106.0 7.50 30.0

1 11 21 31 41

MINGMT MINGML LODMAX LODMIN TSMIN

30.00 30.00 20.00 16.00 5.0

9

OUTPUT DESCRIPTION

The first page of output consists of an echo print of the input

including the S array of scaling factors. Following the array of scaling

factors, NL and NF (value of counters in the optimization procedure)

P (the penalty value), PEHP (the function to be minimized), and a repeat

of tle scaling factors (S array) are printed.

The second page of output consists of an intermediate plot of a body

sectional area curve and strut waterplane outline curve where the symbols

B and S stand for body and strut respectively. At the bottom of the plot,

the values for the strut and body geometry are printed with the effective

horsepower required for the design.

The third page of output lists values for intermediate design

consisting of the symmetric and antisymmetric Chebychev coefficients for

the body and strut and the wetted surface and ship geometry for the optimization

speed. Also a table of speed-length ratios, wave resistance, frictional

and residual resistance coefficients as well as the actual resistances

and the EHP predicted is printed for the range of speeds from 10 to 26 knots.

The fourth page of output consists of a tabular listing of all of

the scale factors of the S array corresponding to the strut and body

geometric variables, as well as the penalty value and PEHP. The second

difference array D(*) is printed out at the bottom of the listing. Also

printed is NL, NF, PEHP and the values of the S array corresponding to

a minimum PEHP so far. These S values are input by the user for the next

run of the program. Often, when the execution time limit is exceeded, the

output will stop in the middle of this tabular listing, in which case

the minimum PEHI' is found and the S array of scale factors are picked

out from the table and used as input for the subsequent run.

The sequence of body sectional area curve and strut waterplane curve,

intermediate design description, followed by tabular output is repeated

until the time limit is exceeded or until convergence of the PEHP function

is reached.

10

When the program has reached a prescribed degree of convergence,

a final page is printed out, entitled "Wave Resistance for Strut and Body

Geometric Characteristics." This page contains the optimum strut and body

geometric characteristics, separation in feet, B/LB, strut centerline

distance from body centerline (STRUT OFFSET), strut offset/LB, the wave drag

for strut and body and the frictional drag of body and strut. The coefficients

of resistance as well as the minimum EHP calculated at certain speeds are also

given.

.1'

OUTPUT EXAMPLE

12

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JOB CONTROL FOR SWATHO

The Fortran code file of SWATHO is stored on a bTNSRDC Computer Center disk

pack. To access the disk pack and make use of the program the following directions

should be followed.

One must first set up an account with the DTNSRDC Computer Center and acquire

a user ID. A Computer Center Reference Manual will provide the user with a

further explanation of the Job Setup Control cards.

1. To get the program from the disk pack to a permanent file:

JOB CARDCHARGE CARDPAUSE. JOB REQUIRES DV4865.MOUNTVSN=DV4865,SN=CHRELIB.REQUESTLOG,*PF.ATTACH,OLDPL,OPTSWATH,ID=CHRE,SN=CHRELIB.

UPDATE(PF)FTN(I-COMPILE,R=3)CATALOG,LOG,USER'S NAME OF FILE,ID=USER ID7/8/96/7/8/9

2. To run the program which is stored on the user's permanent file:

JOB CARDCHARGE CARD

ATTACH,USER'S NAME OF FILE,ID=USER ID.FIRST SEVEN LETTERS OF NAME OF FILE (PL.50000)7/8/9INPUT CARDS6/7/8/9

The user's permanent file which is set up with the catalog card in Part 1 will

be purged after 30 days of inactivity.

34

• ]

MAINTENANCE MANUAL

This part of the report contains a description of the main program,

SWATHO and the subroutines directly called by it as well as descriptions

of how other important subroutines fit together. Also included in this manual

are descriptions of the common blocks, functions and subroutines and

a functional listing of the functions and subroutines.

INTRODUCTION

The main program, SWATHO, first calls subroutine INPUT. Subroutine INPUT

reads the input cards prepared by the user. The strut geometric characteristics,

waterplane area and moment coefficients as well as the height of the CG above the

baseline are read. Next, the body geometric characteristics, separation of hull

centerlines, placement of the strut centerline in relation to the body centerline

and the body priqmatic and moment coefficients are read. Following that the S

array of scaling factors, the optimization speed and limiting values are read.

After returning from INPUT the program calls PRAXIS, the optimization

routine, which is described in detail in the following pages of this report.

Upon the completion of the optimization by PRAXIS, the subroutine SCALE is

called which sets up an array of current working values of X, each member of the

array corresponding to one of the ship geometric characteristics. The X array

is calculated by multiplying S by XI where the S's are the scale factors and

the Xl's are the initial values of the ship geometry. Both S and XI are read

by subroutine INPUT. Initially the members of the S array all have values of 1

and then, on subsequent runs, the S's correspond to the lowest value of EHP so

far encountered by the user in the output from the previous run of the program.

All of the initial variables are left unchanged.

Subroutine CHEB, called by the main program next, calculates the Chebychev

coefficients for the ship as well as the wetted surface of the body and strut.*

It also calls another subroutine, PCHEB, which plots the strut waterplane and

body sectional area curves.

LISTEX is then called and values c' ship geometry, etc. are listed to display

the proximity of the solution to the extreme limits.

See Appendix C for a discussion of the relationship between the Chebychev

coefficients and the hullform coefficients,

35

Function EHP is lastly called at which time final design resistance and power

is output. Whether the program gets this far is contingent upon how much time

the computer is given for the computations.

Subroutine PRAXIS accomplishes the optimization of the objective function,

EHP. PRAXIS finds the minimum of a function using the principal axis method as

implemented by Brent. The arguments of the subroutine are TO, HO, N, IPRIN, X,

F, and FMIN. PRAXIS attempts to find this minimum such that if XO is the true

local minimum near X, then NORM (X-XO)=TO + SQUARE ROOT (MACHEP) *NORM (X), where

MACHEP is the machine precision, the smallest number such that 1 + MACHEP> 1, and

TO is the tolerance.

The values of TO, HO, N and IPRIN are set in the data statement in SWATHO.

The value of MACHEP should be 2**-47 (about 7.105 E-15) on the CDC 6000 series for

single precision arithmetic or 16"*-13 (about 2.23 E-16) for double precision

on the IBM 370 system. HO is the maximum step size, and should be set to about

the maximum distance from the initial guess to the minimum. This value of HO

affects the initial rate of convergence, in the current program HO - .25. N is

the number of variables upon which the function depends, in this case N - 12. N must

be at least 2. IPRIN controls the printing of intermediate results of the optimi-

zation.

X is an array which contains a guess of a point of minimum and an estimated

point of minimum, on entry and return, respectively. F(X,N) is the function to

be minimized, in our case it is PEHP. F needs to be a real function declared

EXTERNAL in the calling program. FMIN is set to the minimum value of F that is

found.

The approximating quadratic form is

Q(X') = F(X,N) + 2 * (X' -X) TRANSPOSE *A* (X' -X).

Here A is INVERSE (V-Transpose) *D* INVERSE (V) where V(*,*) is the matrix of

search directions and D(*) is the array of second differences.

PRAXIS in operation proceeds as follows. PRAXIS calls PRMIN which minimizes

the function along first direction V(*,J). PRMIN then calls function PRLIN which

is a function of one real variable, which it minimizes. Function RANDUM is called

and a random number is returned to PRAXIS to avoid resolution valleys. PRMIN

continues to minimize the PRLIN function in different directions. PRQUAD is then

36

called and this subroutine tries a quadratic extrapolation in case minimization

is being done in a curved valley. PRAXIS then calls PRFIT to find principal

values and directions of the quadratic forms and PRSORT is called to sort the

eigenvalues and eigenvectors. PRPRIN prints out the scaling factor on the first

page of the output as is described in the output section of the USER's manual and

VCPRINT prints the second difference array.

The optimization procedure is performed on the function PEHP. PEHP gets its

value from the EHP calculated by the function by that name and a penalty P, so

that PEHP = (1 + P)* EHP. P is determined from an array calculated in subroutine

CKLIM. CKLIM uses the constraints determined by subroutine INPUT to check the

current ship values for violation of these constraints. The checking procedure

is quantified with an array GG of violation coefficients, in the following way.

The current value of ship displacement is multiplied by a member of a scaling

array, ALPHA, given in a data statement in CKLIM. This results in a violation

coefficient GG(l) for displacement. Nineteen of these coefficients are calculated

and this array of GG(I)'s is used in function PEUP. Since GG(I)>O means that

the constraint is violated whereas GG(I)<O means that the constraint is satisfied,

the function PEHP totals up all of the GG(I)'s greater than zero and this becomes

the value of P, used to calculate PEHP. For further details on penalty functions

see Appendix A.

Other important subroutines have not been mentioned yet. Subroutine RWAVE,

called by Function EHP, computes the auxiliary wave resistance functions T and W

as explained in Appendix B, using the Chebychev expansions calculated in subroutine

CHEB, as discussed in Appendix C. Subroutine RINTEG integrates these auxiliary

functions with subroutine INIT initializing the arrays. The symmetric matrices

of T and W are reflected by subroutine REFLKT. Using the Chebychev coefficients,

subroutine WSURFB determines the wetted surface of the body of revolution and

WSURFS determines the wetted surface of the strut.

Subroutine ROUT is called by function EHP, it computes values of Froude number

and speed-length ratio as well as wave resistance coefficients and frictional drag

coefficients and returns the value of EHP to function EHP. It also prints the

intermediate and final design values. Which one of these is printed is determined

by the value of KEY. If KEY = 0 no output is printed, instead only the compu-

tations in ROUT are performed. If KEY = 1 the optimization is complete and just

the final design is printed. If KEY = 2 when ROUT gets to the output section,

37

the intermediate results are printed. The different values of KEY are set in

the subroutines which call FUNCTION EHP which in turn calls ROUT; KEY is set to

0 in PEHP, 1 in SWATHO and 2 in AUXPLOT.

38

----- . .. . .. . ...-

Figure -TREE DIAGRAM OF SWATH PROGRAM

LI STEX

WSURFB EVAL

CHBWSURFS EVAL

SCALE SCALE

PEHP CHEB

RANDUM - CKLIM -EVAL

VCPRNT -EHP r PLOTIMAPRNT -PLOT 2

PRAXIS -PRFIT PCHEB -PLOT3

-PRSORT ~ - SCALE L PLOT4-QPLOTZ5-PRMIN - PRLIN -CHEB

EPRPRIN -U XPLT---Lj LISTEX

PRQUAD- PRMIN

IEPSWATHO CFITTCf FORMDR YINPT

ROUTsu

EHP BESSJ

RWAVE RINIT

REFLKT NOT CALLED:

SIMPSN POMITi

POMI T2

RINTEG -BESSJ

LINPUT

39

FUNCTIONAL LISTING OF FUNCTIONS AND SUBROUTINES

Subroutine Name Description

AUXPLT Prints titles and labels for body sectionalarea and waterplane curves

BESSJ Evaluates Bessel function

CHEB Determines Chebychev coefficients forbody and strut

CKLIM Checks for violation of geometric constraints

INPUT Reads in initial values of ship geometryand constraints

LISTEX Lists various values of the design to see howclose to the limits the solution lies

MAPRNT Prints columns of matrix

PCHEB Plots body sectional area curve and water-plane outline curve

PLOT1 Sets up spacing and determines values of theaxes for printer plot

PLOT2 Establishes formula for computing locationin the image region where the point will beplotted, in the printer plot

PLOT3 Assigns an alpha character to each point

to be plotted on the printer plot

PLOT4 Prints image of completed graph on theprinter

POMITi Causes certain grid lines on printer plot

to be deleted

POMIT2 Causes values at grid lines to be deleted

PRAXIS Finds the minimum of the objective functionusing the principal axis method

PRFIT Fits points to a quadratic curve

PRMIN Minimizes the objective function, F, in onedimension

PRPRIN Prints intermediate results and calls theroutine which does the printer plotting

40

FUNCTIONAL LISTING OF FUNCTIONS AND SUBROUTINES (CONT)

Subroutine Name Description

PRQUAD Looks for minimum of F along a quadraticcurve

PRSORT Sorts the elements of a matrix to find theeigenvalues and eigenvectors

QPLOTZ5 Calculates scaling information to label theplot

REFLECT Reflects matrices of auxiliary wave resistancefunctions

RINIT Initializes the auxiliary wave resistancefunciton matrices to zero

RINTEG Evaluates the integrand of the auxiliarywave resistance function

ROUT Calculates EHP, wave resistancecoefficients and constants, and alsoprints the intermediate and final designresults

RWAVE Computes the auxiliary wave resistancefunctions

SCALE Calculates working values of ship geometry

SIMPSN Sets up Simpson's multipliers for numericalintegration by Simpson's rule

SUM Computes the components of wave resistanceas a matrix multiplication involving theChebychev coefficients and the auxiliarywave resistance function matrices

VCPRINT Depending on the option; prints thesecond difference array, scale factors,value of quadratic, and value of theobjective function as part of theoptimization

WSURFB Determines surface area of a body ofrevolution given by Chebychev coefficients

WSURFS Determines surface area of strut whosethickness distribution is given byChebychev coefficients

41

FUNCTIONAL LISTING OF FUNCTIONS AND SUBROUTINES (CONT)

Function Name Description

CFITTC Determines frictional resistance

coefficient from ITTC line

EHP Calculates body constants and frictionaldrag coefficients

EVAL Evaluates the Chebychev series

FORMDR Evaluates form drag coefficients

PEHP The objective function to be minimizedby PRAXIS

PRLIN Function of one variable minimized by

PRAXIS

RANDUM Returns a random number which createsrandom steps during the optimization

YINTP Interpolates through a set of discretedata

42

COMMON BLOCK DESCRIPTIONS

43

COMMON/COEFS

PURPOSE: COEFS stores the values of the Chebychev coefficients for thestrut and body

NAME TYPE LENCTH DEFINITIONASM R (3) CoeffIcients of Chebychev Sine Series for strut

BSM R (3) Coefficients for Chebychev Cosine Series

for strut

ABM R (3) Coefficients of Chebychev Sine Series forbody

BBM R (3) Coefficients of Chebychev Cosine Series forbody

MMAX Maximum order of Chebychev Series

44

COMMON/ EXTLIM

PURPOSE: EXTLIM defines and stores physical constants and geometricparameters

NAME TYPE DEFINITION

DISPMN R Minimum ship displacement in long tons

DFAC R Minimum draft/body diameter ratio

DRFTMX R Maximum draft of ship

WMAX R Maximum width of ship

XFWD R Minimum distance from body L.E. to strut L.E.

XAFr R Minimum distance from body T.E. to strut T.E.

MINGMT R Minimum GMT (transverse GM)

MINGML R Minimum GML (longitudinal GM)

LODMAX R Maximum length over diameter ratio

LODMIN R Minimum length over diameter ratio

TSMIN R Minimum strut thickness at 50 percent chord

KG R Height of center of gravity of ship abovebaseline

45

COMMON/INITL

PURPOSE: Store initial values of ship geometry

NAME TYPE DEFINITION

XLSI R Length of strut in ft.

HSI R Draft of strut in ft.

TSMAXI R Max. strut thickness in ft.

CWPI R Waterplane area coefficient

CLCFI R Waterplane moment coefficient

CIYYI R Waterplane inertia coefficient

CLCBI R Body moment coefficient

XLBI R Length of body in ft.

BDIAI R Body diameter in ft. at XLB/2

BSI R Separation of hull centerlines

CPI R Body prismatic coefficient

CSTRTI R Dist. of strut CL fwd of body CL

VBI R Initial displaced volume of the body in

cubic ft.

VSI R Initial displaced volume of thestrut in cubic ft

KBI R Initial height of center of buoyancy abovethe baseline

BMLI R Initial longitudinal metacentric height

IYYI R Waterplane moment of inertia

46

COMMON/NOCHG

PURPOSE: Checks whether certain ship geometry variables change or not.

NAME TYPE LENGTH DEFINITION

LL I 6 LL(I) thru LL(6) represents positions

in WORKVL of XLS, BS, HS, XLB, BDIA

CSTRT. In the data statement in

SWATHO they are assigned values;

LL(l) = 1, LL(2) = 10, LL(3) = 2,

LL(4) = 8, LL(5) = 9, LL(6) = I

SS R 6 Holds array of scale factors correspon-

ding to the six ship geometry variables

in the LL array

SAME L Variable which indicates whether scaling

variables have changed or not.

47

COMMON/OMEGA

PURPOSE: OMEGA stores the values of variables and constants forevaluation of auxiliary wave resistance functions and shipresistance coefficients

NAME TYPE DEFINITION

VMFPS R Speed of ship (feet per second)

GAMAOS R Yos

GAMAOB R Yob

GOSQ R (Y os)2

HSOLS R Ratio of draft to length of strut

HBOLB R Ratio of draft to length of body

WETS R Wetted surface area of strut (ft )

WETB R Wetted surface area of body (ft2 )

WTSURF R Total wetted surface area (ft )

SEP R 2b (Y osL)

PHIS R 2(h)

LsYos

PHIB R 2(hb)

I1 Y ob

RATIOL R Yob /Yos

CFS R Frictional drag coefficient of strut

CFB R Frictional drag coefficient of body

48

COMMON/PHYSCO

PURPOSE: PHYSCO stores physical constants

NAME TYPE DEFINITION

RHO R Density of water

GNU R Kinematic viscosity of water

G R Acceleration due to gravity

PI R Ratio of circumference to diameter of acircle

DELCF R Correlation allowance

KTFPS R Conversion factor for changing ft/secinto knots

49

COMMON / PRCOMM

PURPOSE: Used by optimization subroutines to transmit values

amongst themselves

NAME TYPE DEFINITION

FX R Function to be minimized

LDT R Used in step size computations

DMIN R The square of the machine precision

NF I Function evaluation counter

NL I One dimensional search counter

50

COMMON/PSI

PURPOSE: PSI stores the values of the variables and constants used in

integration routines

NAME TYPE DEFINITION

NPTSZ I Number of integration steps from

Yos to y os+1-

PTSAF R Scaling factor of step size in

integrating from a to amax smax

EXPN R Empirical constant for integration

to stop

NALMAX Maximum number of integration stepsfrom Y +l to a

0s max

NAL I Counter of integration steps

ALFA R Integration variable (a)

ALSMAX R Maximum of a for integration

Comments: Values of NPTSZ, PTSAF, EXPN, NALMAX and ALSMAX are

assigned in data statement in SWATHO

51

COMMON/Q

PURPOSE: Used by optimization subroutines to transmit functionrepresentations amongst themselves

NAME TYPE DEFINITION

V R Matrix to define directions for linearsearch

Q0 RVariables which define the plane for thequadratic search

Q1 R

QA R

QB R Variables which define a parabolicspace curve

QC R

QDO R Variables used in QA, QB, QC to define

QDI R quadratic curve

QFI R Value of function defined by QA, QB, QC

52

• . . . .. . . ..7 -. , ...2 ._ . '

COMMON/SPEEDS

PURPOSE: Stores speeds at which resistance clculations are made

NAME TYPE LENGTH DEFINITION

NSPEED I Number of speeds

SPEED R 20 Speeds in feet per second

.5

53

COMMON /XRPLOTF

PURPOSE: XRPLOTF stores the values of variables for the plotting

routines

NAME TYPE DEFINITION

XL R Value of abscissa at left-most gridline

XH R Value of abscissa at right-most gridline

YL R Value of ordinate at bottom grid line

YH R Value of ordinate at top grid line

XI R Increment of divisions along theabscissa

YI R Increment of divisions along theordinate

YMOV R Ordinate index increment number of

array GRAF

XMOV R Abscissa index increment number ofarray GRAF

54

COMMON/XRPLOTG

PURPOSE: XRPLOTG stores the values and characters of variables forthe plotting routines

NAME TYPE LENGTH DEFINITION

GRAF I (11,204) Array containing the image to be plotted

55

COMMON/ XRPLOTQ

PURPOSE: XRPLOTQ stores the constants and characters for the plotting

routines

NAME TYPE DEFINITION

I, II I Ordinate scale factor is 101

J, JJ I Number of digits following

ordinate decimal point

K, KK I Abscissa scale factor is 1 K

L, LL I Number of digits following

abscissa decimal point

A, NHL I Integer number of horizontal grid lines

G, NSBH I Integer number of spaces beyond eachhorizontal grid line to next grid line

C, NVL I Integer number of vertical grid lines

D, NSBV I Integer number of spaces beyond eachvertical grid line to next grid line

E, HCHAR Horizontal grid character

F, VCHAR Vertical grid character

M, IX, ISX I Number of horizontal spaces

N, IY, ISY I Number of vertical spaces

V L Logical variable = TRUE when maximumand minimum values of the ordinate aredetermined

H L Logical variable = TRUE when maximum andminimum values of the abscissa aredetermined

56

COMMON/WORKVL

PURPOSE: WORKVL stores the working values of the ship geometry

NAME TYPE DEFINITION

XLS R Length of strut in ft.

HS R Draft of strut in ft.

TSMAX R Max strut thickness in ft.

CWP R Waterplane area coefficient

CLCF R Waterplane moment coefficient

CIYY R Waterplane inertia coefficient

CLCB R Body moment coefficient

XLB R Length of body in ft.

BDIA R Body diameter in ft. at XLB/2

BS R Separation of hull centerlines

CP R Body prismatic coefficient

CSTRT R Dist. of strut CL fwd of body CL

AX R Maximum body cross-sectional area insq. ft.

HB R Body centerline submergence

DISP R Displacement of total ship in long tons

AWP R Waterplane area of one strut

NLOC I Switch indicating presence of secondstrut

CSTRT2 R Dist. of strut CL fwd of body CL

OVMFPS R Optimization speed in ft/sec

57

COMMON/WORKVL (CONT)

NAME TYPE DEFINITION

KB R Height of center of buoyancy abovebaseline

BML R Longitudinal metacentric height

BMT R Transverse metacentric height

58

SUBROUTINE AND FUNCTION DESCRIPTIONS

59

NAME: SUBROUTINE AUXPLT

PURPOSE: Prints title and values for ship geometry onthe body sectional area and waterplane outline

curve. Subroutine PCHEB prints the actual curve

CALLING SEQUENCE: Call AUXPLT (S, N, HP)

ARGUMENTS: S - Scaling values for each of the variables

N - The number of variables upon which thefunction depends

HP - Horsepower

COMMON BLOCKS: INITL, NOCHG, PHYSCO, SPEEDS

SUBROUTINES CALLED: SCALE, CHEB, LISTEX, PCHEB, EHP

CALLED BY: PRPRIN

60

NAME: SUBROUTINE BESSJ

PURPOSE: SUBROUTINE BESSJ evaluates the Bessel functionfrom order 0 to order N

CALLING SEQUENCE: CALL BESSJ (X, N, VJ)

ARGUMENTS: X - Argument of the Bessel function

N - Maximum order of the Bessel functionVJ- Array holding (N+l) values of the Bessel

function of order zero up to N, where

VJ(l) = J (X)

COMMON BLOCKS: NONE

SUBROUTINE CALLED: NONE

CALLED BY: RINTEG, RWAVE

61

NAME: FUNCTION CFITTC

PURPOSE: Function CFITTC determines the frictional

resistance coefficient based on the ITTC

correlation line

CALLING SEQUENCE: C f CFITTC (RN)

ARGUMENT: RN - Reynolds number at test condition

COMMON BLOCKS: NONE

SUBROUTINE CALLED: NONE

CALLED BY: Function EHP

COMMENTS: C = 0.075(log, 0 (RN) -2) z

62

- -.. ... -

NAME: SUBROUTINE CHEB

PURPOSE: Determines Chebychev coefficients for theship, and calculates wetted surface of bodyand strut

CALLING SEQUENCE: CALL CHEB (TITLE, KEY)

ARGUMENTS: TITLE- Label printed on outputKEY - Variable used to skip or include the printed

output.

COMMON BLOCKS: COEFS, OMEGA, PHYSCO, WORKVL

SUBROUTINES CALLED: WSURFB, WSURFS

CALLED BY: SWATHO, PEHP, AUXPLT

63

NAME: SUBROUTINE CKLIM

PURPOSE: Checks for violation of constraints andscales penalties by the values in thearray ALPHA

CALLING SEQUENCE: CALL CKLIM (GG, NGG)

ARGUMENTS: GG - actual amount of violationNGG - number of values to be checked for

violation

COMMON BLOCKS: COEFS, EXTLIM, WORKVL

SUBROUTINES CALLED: EVAL

CALLED BY: Function PEHP

64

NAME: FUNCTION EHP

PURPOSE: Calculates body constants and frictional

drag coefficients and calls ROUT to

calculate EHP

CALLING SEQUENCE: EH = EHP (SPEED, TITLE, KEY)

ARGUMENTS: SPEED - Speed of ship in feet per second

TITLE - Label printed on output

KEY - Variable to skip or includethe printed output

COMMON BLOCKS: NOCHG, OMEGA, PHYSCO, WORKVL

SUBROUTINES CALLED: ROUT, RWAVE, CFITTC

CALLED BY: SWATHO, AUXPLT, PEHP

65

NAME: FUNCTION EVAL

PURPOSE: Evaluates the Chevychev iries

MAXF~x) - E A(M)*U(M,X) g(M)*V(M,X)

Min1

CALLING SEQUENCE: EV =EVAL (X,A,BMAX)

ARGUMENTS: X -Arguments of the Chebychev seriesA -Coefficients of the Chebychev

Cosine seriesB -Coefficients of the Chevychev Sine

seriesMAX -Maximum order of the Chebychev series

COMMON BLOCKS: NONE

SUBROUTINES CALLED: NONE

CALLED BY: CKLIM, WSURFB, WSURFS

COMMENTS: UM(X) -COS {(2-)()'

VMCX) - SIN 2MG

e= SIN1 (X)

66

NAME: FUNCTION FORMDR

PURPOSE: Function FORNDR evaluates the form

drag coefficient

CALLING SEQUENCE: FOR = FORMDR (VL)

ARGUMENT: VL = Speed-length ratio of ship based

on strut length

COMMON BLOCKS: NONE

SUBROUTINE CALLED: FUNCTJON YINTP

CALLED BY: ROUT

COMMENTS:

1.0

o 0.9

x

S 0.8

.,-4

0.70

c 0.6

0 0.5

0 .4 I I I I I I

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

V/4-L, Speed-Length Ratio of Strut

Figure 3. FORM DRAG COEFFICIENT

67

NAME: SUBROUTINE INPUT

PURPOSE: Reads iato common b~o,-k/INITL/the initialvalues of ship geometry and constraints,coucurrently defining array XI. It thenpr.ints the stazting point, along with anecho ot the inpat values

CALLING SEQUENCE: CALL INPUT (TITLE, S, OPTSPD)

ARGUMENTS: TITLE - Label printc~d on outputS - Scale factors

OPTSPD - Optimization speed in knots

COMMON BLOCKS: EXTLIM, INITL

SUBROUTINES CALLED: NONE

CALLED BY: SWATHO

68

NAME: SUBROUTINE LISTEX

PURPOSE: Lists various constraint values sothat it can be determined how closeto the extreme limits this solutionlies

CALLING SEQUENCE: CALL LISTEX

ARGUMENTS: NONE

COMMON BLOCKS: EXTLIM, PHYSCO, WORKVL

SUBROUTINES CALLED: NONE

CALLED BY: SWATHO, AUXPLT

69

NAME: SUBROUTINE MAPRNT

PURPOSE: Prints the columns of the NxN Optimizationmatrix V with a heading as specified by OPTION

CALLING SEQUENCE: CALL MAPRNT (OPTION, V, M, N)

ARGUMENTS: OPTION = 1 LABELS NEW DiRECTION2 LABELS PRINCIPAL AXES

V - Logical variable = TRUEM - The maximum step sizeN - Number of variables

COMMON BLOCKS: NONE

SUBROUTINES CALLED: NONE

CALLED BY: PRAXIS

70

NAME: SUBROUTINE PCHEB

PURPOSE: Subroutine PCHEB plots by line printer thebody sectional area curve and waterplaneoutline curve from the given Chebychevcoefficients

CALLING SEQUENCES: CALL PCHEB (AS, BS, AB, BB, NN, TITLE)

ARGUMENTS: AS - Coefficients of Chebychev SineSeries for strut (Symmetric)

BS - Coefficients of Chebychev CosineSeries for strut (Antisymmetric)

AB - Coefficients of Chebychev Sine

Series for body (Symmetric)

BB - Coefficients of Chebychev CosineSeries for body (Antisymmetric)

NN - Dimension of AS, BS, AB, BBTITLE- Array containing the alphanumeric

characters of the title of theproject.

COMMON BLOCKS: NONE

SUBROUTINES CALLED: PLOTi, PLOT2, PLOT3, PLOT4

CALLED BY: AUXPLT

71

NAME: FUNCTION PEHP

PURPOSE: PEHP is the objective function to be

minimized by PRAXIS

CALLING SEQUENCE: PE = PEHP (S,N)

ARGUMENTS: S - Array of scale factors being optimized

N - Maximum order of the Bessel function

COMMON BLOCKS: OMEGA, WORKVL

SUBROUTINES CALLED: SCALE, CHEB, CKLIM, EHP

CALLED BY: PRAXIS

72

NAME: SUBROUTINE PLOTI

PURPOSE: Subroutine PLOT1 sets up spacing anddetermines the values of the axes

CALLING SEQUENCE: CALL PLOTi (NSCALE, A, B, C, D, E, F)

ARGUMENTS: NSCALE - Integer array defined as

follows:

NSCALE (1) - I, if printed, values ofthe ordinate are 10 ** Itimes the actual values

NSCALE (2) - J, if printed, values ofthe ordinate are 10 ** Jtimes the actual values

NSCALE (3) - K, if printed, values ofthe abscissa are 10 ** Ktimes the actual values

NSCALE (4) - L, if printed, values ofthe abscissa are 10 ** Ltimes the actual values

A - Integer number of horizontalgrid lines

B - Integer number of spacesbetween each horizontalgrid line

C - Integer number of vertical

grid lines

D - Integer number of spacesbetween each vertical gridline

E - Horizontal grid character

F - Vertical grid character

COMMON BLOCK: XRPLOTQ

SUBROUTINE CALLED: NONE

CALLED BY: PCHEB

73

NAME: SUBROUTINE PLOT2

PURPOSE: Subroutine PLOT2 examines the minimum andmaximum values of the abscissa and theordinate and establishes an internalformula for computing location in theimage region corresponding to the pointto be plotted

CALLING SEQUENCE: CALL PLOT2 (XMAX, XMIN, YMAX, YMIN, NSCLI)

ARGUMENTS: XMAX - Value of abscissa at rightmost

grid lineXMIN - Value of abscissa at leftmost

grid lineYMAX - Value of ordinate at top grid

lineYMIN - Value of ordinate at bottom

grid lineNSCLI - Logical flag (should be FALSE,

if PLOTI has not been calledand standard grid is desired)

COMMON BLOCKS: XRPLOTF, XRPLOTQ, XRPLOTG

SUBROUTINE CALLED: NONE

CALLED BY: PCHEB

74

NAME: SUBROUTINE PLOT3

PURPOSE: Subroutine PLOT3 assigns an alpha-

character to each point to be plotted

CALLING SEQUENCE: CALL PLOT3 (PCHAR, X, Y, SDATA, FDATA,

DDATA)

ARGUMENTS: PCHAR - Plotting character

X - Array containing the X

coordinates to be plotted

Y - Array containing the Y

coordinates to be plotted

SDATA - Integer position in the arrays

of the first ordered pair to

be plotted

FDATA - =1 if each point from SDATA

to DDATA is to be plotted

=2 if every other point is to

be plotted=3 if every third point is to

be plotted

DDATA - Integer position in the array

of the last ordered pair to be

plotted

COMMON BLOCKS: XRPLOTF, XRPLOTG

SUBROUTINE CALLED: NONE

CALLED BY: PCHEB

75

NAME: SUBROUTINE PLOT4

PURPOSE: Subroutine PLOT4 prints the image of thecompleted graph on the printer, includingthe values of the abscissa and the

ordinate at the grid lines outside thebottom and left edge of the graph

CALLING SEQUENCE: CALL PLOT4 (MCHAR, NCHAR)

ARGUMENTS: MCHAR - Single dimension arraycontaining alpha characters

to be plotted at the left

of the graphNCHAR - Number of valid characters in

MCHAR

COMMON BLOCKS: XRPLOTF, XRPLOTG, XRPLOTQ

SUBROUTINE CALLED: QPLOTZ5

CALLED BY: PCHEB

76

NAME-: SUBROUTINE POMITI

ETo cause certain grid lines on graph to be

deleted, depending upon which arguments

are labeled TRUE

CAI.ING SEQUENCE: CALL POMITI (T, B, L, R)

ARGUMENTS: T = top line on graph

B = bottom line on graphL = left line on graphR = right line on graph

COM2MON BLOCKS: XRPLOTG, XRPLOTQ

SUBROUTINES CALLED: NONE

CALI.ED BY: Main program

COMMENTS: This subroutine is part of a plotting

package and is included in thisprogram only to keep this packageintact

77

NAME: SUBROUTINE POMIT2

PURPOSE Causes horizontal and/or vertical valuesat grid lines to be deleted

CALLING SEQUENCE: CALL POMIT2 (ARG)

ARGUMENTS: ARG ARG = 1 values of abscissa at

grid lines deletedARG = 2 values of ordinate at

lines deleted

ARC = 3 both sets of valuesdeleted.

COMMON BLOCKS: XRPLOTQ

SUBROUTINES CALLED: NONE

CALLED BY: Main program

COMMENTS: This subroutine is part of a plotting

package and is included in this programonly to keep this package intact

78

NAME: SUBROUTINE PRAXIS

PURPOSE: Finds the minimum of the function

F(X,N) of n variables using the principalaxis method. The gradient of the functionis not required.

CALLING SEQUENCE: CALL PRAXIS (TO, HO, N, IPRIN, X, F, FMIN)

ARGUMENTS: TO - is a tolerance

HO - is the maximum stepsize

N - the number of variablesupon which the functiondepends (must be at

least two)IPRIN - controls the printing

of intermediateresults

x - contains oh entry a guessof the point of minimum,

on return the estimatedpoint of minimum

F - the function to be

minimizedFMIN - is set to the minimum

found

COMMON BLOCKS: PRCOMM, Q

SUBROUTINES CALLED: PRPRIN, PRMIN, PRQUAD, MAPRNT, PRFIT,

PRSORT, VCPRNT, RANDUM, PEHP

CALLED BY: SWATHO

COMMENTS: The approximating quadratic form is:

s(X') - F(X,N) + *((X'-X) - Transpose *A* (X-X'))

X is the best estimate of the minimumA is Inverse (V-Transpose) *D* Inverse (V)V(**) is matrix of search directionsD(*) is array of second differences

79

NAME: SUBROUTINE PRFIT

PURPOSE: An improved version of MINFIT restrictedto M - N,P - 0. The singular valuesof the array AB are returned to Q, andAB is overwritten with the orthogonalmatrix V such that U DIAG (Q) = AB.V,where U is another orthogonal matrix.

CALLING SEQUENCE: CALL PRFIT (M, N, MACHEP, TOL, AB, Q)

ARGUMENTS: M - is the maximum step sizeN - the number of variables upon which

the function dependsMACHEP - is the machine precision, the smallest

number such that I+MACHEP>I. MACHEPshould be 2**-47 (about 7.105 E-15)for single precision arithmetic onthe CDC 6000 system or 16**-13 (about2.23 D-16) for double precision onthe IBM 370 system.

TOL - ToleranceAB - Array for which singular values are

to be determinedQ - Where the original values of the array

are to be stored.

COMMON BLOCKS: NONE

SUBROUTINES CALLED: NONE

CALLED BY: PRAXIS

80

NAME: FUNCTION PRLIN

PURPOSE: PRLIN is the function of one real

variable L that is minimized by thesubroutine PRMIN

CALLING SEQUENCE: PR = PRLIN (N, J, L, F, X, NF)

ARGUMENTS: N - Number of variableJ - Defines direction for search/

minimization in the V matrixL - The single variable in the rotated

coordinate system upon which thefunction is assumed to varyquadratically

F - The function to be minimized

X - Coordinate of the initial pointin the quadratic space

NF - The function evaluation counter

COMMON BLOCKS: Q

SUBROUTINES CALLED: NONE

CALLED BY: PRMIN

8|81

NAME: SUBROUTINE PRMIN

PURPOSE: Minimizes F from X in the direction V(*,J)unless J is less than 2, when a quadraticsearch is made in the plane, defined byQ0, Q1, x

CALLING SEQUENCE: CALL PRMIN (N, J, NITS, D2, Xl, Fl, FK,F, X, T, MACHEP, H)

ARGUMENTS: N - is the number of variables upon whichthe function depends.

J - Defines direction for search/minimiza-tion in the V matrix.

NITS - Controls the number of times anattempt will be made to halve theinterval

D2 - is either zero or an approximationto half F

Xl - is estimate of the distance fromX to the minimum along V (*, J)

F1 - PRLIN (N, J, Xl, F, X, NF)FK - Logical operatorF - The function to be minimizedX - Coordinate of the initial point

in the quadratic spaceT - Tolerance

MACHEP - Machine precisionH - Step size

COMMON BLOCKS: PRCOMM, Q

SUBROUTINES CALLED: PRLIN

CALLED BY: PRAXIS, PRQUAD

82

NAME: SUBROUTINE PRPRIN

PURPOSE: PRPRIN prints intermediate optimization

results including the number of functionevaluations, function value, and currentcoordinates

(ALLING SEQUENCE: CALL PRPRIN (N,X, IMPRIN)

ARCUMENTS: N - the number of variables upon whichthe function depends

X - contains on entry a guess of thepoint of minimum, on return theestimated point of minimum.

IPRIN - controls the printing of intermediateresults

COMMON BLOCKS: PRCOMM

SUBROUTINES CALLED: AUXPLT

CALLED BY: PRAXIS

83

NAME: SUBROUTINE PRQUAD

PURPOSE: Looks for the minimum of F along aquadratic curve defined by Q0, Qi, X,

in case minimization is being done ina curved valley

CALLING SEQUENCE: CALL PRQUAD (N, F, X, T, MACHEP, H)

ARGUMENTS: N - Number of variableF - The function to be minimized

X - Coordinate of the initial pointin the quadratic space

H - Step size

COMMON BLOCKS: PRCOMM, Q

SUBROUTINES CALLED: PRMIN

CALLED BY: PRAXIS

84

NAME: SUBROUTINE PRSORT

PURPOS E: Sorts the elements of D(N) intodescending order and moves thecorresponding columns of V(N,N)

resulting in eigenvalues and elgenvectors

CALLING SEQUENCE: CALL PRSORT (M, N, D, V)

ARGUMENTS: M - The maximum step sizeN - Number of variables

D - Second difference arrayV - Matrix of search directions

COMMON BLOCKS: NONE

SUBROUTINES CALLED: NONE

CALLED BY: PRAXIS

85

NAME: SUBROUTINE QPLOTZ5

PURPOSE: Subroutine QPLOTZ5 calculates thescaling information needed to

generate the format to label the left-hand side of the printer plot.

CALLING SEQUENCE: CALL QPLOTZ5 (PDQ)

ARGUMENT: PDQ-scaling factor for ordinate plot

COMMON BLOCKS: XRPLOTF, XRPLOTQ

SUBROUTINES CALLED: NONE

CALLED BY: PLOT4

86

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411111 1.25 -11114 h

MICROCOPY RESOLUTION iEST CHARTNIN(NAl BIIR AU 01 IANDAAN I A

NAME: FUNCTION RANDUM

PURPOSE: To avoid resolution valleys, by makingrandom steps on a periodic basis,Function RANDUM returns a random numberuniformly distributed in (0,1)

CALLING SEQUENCE: RA = RANDUM (NAUGHT)

ARGUMENTS: NAUGHT - number used to initializethe random number generator

COMMON BLOCKS: NONE

CALLED BY: PRAXIS

87

NAME: SUBROUTINE REFLKT

PURPOSE: Subroutine REFLKT defines the lowerhalf of the symmetrical matrices T andW, the auxiliary wave resistance integrals,

by reflection about the diagonal of thematrix.

CALLING SEQUENCE: CALL REFLKT

ARGUMENTS: NONE

COMMON BLOCKS: AUX, COEFS

SUBROUTINE CALLED: NONE

CALLED BY: RWAVE

88

NAME: SUBROUTINE RINIT

PURPOSE: Subroutine RINIT initializes the auxiliarywave resistance matrices, T and W, to zero

t

CALLING SEQUENCE: CALL RINIT

ARGUMENT: NONE

COMMON BLOCKS: AUX, COEFS

SUBROUTINE CALLED: NONE

CALLED BY: RWAVE

89

NAME: SUBROUTING RINTEG

PURPOSE: Subrouting RINTEG evaluates theintegrand for T and W functions.

CALLING SEQUENCE: CALL RINTEG (ALFA, B, D, NLOC2, WTINT,

SEPCOS, SQ)

ARGUMENTS: ALFA - Integrating variableB - CSTRTl(I)/XLSD - CSTRT2(I)/XLSNLOC2 - Flag indicating the

presence of a second strut

NLOC2 = 0 if single strut= 1 if tandem strut

WTINT - Weighting constant

for the integrandSEPCOS - Value of the cosine

function in theintegrand

SQ - Value of the integrand

1

(a2 - Y2 )1/2

COMMON BLOCKS: AUX, OMEGA, COEFS, WORKVL

SUBROUTINE CALLED: BESSJ

CALLED BY: RWAVE

COMMENTS: Integrals used in calculations are of the form:

Yo + /d G da

f + 2 da f(a) + f f(a)YOS C1a 802 fo S + 1 %/ a2 y s2

- -- 1of

For the first integral we use the substitution: a = y + C2 where.d = 2dz.

Therefore the first integral becomes fI o + 2 f(Yos + 2

Where the function f(a) takes the form of E0(a) Jm(a)5n(a )

where the order of the Bessel functions is determined by which auxiliaryfunction is to be evaluated.

90

NAME: SUBROUTINE ROUT

P POSE: Calculates EHP, and wave resistance coefficients,frictional drag coefficients, speed constants,and prints the intermediate and final designresults

CALLING SEQUENCE: CALL ROUT (TITLE, EHP, KEY)

ARGLWMENTS: TITLE - label printed on outputEHP - effective horsepower of the shipKEY - variable used to skip or print output

COMON BLOCKS: OMEGA, PHYSCO, WORKVL

SUBROUTINES CALLED: SUM, FORMDR

CALLED BY: Function EHP

COMMENTS: The type of output from ROUT is determined bythe value of KEY, which is set in the subroutineswhich call ERP. If KEY = 0 (set in PEHP) nooutput is printed and ROUT just does computations;

if KEY = 1 (set in SWATHO) the optimization iscomplete and the final design values are printed;

if KEY = 2 (set in AUXPLOT) the intermediatedesign results are printed

91

NAME: SUBROUTINE RWAVE

PURPOSE: Subroutine RWAVE computes the auxiliary waveresistance functions T and W.

CALLING SEQUENCE: CALL RWAVE (B, D, NLOC2)

ARGUMENTS: B -CSTRTI(I)/XLSD -CSTRT2(1)/XLSNLOC2 -Flag indicating the presence of a

second strut

NLOC2-Jo if single strutI if tandem struts

COMMON BLOCKS: AUX, OMEGA, PSI, COEFS, PHYSCO, WORKVL

SUBROUTINES CALLED: RINIT, SIMPSN, RINTEG, BESSJ, REFLKT

CALLED BY: FUNCTION EHP

COMMENTS :

Integrals evaluated by RWAVE:

T = (2m-1) (2n-1) f S 4,,2 _y 2 2

WS = (2m) (2) 1d_ 2 _ y E2s (C) 3 m- (c) 3m()

Smn "yos C12 os EB (a) J 2m(a) J 2n(a)

T = (2m) (2n) f'o a2 2. .(a) (a) EB()"B m( (n) fy fa 2 -_ 2 EB 2m-1 2n 1os

WB = (2m) (2n) f E c da E2 (s)n Yos a 2 - 2 B

T B m(2mn-1) (2n-1) f 1 da E S(() EB(5) 3 m-(a) Jj(6

os

WSBmn - (2m) (2n) oP doa ES(a) EB(8) J2m(CO J2n( )

M' s 2 -.f '0500 ¥os

92

. . . .. .w e .. . . .. . . .

NAME: SUBROUTINE SCALE

PURPOSE: Creates current working values X, by X - S*XI,

where S is the scale factor, and XI is thevector of initial values.

CALLIk SEQUENCE: CALL SCALE (S, N)

ARGUMENTS: S - scaling value for each of the variables

N - number of variable

COMMON BLOCK: INITL, NOCHG, PHYSCO, WORKVL

SUBROUTINES CALLED: NONE

CALLED BY: SWATHO, PEHP, AUXPLT

93

NAME: SUBROUTINE SIMPSN

PURPOSE: Subroutine SIMPSN sets up Simpson's multipliersfor numerical integration by Simpson's rule.

CALLING SEQUENCE: CALL SIMPSN (NPTS, SIMP)

ARGUMENTS: NPTS - Number of integration steps

SIMP - Array containing the Simpson's multipli~rs

COMMON BLOCKS: NONE

SUBROUTINE CALLED: NONE

CALLED BY: RWAVE

COMMENT: These values are used to integrate the auxiliary

wave resistance integrals, T and W, for valuesof the integral between Yos and Y08 + 1

An example of one of these integrals in the range Y., to Yos+l is

fYos + d_ f (a) 1 2d; + os + C2 )

0s Os9

94

NAME: SUBROUTINE SUM

PURPOSE: Subroutine SUM computes the sums for the waveresistance from the Chebychev coefficients andthe auxiliary wave resistance functions, T and W

M NE (A A T + B B W )

(A ni m n mn m n mn

COMMENTS: where Am and Bm are the symmetric and antisymmetric Chebychev

coefficients for the strut or body, and Tmn and Wm are theauxiliary wave resistance functions for various nu1h-strutcomb inat ions

CALLING SEQUENCE: CALL SUM (SUMIS, SUMIB, SUM1SB, SUM12S,SUM12B, SUMI2SB)

ARGUMENTS: SUMIS - Partial sum for strut 1SUMIB - Partial sum for body 1SUMISB - Partial sum for interaction

between strut 1 and body 1SUM12S - Partial sum for interaction

between strut 1 and strut 2SUMI2SB - Partial sum for interactions

between strut 1 and body 2 orstrut 2 and body 1

COMMON BLOCKS: AUX, COEFS

SUBROUTINE CALLED: NONE

CALLED BY: ROUT

5oU141ZS

SUM2S

Figure 4 -Body-Strut Configuration

95

NAME: SUBROUTINE VCPRINT

PURPOSE: With certain input options, this subroutineprints a vector of given length using two ofseveral formats

CALLING SEQUENCE: CALL VCPRINT (OPTION, V. N)

ARGUMENTS:. OPTION =1 Prints the Second Difference Array- 2 Prints the Scale Factors= 3 Prints the value of the approximating

quadratic form= 4 Prints the value of X

V = Matrix of search directions

N = Number of variable

COMMON BLOCKS: NONE

SUBROUTINES CALLED: NONE

CALLED BY: PRAXIS

96

NAME: SUBROUTINE WSURFB

PURPOSE: Determine surface area of body of revolu-tion whose sectional area is given by

Chebychev coeff icients

CALLING SEQUENCE: CALL WSURFB (AREA)

ARGUMENTS: AREA - Surface area of a body (ft )

COMMON BLOCKS: COEFS, PHYSCO, WORKVL

SUBROUTINES CALLED: FUNCTION EVAL

CALLED BY: CHEB

97

NAME: SUBROUTINE WSURFS

PURPOSE: Determines surface area of a strut whose thick-ness distribution is given by Chebychevcoefficients

CALLING SEQUENCE: CALL WSURFS (AREA)

ARGUMENTS: AREA - Wetted surface area of a strut (ft )

COMMON BLOCKS: COEFS, WORKVL

SUBROUTINES CALLED: FUNCTION EVAL

CALLED BY: CHEB

98

NAME: FUNCTION YINTP

PURPOSE: Function YINTP interpolates through a set ofdiscrete data, using quadratic interpolation.

CALLING SEQUENCE: YTP = YINTP (XZ, X, Y, N)

XA - Point to be interpolatedX - Array of ordinate dataY -Array of abscissa dataN -Number of data points

COMMON BLOCKS: NONE

SUBROUTINE CALLED: NONE

CALLED BY: FORMDR

COMMENTS: Uses linear interpolation

99

REFERENCES

1. Lin, W. C. and W. G. Day, Jr., "The Still Water Resistance and Propulsion

Characteristics of Small-Waterplane Area Twin-Hull (SWATH) Ships," AIAA/SNAME

Advanced Marine Vehicles Conferences. Paper No. 74-325, (1974).

2. Brent, Richard P., "Algorithm for Finding Zeroes and Extrema of Functions

without Calculating Derivatives," Stanford University Report, CS-71-198, 1971

100

APPENDIX A

OBJECTIVE AND PENALTY FUNCTIONS

101

APPENDIX A - OBJECTIVE AND PENALTY FUNCTIONS

The program SWATHO is designed to determine the geometry of a SWATH ship which

yields the minimum effective power for a given speed. Thus the objective function

is the effective power. However, this is an over simplification of the problem,

in that the solution to the problem must also satisfy a number of constraints. That

this is so is intuitively obvious if one considers that the ship with minimum power

for a given speed is that ship with zero length and volume, and therefore zero

resistance. Thus an obvious constraint is that the ship contain at least some

minimum volume.

The subroutine PRAXIS, which is used to perform the minimization, is a program

designed for use in unconstrained minimization problems. This dictates that a

modified objective function be employed, which takes into account the fact that

constraints are or are not violated. To meet this requirement, an external con-

straint method was employed, where the functions g 1(x) were defined to be greater

than zero when the i thconstraint was violated, and less than zero when the i th

constraint was satisfied. Using these g i ', the modified objective function was

defined as:

PEHP - EHP(1 + Max(g (x) 0)).

In all of the above definitions, the vector x is the vector of parameters defining

the ship's geometry.

This definition of the objective function results in the addition of no penalty

to the objective function internal to the region where the constraints are satisfied,

and results in the addition of a penalty exterior to the region where the constraints

are satisfied. This gives rise to the description of this as an external constraint

method.

There are a total of nineteen penalty functions which are applied to the pro-

blem of minimizing the resistance of a SWATH ship. These constraints/penalty func-

tions are as follows:

1. Minimum displacement constraint

91 (DISPMN-DISP)ct 1

where DISPMN is the minimum ship displacement as input by the user and

DISP is the current ship displacement at that iteration of the calculations.

102

2. Minimum draft-diameter ratio constraint

9= [Draft Factor x BDIA - (HS + BDIA)]ca2

where the Draft Factor is the minimum draft to diameter ratio, BDIA

is the body diameter at half of the length of the body and HS is the draft

of the strut.

3. Maximum draft constraint

= (HS + BDIA - DRFTMAX)a 3

where DRFTMAX is the maximum draft of the ship as input by the user.

4. Maximum beam constraint

g4 = (BS + BDIA - WMAX) C 4

where BS is the separation of the hull centerlines and WMAX is the

maximum beam of the ship as specified by the user.

5. Hull overlap constraint

g 5 - (BDIA - BS)Ct 5

6. Strut-hull Nose clearance constraint

96 ' (XFWD + CSTRT + XLS/2 - XLB/2)a 6

where XFWD is the minimum distance from the body leading edge to the

strut leading edge, CSTRT is the distance of the strut forward of the

body centerline, XLS is the length of the strut and XLB is the length of

the body.

7. Strut-hull tail clearance constraint

- (XAFT - XLB/2 + XLS/2 - CSTRT)a 7

where XAFT is the minimum distance from body trailing edge to strut

trailing edge.

8. Minimum transverse GM constraint

g8 a[MINGMT - (KB + BMT - KG)]c 8

where MINGMT is the minimum transverse GM, KB is the distance of the

center of buoyancy from the baseline, BMT is the transverse metacenter and

103

KG is the distance of the center of gravity from the baseline.

9. Minimum longitudinal GM contraint

g 9 - [MINGML - (KB + BML - KG)]ct9

where MINGML is the minimum longitudinal GM and BML is the longitu-

dinal metacenter.

10. Body offsets greater than zero contraint

O= f10 dx[Ib(x)I - b(x)alO

where b(x) represents the body offsets

11. Strut offsets greater than zero constraint1 d tri ,I - t(x)ll

where t(x) represents the strut offsets

12. Maximum length-diameter ratio check

912 ' (XLB - LODMAX x BDIA)a12

where LODMAX is the maximum body length over diameter ratio.

13. Minimum length-diameter ratio check

913 - (LODMIN x BDIA - XLB)a 13

where LODMIN is the minimum body length over diameter ratio.

14. Strut minimum thickness constraint

914 - (TSMIN - TSMAX)a 14

where TSMIN is the minimum strut thickness at 50% chord and TSMAX is

the strut thickness in the current iteration of the calculations.

15. Strut thickness-body diameter constraint

915 = (TSMAX - BDIA)a 15

16. Positive strut length contraint

916 = (-XLS)a 16

17. Positive body length constraint

917 (-XLB)(z17

104

18. Maximum body offset constraint

g18 w(b ma 1.2 BDIA)ct18

where b is the'maximum body offset. This constraint insures amax

reasonable value for the maximum body offset.

19. Maximum strut thickness constraint

g19 , (tmax - 1.2 TSMAX)ct19

where t is the maximum possible value of strut thickness. Thismax

constraint insures a reasonable value for the maximum strut thickness.

In all of the above penalty functions, the constraints a are chosen so that

a constraint violation of ten percent results in a value of g i around 104.

Of the above penalty functions, eleven are constraints imposed by the program

user to insure that the optimum vehicle meets some minimum set of criteria such as

minimum displacement and transverse and longitudinal GM. These user specified

constraints are the constraints numbered 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, and 14.

The other eight constraints are constraints imposed by the program to insure

that the geometry of the vehicle is reasonable. Examples of these would be con-

straints to insure that the lengths of the body and strut are positive, and con-

straints to insure that the offsets of the strut and body are all positive. The

numbers of these constraints are 5, 10, 11, 15, 16, 17, 18, and 19.

105

APPENDIX B

COMPUTATIONAL PROCEDURE FOR SWATH

RESISTANCE PREDICTION

106

APPENDIX B - COMPUTATIONAL PROCEDURE FOR SWATH RESISTANCE PREDICTION

The main purpose of the SWATHO program is to minimize the power required to

propel a SWATH ship in a calm sea at constant speed. For a ship moving at speed V

and experiencing a total drag force RT the effective horsepower required is:

P RTVE 550

The total drag is comprise " three components: frictional resistance (RF),

wave-making resistance (Rr, -' C rw drag (RFM). That is:

• - RF + RW + RFM.

The frictional resistan. and :;-rm drag are caused by the motion of the hull through

a viscous fluid. The wave-making resistance is due to the energy that must be

supplied by the ship tn the wave system created on the free surface. However, wave-

making resistance and form drag are usually grouped together under the title of

residuary resistance.

FRICTIONAL RESISTANCE

Frictional resistance is the single largest component of the resistance of a

SWATH ship. Frictional resistance is calculated in the traditional fashion of naval

architects, using the ITTC model ship correlation line to determine a frictional

resistance coefficient (CF). However, for a SWATH ship, the frictional drag of the

hulls and struts are calculated separately based on their respective Reynolds

numbers. Thus we have that the frictional resistance coefficient is given by the

formula:

C 0.075(loglo Rn - 2)2

and the frictional resistance of the hulls and struts are given by:

RF F 1 S Hull V2 CHull Hull

and

V 2 CRFStrut p S SSru t FStrut*

107

In calculating the total frictional resistance, a correlation allowance (C A) Of

0.0005 is also included in the total, so we have that:

R F =RF Hull + RF Strut + '-2 (5wll + SStt )V2 C A

FORM DRAG

Form drag (R FM) is usually defined as the viscous component of the drag due to

the shape of body, i.e., the difference between the total viscous drag, and the

viscous drag of the equivalent flat plate of the same length. However, in the case

of the SWATH resistance programs, the form drag has been determined in a much more

empirical fashion.

SWATH form drag has been defined as the difference between the experimentally

determined residuary resistance of a hull form and the theoretically derived wave

resistance for that same hull form. Such a difference was calculated for both

SWATH 3 and SWATH 4 based on bare hull resistance results, and plotted as a func-

tion of strut speed-length ratio. A curve was then faired through the data,

Figure 3 (p. 69). Due to the scatter in the data a decision was made not to allow

the form drag coefficient go below 0.0005; this can be seen on Figure 3. The form

drag coefficient, determined from Figure 3, is converted to full scale form drag,

using the traditional formula and the total wetted surface:

RM= p SV2 CFM.

WAVE-MAKING RESISTANCE

The wave-making resistance of a ship is not easily predicted from empirical

relations or from gross ship features. Experience has shown that two ships of

similar form may differ significantly in their measured total resistance due to

differences in the wave-making resistance component. Wave-making resistance is a

function of the fine details of ship form, and is best studied through mathematical

analysis.

Lin and Day 1Investigated the problem of wave-making resistance for twin-hull

ships. The mathematical problem is formulated within the context of the linearized

thin-ship theory. In their investigations, the potential flow around such a ship

is represented by sheet distributions of sources.

The computer program SWATHO documented in this user's manual is devoted

primarily to the implementation of numerical procedures for the prediction of the

108

wave-making resistance of SWATH ships following the theory developed by Lin and

Day. The remainder of this appendix will be devoted to the discussion of the

essential details involved in the numerical procedure utilized in the wave-making

resistance predictions.

Based on the analysis of Lin and Day, the wave resistance is expressed as

follows:

R= ( pgT2Lsyos)

M M

x {Ass ASn Tsmn + BSm B Sn WSmnl 'm=ln=l

(2wPgAO2)

RB = \ yo J a2 a IABm ABn TBmn +B Bn WSBmnm=1 n=1

M M

RSB= (2fgACT) Asm An TSBmn + BS ER WSB n

m=l n=1

where T, LS, and hS are the maximum thickness, length, and draft of the strut,

respectively; A 0 , LB, and hB are the maximum section area, length, and depth of

submergence of the axis of the body.

In the above equations, RS, RE, and RSB represent the wave resistance due to

one strut, one main body, and the intersection between strut and main body, respec-

tively. Hence, the wave resistance of a SWATH ship becomes

RW = 2(RS + RE + R

Also in the above equations, ASm and BSm are the Chebychev coefficients for the

strut, ABm and BBm are the Chebychev coefficients for the hull, and T Sn, TSBmn,

TBmn, WSmn, WSBmn, and WBmn are the auxiliary wave resistance functions. The

auxiliary wave resistance functions are defined as follows:

T smn l f

(2m-l)(2n-l)

j 2 d0 D 2b, YsWSm n CL o vf 'YO

(2m)(2n) os

J2m-1 a) J2n-l1a)

s E J2m(o) J2n(a)

109

2b =1+_cs-S j-jo L2s2

Es 1 2 (h s/L s) (a2 /y OS),

gL sYOS 2U79'

T Bm

(2m-1)(2n-1) 00da a2 D aLb2b

WBn JYOS , Os

(2m) (2n) J

SEB ,)jj2m-1 (a) 12n-l8

x ~~ E()[2m(a) 12n(a)

-2(h B/L B ) a~yOB~

B L

YOB =U 2)

YOs

and

T SBm

(2m-1)(2n-l) 00 a D (a$ b

WS -- LJ~ s/2~ O

(2m) 2n) x E s(a)E B M 2m-1 ( ) 2 -

l3 2m (a) J 2n~a)

110

where Jm () is a Bessel function of the first kind, defined as follows:

) 2 - r- 2 cos(a sin t) cos(mt) dt.

0

NUMERICAL INTEGRATION METHODS FOR AUXILIARY WAVE

RESISTANCE FUNCTION EVALUATION

The range of integration for the auxiliary wave resistance functions is divided

into three intervals of integration: [y os, 'yos + 1], a "central region" and the

"tail" region. A detailed discussion of these intervals and the integration proce-

dure for each interval follows.

INTEGRATION PROCEDURE FOR THE INTERVAL [yos, Yos + l

In the region of yos the integrands do not behave well, but they are neverthe-less integrable. This is due to the presence of in the denominator. This

may be easily handled by utilizing the substitution:

ay o

where

da = 2dr.

This substitution will be utilized in the region y os, Yos + 1]. It eliminates the

square-root singularity by giving

f Y o l d0 2dC f(y os + C2).f os+ d f() = 2yos f2

Os 2 __KI

Investigating the behavior of the integrands as functions of in the region

0 < C < I shows that they are very well behaved and a simple numerical integration

scheme with about 30 points will yield excellent results.

1

* -*

INTEGRATION PROCEDURE FOR THE CENTRAL REGION

The central region of the numerical integration starts at a = y + 1 and con-

tinues until EB() reaches a very small fraction of its initial value,

[e+l)yoB l

EB O init) = EB Y 1The fraction is given as 10- so

exp Y YOB 10- = exp(-2.3026E).

exp 1-2 Yo +)yO

Y osY OB

Hence,

a L ~ = 2.3026ey~max L maxh L- B 's

The justification of this choice for aax and the effect of the choice of E, will

be discussed in the next section.

In order to implement an effective and economical numerical integration scheme,

it is necessary to investigate the behavior of the intagrands within this central

region. The requirement is to provide at least sevcral points of the integrand for

each full cycle of its oscillation. Hence, a constr:,tive estimate of behavior will

be an estimate of more rapid oscillation than actually exists.

The Bessel functions are approximated as

co.13V Z 2 4)

for IZ>>l and lZ>>vj.

112

: - .. .... .. . . .. .. . " m .... ... i..... u~n " . ... ... _L . ... . ..

for smaller Z, the behavior will be less oscillatory, so this represents a con-

servative estimate for all Z. This assumption results in

[ 1 [cos(2ct+T) + 1] -even

- [cos(2+T) - 1] u- odd

2

where the case of (p-v) odd will never occur due to the form of the integrands.

The other combinations of Bessel functions are found as

Jm (cO Jn - cos (Ct+a)

1

JM (n) - -- cos(2),

where the phase angles, a slow oscillation in the first case, and a constant term

in the second case, have all been ignored. Since 8 is generally greater than CL,

but of the same order of magnitude, a conservative estimate gives the variation of

all of the above as

1 1 /2L BJ J cos(20) = cos [2L)

The term [v yI behaves as [a] for cx >> yos and is better behaved for smaller

a. This conservative estimate will be used for the occurrence of this term at the

beginning of each integrand. However, the substitution will not be made in the term

D(c) = cos [ 2 abos SJ

which occurs in all of the two-hull integrands, since that would result in consid-

erably less economical numerical integration.

The terms EB(a) and ES(a) contain decreasing exponentials and are not oscil-

latory and are quite well-behaved, and so will not affect the spacing of points for

the numerical integration.

113

Therefore, the integrand for T12Smn and W 12Smn behave like

do C t)CO LB Cos 2 _1Trot E() (2 co a - -S Yos LSor

those for T12B.n and WI2Bmn behave like

da E 2 BCi) Cos ( 2 LB~ c) cos -L l v W ,nT B L o S L os L S do

those for T 2SBn and W12SBmn behave like

do IL L BE )S~)co-s(2 L! a) COSV . YL7~

(ci Os o

The single-hull terms (TSmn, WSmn, TBmnq WBmn, TSBmn, WSBmn) are similar but

without the cos [ s 2b a"ci 2 y2 term.[Y s LS -'os J t

Based on the behavior shown above, the most rapid oscillation of the integrands

is like

L SoS S.c]os[#~.c V/tW .

In the vicinity of c a , cos(f(ci)] behaves like cos [(fa) and hencedci

--B CO, S _ 2b 2

o L c 2

+

os

232ycos (2L + 2 2b d.os

( yos LS os

114

If the accuracy desired from the numerical integration scheme requires P points per

cycle, then the step size S (distance between points) in the vicinity of c c a is

S B P•

2 LB--- + 2 2b M o

Os S y!___Yoss

This step size will decrease with increasing a but will be smaller for larger hull

centerline spacing b. Hence, if computations are made for various values of b for

the same sample points, the step size should be based on the largest value of b.

The numerical integration scheme chosen for the central region is a modified

Simpson's rule in which all steps are not equal. For this method, a step size is

computed at a given a (starting first with yOs + 1), gsing the last equation. Then

an integral is taken over an interval equal to two of these step sizes, using a

three-point Simpson's rule. A new step size is computed for the a at the end of

that interval, and the process is repeated until a ma is reached or exceeded. The

concept may be visualized as shown below.

-Cos IH3pt 3pi1wroms simpscmRut# utt

This economizes on samples (points) where the integrand does not vary quickly, and

uses more densely spaced points where necessary.

All of the approximations made in this section have been only for the purpose

of choosing the step sizes for the numerical integration. The samples taken, which

are multiplied by appropriate weights added to obtain the numerical evaluation of

the integral, are all found using the actual integrands.

115

INTEGRATION PROCEDURE FOR THE TAIL

It should be noted that the criterion for the end of the "central region," and

hence the beginning of the "tail," is that EB(a) reach a very small percentage of

its original value. This will insure that any integrand which includes E (6) will

have long ceased to contribute significantly to the result. The body-alone and

strut-body integrals (TB, WB, T12B' W12B9 TSB, WSB, T12SB and W12SB) can therefore be

assumed complete at the conclusion of the "central region" numerical integration.

The strut-alone terms (Ts, WS, TI2 S, and W1 2S) are not complete, however. The

integrand for these terms decreases only like a- . The utilization of the "central

region" numerical integration scheme until this integrand were very small would be

quite accurate but exceptionally expensive, especially due to the constantly de-

creasing (with increasing a) step size required for that scheme. Hence, a more

economical scheme must be developed, accepting the resulting loss of accuracy.

The single-hull strut-alone terms (Ts and WS) are easily handled since they do

not include the highly oscillatory term

L S os

The behavior of the integrands for these terms is like

d.E 2 (OL) cos(2),

which allows the use of a simple numerical integration scheme with constant step

size

P

where P is the desired number of points per cycle.

The above method is still not very economical due to the large region over

which the integral must be taken, due to the relatively slowly decreasing a-4

envelope in which the integrand oscillates. A more economical method is to approx-

imate the integrand asymptotically, and analvtically integrate the approximate

integrand from a given value to infinity. The obvious simplification is to assume

that ES(a) =1 , which is an excellent approximation, becuase ES(a) approaches unity

as EB(0 ) approaches zero. The approximation is hence assured by the condition that

116

determined the end of the "central region." In addition to ES(a), the Bessel

functions must also be approximated. We use the approximation

J (a) .- cos Z 2

for Jul >> 1 and Jul > lVI.

If Jul is not enough greater than IvI at the beginning of the tail, a special

"base of tail" numerical integration is done for TS and W S using step size

S = Tr/P (as discussed on the previous page). This is done over a region from the

start of the "tail" until a is large enough for the above asymptotic approximation

to yield an accurate result. The above asymptotic approximation gives

J(V)JV(1) [ - cos(20+T) + 11 -2 even

1 [cos(2-+T) - 1] U:- odd7TO1 2

where the case of (V-v) odd will never occur due to the form of the integrands.

The integrands therefore consist of a constant term and an oscillatory term. The

contribution, of the constant term to the integral, using the above approximations,

is

f 2da s = arc sin~ys 2'AsA

~~A~ 1r3 /7 ry 5 A 2Try2SciA2

When aA is much greater than y os, this expression will not provide an accurati

solution since it will be the difference of two very similar values. For

R = os /A >> 1, the result is

1 ~ [R + I + - + R • '+2 R R 3 - R5 +"""27y3 6 40 2r

31 [ + R +. •

117

The simplified form

3 3

will therefore be accurate to about 2n decimal places if R is 10 -n. Likewise, the

exact solution will lose 2n decimal places in the subtraction of the two parts when

R . 10-n . Hence, for a computer with about 16 decimal place accuracy, the simpli-

fied form is used whenever R < 10- 4.

The analytic integration of part of the integrands for TS and WS shown in the

preceding paragraph depends on a large enough aA for sufficiently accurate approx-

imation of the Bessel functions. Hence, the simple numerical integration shown

must be utilized to fill in the gap if the "central region" ends before such a

sufficiently large aA is reached.

The oscillatory components of the integrands for TS and WS in the region of

analytic integration of the asymptotic approximations have been ignored in this

formulation, as their contribution is assumed to be negligible.

The integrals for T12S and W12S were not carried past the end of the "central

region" due to their highly oscillatory integrands, which made the contribution

from further computation both uneconomical, and not critical to the final values of

T and W This is, however, not as good an approximation as those previously12S 12S*

noted in the above development. As a result, errors of as much as 2% can be

expected in the computation of wave drag due to interference between the two struts.

118

APPENDIX C

RELATIONSHIPS BETWEEN CHEBYCHEV SERIES AND SWATH

HULLFORM COEFFICIENTS

119

APPENDIX C - RELATIONSHIPS BETWEEN CHEBYCHEV SERIES AND SWATH

HULLFORM COEFFICIENTS

CHEBYCHEV SERIES

In the wave resistance integrals, geometry of strut and body are represented

by a special form of Chebychev series. In SWATHO, the Chebychev coefficients are

.pproximated from various statistical moments of strut and body. The Chebychev

series representation of hull and body geometries are discussed below. Let

x = sin 0,--1- <0 1

Define the fundamental functions of the Chebychev series as follows:

Um (x) = cos(2m-1)0

= cos[(2m-l)sin-tx]

V (x) = sin 2m0

= sinj2m sin-x], m = 1, 2, . , m.

The strut half-thickness and the body sectional-area functions can be repre-

sented by finite sums of the fundamental Chebychev series as follows:

M

t(x) =E [AsmUm(x) + BsmVM()]m-1

M

=E [ cos(2m-l)0 + B smsin 2m0]m=l

and

M

A(x) =E (AbmU(x) + BbmVm (x]m=l

M

" [bmcoS(2ml0 + Bbmsin 2m0]m-1

120

Through use of the orthogonality of the series, the following inversion

formulas are obtained:

2 fl t(x)U (x)A -= J dx mASm i -I "l-x2

X2/2

= f 7-/2 det(sin 0) cos[(2m-l)81,

2 t(x)V (x)B f dx mSm -1 /lx2

2J 1/2

= f--/2 d~t(sinO) sin(2m0),

AI2 f/2

Bm =7T d6 A(sin 0) cos(2m-1)6,

and

2 f /2

BBm =2 7 dO A(sin0) sin 2m0.

-n/2

INVERSE PROCEDURE FOR DETERMINING THE CHEBYCHEV COEFFICIENTS

The geometric coefficients of a SWATH ship's form can be shown to be closely

related to the Chebychev coefficients of the strut and body.

These coefficients of form are easily obtained at the preliminary design stageand, thus, can be used to approximate the Chebychev coefficients. In the following

example, the waterplane coefficient, Cwp,can be used to find AS1 .

By definition, the waterplane area is

Ls/2

AW 2 / dx t(x)

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where t(x) is the thickness function of the strut. Let

L L

., dx = - - = x, t(x) tmax t(C).

Thus,

L-I

2w + max fl ~t~

1-L I' M.±S j ,4[ A U (C) + B V (E)

2 tmax '~[ dsm m sm m J

m As U (E)dE + B V (E)dC•2 tmax As Bsm

Making the substitution,

x = sine, dx = cos dO,

we obtain:

AW = 2 - tmax Asmf de cos(2m-l)Ocos 01

M /2

+7Bsm f dO sin 2m0 cos 81 -'r/2coj

The second integral is odd and, therefore, it is identically zero. Hence

L t M 7T/2= 2 ma Asm f dO(cos(2m-l) 6 cos 0

L t M r/2=s2max . A sm f-Tr/2 dO[cos(2m-2)0 + cos 2m0]

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L t M T/

max Asm sin(2m-2)0 + - sin 2m1

2 2m-2 2m 1

Lt M [s t max ,A' sin(m-l) + sin mir

2 m 2(m-l) 2m

These integrals are identically zero except for m=l where the first term contri-

butes Tr/2,

Lts max U

AW= 2 2ASi.

The waterplane coefficient is defined as

C AwCWp =L t

s max

Therefore

A - 4CWPS1 7

This derivation shows how the waterplane coefficient is related to the Chebychev

coefficient, A S. Similarly we can prove,

4CpAB1 U

where C is the body prismatic coefficient.

In the SWATHO program the maximum number of terms of the Chebychev series is

three. The Chebychev coefficients are determined by the following formulas for the

strut:

4CWP

AS2 ASI(I - 16 CIW),

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AS3 1 ASi S2'

BSi 4 LCF 'A S1,

B S2 =0,

B =0,

and the body coefficients are determined as follows:

A 4-C

A =1 -AB2 BI

AB3 0

VB =4C *ABi LCF li'

BRB2 =0,

B B3 0,

where

=W Waterplane area coefficientIW = ArW/(LS T s), where AW is waterplane area of one strut.

CF aterplane moment coefficientL F C L F = M x ( W -LS) *

C 1W =Waterplane inertia coefficient

C = IWY/(AW * L,)

CP = Body prismatic coefficient

CP=VB /(x L B) where V B is displaced volume of one body.

C = Body moment coefficientLCB*

All moments and the moment of inertia are taken about the mid-length of therespective strut or body.

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