Computer aided input/output for use with the finite element method of structural analysis

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12 DEC. 1972

NAVAL SHIP RESEARCH AND DEVELOPMENT

iotheek van

C)nderafdelin

sbouviltunde

mc e Hogeschoo

DOCUMENTATIE

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DATUM:

IDOCUMENTATIEl oactVr.

Washington D. C. 20034

Lab. v. Scheepsbouwkunde

COMPUTER AIDED INPUT/OUTPUT FOR USE

WITH THE FINITE ELEMENT METHOD OF

STRUCTURAL ANALYSIS

by

Robert D. Rockwell

and

Daniel S. Pincus

Technische liogesc,h4:0

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-,e4.zrz

This document has been approved for

public release and sale; its distri-bution is unlimited.

DEPARTMENT OF STRUCTURAL MECHANICS

RESEARCH. AND DEVELOPMENT REPORT

co...p..4

The Naval Ship Research and Development

_{Center is a U.S. Navy center for laboratory effort}

directed at achieving improved sea and air vehicles.

_{It was formed in March 1967 by merging the}

David Taylor Model Basin at Carderock, Maryland and the Marine Engineering Laboratory (now

Naval Ship R _{D Laboratory) at Annapolis, Maryland. The Mine Defense Laboratory (now Naval}

Ship R 8s D Laboratory) Panama City, Florida became part of the Center in November 1967.

Naval Ship Research and Development Center

Washington, D. C. 20034 *REPORT ORIGINATOR SitIP CONCEPT RESEARCH OFFICE OHM DEPARTMENT OF ELECTRICAL ENGINEERING A600 DEPARTMENT OF MACH DE NY TECHNOLOGY A700 DEPARTMENT OF MATERIALS TECHNOLOGY A800 DEPARTMENT OF APPLIED SCIENCE A900 SYSTEMS DEVELOPMENT OFFICE 01101 NSRDL ANNAPOUS COMANDING OFFICER TECHNICAL. DIRECTOR

MAJOR NSRDC ORGANIZATIONAL COMPONENTS

DE VE LOME N T PROJECT OFFICES OHM, SO, 50. 90 NSRDC CARDEROCK COMIANDER TECHNICAL DIRECTOR DEPARTMENT OF ACOUSTICS AND VIBRATION

901:1 NSRDL PANAMA CITY COMMANDING OFFICER TECHNICAL DIRECTOR

d

DEPARTMENT OF COUNTERMEASURES AIRBORNE MINE P730 H,DEPARTMENTOF INSHORE

WARFARE AND TORPEDO DEFENSE P740 NDW-NSRDC 3960/43 (3.70 DEPARTMENT or HYDROMECHANICS SOO DEPARTMENT OF AERODYNAMICS SOO IDEPARTMENTOF OCEAN TECHNOLOGY P710 * DEPARTIENT OF STRUCTURAL MECHANICS 700 DEPARTMENT OF APPLIED MATHEMATICS 500 IDEPARTMENTOF MINE COUNTERMEASURES P720

DEPARTMENT OF THE NAVY

NAVAL SHIP

_RE'SEAR.6

_{AND DEVELOPMENT CENTER}

WASHINGTON, D. C. 20034

COMPUTER AIDED INPUT/OUTPUT FOR USE

WITH THE FINITE ELEMENT METHOD OF

STRUCTURAL ANALYSIS

by

Robert D. Rockwell

and

Daniel S. Pincus

This document has been approved for

public release and sale; its distri-bution is unlimited.

TABLE OF CONTENTS

Page

ABSTRACT

1

ADMINISTRATIVE INFORMATION

1

INTRODUCTION

1

COMPUTER PROGRAM IDLZ

-3

STRUCTURE SUBDIVISIONS

- 4.

NODAL NUMBERS

5

ELEMENTS

NODE LOCATIONS

7

IDLZ OUTPUT

7

COMPUTER PROGRAM OSPL

8

RESULTS AND DISCUSSION

10

'ACKNOWLEDGMENTS

REFERENCES

105.

APPENDIX A - USER'S MANUAL FOR

COMPUTER PROGRAMS

1DLZ AND OSPL

37

.APPENDIX B - PREPARATION OF

INPUT DATA. FOR IDLZ

43

APPENDIX C - PREPARATION OF INPUT DATA

FOROSPL

49

APPENDIX D - AUTOMATED

DETERMINATION OF CONTOUR

SPACING IN COMPUTER PROGRAM

OSPL

. 52

APPENDIX E- FLOW DIAGRAM AND

SOURCE LISTING OF

COMPUTER PROGRAM IDLZ

53

APPENDIX F - FLOW DIAGRAM AND SOURCE LISTING OF

Figure

Figure

Figure

Figure

Figure

6

-Figure

7

-Figure

8 - Idealization of

LIST OF FIGURES

Figure

1 - Idealization of Internally Reinforced

Glass Joint

2 - Rectangular Subdivision

...

3 - Trapezoidal Subdivisions

_{- Initial}

Representation

4 - Trapezoidal Subdivisions

_{- Initial}

Representation

5 - Trapezoidal Subdivision

. .

Idealization of Glass Viewport Juncture

17

Idealization of DSSV Viewport

₁₈

DSSV Viewport and

iii

Page

12

12 - Typical Output Values from

_{Analysis and}

Resulting Plot from Program OSPL

25

13 - Plot from Program

_OSPL

_{of Effective}

Stresses in DSSV Bottom Hatch

₂₆

14 - Plot from Program

_OSPL

_{of the Temperature}

Distribution in a T-Beam Exposed to

a

Thermal Pulse

₂₇

Figure 15 - Results of Use of Programs

_{IDLZ and OSPL}

for a Stiffened Orthotropic Cylinder

_and

Titanium End Closure

₂₈

Figure 16 - Results of Use of Programs

_{IDLZ and OSPL}

for an Unstiffened Orthotropic Cylinder

and Titanium End Closure

₃₀

Transition Ring

_{. . 19}

9 - Idealization of DSRV Hatch

₂₀

10 - Idealization of Typical Shape

₂₂

11 - Optional Plots Available from

_Computer

Program IDLZ

₂₃

Figure

Figure

Figure

Figure

Figure

Figure

Figure 17 - Results of Use

of Programs IDLZ and

OSPL for an Internally Reinforced

Glass Joint

. . . .

. ... . ...

Figure 18 - Results of Use of Program

IDLZ and

OSPL

for theHemispherical hatch

of a Glass Sphere

. . . .

.

LIST OF TABLES

Table 1 - Numerical Restrictions

in the Use of

Program OSPL

38

Table 2 - Numerical Restrictions

in the Use of

Program IDLZ

40

iv

32

ABSTRACT

The enormous computational ability of modern

computers has encouraged development of the finite

element method of structural analysis.

However,

preparing the large amount of input data and

in-terpreting the large amount of output data

gener-ated by the analysis can be very time consuming

and costly.

Programs IDLZ and

OSPL

were developed

as aids to computer input/output.

IDLZ divides a

plane surface into triangular elements and

gener-ates required input data for the analysis program.

OSPL plots the output data in a form which can be

quickly interpreted by the analyst.

ADMINISTRATIVE INFORMATION

Development of the programs reported herein was

author-ized and funded under the In-House independent Exploratory

Development Program, Task Area ZR 011 0101.

Documentation

of

the programs was funded under Task Area SF 35.422.305,

Task 14665.

INTRODUCTION

Users of today's third generation computers are hard

pressed to exploit their full potential.

That this is true

is indicated by the proliferation of devices for aiding the

user, particularly in the areas of data input and output.

The high computational speeds and large storage capacities

of the latest machines have led to a push to solve complex

problems requiring considerable amounts of data.

However,

just as the use of huge passenger airplanes is limited by

input/output facilities (terminals), so is the use of modern

computers limited for problems requiring a sizeable amount

of data.

For such problems the user must spend much

valua-ble time preparing and checking input data as well as

inter-preting output data.

For the structural engineer, this "data problem" can

be significant.

For example, the finite element method of

structural analysis, which is fast emerging as a most

power-ful tool, requires much data.

In this method of analysis,

a structure is divided into a large number of small element,

each having a very simple geometry.

By determining the

re-lationship between force and deformation for a typical

ele-ment and then properly connecting the eleele-ments together, a

solution for the entire structure can be obtained.

A

problem of moderate size requiring 500 elements would need

almost 2000 input data values and produce nearly 2000 output

data values.

Obviously, a sizeable amount of data must be

handled.

In order to alleviate this "data problem" for finite

element computer programs, the approach in the Department of

Structural Mechanics of the Naval'Ship Research and

Develop-ment Center has been to automate, where possible, both the

preparation of input data and the interpretation of output

data.

One program developed for this purpose, called IpLz,

divides a two-dimensional surface of any shape into

trian-gular elements and then generates necessary geometric and

bookkeeping data for finite element analyses.

Another

production program called OSPL reduces the output data to

plots of stresses, temperatures, etc., which speed

consider-ably the interpretation of such data.

This report describes two computer aids to input/output.

The logic of the programs is explained and results are

pres-ented which indicate the range of applicability and the

potential of these aids.

Information for users of the two

programs is included in Appendices A, B, and C.

Source

listings and overall flow diagrams are also included.

COMPUTER PROGRAM IDLZ

A prime step in the use of the finite element method of

structural analysis is that of "idealization," in which the

structure is divided into a large number of small elements

and the necessary data defining these elements is generated.

The analyst decides upon the size and location of elements

in this step, with the knowledge that very, small elements in

a critical area of the structure will result in more accurate

infotmatiOn for that area from the analysis program.

Data

defining the elements generally include the size and location

of each element, specified by coordinates at each corner or

node of the element, and bookkeeping information, comprised

of numbers indicating which nodes belong to each element.

For the axisymmetric analysis in Reference 1, for example,

idealizing a structure by hand can take as much as three

to four mandays of effort.

Therefore, the computer program

IDLZ was developed to automate the idealization process.

The user of IDLZ must establish the number of nodes

along the external boundary of the surface to be idealized

and locate that boundary.

First, the analyst represents the

surface by an assemblage of rectangular and trapezoidal

subdivisions (see Figure la).

The number of nodes along the

boundary can then be established by integer coordinates given

as input data at opposite corners of each subdivision.

The

actual location of the boundary (see Figure lb) is defined

by rectangular coordinates specified for each boundary node.

These are input collectively for boundary no,des forming a

straight line or a circular arc.

In general, the amount of

input data required for IDLZ is less than five percent of the

data produced by IDLZ for the finite element analysis.

STRUCTURE SUBDIVISIONS

Representing the surface to be idealized by an

assem-blage of rectangles and trapezoids is a most important step

in the use of TDLZ.

It is here that the analyst specifies

the number of elements to be placed within a certain area

and thereby fixes their size.

In Figure 1, for example, the

critical area of the structure requiring many elements is

near the joint at the third and fourth rows from the bottom

of the figure.

By combining rectangular subdivisions

(see Figure 2) with isosceles-trapezoidal subdivisions

oriented in different ways (see Figures 3 to 5), the user

is able to crowd many elements into areas requiring close

scrutiny.

The trapezoidal subdivisions in Figures 4 and 5

are especially suited for that purpose as indicated in

Figure 6.

Special care has been taken in IDLZ to simplify the

idealization of unusual shapes.

The trapezoidal

sub-division, for example, can effectively be made three-sided

by the user through a judicious choice of integer

coordi-nates establishing the number of nodes on its boundary.

If

the short side of the trapezoidal subdivision has only one

node, the subdivision is a triangle.

Several such

sub-divisions were used in the idealizations shown in Figures 7

and 8.

NODAL NUMBERS

Given the assemblage of subdivisions representing the

surface., IDLZ assigns numbers to the nodes and creates

elements.

Points in the grid of integer coordinates across

the surface of the assemblage

represent nodal points.

These

are first numbered

arbitrarily from left to right

and bottom

to top with programming

convenience being the prime

considera-tion.

Since the size of the coefficient matrix bandwidth,

which is obtained subsequently

in the finite element

analysis,

is directly related to the numbering scheme used

here, a more

than arbitrary scheme is

usually necessary.

Therefore, if

the user desires, the

numbering scheme of Reference

2 is

applied to ensure a narrow bandwidth.

ELEMENTS

Elements are created

by grouping three adjacent

nodes

together.

The first elements,

like the initial node

'numbers,

are the result of a Convenient arbitrary

procedure.

This

procedure often produces

elements having shapes

quite

different from the most

desirable equilateral

shape.

Several

of the elements in Figure

9b, for example., have needle-like

corners.

For this reason,

the elements are reformed by IDLZ,

where necessary,

following the "shaping" process in which

each node is given its

rectangular coordinates.

Figures 9b

and 10a are examples

showing poor elements

which were

reformed to Figures 9c and 10b respectively.

NODE LOCATIONS

After the nodes are numbered and elements formed,

"shaping" takes place.

The user must first.establish for

IDLZ the location of each boundary node on two opposite

sides.

This tedious-sounding task is actually quite simple..

Adjacent boundary modes forming a straight line

or circular

arc need only have the coordinates of the two end nodes

specified, along with the radius, if any.

Hence, the

amount of data required to "shape" a surface is relatively

small.

For example, the complex shape shown in Figure 9,

which contains 100 boundary nodes, needed coordinates of

only 24 nodes and the radii of eleven circular

arcs in

order to have its boundary completely established.

Each subdivision is shaped separately.

_{The user}

specifies the location of nodes on any two opposite sides

of the subdivision and IDLZ locates the rest of the nodes

through linear interpolation.

It should be noted that

with this procedure, two opposite-sides in every

sub-division will be straight lines.

IDLZ OUTPUT

At the option of the user, output from IDLZ can include

besides a printed listing, plots of the idealization and

punched data cards describing it.

Optional plots produced

with the Stromberg-Datagraphic 4020 Plotter include

X-Y plots of the surface with the elements shown, before

and after shaping, and plots of each subdivision after

shaping with the node numbers labeled (see Figure 11).

Data cards with the required geometric and bookkeeping data

suitable for input to the finite element analysis program

are produced in the form specified by the user.

For problems of moderate size, IDLZ requires less than

five minutes of IBM 7090 computer time to idealize the

structure and generate the output.

Since less than one

hour of the user's time is needed to set up a problem

for

IDLZ, including subdividing the structure and preparing

data, significant savings can be realized through its use.

COMPUTER PROGRAM OSPL

Output from a finite element analysis

genetally includes,

at

every node, one or more (depending on

the complexity of

the analysis) values

of

stress, strain, etc.

Since a problem

with 500 or more nodes 'is not unusual, delays

interpreting.

such data are to be expected When they are in the

form of

printed output..

For this reason, computer program OSPL

Was developed to reduce such output

data to plots of lines

called

tsogreme,

along each

of

which there is

e

constant

value (as

of stress).

Such plots resemble contour maps on

which the physical features of the

earth's surface are

indicated through use of cOntOur lines connecting- all

points

of the same elevation.

These "iso-plots" produced by OSPL

on the Stromberg-Datagraphix 4020 Plotter present a means

by which interpretation of such data is much more rapid.

The conceptually difficult problem of developing the

"iso-plots" may be simplified by considering only a small

surface area at one time.

For example, triangle ABC of

Figure 12a is to be a small part of the surface to be

plotted.

It is relatively simple to draw lines of equal

value through this triangle.

Assuming an interval of 10

between lines, and beginning with 10, it is seen that

lines of value 10, 20, and 30 pass through ABC.

Linear

interpolation results in the plot shown in Figure 12b.

This method is essentially the one used in OSPL.

The

size of the contour interval and the value of the lowest

contour are initially set by the user or by considerations

for proper spacing of lines between the smallest and the

largest value to be plotted (see Appendix D).

Then, taking

one element at a time, the steps below are repeated

until

the plot is complete.

The number and size of the contours passing through

the element are determined.

For each of these, steps 2-4

are completed.

Two pairs of adjacent corners are found, each of whose

values bound the subject contour.

End. paints of the subject contour in the element are

found by Interpolating linearly between the values at the

adjacent corners of each pair.

A straight line is drawn between these end points.

The value of each contour is printed next to its

intersection with the boundary of the

plot unless adjacent

labels overlap.

All-contours of zero value are labeled

.

Since adjacent contours are either one

interval apart or of equal value, these labels sufficiently

specify the value at any point inside the boundary.

In Figures 13 and 14, typical examples of stress

contour plots produced by

OSPL

are shown.

Figure 13 is a

plot of effective stresses on the cross section of a

hatch-shell intersection analyzed by the method of

Reference 1.

In Figure 14, the isograms represent constant

temperatures in one-half of a Tee-frame which were

deter-mined with the analysis of Reference 3.

In both examples,

the amount of data'represented by the plots is significant.

However, because of the manner in which the data is

presented, it may be interpreted very quickly.

RESULTS. AND DISCUSSION

Shown in Figures 15 to 18 are results of problems for

which program

IDLZ

has been used to idealize the structure

and then program OSPL used to plot results from the finite

element analysis of Reference 1.

In each case these input/

output aids have reduced significantly the time required to

prepare input data and interpret output

data.

The idealizations shown in Figures 1,

6 to 9, and

15 to 18 indicate that a variety of shapes are easily

handled by IDLZ.

For problems in which many elements are

desired at the joints between different materials

(Figures 1 and 6), trapezoidal subdivisions are available.

For problems in which the shape to be idealized is unusual

(Figures 7 and 8), triangular subdivisions may be used.

For more simple shapes (Figures 9, 15, 16, and 18),

rectangular subdivisions are aatisfactory.

In all cases

the output data is easily plotted by OSPL.

It should be

noted that while only axisymmetric problems have been

shown here, IDLZ and OSPL work equally as well with any

plane stress or plane strain analysis program.

ACKNOWLEDGMENTS

The authors wish to acknowledge the assistance of

Messrs. L. N. Gifford, Jr., H. P. Gray, F. Koehler,

R. P. Lerner, K. Nishida, and T. N. Tinley during the

development of this work.

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Figure la - Initial Representation of

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5555

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INTERNALLY REINFORCED GLASS JOINT

Figure lb - Final Idealization of

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Figure 1 - Idealization of Internally

Reinforced Glass Joint

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Figure 8b - Final Idealization of

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Figure 8 - Idealization of DSSV Viewport and

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by User

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Figure 9c - Final Idealization of Surface

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Figure 11 - Optional Plots Available in Computer

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11/

CIRCULAR RING IDEALIZED WITH TRIANGULAR Su3MINS

Figure ha - Plot of Initial Representation by User

USA

U

CIRCULAR RING IDEALIZED WITH

TRIANGULAR SUBDVNS

Figure llb - Plot of Final Idealization by IDLZ

CIRCULAR RING IDEALIZED WITH TRIANGULAR SUBONS

MOO* I

I OA

Figure llc - Plots of Each

Subdivision with

Nodes Numbered

Figure 12a - Typical Output Values

Figure 12b - Typical Plot

Figure 12 - Typical Output Values from Analysis and

Resulting Plot from Program OSPL

DSSV BOTTOM HATCH

1001FIE0 FOR

CONTACT.

SECOND IDEALIZATION

CONTOUR INTERVAL IS 251311. '2250G. 175:15. .27sco 4225w. +37$

\\\

o3

\

_\

exam1--N

\

+1505O. +3 ,...--...,..., -, ,

_.

: ,

_---"N

....-

____

-- ....---

--atT j.

--4

+2U"" ---.

---412500.

---''' .... -..-"-. +175

,

.... I '1134./ / +15 ..,' 15007.

7...--ZP24

CONTOUR PLOT

*

EFFECTIVE STRESS

*

INCREMENT NUMBER

I

Figure 13 - Plot from Program OSPL of Effective Stresses

TINTONTITRE DISTSIDUTION IN 1-SEAN EXPOSED 10* TNERNAL RADIATION PULSE OUTER DIN. 11.5 IN. VINE I G.S ICS DI

IN. DP. - WOO ST NUM

CCAITOUR INTERVAL IS I.

*5011.

TENKTIATURE DISTRIBUTION IN 1-SCAN EXPOSED TO A hiLANAL RACIAIION PULSE CUTER DIN. 6.1 IN. MIDI I 6.0 TM II IN. DP. - IWO ST KANN

coseroult imravAL It 00.

30.

.40. .S0. 00. 111. so. 710

.50. los. I 111.

4111:.

*40. 50. *PO. .10. sta. teu

-.50. *SO. 1/817=z-zz-T,G, Po, SO. ESt. *EP.

\

\s. \ N.

\\\

_\

ttootto. Iso. tiltl. *050 ..040. .510. . Figure 148

- Time Equals Two Seconds

Figure 14b - Time Equals Three

Seconds

Figure 14 - Plot from Program OSPL of

the Temperature

Distribution in a T-Beam Exposed to

Figure 15 - Results of Use of Program IDLZ and OSPL for a

Stiffened Orthotropic Cylinder and Titanium End Closure

STRUCTURAL IDEALIZATION

REDESIGN STIFFENED OF OCT 1969 WITH FULL H EMISPHERE AND DELTA ZERO

Figure 15a - Initial Representation

by User

1111.1

STRUCTURAL IDEALIZATION

REOESIGN STIFFENED OF OCT 1969 WITH FULL H EMISPHERE AND DELTA ZERO

Figure 15b - Final Idealization of

GRP RING F.TIFFINED ElLINDER-ANIFENO

CLOSURE-GEOMETRY TIMER ONE

1

CONN IIMPAL IS CP

oNTIJOR FL,OT *iIJkIJWERENTIAL

* INCREMENT, NUMEERIlim I j

Figure 15c - Plot of Circumferential Stresses

by Program SPL

z.

2

5

GRP RING -STIFTENED

ETITIMIEVANDIND-

nosuRr-GEOMETRY NUTTER ONE

cr.N,IulIaNLIl 11.11 I IND _ lk. r LIT * 4EAR INCREMENT t.JtERIIlIllI

Figure 15d - Plot of Shear Stresses by

Figure 16 - Results of Use

of Programs LUZ and OSPLJor an

Unstiffened Orthotropic Cylinder and Titanium

End Closure 44444444444 &&&&&&&& &FA 4444 44 &&&&&&&&&& STRUCTURAL IDEALIZATION

11 69 RE-DESIGN FOR UNSTiFF CYL Figure 16a - Initial Representation

by User

SS I

SO:

ViMr4

TM

11014 iiu iii rAMI4 FAW WWI ;1111101

STRUCTURAL IDEALIZATION

11 69 RE-DESIGN FOR UNSTIFF tYL _{Figure 16b - Final Idealization of}

Surface by Program IDLZ

44 AAJAAA

...

?4 44

4 44444

44444

FAFAI

r

ipp-UNITITITIEW mien-ig ENEITLOWIE Clo 1111M11. IS CIO

L CONTCUR PLOT *_ EFFECTIVE STRESS *. INCREMENT MUMBER0001

Figure 16c - Plot

of

Effective Stresses

by

Program $SPL

r

wuterirmiu

MINDEN AND END ROME SEINETNT MEN TIO COMM 111111M U 0.111

31

KONETNY MOT MO -I

L CONTOUR FtOT * CIRCUMFERENTIAL STRESS * INCREMENT NUMBER0001_1

Figure 16d - Plot of Circumferential

Figure 17 - Results of Use of Programs IDLZ and $SPL

for an Internally Reinforced Glass Joint

3031 SO 30 ZIA 0.133 32 &&&120

&&&&&

&&&&i&

&&&&0

WW2&

WAFAM

&&i&M*

WA&&& rAW2Cd& &&&Cd& A&&&&12L

A&&&&&&&

A&&&&&&02&12L A&M&M&&&IM1212&L ArAMM&&&*&&&&&&L

AOOMOMMOM5IMMIL

AW2M5MMWMTAM&&M&&&L AMIOCMOMM&&&&&&102&12MPik

SO AM&00&0&0000&0&&&&WWW. &&

0& MOMOMMO&O&M&00000&&&

rdrA 0,2&&0&&&&&&&0&&&&&&M&1212 FAO

&& WriiMO&OW2MOWFAMOWSOWOA && 0012112MIM&&&&0&0&&%0&&&M&FAMMIM&&&

0120&&1212M510&00&!&&&12,212&&&*&&&

_{loommoomoomoomomm0000r}

Nomoomoomoommoomocom,

_{loommovocomoomoomoor}

INTERNALLY REINFORCED GLASS JOINT

A"

Figure 17a - Initial Representation

by User

IOU

h

Figure 17b - Final Idealization of

COMM tO,11110101.11111/1101111111. 0111C1101I

111.11111111NLLT

It111101001 SUN 101In

Wine !UMW IS 0.10

Figure 17c - Plot of Meridional Stresses

by Program $SPL

COMM MUM IN MI& 01littION I

SI 11011111filLt IttINFONNIN

PAN .001?

corm

II1D0S

IS 0.10

Figure 17d - Plot of Radial Stresses

Figure 18a - Initial Representation

by User

Figure 18 - Results of Use of Programs IDLZ and OSPL for the

Hemispherical Hatch of a Glass Sphere

P2 1/4

WA WA WA

WA Win WA WA 02 02 WA WA WA WA WA WA

WA

WA WA

WA

mme.

WrIl

ZdKIWArAVIIM

Ir4r41150

rArigiWirAIFAIA

WA,

&Inn

WA

'Ann

FAMMar.oriramorAM

MOSOMM4.2*

2/0202/02/4011MA SCII/M4/4101goseirir. U12,41414IMAIMAK unirecommoraoriteirmis WerArerArer4r.9.../.02/402/4/4

WeitemoromannoriteinmormeArMace,02

GOMMAKKOMMArIr4r04/4/4/4/4/002/MAIAra

irAr412,4002/4/41WOMMMArOMMMA

orimarMdrAI.94/4/4/405/41WMACOMAIMA

ormogrAWOMMour

WWWWWWWWWWWWWAgI

Ir/ArIMIAW

WO/

_STRUCIURAL IDEALIZATION'

NEW 'HATCH

STRUCTURAL IDEALIZATION

NEW HATCH

Figure 18b - Final. Idealization. of

BUDT'S NENINATCH 1/13/70 LERNER CODE 721

Maga MM. II ao.

ZP26

CONTOUR PLOT CIRCLPFERENTIAL STRESS ItiCREMENT NuMBER 100 Figure 18c - Plot of Effective Stresses

by Program SPL

DUEIT'S NEN1NATCH 1/13/70 LLRNER CODE 721

conal zutavat ices.

ZP26 CONTOUR PLOT

Figure 18d

EFFECTIVE STRESS INCRDINT

WIER

100=

- Plot of Circumferential Stresses

by Program $SPL

BLANK

APPENDIX A

USER'S MANUAL FOR COMPUTER PROGRAMS IDLZ AND

OSPL

The computer programs

OSPL

and IDLZ were developed to

ease the data preparation and interpretation problems

asso-ciated with finite element analyses.

Of course, correct

results will be obtained only when these input/output aids

are properly used.

For this -reason, a detailed and perhaps

simple-minded approach to their use is presented.

USE OF COMPUTER PROGRAM

OSP1

When

OSPL

is attached to the analysis, as it is to the

analyses of References 1 and 3, its use is generally a

matter of specifying the proper option.

To attach OSPL to

an analysis, one must set up internally the data values

which would otherwise be read by OSPL as input data.

The

statement "CALL CONPLT (

)," which appears in the main

routine of

OSPL,

must be placed in the analysis program at

a point where these data values are available.

The main

routine of

OSPL

is listed on the first page of the

OSPL

Source Listing in Appendix F.

All the subroutines and

functions of OSPL must be attached to the analysis program

except this main routine.

Of course,

OSPL

can also be used

by attaching the proper data described in Appendix C to the

program as listed in Appendix F.

Note the limitations on

the size of various arrays in Table 1.

TABLE 1

Numerical Restrictions in the Use of Program OSPL

38

Total number of elements allowed

1000

Total 'number of points

data may be given

____

USE OF COMPUTER PROGRAM IDLZ

Experience has shown that use of .computer program IDLZ

requires some mental effort.

For best results, the

step-by-step procedure outlined below should be followed in setting

up a problem for IDLZ.

Become familiar with IDLZ as described in the body of

this report.

Determine, in a general way, which areas of the surface

being idealized will require close scrutiny.

Many elements

should be crowded into these critical areas.

Read the hints and the restrictions listed below and in

Table 2.

Represent the surface being idealized by an assemblage

of rectangular, trapezoidal, and triangular subdivisions.

See Hint Numbers 2 and 3 below.

Define the lower left and upper right corners of each

subdivision by integer coordinates.

Note the restrictions on

these coordinates in Table 2.

See Hint Number 1 below.

Count the number of nodes, number of elements, and

num-ber of subdivisions.

Note the limits for these in Table 2.

Locate every node on any two opposite sides of each

sub-division.

See Hint Numbers 4, 5, and 6 below.

TABLE 2

Numerical Restrictions in the Use of Program

IDLZ

40

Total number of subdivisions allowed

50

Total number of elements allowed

850

Total number of nodes allowed

500

Maximum horizontal integer coordinate

used to define a subdivision

40

Maximum vertical integer coordinate

used to define a subdivision

60

HELPFUL HINTS FOR THE USER OF PROGRAM IDLZ

Integer coordinates used to define subdivisions are

limited to 40 in the horizontal direction and

60 in the

vertical direction.

Therefore, turn the surface to be

idealized so that its longest dimension is in the

vertical

direction.

Better elements result and less input data is

required

when each subdivision retains close to its original appearance

after shaping.

Therefore, try to fit different shaped

sub-divisions into the surface being idealized just as one

would

fit the pieces of a jigsaw puzzle.

Use the trapezoidal subdivisions with slopes greater

than one for two purposes:

(a) to change quickly from many

nodes on one side of a subdivision to few nodes on theother

side; and (b) to fit as closely as possible (Hint Number

2)-a

surface whose cross section changes rapidly.

When necessary, the nodes on all four sides of a

par-ticular subdivision can be located by breaking it into

several

subdivisions each having only one element across its minimum

dimension.

-Use as many line segments as needed to locate nodes on

two opposite sides of a subdivision.

If several different

spacings of nodes are required along one side of a subdivision,

break that side into several line segments, each having a

different node spacing.

6.

It is possible to shape simple subdivisions with only

one line Segment.

In such cases, the nodes on one side of the

subdivision are located as part of another subdivision which

has already been shaped.

GENERAL RESTRICTIONS IN THE USE OF PROGRAM IDLZ

The two parallel sides of trapezoidal and triangular

subdivisions (a triangular subdivision is an isosceles

trape-zoid with its short parallel side reduced to a point) must

either be horiZontal or vertical.

When circular arcs are used for shaping, the angle

subtended by the arc must be less than or equal to 90 degrees.

At least one line segment must be uSed to deform each

subdivision.'

aee Hint Number 6 above.

For purposes of shaping, the triangular subdivision,

which is a trapezoidal subdivision with its short parallel

'

side reduced to a point, is considered to have four sides.

If the two parallel sides (one side to a point) are

being

located by the user, the point is located as if it were a

line.

-APPENDIX B

PREPARATION OF INPUT DATA FOR IDLZ

The computer program IDLZ divides a plane

surface of any

shape into triangular finite elements and

produces the

necessary geometric and bookkeeping

data.

At the option of

the user, IDLZ will:

(1) Generate plots of the cross section

with elements shown; (2) Number the nodes in a manner

which

minimizes the bandwidth; and (3) Punch the

data on cards

suitable for input to a finite element program.

The user of IDLZ must establish the external boundary

the surface and define the number of nodes on

the boundary.

This is done by first representing the surface by an

assem-blage of rectangles and trapezoids

(subdivisions) whose

corners are defined by integer

Cartesian coordinates.

Then,

the assemblage is deformed into the desired shape.

The

integer coordinates define the number of nodes on

the boundary.

The desired shape is established by actual values of

coordi-nates for each boundary node which may be given collectively

for surface nodes forming a straight line or an arc of

constant curvature.

In general, this data required for IDLZ

represents only about five percent of the data which would

otherwise be necessary for the finite element program.

In all, there are seven different types of data

cards.

Each type is described and its occurrence and location

given

in the following outline.

Type 1:

NSET

FORMAT (15)

The first data card must specify the number of data

sets there are to follow.

Type 2:

(ARRAY(1), I = 1, 12)

FORMAT (12A6)

The second data card, which is the initial card of

a

set of data applying to one shape, allows the input

of alphanumeric data to be used as a title for that

set.

Type 3:

NOPLOT, NONUMB, NOPNCH, NSBDVN

FORMAT (415)

One data card of this type is required as the second

card in each data set.

NOPLOT:

Indicates whether plots are desired.

NOPLOT = 0, plots will not be produced.

NOPLOT = 1, plots will be produced.

NONUMB:

Indicates whether the user desires the

nodes to be numbered so as to ensure a

narrow bandwidth.

NONUMB = 0, nodes will not he renumbered..

NONUMB

='1,

nodes will be renumbered.

NOPNCH:

Indicates whether punched output is

desired.

NOPNCH = 0, punched cards will not be

produced.

NOPNCH = 1, punched cards will be produced.

NSBDVN:

Specifies the total number of

sub-divisions which represent the structure.

Type

4:

(I, KK1(I), LL1(I), KK2(I), LL2(I),

NTAFRW(I),

NTAPCM(I), N = 1, NSBDVN)

FORMAT

(515, 5X, 215)

One data card of this type is required for each

subdivision.

I:

Number of the subdivision.

KK1(I), LL1(1):

Integer X and Y coordinates of

the lower left dorner of the subdivision.

KK2(I), LL2(I):

Integer X and Y coordinates of

the upper right corner of the subdivision.

NTAPRW(I):

Indicator for isosceles trapezoid

with top and bottom sides horizontal and

parallel.

If NTAPRW(I) is positive, the

top side is longer than the bottom side.

If NTAPRW(I) is negative, the top side is

shorter than the bottom side.

The value

of NTAPRW(1) specifies one half of the

. .

change

in

the number of nodes from one

row to the next.

For example, if

NTAPRW(I) =

-2,

the bottom horizontal

side is longer than the top side and the

number of nodes decreases by

four from row

to row going from the bottom to the top

side.

NTAPCM(I):

Indicator for isosceles

trapezoid

with left and right sides

vertical and

parallel.

If

NTAPCM(I)

is positive, the

left side is shorter than

the right side.

If

NTAPCM(I)

is negative, the left

side

is longer than the right

side.

The value

of

NTAPCM(I) specifies one half of the

change in the number

of nodes fro t one

column to the next.

For example, if

NTAPCM(I) =--3,

the left vertical side is

longer than the right

side and the number

of nodes decreases by six

from column to

column going from left to

right.

Type 5:

I, NLINES

FORMAT (215)

One data card of this type

must precede the data

cards of type 6

for

each subdiVision.

I:

The number

of

the subdivision.

NTAINES.:.

The number of .straight

iineS or

cir-cular arcs which will

be used to establish

the shape of the boundary of the

subdivision.

Type 6:

Kl, Ll, K2, L2, Xl, Yl, X2, Y2, RADIUS

FORMAT (415, 5F8.4)

Data cards of this type, equal in number to the

value of NLINES on the preceding card of type 5, are

necessary for each subdivision.

Kl, Ll:

Integer X and Y coordinates specifying

end 1 of the line being deformed.

K2, L2:

Integer X and Y coordinates specifying

end 2 of the line being deformed.

Xl, Yl:

X and Y coordinates specifying actual

location of end 1 of this line on the

boundary of the subdivision.

X2, Y2:

X and Y coordinates specifying actual

location of end 2 of this line on the

boundary of the subdivision.

RADIUS:

The radius of curvature of the line.

If the line is to be straight, set

RADIUS = 0.

The center of curvature is

located such that moving from end 1 to

end 2 on the arc is a counterclockwise

motion.

Type 7:

(FMT1(I), I = 1, 12)

FORMAT (12A6)

Two cards of this type must be included.

On the

first of these two cards, the format to be used in

punching "nodal cards" is required.

Each "nodal

card," one for each node, will contain

X and Y

coor-dinates of the node, an integer

specifying whether

the node is on the boundary of the area to be

plotted and the integer node

number.

On the second

of these two cards, the format

to be used in

punch-ing

"element cards" is requited.

Each "element

card," one

f.

r

each element-, will Contain the node

numbers to be associated with that element and the

integer element number..

The FORMATs compatible

with the finite element analysis

program of

reference I are:

(2F9.5, 51X, 13,. 5X,

13) for the

nodal card and (315, 62X,

13) for the element card.

'

APPENDIX C.

PREPARATION OF INPUT DATA FOR

OSPL

The computer program

OSPL

develops two-dimensional plots

of lines called isograms, along each of which there is a

constant value (as of stress, strain, temperature,

etc.).

Such plots resemble contour maps on which the physical features

of a part of the earth's surface are indicated through use of

contour lines connecting all points of the same elevation.

In order to facilitate the conceptually difficult problem

of developing these "iso-plots," OSPL considers only a small

element of the total surface area to be plotted at one time.

Data for one element, which must be three cornered, include§

the value being plotted and the Cartesian coordinates at each

corner or node of the element.

Also requited for each node is

an integer which indicates whether the node is on

the boundary

of the area to be plotted.

Adjacent boundary nodes are

connected by straight lines by OSPL.

For each node, then, one

data card

is

required

on which

this data is presented.

The

order in which these "nodal" cards are received by the computer

is the order in which the nodes are given nodal numbers.

A means by which the proper nodal data can be associated

with each element is also necessary.

For each element, one

data card is required on which the node numbers to be

assotiated with that element are given.

Additional data cards supply alphanumeric data for plot

titles and define the total number of nodes and elements as

well as the extent of the plot and the interval between

isograms.

Since it may be desirable to "zoom-in" on a critical

area even though some nodes in the data set are outside that

area, the desired extent of the plot must be a part of the

input data.

In all, there are four different types of data cards

required for OSPL.

In the following outline, each type is

described and its occurence and location are given.

The total

number of nodes and elements is limited to 800 and 1000,

respectively.

Type 1:

NN, NE, XMX, XMN, YMX, YMN, DELTA

FORMAT (215, 5F10.4)

One data card of this type is required at the

beginning of the data set.

NN:

Total number of nodal cards In the data

set.

NE:

Total number of element cards in the data

set.

XMX:

Maximum X-coordinate to be plotted.

SMN:

Minimum x-coordinate to be plotted.

YMX:

Maximum Y-coordinate to be plotted.

YMN:

Minimum Y-coordinate to be plotted.

DELTA:

Specifies interval between adjacent

isograms.

If DELTA = 0, this interval

will be determined automatically.

Type 2:

(TITLE(I), I = 1, 12)

FORMAT (12A6)

The second and third cards of the data set must be

of this type.

They allow the input of alphanumeric

data which is used to title the plot.

Type 3:

(X(I), Y(I), S(I), N(I), I = 1, NN)

FORMAT (2F9.5, 22X, F10.3, Ii)

One data card of this type is required for every

node.

The order of these cards specifies the order

in which the nodes are numbered.

X(I):

The X-coordinate of the node.

Y(I):

The Y-coordinate of the node.

S(I):

The value at this node which is to be

plotted.

N(I):

Integer specifying boundary nodes.

N(I) = 0, node is not on the boundary.

N(I) = 1, node is on the boundary and is

in more than one element.

N(I) = 2, node is on the boundary and is

in one element only.

Type 4:

(N1(I), N2(I), N3(I), I= 1, NE)

FORMAT (315)

One data card of this type is required for every

element.

N1(I), N2(I), N3(I):

The three node numbers to

be associated with this element.

APPENDIX D

AUTOMATED

DETERMINATION OF

CONTOUR SPACING IN COMPUTER PROGRAM

OSPL

At the option of the user, the interval between adjacent

contours is either specified as input data or determined by

OSPL.

In this Appendix, the method used to determine the

interval in the program is described.

The size of the interval depends upon the largest and

smallest values to be plotted and upon the desired spacing of

lines.

After examination of many hand-drawn plots, it was

decided that in order to achieve good spacing, an interval

should be used which is about 5 percent of the difference

between the largest and smallest value.

Using base intervals

of 1.0, 2.5, and 5.0,

OSPL

chooses the interval which is the

Product of a base interval and a power of ten and which is

closest to, but not greater than, 5 percent of this difference.

The procedure results in intervals of 1.0, 2.5, 5.0, 10.0,

25.0, 50.0, etc.

For example, if the largest and smallest

values to be plotted are 50000 psi and 10000 psi, the

deter-mined interval would be 2500 psi.

APPENDIX E

FLOW DIAGRAM AND SOURCE LISTING OF COMPUTER PROGRAM IDLZ

FLOW DIAGRAM OF COMPUTER PROGRAM IDLZ

Start

Read

data

Assign

nodal

nuMbers

Create

elements

Plot

structure

:\/)

before

shaping

_

K

Shape the

structure

(Locate the

nodes)

54

(

Reform

elements

with

needle-like

corners

:* Renumber'

nodes to

ensure

bandwidth

narrow

Print

output

Punch

output

(

after

(lc

structure

Plot

shaping

(

End

)

* Optional

ALPHABETICAL INDEX OF ROUTINES IN COMPUTER PROGRAM IDLZ

Routine Name

Page Number in Listing

ANGMIN

83

BTTREL

. . . 82

CURVE

75

DFRMRG

80

DFRMTP

72

DLOOP1

77

DOLOOP

76

DRWSB1

92

1

ELMENT

. . ,

...

1 !

ELMNTO

₆₉

ELMNT1

.

...

69

ELMNTS

67

ENGTH

₇₉

FIRST1

₈₅

GETNMB

61

GETXY

79

GETXYS

₇₁

GPLOT

₉₀

GRID

91

MAIN (Main Program)

57

NFNC

₅₈

NLCFNC

68

NSRFCE

65

NUMBEL

₆₄

Routine Name

NUMBND

OUTPUT

Page Number in Listing

63

88

PLOTS

89

PRFRST

84

RDINLZ

a

60

REGLAR

70

RENUMB

87

REVRSE

59

SHAPE

. . . .

59

SLOPSD

. . . . 77

SUBPLT

. . .

..

. . .

90

70PBOT

XYDIST

XYFIND

-56

78

78

76

REVISED LISTING OF DECK MAIN

EFN

FORTRAN

STATEMENT

I.D.

SIBETC MAIN DECK MAIN 10

DIMENSION ARRAY(I2) MAIN 20 DIMENSION KKI(50),KK21504,IL14501,LL2(50),NTAPCM(50),NTAPRN(50) MAIN 30 DIMENSION NEW(506) MAIN 40

DIMENSION NODE1(850),NODE2(850),NODE3(850),NUMBER(41,61) MAIN 50 DIMENSION XORD(41,61),YORD(41,61) MAIN 60

DIMENSION RORD(560),ZORD(500) MAIN 70 DIMENSION NSRF(41,61) MAIN 80 DIMENSION NSURF(500) MAIN 90 DIMENSION IEL(856)00(500) - MAIN 100 EQUIVALENCE (NSRF(1,1)4IEL(1)/i(NSRF(1,31),N0(1)) MAIN 110 MAIN 120

READ (5,50) INUM MAIN 130

-J=0 MAIN 140

DO 40 I=1,INUM MAIN 150

C MAIN 160

C * *****************************************************************omAiN 176

C GO GET INPUT DATA MAIN 180

CMAIN 190

CALL RDINL2(ARRAY,KKIAK2AL1,LI2,NOPLOTiNONUMB,NOPNCH. ' MAIN 200

A N&BDVNOSRF4NSURF,NTADSMOTAPRWOUMBER,IOR0OORD) MAIN 210

C MAIN 220

C

4

***************************************************************.**MAIN 230 C THI S SECTION NUMBERS THE NODES AND FINDS THEIR COORDINATES MAIN 440

C THI S IS DONE SUBDIVISION BY wamylsIoN MAIN 250

C MAIN 260

CALL GETNMBIKK1,KK2,LL1.LL24NCOUNT,NELCNT.NODE14 MAIN 270 NODE2OODE3,NOPLOI.NSBOYNOSRF,NTAPCM4NTAPRW,NUMBER4R0RD,ZORDI MAIN 280

IF (NOpLOT.E04) GO TO 10 ' MAIN 290

C MAIN 300

C * ******************************************************************mAN 310 C PLOT: IDEALIZATION BEFORE SHAPING MAIN 320

C , MAIN 330

J=J+I MAIN 340

CALL PLOTS(ARRAY,J,I,KR1fKK2,LLI,LL2,NCOUNT,NELCNT,NEN,N00E1, MAIN 350

A NUDE20100E3,NONUMBOSBOVNOTAPCM,NTAORW,NOMBER,KORD,YORD,RORDI MAIN 360

B ZORD,IEL,ND) . MAIN 370

10 CONTINUE MAIN 380

O MAIN 390

C

4

******************************************************************NAIN 400

C SHAPE THE IDEALIZATION MAIN 410

C . MAIN 420

CALL GETKYS(KKI,KK2,LL1iLL2,NSBDVN,NTAPCM,NTAPRM,X0RD,IORD) MAIN 430

oo 20 K=1,4I MAIN 440 DO 20 L=1,61 MAIN 450 IF (NUMBER(K,L).E0.0) GO TO 20 MAIN 460 N=NUMBER(K,L) MAIN 470 RORD(N)=XORD(K4L) . MAIN 480 ZORD(N)=YORD(K.L) . MAIN 490 20 CONTINUE MAIN 500 CALL.BTIREL(NELCNT,NODE1000E2,NODE3AORD4ZORD) MAIN 510 IF (NONUMB.E0.0) GO TO 30 MAIN 520 57

REVISED LISTING OF DECK MAIN

EFN.'

FORTRAN

STATEMENT

_{1.0. k}

-c MAIN 530

C ******Mi*****************#4********104,44**********************MAIN 540 C RENUMBER THE NODES TO ENSURE A NARROW BANDWIDTH MAIN 550'

C _{MAIN 560}

CALL PRFRSTINCOUNTAELCNTOODEI.HODE2OODE3,NEW.NO/ MAIN 570 IF (NO.EQ.1) GO TO 40 7 ,. ', -. MAIN 580

CONTINUE - MAIN 590,

. . MAIN 600

******************************9*********************4**********HAIN 610

CHANGE FROM DOUBLE SUBSCRIPTS TO SINGLE SUBSCRIPTS MAIN 620' MAIN 630 CALL RENUMB(NELCNT,NEW,NODEI,NODE2,NODE3,NONUMBOSRF,NSURF4NUMBER.MAIN 640

RORD.XORD,YORD,ZORD) -MAIN 650

. MAIN 660

3*********************M4*****************************************M6/14 670 GO WRITE OUTPUT . - MAIN 680

MAIN 690 CALL OUTPUT(ARRAYIACOUNTIAELCNT.NEW,NODE-1,N0DE2:00DE3,NONURB, MAIN 700 NOPLOT.NOPNCH.NSURF.RORD.IORD) MAIN 710

IF INOPLOT.EQ.0) GO. TO 40 MAIN 720

MAIN 730

0111.*4**.*************************#'4***********te*******************416MAIN 740

PLOT IDEALIZATION AFTER SHAPING,- . MAIN 750

. . . MAIN 760

CALL PLOTS(ARRAT.J.1.KKI.KK2,LLIAL2,NCOUNT.NELCNT.NEW,NODE19 MAIN 170'

A _{NODE2,NODE3,NONUMBOSBIWN.NTAPCMOTAPRW,NUMBEROCORD.VORDIAORD, MAIN 780}

e. ipAcp,tEL,Nor MAIN 790 I

CALL PLOTS(ARRAii..1,2.KKI,KK2,LLI.LL2.NCOUNT.NELCATINEW,NODELP MAIN 800 NODE2A,NODE3.NONUMBoNSBOVNINTAPCM,NTAPRW.NUMBER,RORD.IORD.RORD, MAIN 810 ". ;8 ZORD,IELOD) _{MAIN 820} - 40 CONTINUE: MAIN 830 50 FORMAT(15) MAIN 846 STOP MAIN 850 END _{MAIN 860}

REVISED LISTING OF DECK AFCN.

EON

F ORTRAN

STATEMENT

1.0.

SOFTC

NFCN DECK _{NFCN 10} FUNCTION NFNCILI.L2OTP,IDUMI.IDUM29NDUMIODUA21. NFCN 20 NFNC=-1 NFCN 30 NDUM1=0 _{NFCN 40} NDUM2=1 NFCN SO IDUMI=IABSINTP*IL2-!L1) _{NFCN 60} IDUM2=0 - :NFCN 70 ' RETURN ' NFCN 80 " END NFCN 90 58

REVISED LISTING OF DECK SNAP

EFN

FORTRAN

STATEMENT

1.04

4IBFTC SHAP DECK SNAP 10

SUBROUTINE SHAPE ( KBEGINsKEND9LBEGIN,LENOrNTPCM,NTPRidg Mg L oN oNCIAR ) SHAP . 20

C

SNAP 30

C THIS SUBROUTINE. EFFECTIVELY ROTATES TRAPEZOIDAL SUBDIVISIONS SHAP 40

C SO THAT SIMILAR SHAPES (FOR EXAMPLES NTPR/02-1 AND NTPCM=..1( SNAP 50

C ARE TREATED SIMILARLY . SNAP 60

C SNAP 70

IF (NTPRH.E0.0) GO TO 10 SHAP 80

ISH=2 ' SNAP 90

IF ANTPRB.LT.0) 'ISH=3. SNAP 100

GO TO 20 SNAP 110 10 CONTINUE SHAP 120 ISH=4 SHAP 130 IF (NTPCM.LT.0) ISH=5 ' SHAP 140 20 CONTINUE SHAP 150 GO TO 170,30,40.50,601pISH SNAP 160 30 CONTINUE SHAP 170 N=NFNCILBEGINgLEND,NTPRWA,MINC6NR/ SNAP 180 GO TO 70 SNAP 190 40 CONTINUE SNAP 200

N=NFNC(LBEGIN,LENDIATPRB,M,L,NR,NO SNAP iiii

GO TO 70 SNAP 220 50 CONTINUE SNAP 230 N=NFNC(KBEGINgKENDOTPCM.L.M.NQ,NR) SHAP 240 GO TO 70 SNAP 250 60 CONTINUE SNAP 260 N=NFNC(KBEGINgKEND,NTPCM,M,L,NR,NO SNAP 270 70 CONTINUE SNAP 280 RETURN SNAP 290 END SNAP 300

REVISED LISTING OF DECK RVRS EFN

FORTRAN

STATEMENT

SIBFTC RVRS DECK SUBROUTINE REVRSE(IRV.11,JJ.I.J/ GO TO (10,20).URV' 10 CONTINUE 4=II J=jJ GU TO 30 20 CONTINUE J=II I=JJ 30 CONTINUE RETURN END 59 I.D. RVRS 10 RVRS 20 RVRS 30 RVRS 40 RVRS 50 RVRS 60 RVRS 70 RVR4 go RVRS 90 RVRS 100 RVRS 110 RVRS 120 RVRS 130