COHPUTER-AIDED DESIGN OF EXTRUSION DIES FOR THERMOPLASTICS A thesis submitted in fulfilment of the requirements for the Degree of Master of Engineering at the Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand. by JULIUS YAO BADU B.E. (Hons) University of Canterbury March 1975 - June 1976
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Computer-aided design of extrusion dies for thermoplastics
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COHPUTER-AIDED DESIGN
OF EXTRUSION DIES FOR THERMOPLASTICS
A thesis
submitted in fulfilment of
the requirements for the Degree
of
Master of Engineering
at the
Department of Mechanical Engineering,
University of Canterbury,
Christchurch, New Zealand.
by
JULIUS YAO BADU
B.E. (Hons)
University of Canterbury
March 1975 - June 1976
TO MY PARENTS
"NUSIANU MIATENU AW'J LA, MI ~t\f'JE,
GAKE NUSI M~E MATENU AW'J
HADE 0 LA, MILE WOGE."
YULIUS KAISAR.
"ANYTHING WHICH IS POSSIBLE HAS
BEEN DONE, ANYTHING IMPOSSIBLE
WILL BE DONE."
GAlUS JULIUS CAESAR.
i
ACKNOWLEDGEMENTS
I wish to express my deepest gratitude to
Doctors K. Whybrew and R.J. Astley, my supervisors,
for their valuable suggestions and assistance throughout
this work and for showing great interest in the research.
I am very much indebted to the Staff of the
Computer Terminus (University of Canterbury) for their
help during the course of this work.
An inestimable debt is owed to several colleagues
who, by personal kindness, have cheerfully shared with me
their wisdom. Thanks are also due to my friends who
helped me in several ways and who inspired me to complete
this work.
The author wishes to express gratitude to
Helen Oteng for her motherly care and constant
encouragement given to me during the course of study.
My sincere appreciation goes also to Mrs A.J. Dellow for
her expert typing of the manuscript.
June 1976.
J. BADU
B.E. (Hons).
ii
iii
SUMMARY
This report presents a survey of the literature on
the design of extrusion dies for thermoplastics. The need
for scientific die design and the major problem areas are
discussed.
In the literature survey, the major problems
isolated were:
1. Material Properties
2. Die Flow Analysis
3. Flow Instability and Melt Fracture
4. Melt Elasticity.
Attention was then confined to Die Flow Analysis.
The finite element method was proposed for solving two
dimensional flow problems in complex geometrical
configurations commonly encountered in polymer extrusions.
The two finite element methods adopted used the
variational principle in the formulation of the problem.
Also, different variational functionals and nodal variables
were used.
The results of the two methods were successfully
compared with finite difference, analytical and existing
numerical solutions.
The flexibility of the finite element methods makes
them very suitable for problems involving complex boundary
geometries. It is shown that the finite element method has
great potential for use in flow problems, and represents a
powerful new tool for the analysis of viscous flows.
QUOTATION
ACKNOWLEDGEMENTS
SUMMARY
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
NOMENCLATURE
ABBREVIATIONS
INTRODUCTION
CHAPTER
TABLE OF CONTENTS
SECTION 1 (LITERATURE SURVEY)
1 INTRODUCTION
iv
Page
i
ii
iii
iv
vii
ix
xi
xiii
xiv
3
(a) Description of the extrusion process 3
2
3
4
5
(b) Regular practice in die design 4
PROBLEMS OF EXTRUSION DIE DESIGN FOR THERMOPLASTICS
MATERIAL PROPERTIES
Introduction
(a) Non-Newtonian fluids
(b) Flow behaviour of thermoplastics
DIE FLOW ANALYSIS
Introduction
(a) Entrance effects
(b) Flow channel design
(c) The effect of die design on the quality of products
FLOW INSTABILITY AND MELT FRACTURE
(a) Die entry effect
(b) The concept of wall slippage
(c) Die exit effect
6
8
8
9
1 1
15
15
16
17
22
23
23
25
26
CHAPTER
6
7
8
9
1 0
1 1
12
v
Page
(d) Behaviour of linear polyethylene (L.P.) and branched polyethylene (B.P.) 28
(e) What is to be done? 29
(f) What can be done? 29
MELT ELASTICITY 31
(a) Cause of die swell 31
(b) Various studies carried out on die swell 32
(c) What can be done to reduce die swell? 34
CONCLUSION AND RECOMMENDATIONS
SECTION 2 (ANALYSIS OF FLUID FLOW)
BASIC THEORY
(a) Introduction
(b) Equations governing flow
STREAM FUNCTION ANALYSIS
(a) Introduction
(b) Finite element formulation of the problem
(c) Problems in this analysis
ANALYSIS USING VELOCITIES AND PRESSURE
(a) Introduction
(b) Finite element formulation of the problem
(c) Advantages and disadvantages of this analysis
RESULTS AND DISCUSSION
(a) List of results
(b) Discussion
GENERAL DISCUSSION
38
45
45
45
49
49
49
54
59
59
59
65
66
67
86
96
(a) Constitutive ~quations 96
(b) Problems encountered in the solution of the "stream function analysis" 98
(c) Problems encountered in the solution of the "velocities and pressure analysis" 101
(d) Comparison of both analyses 106
CHAPTER
1 3
REFERENCES
APPENDICES
A1
A2
A3
A4
AS
A6
B1
c
D
P1
P2
CONCLUSION AND RECOMMENDATIONS
(a) Conclusion
(b) Recommendations
Dimensional analysis
Finite element analysis of fluid flow within a converging channel using w and ¢
Dimensionless boundary conditions (W and ¢)
Evaluation of u (the x-component velocity) of an element
Pressure drop (6p) across the channel
Theoretical velocities and viscosities
Finite element analysis of fluid flow using u, v and p
Part of the initial computer programme
Estimation of costs involved in the research work
Resulting computer programme from the first analysis
Resulting computer programme from the second analysis
vi
Page
108
108
1 1 2
1 1 5
120
123
129
130
132
135
1 38
145
153
154
184
vii
LIST OF FIGURES
FIGURE Page
1 Elements of an extruder 2
2 General flow curve for a pseudoplastic fluid 10
3 Obstruction introduced by a spider leg 18
4 A sketch of recoverable shear strain versus shear stress for "Styron" 24
Sa Shear stress versus apparent strain rate for L.P. 27
Sb Shear stress versus apparent strain rate for B.P. 27
6a The shape of the required die by the compression method 35
6b Die profile 37
7 Converging channel flow 44
8 Triangular element showing nodal points 48
9 Mesh used in the analysis 50
10 Initial mesh 57
11 Typical finite element 61
12 Velocity profile for Newtonian flow in a parallel channel 88
13 Velocity profile for non-Newtonian (n = 0. 5) flow in a parallel channel 89
14 Dimensionless pressure drop versus taper angle 92
15 Initial layout of the system matrix equation 97
16 A portion of the mesh used 99
17 Unfavourable way of labelling the nodal variables 102
18 Favourable way of labelling the nodal variables 103
FIGURE
1 9
20
21
22
Channel boundary division for the forward difference formula
Channel boundary division for the central difference formula
Channel boundary division for the backward difference formula
Flow region showing the nodal points along the bottom boundary
viii
Page
131
131
131
133
TABLE
1
2
3
4
5
6
7
8
9
1 0
11
12
LIST OF TABLES
Number of cycles of iteration for non-Newtonian (n = 0. 5) parallel channel flow using a (9X9) mesh
Dimensionless flowrate for parallel channel flow for a Newtonian fluid
Dimensionless flowrate for parallel channel flow for non-Newtonian fluid (n = 0.5)
Dimensionless flowrate for converging channel flow at x = 0. 5 for a Newtonian fluid
Dimensionless flowrate for converging channel flow at x = 0. 5 for a non-Newtonian (n = 0. 5) fluid
Dimensionless velocities for parallel channel flow (first analysis) for a Newtonian fluid
Dimensionless velocities for parallel channel flow (second analysis) for a Newtonian fluid
Dimensionless velocities for parallel channel flow (first analysis) for a non-Newtonian (n = 0. 5) fluid
Dimensionless velocities for parallel channel flow (second analysis) for a non-Newtonian (n = 0. 5) fluid
Dimensionless pressure drops for converging channel flows (first analysis) for a Newtonian fluid
Dimensionless pressure drops for converging channel flows (first analysis) for a non-Newtonian (n = 0. 5) fluid
Dimensionless pressure values for flow in parallel channel (second analysis) for a Newtonian fluid
ix
Page
68
70
70
71
71
73
74
75
76
78
78
79
TABLE
1 3
1 4
15
16
Dimensionless pressure values for flow in parallel channel (second analysis) for a non-Newtonian fluid
Dimensionless nodal viscosities for parallel channel flow using a (9x9) mesh for a non-Newtonian (n = 0. 5) fluid
Dimensionless nodal viscosities for parallel channel flow at x = 0. 5 (first analysis) for a non-Newtonian fluid (n = 0.5)
Cost and storage requirements for the computer programmes resulting from the analyses
X
Page
80
82
83
85
xi
NOMENCLATURE
Symbol Definition
a, b local nodal point coordinates.
A area.
c1 to c 6 constants in stream function distribution.
F
H
H1
I2
i, j '
[K]
[Ke]
L
m
n
k
viscous dissipation functional.
depth at any section along the axial length.
channel depth at inlet.
second invariant of rate of deformation.
subscripts referencing nodal points.
system viscous stiffness matrix.
elemental viscous stiffness matrix.
length of converging channel.
subscript referencing element number.
flow behaviour index of a polymer; power-law index.
N shape functions. with subscripts
P pressure.
6P overall pressure drop.
Q volumetric flowrate.
u velocity in the x direction
u' dimensionless value of u.
v velocity in the y direction; volume of an element.
v' dimensionless value of v.
vc characteristic velocity.
w width of flow normal to x-y plane.
x, y Cartesian coordinates.
X', Y' dimensionless Cartesian coordinates.
Greek letters.
]1
Tw1' th b · t su scr1p s
cp'
X
l)J'
channel taper angle.
constants in the velocity distribution.
reference shear rate.
characteristic shear rate.
elemental area.
viscosity.
effective viscosity at y . 0
cha~acteristic viscosity.
viscous stress.
characteristic viscous stress.
sum of velocity components u + v.
dimensionless ¢ •
integral of functional F.
stream function.
dimensionless stream function.
overrelaxation factor.
optimum overrelaxation factor.
dimensionless pressure drop
dimensionless flowrate.
xii
xiii
ABBREVIATIONS
The following abbreviations were used in this work.
FD Finite Difference
FE Finite Element
FEM Finite Element Method
LA Lubrication Approximation
LP Linear Polyethylene
BP Branched Polyethylene
xiv
INTRODUCTION
Many flow problems in polymer processing are regarded
as difficult to solve without gross over-simplifications or
excessive labour. The two main sources of difficulty are
the intricate geometrical configurations which are
encountered and the complex rheological properties of
polymer melts. Consequently one finds that theoretical
treatments are often restricted to simple, sometimes
unrealistic, geometries and Newtonian fluids.
Industrial processes involving viscous fluids are
invariably associated with geometries of a complex nature;
for example, mixing vessels, screw extruders and extrusion
heads. Herein lies the potential of the finite element
method as a design procedure, because it is not limited to
geometries associated with the major coordinate systems
which, in practical terms, tends to be the case with other
methods. Furthermore, because of the way modifications to
the geometry can be effected the potential also exists for
"Computer-aided Design" in its entirety.
The present study has been restricted to the analysis
of two-dimensional, creeping, non-Newtonian flows using the
finite element method.
Finite element method approximates the flow by
dividing the flow region into small subregions or elements
and analysing the flow in terms of, say, velocities at the
corners of these elements.
XV
The FE methods are well established in the fields of
structural and solid mechanics (Zienckiewicz (60)), but
have not been widely used in solving fluid mechaniQS and
heat transfer problems. General examples include Martin
(25) and Oden and Somogyi (13). A general FE formulation
for fluid flow problems has been developed by Oden (35).
Zienkiewicz (60) has also described a FE formulation of
two-dimensional flow problems in terms of u, v and p.
Various authors (Oden (35), Martin (25) and Card (12)) used
the variational principle and demonstrated the FE analysis
of two-dimensional fluid flow. Atkinson et al. (1,3) applied FE
methods to two-dimensional Newtonian flow problems. Palit
and Fenner (37) also applied the FE method to two-dimensional
slow non-Newtonian flows.
The object of this work is to compare the type of
analysis used by Palit and Fenner (37) with the results
using a new type of analysis; a different variational
functional and nodal variables.
The present work contains two major sections:
Section 1 contains the literature survey. It
critically reviews the published literature on extrusion
die design for thermoplastics and lists the major problem
areas.
Section 2 presents the analysis of the flow of a
Newtonian and non-Newtonian fluid within a two-dimensional
channel. Two FE methods were adopted and the differences
would be pointed out in the course of this work.
1
SECTION 1
LITERATURE SURVEY
2
Fig. 1 Elements of an extruder.
CHAPTER 1
INTRODUCTION
Little has been written about the detailed design
of extrusion dies and it is unlikely that this state will
change appreciably for some years to come. The die maker
is a craftsman who depends on personal experience to an
extent which is uncommon in modern industry. The mystique
which appears to surround the craft is increased by the
reluctance of companies to talk about their success - the
solution to the problems of a difficult die may give a
distinct advantage over a competitor.
(a) DESCRIPTION OF THE EXTRUSION PROCESS
Fundamentally, the process of extrusion consists
of converting a suitable raw material into a product of
specific cross-section by forcing the material through
an orifice or die under controlled conditions. This
definition, as applied to the extrusion of thermoplastic
materials, covers two general processes - screw extrusion
and ram extrusion.
Thermoplastics are extruded predominantly through
screw extruders; more specifically, through single-screw
extruders. The elements of a typical single-screw extruder
are shown in Fig. 1.
3
4
The plastic material is fed from a hopper through the
feed throat into the channel of the screw. The screw
rotates in a barrel which has a hardened liner. The screw
is driven by a motor through a gear reducer, and the
rearward thrust of the screw is absorbed by a thrust bearing.
Heat is applied to the barrel from external heaters, and the
temperature is measured by thermocouples. As the plastic
granules are conveyed along the screw channel, they are
melted. The melt is forced through a breaker plate which,
in some cases, supports a screenpack. The melt then flows
through the adapter and through the die, where it is given
the required form.
When the thermoplastic material leaves the extrusion
die it is usually in the form of a melt or a very soft mass.
It has often little decisive form at this stage and even
this would rapidly be lost unless it were handled in a
suitable manner as it leaves the die. The product from the
extrusion die therefore must, in a large number of cases,
be looked upon as a semi-finished raw material which may be
given its correct form and dimensions by subsequent
processing whilst it is still mouldable. The methods of
doing this naturally depend on the thermoplastic being used
and on the desired product. This subject will not be
discussed here. The reader is referred to a suitable work
on this topic.
(b) REGULAR PRACTICE IN DIE DESIGN
In most extrusion shops, it is still a regular
practice to shape the die cross-section as judged
appropriate from previous experience and to proceed from
there by trial and error. The die is then installed, and
5
a run is begun. Samples of the chilled extrudate are taken,
and the dimensions are checked. Where the thickness is
excessive, the corresponding points along the die lips are
peened with hammer and punch while the extruder is running,
so as to close them slightly and reduce the section
thickness. It is hoped that some day plastic flow would
be understood well enough so that the crude method of
peening extrusion dies would be eschewed by die makers
forever.
The work presented here reviews the present state of
scientific die design and investigates methods of analysing
flow in extrusion dies.
CHAPTER 2
PROBLEMS OF EXTRUSION DIE DESIGN FOR THERMOPLASTICS
INTRODUCTION
The rheologist makes measurements of viscosity and
elasticity under carefully defined steady flow conditions;
the practical processor commonly works in less ideal
circumstances. An example of this contrast is extrusion,
where the rheologist measures the flow through a lon~
cylindrical die, but the practical processor operates with
relatively short, tapered dies, often of slit or other
profile.
6
A typical tubular extrusion die of 150mm diameter
costs about a thousand dollars to make. At this price the
processor has a problem which is expensive to solve by trial
and error, and he legitimately turns to the rheologist for
assistance. The processor needs to know what will be the
swell ratio as the melt leaves the die and what will be the
pressure drop through this die. He also needs to know at
what throughput rate non-laminar flow will occur. The ideal
die will maximise output rate of smooth extrudate and
minimise pressure drop and swell ratio, and such
optimisation commonly requires an accurate choice of taper
for the converging flow regions in the die.
A survey of all available literature on die design
was carried out to ascertain the various problems. The
following major problem areas were isolated.
These are:
(1) Material Properties.
(2) Die Flow Analysis.
(3) Flow Instability and Melt Fracture.
(4) Die Swell.
The above areas are further discussed in the next
four chapters.
7
CHAPTER 3
MATERIAL PROPERTIES
INTRODUCTION
Ideally, during extrusion, different screws are
required for each material because of the change in
extrusion viscosity values from one material to another.
The same requirement applies to die design, although
the necessity is, perhaps, not so stringent. A change
in material viscosity brings about a corresponding
change in flow properties, with the result that a die
arrangement suitable for a polyamide, for example, would
be completely inadequate for an unplasticised vinyl
material. In general, the higher the extrusion viscosity
of a material the greater the necessity for streamlining
the interior of the die.
The dependence of die design on material
characteristics need not be overemphasised. This part
of the work therefore briefly looks at non-Newtonian
fluids, flow behaviour and properties of thermoplastics.
The rheological equation describing thermoplastics and
its limitations are also discussed.
8
(a) NON-NEv'iTTONIAN FLUIDS
Classification
The Newtonian viscosity, ~, depends only on
temperature and pressure and is independent of the rate
of shear. The diagram relating shear stress and shear
rate for Newtonian fluids is the so-called "flow curve",
and is therefore a straight line of slope~.
Non-Newtonian fluids are those for which the flow
curve is not linear. The viscosity is not constant at
a given temperature and pressure, but depends on other
factors like shear rate, the apparatus in which the
fluid is contained, or even on the previous history of
the fluid.
These fluids can be classified into three broad
types:
(1) Fluids for which the shear rate is a function of
the shear stress only.
(2) Those for which the relation between shear rate
and shear stress depends on the time the fluid has
been sheared or on its previous history.
(3) Systems which have the characteristics of both
solids and liquids and show partial elastic recovery
after deformation - viscoelastic fluids.
Viscoelasticity results in stresses normal to
the dir~ction of shear stress. When viscoelastic jets
emerge from capillaries, the normal stresses result in
an expansion of the jet as opposed to the usual vena
contracta in Newtonian fluids. This is the Barus effect.
9
Shear stress
Fig. 2
pseudoplastic •n T = ky
k = constant
n = flow behaviour index< 1.
Strain rate
General flow curve for a pseudoplastic fluid.
___,.
0
11
Rheological Equation and Limitations
In this work, attention is confined to pseudoplastic
fluids which are described by the equation
·n T = ky
where k is a constant and n (less than one) is the flow
behaviour index.
A typical flow curve for this fluid is as shown
in Fig. 2. It can be seen that n is not constant over
the whole range of the shear rate. This is not a serious
drawback because all that is required is the value of n
which describes the flow over the particular range
encountered in a particular problem. The chief limitation
of this equation is its inability to portray correctly
flow behaviour at shear rates in which the fluid is
approaching Newtonian behaviour. Thus, extrapolating of
data taken over a modest range of shear rates from the
non-Newtonian into Newtonian region may result in
appreciable errors.
Unfortunately there is no ready solution to this
problem and hence one must obtain rheological data at shear
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c c ************************************ C THIS IS(]li\TI:~; l(l,f::>,~·JF.LTl,OEU?,FfWI\ C SYSr~il 11/\TIUX F.OUAT!OII c ****************************~~****** c
c
REAL K ( ll!(ll!l() II( I ( l!llF ,IIDF) I
TK 2 (II[) F I I!) lJiiiEIIS!Oil DELT/1(111<) 1 0F.LT1 (tiDF),lJELT:?.(II)
C I< 1 ISOLATED, c
c
DO 28 !=1 ,ffDF DO 2 8 J:: 1 ! 10 F f( 1 ( 1 , J ) = rd 1 , .J )
2 8 C :J liT I lllJf:
C K2 ISOLATED. c
c
Dfl 29 I=1 ,IIDF o:; 2:) .J=IIDF+ 1, 'IOF+fl f( 2 ( I , .J., I ![)F) ::f( ( l , J )
c c *************************************** C SUBROUTINE TO TAKE THE UPPER TRIANGULAR C ELEMENTS OF K1 AND PUT THEM INTO K3. c *************************************** c
REAL K(NK,NK) ,K1 (NDF,NDF), TK2(NDF,N) ,K3(NDF,IBW)
DIMENSION DELTA(NK) ,DELT1 (NDF), TDELT2(N)
DO 27 I=1,NDF DO 27 J=1,IBW K3(I,J)=O.O
27 CONTINUE c C K1 REQUIRED. c
c
CALL ISOLAT(K1,K2,DELT1,DELT2,K,DELTA, TNDF,N,NK)
DO 127 J=1,IBW DO 127 I=1,NDF+1-J
C K1 INTO K3. c
K 3 ( I , J ) = K 1 ( I , I +J- 1 ) 127 CONTINUE
RETURN END
1 5 1
c Slltir;:wn 11::: f'Wnur( c, :~:2,:F.I.T2, ·:~)F, :1, II)
c *********************************** C T!!IS i!'.ll.Tlf':.TFS T\f'l rL'\T!:rt:F') !~? ;\'I'J C Dr::LT? ;\~1'1 :)w~n:s 1'1![ !'[SlJI.'I !II C c ********************~************** c
c
R E 1\ L f( 2 ( t I D F , l·l) 1 J I! 1 E: IS r n11 c ( 1 ID F, II) , n F. LT?. (: 1, i 1) Dr 1 32 1=1 ,f!IJF f)() 32 ,)::1,11 Y=o.o
C 013T?I It! TilE PRODUCT OF TH[ Tl/0 lli1.TR ICES. c
D D 3 3 K:.: 1 , 1·1 33 Y=Y+I(2( I,I~)":DELT?.(I<,J) 32 C(I,J)::::Y
r<F.TlJfHI F.i!D
152
APPENDIX D
Estimation of costs involved in the research work
The various costs encountered during the course of
computation are as detailed below.
B6718 CPU Time
I/0 Processor Time
Memory Integral
Slow IO Operations
File Open Time
Total
$210.58
67.90
111.82
170.65
1 8. 2 8
$578.23
153
154
APPENDIX P1
Resulting computer programme from the first analysis.
155
,',II rJT 1\ T I rJ i IS .,., *******"********~***~*********************************************
..... -.':·.':;': ... ·:
l\J, ll.J, ~1\, iH~::LOCr'\t.. COCli\D I ilt\'i"ES OF /\II ELEI~ liT \/IT II i \ E S I' E (: T T ,) r:1 E l 1 : i d D l; • I\ t I ELE! lUll I·; l. .\_)I·:U ::: l [ 1, [', '.· ':1 :1·1 'diT I CLOCI\\IISE FtiSfl I Oil.
A:lOl.H:(;III,l;)::J\r(RI\Y Ct'JiiTl\IIIliiG I I·IFOI~!li'ITIOI! ABDUT ;\LL TilE llllDES. Tilt: FiRST H/0 PGSITIOIIS F C~ T I IE(;<) ,·\: !D ( Y) CO:JR U I: l/1 TE S ( GLOGi\1.) J\1 iD TilE THIRD 1\110 FOURTI~ Fa\ TilE 0. 0. F. I· I OS.
DXDVY( )::(AHRAY) DERIVATIVE OF (OVY) \.J •. R."f.X
OYDUY( )=(MRAY) SECOND DERIVATIVE OF (U)H.R.T.Y
ELEtlE T (II~, 4) = /l.RRAY C otHA Irll tH1 I IIFORI·iA TI OH ABOUT ALL THE ELH1E IHS. THE FIRST THREE POSITIOtlS FOR TilE ~JOOAL liDS. i\110 THE FOURTH FOR THE EL Et1EIIT VISCOSITY.
PS=COUHTER; DEGREE OF FREEOOI·1 1Hlt1BERS; tlOOE tlUt·lBERS
I B\/=B.l\ NOH I D TH
;': i'n': ;': 'i':-
JfS:=COUtiTER, DEGREE OF FREEDOI·l Nut·IBERS; HODE llU1·1BERS.
KE(6,6)::ELE11EIHAL STIFFI·JESS t1ATRTX.
156
K(l~OF, IO~/)=SYSTE11 STTFFIIESS 11ATRIX HI f1AIWED AIIO RECTMIGULAA FORt·t
-.,'( t·l ';'( ..,., ~·:;': ';'('i'(
r~tU=ELH\EIJT VISCOSITY.
fllJIHJDE(I~II)=MRAY COflTAIIIII·IG THE VISCOSITY OF EACH NODE,
f~=!Wtl ZERO flO OF 0, Od,
NDELTA(6):::fiRRAY COI'ITAINII~G THE o.o.F .. IWSd DF AH ELUIE Iff
NDF=UIIKI'Ir.Mtl ~lO. OF 0. b. F •
NE:::fW OF ELHIEHTS
IIK=HIITIAL 110. OF D.O.F(tiO. OF f•IODES ~~2)
NtJ:::t-10 OF NODES
;': Q ";'( ·::-,':·.':'i'n':
Ql ,Q2,Q3=FLO\.JRATE AT HlLET,f·HDHAY AND OUTLET OF THE DIE,
i'( ;': ;': ;': ')'(
T=TANGEiJT OF l\1~ AHGLE
..,•: ;': i': ..,., ;':
U(NE,J)=X-VELOCITY ARRAY OF ALL THE ELEMEHTS. THE FIRST EIHRY REFERS TO THE 11 POSITIDtl OF THE ELEIIEtlT,THE SECmiD TO THE I2 POSJ"I'Iotl AND THE THIRD TO THE I3 POSITION~
UNODEO!tn :::X-VELOCITY ARRAY OF ALL THE NODES
;': v ·:: ..,., ;'('";1( ";'('","'!
V(tiE,3)=Y-VELOCiTY ARRAY OF ALL THE ELEIIENTS THe FIRST EIHRY REI-ERS TO IHE Il POSITIOtl OF THF. ELEI-1EI~T, THE SECmlD TO THE 12 POSITion AI~D THI!. THIRD TO TH~ I3 POSITION
VNdDE(Nti):::Y-VELOCITY ARRAY OF ~\LL ·rHE NODES
'vl~\·1 t D TH DF [} t g
157
S U BR OU TI fJE I NP U T ( ~~ K, NE , NN, NO F, N, I 8\.f, D L, H, \.f, TE LU1lT, AI WOE, DELTA, OH, ALPHA)
c ~*******~~*****~***************** C THIS INPUfS ALL THE AVAILABE DATA c *********~***************~******* c
D II·1E NS ION EL Ef1E T ( NE, L+) , ANODE( W~ ,4) , DELTA( NK) c
158
C DATA /\BOUT f~LL THE ELEI-IEI'ITS. THE FIRST THREE POSITIONS· IS FOR THE C NODAL rHJt1BERS AI~D THE FOURTH FOR THE VISCOSITY. c
c
viR IT E ( 6 , 8 )
8 F 0Rt1A T (31 X, 50H EL Et1E NT I l·lF ffit!A TI ON•'n':;':;'n'n':;':;'n'o':;': EL Et-lE NT I f.IF ORI'IA T I ON, I TJ.It Tj1X,10H El.Et,IENT ,10H tlODE 110 01 101: NODE t·JO., IOH tlODE ~10,,10H V r I seas ITY)
DO 9 I= 1 , NE READ(S,/) P,I2,I3
E L E f1ET ( I , lf) :: 1 • 0 \.JRITE(6,10) I,I1 ,I2,I3,ELHIET(I,4)
10 FORt'lA'f(31X, !5,4X, IS,SX, IS,SX, IS,S:<,F10.5) El. Et1ET ( I, 1 ) :: I 1 + 0. 1 E L D 1 ET ( I , 2 ) = I 2 +0 • 1 E l. E t1E T ( I, 3) = I 3 +0. 1
9 GotH I NUE
C INFORI1ATION ABOUT ALL THE ~lODES IS READ IN c
c
\.JR IT E ( 6 , 1 1 ) 11 F0Rt1AT ( 1 HO, 30X, 1 NODAL I i~FOR~IAT I ON•':.'n'n':.':·.'n>:,·n•n•:tWDAL I NF OR~1A T I Otl 1 II
T31X, 1 tlODE ~lO. 1 , 1 :<COORD 1 , 1 Y COORD 11
1 DOF NO. 1
T 1 x, I DOF NO. I/)
DO 12 I=l,t,ltl R E A 0 ( 5 , /) A t W D E ( I , 1 ) , AN 00 E ( I , 2 ) , J 1 , J 2 ANODE( I, 1 )=MIODE( I, 1) /DH ANODE( l,2)::AIWDE(I,2)IDH \·IR IT E ( 6 , 1 3 ) I , AN 0 D E ( I , 1 ) , AN 0 D E ( I , 2 ) , J 1 , ,J 2
13 F0Rf1AT(31X, I5,4X,2F10.5, I5,5X, 15) ANODE( I,3)=J1+0.1 ANODE( I ,4)=J2+0,1
12 CONTINUE
C READ IN INITIAL NODAL VALUES c
~/R I TE ( 6 , Jlf ) 14 FORt·lA T( IHO ,30X, 1 BOUtlD ARY CDi'lD IT IONS 1 I
15 COIHINUE DO 16 I=tiDF+l ,NK,2 \·IR IT E ( 6 , 1 7) I , 0 E L T A ( I) , I+ 1 , D E L T A ( I + 1 }
117 FOW1AT(31:<, IS,I-IX,F9.6,2:<, I5,4X,F8.6) 6 COIH I tJUE
RETURtl END
SUl3ROUTI !,IE COORD( I, MJODE,Nli,DL, ALPHI\) c c *******~****~********************* C THE GLLJQJ\l CJOi\DII!i\TES GIVCII Iii C T II E I , II' U T S lJ lJ f( ::-1 U T I II E i\:\ E F OR 1\ C Uil! FURil r:ECTAIIGUL/\;\ i;CSII, i\i.JD!FIElJ C 3ELU'./ TJ ,\CC:Ju;;y F:ii~ liiE Tfd't:i~Ii!G C OF TilE DIE. TilE X COOI\D!IIATES C FEIIAill TilE SAllE. c ***~****************************** c
0 U\Eil~ lOll /\[lODE( llil,4) T=T.I\11\ 1'\LPI-Ii\J R=T/ DL MIOD E ( I , 2) = MIOD E ( I , 2) ~._. ( 1 • 0-A 1100 E ( I , 1 ) ~·:R) Jl ::AI~ODE( I ,3) J2=AiWDE( 1,1+) RETURH E HO
159
SUBROUTIIJE UPDATE( ELEI-IET,AIWDE,DCLTA, I, HIE,HK,IHI, ITEI~)
c c ********************************** C THIS SUBROUTINE UPDATES THC VALUES C OF VISCOSITY FOR EACH E.LEt1ElH c *********~~*~*********************
c
o II IE r~s I otl ELEI-IET (tiE, Lf), ArJODE(tltJ, 4), TDELTA( 1,1~:)
REAL l·lU
C NODAL lWllBERS c
c
I 1 :: E L E t1E T ( I , 1 ) I2=ELEI-1ET( I, 2) I 3:: E L E t1E T\ I ,3 )
C D.O.F. HU11BERS c
c
J 1 :: /'\ N 00 E ( I 1 , 3 ) J2=AiJODE( I 1 ,4) J3=ANODE( !2 ,3) Jlf::ANODE( !2,4) JS=MIOOE( I3,3) J6=ANOOE( 13 ,L:)
THIS CALCULATFS TilE STI FFIJESS tiATfU X OF EACH ELEt-lEtH /\flO IIISERTS IT IlrfO THE SYSTEil STIFFIJF.SS I'~ATHIX. TilE ltATRIX IS 81\I~DED AIJD TilE LOAD!t~G VECfOR OBTAHIED IN THE PROCESS ************************~****************
RE.i\L K(IJDF, !Oii),i<.E(6,6) D 0 7 0 I~~ 1 , NO F DO 70 J:::1, IB\4 K ( I, J) =0. 0 C(l~1>=o.o CONTI f·lUE DU 50 1=1 ,NE DO 71 lC=1,6 DO 71 J::: 1 , 6 . KE( IC,J):::O.O
CONTI HUE
ELEMENTAL GEotiETRIC COi'lSTANTS REQUIRED
CALL GM1BET(G1 ,G2,G3,GI~,G5,G6,81 ,82, T83,:\REA,AJ~ BJ,AK, BK,.JAY ,KAY, F, TELEI-H:. T' MIOOE, I I FARIIU I FAR I fJE I l·lK, Nil I \1' H)
c C CALCULATE ELHIENTAL STIFFNESS f1ATRIX N0\4 c
F 1 =FARI1U>':GJ F2= FARIIU>':G2 F 3= FARI·IU~• G3 K E ( 1 , 1 ) :::F P G 1 K E ( 1 , 2 ) ::: F P G4 K E ( 1 , 3 ) ::: F 1 ~' G2
~Hl:~l~~n~a~ K E ( 1 I 6) =F 1'' G6 KE(2,2)=FARtlU>':( ( 81>':81 )+(G4>':Gll)) KE(2,3)=F2>':G4 K E ( 2 , L~)::: F ARI'IIJ>': ( 81 ,., 82 + G4 -:: G5) K E ( 2 , 5 ) :::F 3 >': G4 KE(2,6):::F/\RMU*(B1*83+G4*G6) KE ( 3, 3) =F2>':G2 KE ( 3, Lf) ::F2>':G5 KE ( 3 I 5) =F2>'1 G3 K E ( 3 , 6 ) :: F 2-:: G6 KE(4,4):::FARMU*((B2*B2)+(G5*G5)) K E ( 4 Is)::: F 3 >': G5 KE ( 4, 6) =FARHW ( B2~1 83+G5>':G6) K E ( 5, 5) = F 3 ,., G3 K E ( 5 , 6) = F 3 >': G6 KE(6,6)=FARMU*((B3*B3)+(G6*GG)) 00 26 12=1 ,5 DO 26 ,J:::I2+1 ;6 K E ( ,J , !2 ) ~: I(E ( I 2 , J)
26 CotHI NUE
1 61
c C ELE~·IENTAL STIFFNESS t·1J\TRIX IS AVI\!LABLF.; NO\-/ c C 08Tf.\IN TilE o.o.F. NOS. c
c
II =ELEIIET( I, 1 ~ I 2 = E L E I lET ( I I 2 ) I 3 = E L El 1 E T ( I I 3 ) IIDELTA( I )::AI~ODE( 11 ,.3) N 0 E L T A ( 2 ) ::A I WD E ( I 1 , If ) IWELTA(3)=MJODE( 12,3) NDELTA(lf)=I\IWOE( I2,L+) I JOEL TA( 5 ):::AIJODE( 13,3) NDELTA(6)=ANODE( 13,4)
C CHtCK ALOI-IG TilE RO\-/ OF tl/\TRIX. C IF IT IS BEYOND THE RAI~GE OF (K) IIEGLECT C IT. IF IT IS \IITHII~ THF RAI1GE, GO C ALONG THE COLUMN. c
DO 2 KI=I,6 c C IF BFYOND K1 S RANGE, TRY NEXT ROW c
c c c
c c c c
IF(NDELTA(KI).GT.NOF) GO TO 2 I~K!=NDELTA(KI)
GO COLUI·11~/ISE
DO 3 KJ=l ,6 NKJ=NOEL TA( KJ)
IF BEYOND K 1 S RANGE, OBTAIN THE LOADING VECTOR
IF(NKJ.GT.NDF) GO TO 4
C HITHIN K' S RI\NGE. NO'v/ IN BAI~DED FORt1 c
NKJ1 =I~KJ-I·!Kr+ 1 IF(NKJl.LT.I) GO TO 3 K ( NKI, NKJ 1 )=K( NKI, HKJ I )+KE( KI, KJ) GO TO 3
4 C ( HKI, 1 )::C(NKI, 1) -KE(KI ,KJ)~':OELTA( I~KJ) 3 com I NUE 2 CmlT I NUE 50 CONTINUE
RETURN EI~D
162
c c c c c
c
c
163
SUBROUTINE SYMSOL(NN,M1.NDIM,KKK,A,B)
BANDED SYI111::.TRIC t·\ATRIX EQUATIOI~ SOLVER CROlJT t·1ETHOD . ****************************************************************** D Ir1E NS I Ol ~ A ( 1 ) , B ( 1 )
LOC( I, J)=I+ (J-1 )~':IWH1 GO TO (1000~2000),KKK
C REDUCE 1-tl\TRIX c
c E
c c c
c
1000 DO 280 N=1,NN ~I i :: LO C ( l·l, 1 } DO 260 L= 2, t-M NL=LOC( N, L) C=A(NL)/A(I~1) I=N+Lw1 I F ( r·l~ I • L T • I) G 0 T 0 2 6 0 J=O DO 250 K=L,l't1 J=.J+1 I J=LOC( I, J} NK=LOC(N,K)
250 A( I.J)=A( IJ)-C1':A(NK) 260 A(NL)=C 280 GOtH I NUE
2000
285 290
300
400
GO TO 500
REDUCE VECTOR
DO 29CJ N= 1? I~N N 1 =l.OC( II, 1 J DO 28 5 L= 2 t-1M tJL::: LOG (II, L) I=ll+ L-1 IF(l~l·l.LT. I) GO TO 290 8 ( I } = B ( I ) - A ( I H.) ,·: B ( N) B( t~) =B( II) /A( 1~1)
8,\CK SUBST I TUT I Oil
II=Nt·l !1=11"1 IF(Il.EQ.O) GO TO 500 DO 400 1(::2,11'1 IJl(::J.OC( II,K) L=II+K-1 I F ( ll; 1. L T • L) G 0 T 0 1 f 00 D ( i:) ::2 { II) -~~ ( t li() ~·: B ( L) c.; i I T I IJlJ [ GCl TO 300
500 RETURII Ell()
SUEllUJUT I II[ COIIV[(i( lJEl.TI\, C, IIDF, KODE, iH<) c c ************************ C Tl~[ TEST FIJH COIIVERGEIICY c ************************ c
C STORE THE X-VELOCITY VALUES IN THE (U) ARRAY C MID THE Y"VELDCITY VALUES IH TilE (V) ARRAY c
U(I,l)=Ul V( I, 1) ;:O?.-Ul U ( I, 2) =U2 V ( I , 2):: DL> -U2 U( I,3):::UJ V ( I ~ 3 ) :: 06 ~ U3 f{ETURtl END
165
c
SUBROUT I I~E VELfJC(UNODE, VIIOOE,U, V, ELEI·IET, DELTA, T A IJOO E, HE , liK, N~l, \-1, H)
c ********************************************** C THIS rALCULATES THE X ArlO THEY VELOCITY C COHPOilEIHS nf: EACH tWOE. THE COtHRIBUTIONS C OF VARIOUS ELEI·\S NTS TO THAT I lODE IS TAI~E tl rrno C ACCOUtlT. c ********************************************** c
0 !liENS I 0! I Ul l 00 E ( HN) , V~l 00 E ( l·l~l) > U ( HE , 3 ) , V ( I·IE ,3) , TEL El IE T ( I JE , 4) , 0 E LT A ( t1K) , A IWD E ( liN, 4) , ~~~~ 00 E ( 81 )
DO 7 3 IH:: 1 I HN UI,IODE( N1) =0.0 VI·IODE( Nt·I)=O.O ~JI~ODE( HI) =0
7 3 COIHI NUE c C NODAL NU~IBERS c
c
DO 772 J::J,NE I 1 :: E L E t ·IE l( I I 1 ) 12=t::LEt·IET( 1,2) l3=ELEt1E T( I ,3)
c *****~******~********* C VISCOSITY OF EACH NODE c ********************** c
0 IHEI'lS I uti ELE: lET ( NE, If), DELTA( l·lK), AIJODE( W·l, If), TUII ODE ( HI~) , V rJGD F. ( 1~11) r DUX ( 8 1 ) , DVX ( 8 l ) , DUY ( 8 1 ) , OVY ( 8 1 )
REAL t·\UJJODE( tHn c C I~E\HOHIAII FLO\~ THEFO~E VISCOSITY EQUALS 1.0 c
c
GO TO 560 DO 4 9 t I I :: 1 , l·li'l 1\UNODE( IH )=0.0
lf9 COt·ITI NUE
C VISCOSITY 1\T EACH fiOOE IS EV.£\LUATED C fiY 08 TA I tH I~G THE VALUE OF If! 2 AT C EACH PO I llT c c C FINITE DIFFERENCE METHOD IS USED TO C OBTAI!~ THE DIFFEREHTIALS c c C OBTAIN DERIVATIVES W.R.T.X c
DO 50 I:: 1 , 73 , 9 A=AI~OD£(1+1,1)-Af.IODE( !,1) B=AIIODE( 1+2, 1 )-AIIOOE( I, 1) DUX ( I ) :: ( IJ~': s~·:u t·J ODE ( I+ 1 ) - M: A~': W·l ODE ( I+ 2) ~ ( s~·, B- N: A)
T~··u rwo E < I ) ) 1 ( A~·· s~·, ( s-A) ) DVX ( I):: ( B~'•B~'•VtlODE( I+ 1)- A~'•M•Vf.JODE( 1+2)- ( s~·:s-N: A)
T~'•VIWDE( I)) I( A~':B~':( B-A)) 50 CONTI HUE
DO 51 1=9,81,9 A 1 =ANODE ( I, 1 ) -ANODE ( I-1 , 1 ) Gi =ANODE( I,1 )-ANODE( I-2, 1) DUX (I)::( APAPUt·lODE( l-2) -BP8J:':UHODE( I-1)
T+ ( ( 8 1 ~·, B 1 ) - ( A 1 ~·,A 1 ) ) ~·:u II ODE ( I ) ) I (A 1 ~·, B 1 ~·, ( B i -A 1 ) ) DVX( I)=(A1~'•At~'•VNODE( I-2)-B1~•81~':VNOOE( I~t)
T + ( ( 8 P B 1 ) - (A 1 ~·,A 1 ) ) ~·, VN 00 E ( I ) ) I (A P 8 P ( B 1 -A 1 ) ) st cmnr NUE
DO 52 ,1=2,8 DO 53 IL=0,8 I=J+ I U:9 A2=ANODE( I, I )-ANODE( 1~1, I) B2=AIWDE( I+ 1, I) -AtJODE( 1··1, 1) DUX ( I) = ( A2 ~: A2 ~·:u f.l ODE ( I+ 1 ) - ( 82- A2) ,., ( 82- A2 ) ·::u r JOD E ( I -1 )
T - ( ( A2 -::A 2 ) - ( ( 82 - A2 ) ~·: ( B 2 - A2 ) ) ) ~·, W J 0 D E ( l ) ) I ( A2 ~·, R 2 ~·: ( B 2 - A2 ) ) D V X ( I ) :: ( A2 ·:: A 2 ,., V N D D E ( i + 1 ) - ( B 2 -A 2 ) ~~ ( !3 2 - l\2 ) ~~ V r I D D E ( I - 1 )
T- ( ( B'(l': BY)- ( AY•< A Y) ) ·::LJ t lODE ( I) ) / ( AY·.'dJ'(l': ( BY" AY) ) DVY ( I) :: ( 8Y•': 8 y·:: VllOD E ( I+ 9)- AY•': AY•'<V II ODE ( I+ 18) - (( B Y•': [l Y)
T - ( A Y ,·,·A Y ) ) ,·,v l·lO D E ( I ) ) I ( A Y ,., 8 Y '~ ( 8 Y ~ A Y )) · 54 Cat-IT! NUE
DO 55 I=-'73,H1 AY=MIODE{ 1,2)·ANOOE( l-9 2) BY=AIJODEt I ,2) -ANODE( I-JS,2) DUY( I)::((AY•':f\Y•':LJtiODE( 1-IB)wBY•'•BY·.':UtJODE( 1-9)
T + ( ( BY,., BY) - ( A Y•': AY) ) ·::u tl ODE ( I) ) ) I (A y,·, B Y•': ( IW- AY)) DVY( I)=(AY•':AY•':VNODE( !-18)-BY·::sy·::VtWDE( 1-9)
Tr ( ( G y,·, BY) - ( A Y•': A Y) ):': V NOD E ( I) )/ ( A Y•': By,·, ( C '(-A Y} ) 55 corn I HUE
DO 56 ,J:::J0,64,9 DO 57 I L=0,8 I=J+IL-'•1 A Y = 1\ N 0 D E ( I , 2 ) -AN OD E ( I -9 , 2 ) BY=ANOD~:( I+9,2)-Ail0DE( I-9,2) DUY( 1):: (AY:':Ay·::UfiODE( I+9)-(BY-AY) .'•(BY-/W)
P U llO D E ( I - 9 ) - ( ( A Y \., A Y ) - ( ( BY - A Y) ,., ( 8 Y -A Y ) ) ) ,., U ~lO 0 E ( I ) ) V (AY*BY*(BY-AY))
su:m:JUT!Il:~ DELT.4P(DP,CLF.IET,OELT1\,AWH1f:,U,V,IIE, T: II~ I IIi I I I I ' II )
169
c c c c c c
TillS C/\LCULJ\Ti::S Tilt Clli\IIGE !II PHF.SSUf\E UET\/[[11 !liLET ~'\110 OUTLET t\LlWG THE X"I~XI S
c
D lt1E !JS I 0! I E L E! lET ( liE , 4) 1 DElTA ( til~) , AtWU E ( t,H I~~.) , T u t 1 oo E < 3 1 ) , u < 1 !E , 3 ) , v 11 o o d n 1 ) , v < r 1 E , 3 ) , oux ( 9) , TGXDUX(9) ,D!IUX(9) ,OUY(9) 1 0YDUY(9) ,DVY(9) 1
TOXD'.'Y ( 9) I DVX(9) IF( 9), Dl!UY ( 9) REAL l·IUtlOOE( B I)
CALL VI SC 0( t1Ut!OD E1 EL E11E T 1 OELT A 1 A!~OD E, U tWO E, VIIODE 1 i~E, NK, Ntl, H, H) c C FINITE D I FFE NCE t·IETHOD IS USED HEr-E C THE FORi/ARD Arm BACI<I.JARD D IFf-EREtiCE Fo:\l·IULA C IS USED FOR THE Et!D POI tHS C CE lHHAL DIFFERENCE F CRt1lJLA FOR OTHERS c c C DBTA HI FIRST Atm SECOt!D D IFI:"EREIHIAL OF C U 1·1. R. T • X c
A=ANDDE(2,1)-ANODE(1,1) B=AtWDE(3 I 1 )-AI~DDE( 1 I 1) . DUX ( 1 ) = ( 8 •': B•':Ut·l ODE ( 2)- A•': f-\·:: U~IOD E (3 ) - ( 8•': 8- A•': A) •':UN 00 E ( 1 ) )
T/ ( A•':B":( B-A)) D XDUX ( 1 ) :: ( 2. 0•': ( B•': Uti ODE ( 2) - A•': UI·J ODE (3 ) ~ ( 8- A) •':LJ tlOO E ( 1 ) ) )
T/ (A•':B•':( A-B)) A 1 =A IWO E ( 9, 1 ) - Al-100 E ( 8 1 1 ) B 1 =·AIIOD E(9 1 1) -A!-IDDE(7 i) DUX(9)=(A1~AI*UNODE(7~-8i*B1*UNODE(8)+((B1*R1)-(Al*A1))
PUtJOOE(9)) /( APBJ:':( Bl-Al)) D XOUX ( 9):: ( 2. Q;': ( A PU I·WD E (7 ) -B 1 1':Ut JOD E ( 8) - (A 1 - B 1 ) •':
T ( A P A 1 ) ) ·:.-ov Y ( 9) ) I ( A l ,.,. B P ( B 1 -A 1 ) ) Oo 52 I= 2, 8 A2=Ai~ODE( I, 1) -MIOOE( l-1, 1) B2=AtWDE( I+ 1,1) -AI lODE( I-1, l) DXOVY (I):::( A2''1A2>':DVY ( !+ 1) -(( 82-1\2) ''1 ( 82-t\2) )>'•DVY (I -1 ~
T- ( ( /\2 ,.,1\2 ) - ( ( f32- A2 ) ·:: ( l32 - /\2 ) ) ) ,., D V Y ( l ) ) I ( i\2 ,.,. 82 ·:: ( 132- A2 J )
52 COI:T I NUE
C OBTA HI THE D l FFERE t~T IAL OF U MID V c
c
. 0 VX ( 1 ) = ( 8>': 8''1 Vt JOD E ( 2) ~A,., A;'1V I I 00 E (3) - ( [3'.'1 B- N· A) PVIIODE( 1)) I(A'''8'''(8-A))
DV X ( 9)::: (A 1 ,.,. A PVI·J ODE ( 7 ) - B P R I ,.,.VI I 00 E { 8) + ( ( 8 1 ,.,. B 1 ) T - ( A 1 ,., A 1 ) ) ,·,.v I JO D E ( 9 ) ) I ( A 1 ·:: 8 1 ,.1 ( B 1 - A I ) )
DO 54 1=2,8 -A2=ANODE( I, 1 )·ANODE( I-1, 1) B2=AHODE( I+ 1,1) -ANOOE( 1··1, 1) DVX ( I)::: ( 1\2 ,.,. A2'':Vtl 00 E ( I+ 1 ) - ( 82- A2) ·:: ( 82 ~A?.) ·::Vtl GO E ( I -1 )
T- ( ( A?.,.,. A2 ) - ( ( 82- A2 ) ,., ( 82- A2 ) ) ) ,.,. V N 00 E ( I ) ) I ( A2 ,.,.112 , ... T ( B2 -A2))
55 COIH I NUE SUI 1:::0.0 DO 56 I::: 2,8 SUII=SUH+F( I)
56 COI·IT I NUE
C IIHERGRATE llSHIG TRAPEZOIDAL RULE TO OBTAI!~ THE C CHAIIGE It~ PRESSURE c
D P:::- ( F ( 1 ) + 2. 0•': S Ul·1+ F ( 9 ) ) I 1 6 • 0 RET URI~ END
171
c
SUOROUT It IE FLORET ( Q I ,Q2 1QJ , ELEI·E T, ToE LT A , AtHJD E, u , LJI'l no E, 1 j[ , I~K, t Hl, 11, H)
c **********~~***************************** C TillS CALCULATES THE FLOWRATE AT D IFFERF.IIT C SECT! OtiS OF THE CHAIU•IEL C Ql FOH TilE ii-ILET C Q2 FOR TilE IIIDDLE SECTION C Q3 FOR TilE OUTLET c **********************************~****** c
c
0 !fiE tiS I Otl ELEt·l£ T( HE, 4), DELTA ( NK), AIIOD E( I~N, 4), TU ( I IE , 3 ) 1 UtI ODE (lit I)
Sut\1 =o. o SUt\2=0 .0 SUI\3=0.0 DO 7 1::2,8 SUI 1 I = SUI\1 +U IWD E ( ( I~·: 9) -8) SUt12=SUI12+UIIODE( ( 1":9)-4) SUi13::SUI13+l!IIODE( 1'':9)
7 COIITI NUE
C TO OBTA!IJ THE FLCJ\./KATE, THE PRODUCT C (U~':DY) \·lAS HITERGRATE ACROSS THE CHAi~NEL c
Ql ::( Ul!OOE( 1) +2.Q·.':SUI11 +UIIOOE(7 3)) It 6.0 Q2:: ( U 1'1 00 E ( 5 ) + 2. o~·: SUf\2 +UtFJD E ( 77) ) I 1 6. 0 Q3 = ( u ttoo E ( 9 > + 2. o~·: sur 13 +U t·l oo E ( 8 1 ) ) 1 1 6. o RETURN END
172
c c c c c c c
c c c
c
SUOROUT! 1/E Gi\IHJET( G 1, G2, (13 'GLf I G5 I G6 I f.ll, 82 I 133' TAR EA, .t\J, L3 J, AK 1 BK, .J/\ Y, KAY t F, E lEI \E T, ANODE 1
T I ' FAR llU ' F 1\f{ ' HE ' t~K' I Ill)\/ I H}
THIS EVALUJ\TES VARIOUS ELEI\Et!TAL COIISTAIITS THUS TilE GJ\ti:IAS(G'S) TilE 1JlT1\S(3 1 S), /~liD THE 1\RE A lJF ,\I I t: L El IE trr *********************************~*******~**
Dii\EiiS !llil ELEI\ET( IIE,If) ,· i\i:ODE( fl:t,Ir) REAL I ·,u , J/i Y, 1(;\ Y
IIOuAL liU:It3ERS AI!D VISCOSITY
I 1 = EL E:\E T ( I, 1 ) 12=ELEI\ET( 1,2) 13=ELE:'iET( I,;::) IIU::[L EI·IE T (I, 4)
C OBTAIII TilE LOCAL COORDHIATES 110\·/ \IITII RESPECT· C TO THE 11 NODE. c
c
AJ:::MIODE( 12, t )-MIODE( It, 1) BJ::AIWDE( I2,2)-AI~ODE( It ,2) AK=L\NODE( 13, t )-MIODE( It, 1) BI(=ANOD[( i3,2)-AIIODE( It ,2)
G 1:: f FAR~: ( JAY-KAY)~: ( H'':H) ) /F G2= ~ ( FAR'':KAY) ,·:( W:H)) /F G3=(-1 .O*(FAR*JAY)*(H*H))/F . ALPHA I =-KAy,·, JAY'': ( BJ,': J\J+BK'': AK) -AK'':BJ,':
T (KAy,·, ( -2. 0~• AJ- B J) +JAy,·: AK)- AJ,': BK'': (KAy,·, TAJ+JAY*(-2.0*AK-BK))
ALPHA2 =KAy,·, ( JAy,·: AK~• BK- ( BK,': ( AJ~: AJ) + AK,;• ( BJ~: B J) ) ) ALPH A3= JA y,•: ( KA y,•: B J'': AJ- ( B J'': ( AK'': 1\K) ) - ( 1\J,': ( B K'': B K) ) ) G4=ALPHAI/F GS=ALPH/\2/F G6=ALPIM3/F Bl=(((AK+BJ)-(AJ+BK))*H)/FAR B2=((BK-AK)*H)/FAR B3=((AJ-BJ)*H)/FAR RETURtl J::HD
173
174
c *******************~*********************~*~********************~* C ,., F I i l I T E E L U t: liT 1\ i ~~\ L Y S 1 S ,., C :': : lA Ill fJI\lH11,111 I .. C ,., IIESH= (~lX:J J ·:: c *****~****************~************·'****************************** c c c C LH IE I:;, l ~JI I ALL NU{i\ Y S
,... I..
Dlili:ilS!OI! /\IInDE (81,1f), C(9il,1), Dl::l.T~\(162); TEL E I IE I ( 1 20 , Lf) I lJ ( I ;~::;I 3 ) ' ! !(! [ L T A ( 6 ) , u TJD E ( 8 I ) , TV:JODEt81), V(123,3)1CEl.iA(98)
f<EAL K(~l~J,JG) ,I<.E( 6,6) 1 lllJIIODE(81) \li< I TE ( 6, 998)
99 8 F OR.i1A T ( 1 flO I 3 OX' I :'::';;';:'::';:'n'::',·:'::'<:'::'::'<:'::'::'n'::'"'::'::'::':;';:'::':\'n':>n':·.'::'::'::'::'::'n'.-:'::'n':·.'::':;';;'n'::'r:'n'::'::'d: T:';·.': I ,
T'*********************'l T31X, ,,., 1\llf\l.YSIS FOR llE\/Totiii\11 FLQI./; FLO\-/ ElEHAVIOUR T I II D E;\ = 1 ;': I I T31:<,':': THROUGH 1\ DIE OF 1\il t\IIGL.E SIID\/!1 1
T ' 13 E L mt ,·:~ TI31X 1 '***********************************************************' T, T'**u***********'ll) .
C ,·..,·:·::·::·:: C Oi I TiW L I i lTE R (1ER S •':·:.-:'::'::': A llD 0 T H El\ I i lF OF;;~!\ T I Oil C READ I i I ,., ,.,., . ..,.,,., c
READ ( 5' /)Ill~ I I IE I tnl, liD F' I l, I f3\ I' D L' HI \I I Oil \ /R I T E ( 6 , 7 ) i lf(, liE , II! I , I W F 1 I i, IG\·1, D l , H , \I, 0 H
7 FOR11AT(.31X,l;I}H COIIHWL IIITEGERS , .. ,.,.,.,,.,.,.,,.,., . ..,.,,.,,.,CDIITRDL Ii'ITEGERS,I,I, T31:<,281i IIHTIAL tiD, OF D.D.F. tlK=,I5,1, T31Xt23H tiD. OF ELEI\EHTS IIE=,I5,1, T31X,28H 110, OF I'IODES 1111=,15,1, T31X,2811 REDUCED HO. Of o.o.F. IIDF=,I5,1, T31X,28H tlO. OF KNO\·HI D.D.F. II=, 15 1 1, T31X,28H l3AIID\·/IDTH IB\.1=, 15,1 T31X,28fl DIE LEI'IGTH DL=,Fio.4,1, T31X,28H HEIGHT OF DIE AT It!LET fi::,F10,4·,1, T3 1 X, 2 811 \II D Tf I OF 0 IE I.J::, F 1 0. Lf , I, T3 1 X' 2 8 H II 01,1- D HIE ,,IS I OI·IAL I z ER DH=, F 1 0. 4 I I' I, I)
DO L17I:::J IWF,2 \ 11\ I T E ( 6 , ~ d ) I , D E L T A ( I ) , I+ 1 , 0 E LT 1\ ( I + 1 )
1+8 F Ofz:\.L\T (3 1 X 1 I 5 , 5 X, F 1 0. 5 ; 15 , 5X, F 1 0. S) 4 7 C u i lT I i ;u E
176
C CALCULATE TilE ilcJOi\L VELOCITIES DF EACH C~.u:;:IIT c
c
0 J Lt ') l -~ 1 , I it: C i\ L L V El ( U , V , I , E LEI r T , D [ L T A , i~: I 00 E 1 IE , I If~, lilt ,
T\.',H) 49 COIITI IIUC
C U\l..CULATE THE VELOCITY uF EACH !lODE c
c
CALL VELOC(UIIODE,VIJODl:,U,V,El.EI'ET,DF.LTI\,MIOOE, HIE , t I K , I -HI , i I , H )
C I·IOOAI. VISCOSITY
c c c
c c c
c
C A L L V I S C 0 ( ; \U II OD E , E L E 11 ET , 0 E lT A , A: I GD F. , lJ i I 0 D E , '.!i I U D E , 1· I E , II K , Ill l , 1-1, H ) 1/iziTE(G,If:JJ) ITER
Lf 50 F 01~1 \AT ( 1 HO' 3 O:<, I ITER A TI 01~·.'.:'; 110. -:I I IS , /) \.'RITE( ),I,:J1)
1+51 F;J,;: i/-1 T (31 X 1 1 : llJ:J [;'n':·:::IIJ. 1
1 5X 1 1 X- VE LOCI TY 1
1 SX, T1 Y·· 1.'ELGCITY1 ,5X, 1VISCOSITY 1 )
DO If 52 I:: I , i 1: I 1·/f\ I TE ( 0,G53) I ,UIIDDE( I), VI lODE( I), t\U:IODE( I)
453 FOI~IIAT(31>:, IS, 10X,F10.5,5X,F10.5,5X,F10.5) 452· CONTI liUE
CALCULATE THE CHAIJGE I II PRESSURE .
CALL DEL TAP ( DP, ELF. 1\E T, DELTA, I\ I JDlJ F, U, V, I IE 1 ilK, I HI, \I, H) \liUTE(6,53) ITEI(,lJP
DO 59I=1 ,i!DF CELTA( !)=DELTA( I) Df::I.TA( I)=C( I, 1)
59 COi !TI iiUE !F(I(DDE.IIE.l) GO TO 39 \/RlTE(6,(>6) TEST
66 FORI~T(11ID,30X, '************************************************** T-::·::t, ~ ********************'/ T31X, 1 ;': VALUE OF COIIVERGEIH CRITERlOtl IS',F10.5, T 1 X, I ;':I
I·IR I TE ( 6, ~; 6) ITER 56 F ORr I/\ T ( I 110, 3 o:<, 5 91l ITEI~ATI at I NUt H3E R •'n'n'n': ITERATION 11Utt8ER ,.,.,,.,.'! TEHA T I
c
Tml I~U:\BER =,15,/,/, TI;2X,J9H PRESE 11T PREV!OUS,11X,I9H PRESEtiT PREV!OUS,/ 1
T31X,57H DOF IW. PSI! PSI! OOF fiu. PHil PHII) 00 571=1 ,IIOF,2 1/R I TE ( 6, 58) I 1 C ( I , 1 ) , CELT A ( I ) , I+ 1 , C ( I+ 1 1 1 ) , C E L TA ( I+ 1 )
58 F ORI·1A T ( 3 1 X, 15 , 5 X , 2 F 1 0. 5, I 5 , 5 X, 2F 1 0. 5) 57 COII'fl tiUE
C CALCULATE TilE ~IOOAL VELOCITIES OF EACH C ELEt·1EtH c
c
DO 60 1=1, tiE CALL VEL ( U , V, l , E L E t1E T 1 0 E L T A; AI·! ODE., I~E , IH~, N~l,
\.JR I TE ( 6 I Sl+ ) IT E R , Q 1 , Q 2 ' 03 IF(KOOE.EQ.1)GO TO 100
GD TO 39 \·IR I TE \ 6 , Lf 5) ITER HR!TE{6,4f)) DO 631=1 ,IWF+N,2 1-IR I T E ( 6 , 6 If ) I , 0 E LT A ( I) , I + 1 , 0 E L T A ( I + 1 )
6L: F OR~1A T (3 1 X, I 5 , 5 X, F 1 0. 5, I 5 , 5 X, F 1 0. 5) 63 COI'ITI NUE
101 103
\.JR ITE ( 6, 1 0 1) KOD E FOR1'1AT(31X 1 'KODE =', Il , 1THEFORE COI·IVERGEIICE TOOK PLACE') STOP END
,·: J\tlA L Y S I S ·:: THROUGH
FOR tiC\HmiiJ\tl FLO\-/; FLO\-/ BEHAVIOUR A DIE OF All U.llGLE SH0\-111 BEL0\1
C OH TRO L I liTE GER S ·::,·n·:·:n·:,'n':,·:,'n': C OIIT RO I. I tHE GE r- S
II~ITIAL tiO, Of D.O.F. 110, OF ELEI\E IHS 110, OF NODES REDUCED 110. OF O,O.F. 110, OF KII0~/~1 D. U. F. BAIJD~/!OTH DIE LEI~GTH HF!GHT OF OlE .L\T INI.ET II IDTH OF 0 IE 1101'1-D It\EI~S IOHt\L IZER
NK:: NE:: Nil=
NO F= tl=
I BH= DL=
H:: \·/=
DH=
162 128
81 98 Lf6 13
1. 0000 1 .oooo 1. 0000 o.oooo
178
I NDEX=1 1':
***************************************************************************** ,·:THIS DIE IS TAPERED. /\T AN AIIGl.E= 0, RADIAilS ·:: *****************************************************************************
ELEt-\E NT I NF 0Rt1A T I mt,•:,'n'n'n>n':;•:,•:,·:·::,': EL EME HT I NF OR 1·1A T 1 OH
Resulting computer programme from the second analysis.
****************************************************************** :~NOT AT I OtiS ******************************************************************
:'r A :'r
A(3)=COIITAitiS THE FIRST THREE GEOIIETRIC COtiSTAIITS OF At~ ELEIIF:HT,
ABC(NE,9)=/\RRAY OF ELEI1EI·ITAL GEot1ETRIC COIISTIHHS.
IHIODE(1~11,5)=ARF\AY Cm!TAIU!I1G ll·lFORIIATint~ Al30UT 1\LL THE ~lODES, THE FIHST H/0 POSITIOIIS FOR THE (X) 1\tlD (Y) COORDII~ATES(GLOGAL) MID THE LAST THREE ~OR THE ll,O,F, I·IOS, I.E. X-VELOCITY,Y-VELOCITY,AI~D THE PRESSURE RESPECTIVELY,
ALPHA=AilGLE OF TAPER OF CHAt·II~EL,
B(3)=COI'IT,'\INS THE SECOI10 THREE GEOtiETRTC CotiSTMITS OF 1\H ELEI1ENT,
'",'( c 'i't
C(3)=COtiTA!NS THE THIRD THREt GEOf-1ETRIC CotiSTAtiTS OF AN ELEfviEiH.
CE(I~K,l)=ARRAY H! 1/HlCH THE SOLUTiotl FOR THE UtiKIIO\ItiS IS STORED.
CELTAUIK)=PREV!OUS VALUE OF THE SDLUTIOtl VECTOR(CE) IS STORED II~.
:': D :':
DELTA(IIK)=ARR/\Y OF REQUIRED HODAL V/\LUES, THESE ARE THE (X) MID (Y) VELOCIT!i=:S, MID THE PRESSURE.
Dl=DIE LEI·IGTH,
".'( ·:: ..,·: ..,•: ;'r
:·: E :': 'i'('i':..,·:-.'(".'(
ELEiiET{!lC,S):::Af=~RAY CotiTA!llltiG TIIFORt1/\Tir:tl AROUT ALL THE ELE11Et1TS, THE FII~ST THREE POSITIO!:S FOR THE tiUDAL f·IOS, A liD THE LAST HJO FOR TilE IJ I SCOS ITY !1110 THE AREA RESPECTIVELY,
1 85
~·: F ~·t
-;'r .,.r..,•r..,·:~'t
FBl=FL0\·1 BEHAVIOUR li·IDEX OF THE POLYIIF.R.
F OIJ R i 2 ::VAL lJ E 0 F ld 2
H=HEIGHT OF CHM!NEL AT ItiLF.T.
I 1 S=COUtiTER; DEGREE OF FREEDot1 t1Ut1BERS; NODE NUI·IBERS
I B\I=BA l~D\-1 I 0 TH
ITEP=COUtlTER lJF THE 110. OF ITERAT!Oi'IS OONE.
~·( J >'t
J' S=COlHHER 1 DEGREE OF FREEDot1 NUt1BERS; ~lODE 11Ut1BERS.
K(I~DF, 18\·I)=SYSTEII STIFnlESS tiATRlX HI RECTi\NGULM A NO 8 A 1~0 E 0 F OR t·\,
KE(9,9)=ELE11EtHAL STIFFtiESS nATRIX.
KDDE=KEEPS CDUHT OF THE liD. OF UI~K\IOI·IS ~lOT. COtiVERGED.
K1(6,6l=A SUB-MATRIX OF 412.
K2(6,3)=A SUB-MATRIX OF KE(9,9)
LCOUNT(I·Iti)=KEEPS COUIH OF THE t·IUI1BER OF ELEI1EIITS COtlTRIBUTHJG TO A t!ODE,
11U=V!SCOSTTY OF Atl ELEI\EIH
1·\U~IODE(tlll):::ARRAY OF THE VISCrJS ITY OF Ei\CH tlODE.
186
N=tiO, OF IJOil ZERO O,O,F.
NDELTA(6):::.f\HR~\Y CClllTAIIHIJG THf O,O,F, tto,; OF E1'.CH !·lODE,
HDELT(9)=ARRI\Y CCIIITAirlltK1 THE O,O,F, tiOS OF 1\N El.Ht::NT.
NDF=UIH<rW\-/H 110, OF 0 ,0, F,
NDOF=COUTER FOR THE O,O,F. NUMBERS
NE:::tlO, fJF ELEI\OlTS,
NK=INITIAL NO, OF D,O,F,
NKI,NKJ=D,O,F, NO, OF AN ELEMENT.
NN::NC, OF IIDOES.
NNODE=IWDAL liUt\BER •
..,., ..,·: .. ,., ;': -,': ·:: Q ~':
Q(l)=FLO\/RATE AT VARlOUS SECTIONS OF THE CHAllllEl.,
;': T ..,·: ~~ l': "i'( "i': ~·,·
T=TANGENT OF AN ANGLE,
~·: "\~ ~·: ..,.,..,.( ",'( .. ,.( .., ...
\./=\.flDTH OF DIE
X= X- COORD I I~ATE (GLOBAL) OF A tlODE, XM=X-COORDII,IATE OF THE CHITROIO OF Ml F.LH1ENT.
XI ,X2,X3=THE X LOCAL COORDHIATES OF THE VERTICES Of At~ ELEt1EIH HITH RESPECT TO ITS CENTROID,
..,·:..,·:;': "i':-,':
Y=Y-COORDJt.lATE(GLOBAL) OF A NODE.
nl=Y-COORDWATE OF THE CENTROID OF All F.LH\ENT,
Yl, Y2 Y3=THE Y LOCAL COORD! IIAHS OF THE VERTICES OF AN ELEtldiT \liTH Rf::SPECT ro ITS CEIHRtJID,
187
SUBROUTINE CONTRO(NK,NE,NN,NDF,N,IBW,DL,H,W,ALPHA) c c ********************************* C THIS READS IN CONTROL INFORMATION c ********************************* c
3 FORMAT(1HO, 'ITERATION***NO.=',IS// T1X,'NO. OF UNKNOWNS NOT CONVERGED YET=' ,IS)
DO 450 I=1, NDF
C PREVIOUS VALUE OF THE SOLUTION IS STORED C IN (CELTA) ,AND THE PRESENT VALUE IN (DELTA). c
c
CELTA(I)=DELTA(I) 450 DELTA(I)=CE(I,1)
IF (KODE.NE.1) GO TO 200
C NON-NEWTONIAN SOLUTION IS NOW AVAILABLE TO BE C PRINTED. c
CALL FINALE(CELTA,DELTA,ANODE,ELEMET,NK,NN,NE,NDF,N,Q,ITER) 550 WRITE (6,600) KODE 600 FORMAT(1X,'KODE=' ,I1,2X,'THEREFORE CONVERGENCE TOOK PLACE') 650 CONTINUE
RETURN END
~; LHll( OUT I liE I liP LJT ( l:l. U 1 F. T, t\ IIOD E , D F LTJ\ 1 !\11 C, T i 11< I t If. ' 1111' I J[) F' t I' I !3\ I, D l. ' i I' \·I' i\L pIll\)
c c *********~************************ C T II IS I IIPIJT S :~.LL Tlll: /W 1\ l L1\llLE D 1\T/\ c ********************************** c
c
llim::1s ror1 r::t. F.rtET{ IW I 5), AilODE( 1HI 1 5), on.TA( tHO, TAG c ( liE I 9) , 1\ < .1 ) , n ( 3 ) , c ( ;1 )
C I·IODAL IIIFDnlti\Tiotl IS OllTI\I:IEO c C 111E EtiTRIES OF MIOD[ i\RF. GEIIERATED. c C THE FIRST 1\I~D SECDIIIl CIITR!ES 1\RE' Fm THE C COORD I H.I\TES OF TH't II ODE i\!10 THE OTHF.Jl.S C fiRE THE OF.GI\Ef: OF Ff\UDOI1 IIUIIBEHS C AS SOC I 1\TF.D TO THE tiOOE. c
c
C'\LL GE IIODF.( 1\IIODE, II! I, tl, IIDF) \ IR I TE ( (> , ?. 0)
20 F mr tAT ( I J 1 X J I 1100 AL I !.JF OR II!\ T I OJ pH:l';-.':,·:,':l'tl'n':l': I '
11 1l·XLii.L ItJFor.:ll/l. r r or,,.,.,.,.,.'"'·:·::-.· ....... , ... , ........ ~ !OD 1\L. r ~JF or~·v\ rIot!' I 1 TJ1X, 1 1100[ 110. 1 ,11X, 1X Cllflf\0 1 ,3X, 'Y COORD' ,5X, 1 00F 1 ,
T' ilfJ. 1 ,?.X, 1 00F 110. 1 ,2X, 1 DOF 110 1 )
D 0 ?. 1 l :: 1 , I·H I
C Q.O.F. 11Uf18ERS c
c
I 1 =!\II Of) E( I, 3) I 2 =AI-IIJO r( I, If) !3=!\!100~( !,5) \F; I TF. ( 6 1 ?.2 ) I I /\, 110[) E ( I I 1 ) I /\1 JCD E ( I • 2) I I 1 ' I 2 ' I3
GO TO 260 c C 'THE; GLOBAL COORDI!!,\TES f1IVE!I ABOVE ,'IRE FOR 1\ UtiiFDRII C RECTM!GIJLI\R IIESII. IIODIF!f:D tlEL0\-1 TD M:COUIIT FlJR TliE C TAPERIIIG OF THE DIE. X CL1Cli\OlllATES REI\,\!11 TilE S;\r~tE c
I·IR I TE ( 6 , ?. 3 ) 2 3 F nr; t \AT ( 1 11n, ' c m R E CTF n 11 oo AL I IIF mr tCI TT nr 1,·,·:.·:,·: 1 ,
T 1 ,·:·.':·.':·.'~ ,·..,·n•:;';,':;':,•:-.•..,•:;•: CDR. R F. CT F. D I I 00 AL I 11F OR i \!\ T I 011 1 I I T1 X, 1 HOOE 110, I ,lfX, 'X COORD' ,3X, 'Y CrJr1f;0 1 ,sx, 1 DOF 110, I
T ,2X, 1 00F 110. I ,?.X, 1 DOF tJO I)
T=TA I~( ALPHA) R=T/ DL DO 21+ I= 1 I lltJ 1\! I 0 0 E ( I 1 2) :: /HJ 00 F. ( I 1 2 ) ,·, ( 1 - MIOO E ( I , 1 ) •': !1) 1·1 =AI,JOOE( 1,3) I 2=AIIOD F-:( I ,Lf) l.3=AIIODE( l, 5) IN~ ITE (6 ,?.2) I I AI·IODE( I J 1), /\IIODE( I, 2)' I1, !2, !3
2Lf Cnt!TI ti!JE c C DATA ABOUT TilE ELEIIEIITS IS READ II! c C THE FlflST THREE POSIT!nt1S FOR THE 1!09AI. 11Ut1f3F.RS, TilE C FOURTH FDF\ THE VISCOSITY AIID THE FIFTH FrlR THE i\J~Et\ c
A.REA::J,5·::(X2·::Y3-X3'''Y2) E L E!· IE T ( I I 5 ) :: J4R E A
C GEOtiETRTC CDIISTAIITS OF EACIJ ELE!1EIIT IS C OBTAI I·IEO , E/\Cfl QUMITITY IS D IV IDEO l3Y T\/TCF. THE C AREA FOf~ C OliVO! IE I ICE c
c c c
27
28 26
260 32
A(1)=(X2*Y3-X3*Y2)I(?.O*AREA) A(2)=(X3*Y1-Xl*Y3)1(2,0*AREA) A( 3)::: (X J>':Y2 -X2'''Y 1) I ( 2 .o·:: AREA) 8( 1 )::(Y2-Y3 )I(2.0>':AREA) B ( 2 ) :: ( Y 3 - Y 1 ) I ( 2 , 0>': ARE A) B(3)=(Y1-Y2)1(2,0*AREA) C ( 1 ) ::: ( X3- X2) I ( 2, 0>': ARE A) C ( 2 ) ::: ( X 1 •• X3 ) I ( 2 , 0 >'q\R E A) C (3 ) ::: ( X 2 •• >: 1 ) I ( 2 , 0 ,., ARE A) DO 27 J::1 ,3 Acc( r, J) =A( ,J) ABC( I, JH)=8(J) ABC( I, J+6)::C(J) COIITI I'IUE \.JR r T E < 6 , :~ n) I , I 1 , I 2 , I J , E L H t E r ( I 1 tf > FOR I"IA T ( 1 X 1 I 5 1lf X • I 5 ' s X I I 5 I 5 X , l5 ' 5 X J 2 F 1 0 • 5 ) emiT!. I'IUE
OOUIIDARY COIIDITIOIIS STORED Ill DELT/'.().
1·/R IT F. ( G , 3?.) FORt1AT(ltl0 1 32:<, 1 f30UIIDNW COIIDITIOiiS 1 /
3 ~ \ IP, I T E ( () 1 3 If ) I 1 D F. LT 1\ ( I) 3 -f F rn r lA T ( :ll :<I I~' I 5 ~<, F 1 o, s J
IU~ TURII E rID
1 91
SUBROUTINE GENODE(ANODE,NN,N,NDF) c c ************************************ C THIS GENERATES THE NODAL COORDINATES C AND THE DEGREE OF FREEDOM NUMBERS c
DIMENSION ANODE(NN,S) c C IN THIS PROCEDURE, THE UNKNOWN NODAL VALUES C ARE NUMBERED FIRST, FOLLOWED BY THE KNmt\TN C NON-ZERO VALUES AND THEN THE ZERO VALUES. c
c
NDOF=1 DO 1 I=1,9 DO 1 JJ=1,9 J=10-JJ I1=0 I2=0 I3=0
C OBTAIN NODAL NUHBER c
NNODE=1+9*(J-I) c C IF NODE LIES ALONG THE TOP OR BOTTOM C OF THE CHANNEL WHERE THE X-COMPONENT OF C VELOCITY IS ZERO, THE X-VELOCITY C D.O.F. NUMBER IS LEFT BLANK IN THE MEANTIME c
IF((J.EQ.1) .OR. (J.EQ.9)) GO TO 2 c C NODE IS NOT ALONG THE TOP OR BOTTOM OF C THE CHANNEL. ASSIGN A DEGREE OF FREEDOM NUMBER C TO THE X-VELOCITY. c
c
I1=NDOF NDOF=NDOF+1
C IF NODE LIES ALONG THE ENTRANCE OR EXIT C OF THE CHANNEL, WHERE THE Y-COMPONENT OF C VELOCITY IS ZERO, THE Y-VELOCITY D.O.F. C NUMBER IS LEFT BLANK IN THE HEANTIME. c
IF ( (I . EQ. 1 ) . OR. (I . EQ. 9) ) GO TO 2 c C NODE IS NOT ALONG THE BOUNDARIES OF THE C CHANNEL. ASSIGN A DEGREE OF FREEDOM NUMBER TO C THEY-VELOCITY. c
c
I2=NDOF NDOF=NDOF+1
C IF NODE LIES ALONG THE ENTRANCE OR EXIT C OF THE CHANNEL, WHERE THE PRESSURE IS KNmt\TN,
1 92
C THE PRESSURE D.O.F. NUMBER IS LEFT BLANK IN THE MEANTIME. c
2 IF ( (I . EQ. 1 ) . OR. (I . EQ. 9) ) GO TO 3 c C NODE IS NOT ALONG THE ENTRANCE OR EXIT C OF THE CHANNEL. ASSIGN A DEGREE OF FREEDOM C NUMBER TO THE PRESSURE. c
C THE ZERO I·IOOAL VALUES ARE 1101·/ ASS IGtiED C DEC1f1EE OF FREEDOII I!LH1BERS, c
005!=1,9 DO S J::l ,9 !JilODE:: !+C);'<( J-1) !1 =AIIfJOE( IIIIODE,3) !2=1\IIODE( I~IIODE:,I+) I 3:: A liDO E (I if I onE, 5) IF(I1,GE.1) GO TO 6 AI' I il D E ( i I i I 0 D E , 3 ) :::I i~ 0 • 1 il=l\1· 1
6 IF(I2,GE.1) GO TO 7 AliOfJE( IF lODE, 1;)::!·1+0, 1 1·\=11+1
7 IF(I3.GF..1) GO TO 5 AIIODE( I·I!IODE,5)=!1+0. I 1·1=11+ 1
~; COIITI I·IUE RE TlJR!\1 END
193
194
sun R n lJT r 1 IE /\S :; E llll ( 1-:. 1 I( E , r:: LEt 1 E I' , 1\ll onE, n E u /\ , c c , AD c I 1~ 1 , K 2 I IlK, t 1 E , lH 1, T I I D F , I ·I 1 1 !31 I , fJ L, II , I I , A l. PI It\, I T ER)
c c ~************************************~****************** C TillS CI\LClJL/\TES TilE ST!FFIIESS 111\TRI:< or EI\CII ELEt\EIIT AIIO C IW,ERTS IT lllTrl THE SYST!.-:t\ STIFF!IESS i\1\HUX. TilE C 1\1\Tf~I:< iS n/\tiOI:O 1\ilD THE L\ltiDI<It; \iF.CTr1r. DflTAI!IED Ill C T II [ r fW C E S S c ***********************~********************~*********** c
c
D I! \ E I IS I 0 II E I. [11 E T ( ! l E 1 ) ) , /\ ll 11 0 [( II ! I , S ) , D C L T 1\ ( ! I 1\ ) , C [ ( II D F , 1 ) 1 ~l\l3C(!lE 1 CJ) 1 !111ELT(C))
f, ':: 1\ L V, ( l I[) F 1 I ?,I) , K!: ( C) 1 ~)) 1 I( 1 ( 6 1 r; ) 1 \(2 ( 6 1 3 ) fJ 'l I.L f) I:: 1 ' : ) D F J .1 I~O J = 1 , I ~)\I l<(l 1 J)=O.O Ct.( l, l )=0.0
L:-0 CDtiTI iJUE
C Ei\CH ELE!\EIH IS BROUGHT HI FOR PROCESSIIJG c
c
D () If 1 I:: 1 I I·IE D 0 Lf 2 I c :: 1 I 9 DO LL2 J::1 1 9 !( E ( I c I j) ::o. 0
1+2 CCJ!ITI IHJE
C OllTAIII THE I\1~TfHCES K1 f\'10 K?. FD>\ ELEtlF.iiT (I) c
Ci~LL 111\TRIX (1:1,K2,i\flCd,!·IE) IF ( ITER. GT , 1 ) CALl. UP D 1\ TE ( E L Ell E T 1 AWl:l E 1 11 E L T A 1 K 1 1 1(7. , I 1 ~IE , I II I, !·IK)
c C EUJ101T/\l. ~.lTIFFIIF.SS 1\f\TRI:< ·Is C/\LCUL1~TED
c DO !13 IC=1 1 6 DO 43 J::l ,() 1\ E ( I c , J ) = 1-: 1 ( I c I J ) ,., E L E r 1 E T < 1 1 1; ) ,., 2 • u ,., E LEt 1 ET ( T , s )
4 3 Cot I Tl t IU l:: DO 4L; IC::1,6 on L!L; J=l 3 K E ( I C 1 J + 6 ) = K 2 ( I C , ,J) ,., 2 , 0 ,., EL El \ E T ( I , 5 )
/+4 COIITI i·IUE Wl L; 5 I C::: 1 ,3 on 1f5 J::l 16 I< E ( I C+6 , J FI(E ( .J, I C+ 6)
L;5 COIITI IIUE
C ELD\EilTAL ST! FFI!F.SS ~ll\TrUX IS /\Vi\. !L/\!.lLE; !10\/ c C 0 8 T 1\1 1-1 Tfl E ll. 0, F, ~lOS, c
c
I I ::[LEIIO( I I 1) I?.::: r:: L Ell ET ( ! , 2 ) I3=ELE11CT( !,.3)
1,1 0 F. LT ( 1 ) ::: f\! HJ 0 F ( I l , 3 ) . llOELr(2)=1\llflnt:( !?. 1 3)
II!' E LT ( 3 ) :: i\ W!D F. ( I3 , 3 ) i I D E LT ( 4 ) ::: J\tllJIJ F. ( II , If ) I I[) F. L T ( 5 ) = t\ II 11 D E ( I ?. , If ) IPlF.LT(f)):::,\1/nOE( I3,lf) lllFLT(7 )=t\IHlOF.( II ,5) I JOEL T( rl) ::J\tiODr::( I2, 5) :!DEL T( 9) =1\lliJDF:( !3 1 5)
C CIIECK ALOIIG THE Rnll OF t\ATRIX, C IF IT IS f1F.Yr1~1D THE f~i\1 !GE OF ( K) I\Er1LECT IT C IF IT IS IIITflll·l THE r{1\IIGE, GO i\LOIIG TilE UJLltrHJ c
00 2 1<!=1,9 c C IF BEYOIIO 1< 1 S RAIIGE, TRY !IEXT Rm/ c
c c
c
IF(IlDELT(I\I) ,GT .tiOF)GO TO 2 I i K I :: I l 0 E L T ( 1\I )
DO 3 KJ=l ,9 I·IKJ=I,IOELT(K,J)
C IF BEYGriD f: 1 s RMIGE, OBT/1!11 THE LOMIIIG VECTOR c
IF (IIIU.GT,I!OF)GO TO 4 c C \.f!THHJ K'S RIHIGE. 110\/ !~I OAIJDEO FORI1 c
IJKJ 1 =NI\J- HK I+ 1 IF(Ili~J1.LT.1)GO TO 3
K (I If\ I 1 I·JI(J 1 ) =I< (Ill< I, I 11\J 1 ) +KE ( K I, KJ) G'l TCl 3
I~ C E ( II K I , 1 ) =C E ( I II\ I , 1 ) - K E ( 1\ I , 1\J )":DE LT 1\ ( I II< .I) 3 CDIHII,IlJE 2 CotiTI HUE
Lf 1 C 0 l'lT I IIU E RETURIJ E r JD
195
SUf1ROUTJ fiE f IMR I X(~~ 1 I I<<' i\[JC, l I liE) c c *********~*********~************************ C THIS 08TATfiS THE IV\TRJCES 1\1 A:IO !(2 FOR EACH C E L. E: 1 E. JT ( I ) c ******************************************** c
c
o I r 1 o1s r n 11 A r1 c <t 'E I 9 > , .l'l. C3 > , G < 3 ) I c < 3 ) REAL K1(6,6),1<2(6,3)
C GE011ETR!C Cm!STAIITS OF EACH El.E!1EilT c
on. L~9 J=l ,3 fl( J) =M3C( I, J) G ( J) =A f3 C ( I , J+ 3 ) C ( J) =t\BC( T, J+6)
1~-9 COIITI IJUE c C 0 f3 T A T f I 111\T H I X K 1 c
DO 50 11=1 ,3 DO 50 J::J,3 K 1 ( I I , J ) ::: 2 • 0 ~·, 8 ( I ·1) ~·: f3 ( J ) + C ( f 1) ~·, C ( J ) I( 1 ( H+ 3 , J + 3 ) :: 2 ~ 0 ,., C ( I 1) ,., C ( J ) + f3 ( 11) ,., f3 ( J ) I< 1 ( f 1, J+ 3 ) :: t ( II; ~·: fl ( J ) I< 1 ( Ill-] I J ) =B ( 11) ..... c ( j)
SUBROUTINE UPDATE(ELEMET,ANODE,DELTA,K1,K2,I,NE,NN,NK) c c ********************************************* C THIS SUBROUTINE UPDATES THE VISCOSITY OF EACH C ELEMENT c ********************************************* c
c
DIMENSION ELEMET (NE, 5) , ANODE (NN, 5) , DELTA (NK) , NDELTA ( 6) REAL K 1 ( 6 , 6 ) , K 2 , ( 6 , 3 ) REAL MU
L oc < r I J) = r + ( J -I ) qJo HI GO TO (1000 1 2000);1\Kf<.
REDUCE 11/\TR I X
DO 280 tJ::J 1 1·lt~ 1·11 =LOC ( 1! 1 l) DO ?.60 L=2tlll-1 ~IL:: LOC (If, LJ C=A( IlL) /A( Ill) I=!I+L-1 lF(NN.LT.I) GO TO 260 J::O 0 0 2 50 K =L I ~tl J=J+l I J::: l.OC ( I, J) IIK=I.OC ( f·l 1 K) A( I j) =A (I J) -c~·: A( IlK) ,'\(IlL) =C COIITI I·HJE GO TO 500
REDUCE VECTOR
DO 290 t-1=1 INN III=LOC(Il,l) DO 285 L=2 1 1·i'1 tiL=LOC( 1~, L) I==I·I+L-1 IF(tiii.LT. f) GO TO 290 B (I ) ::: R ( I ) - A ( !·I L ) ~·• B ( I· I) B( 1·1) =B( II) /A( Nl)
BACK SUBSTITUTION
r·r=~rN H=t·l-1 IF(N.EQ.O) GO TO 500 Oil h,OO K=2,1·1t1 111\=LOC (H. K) L:::fi+K-1 IF(Ilt~.LJ,L) GO TO lfOO B ( t I) ::: f3 ( ! I) - 1\ ( NK) ~·• 8 ( L) CmiTI I·IUE GO TO 300
500 RETURN EtiO
c c c c
c
s !JEll\ ourr 1 IE our PUT c r TE F 1 o E LT A, MiuD E 1 r: L Er 1 u I o, u: I r ~~~I r 11: I~~~ i, 110 F 1 'I)
TillS OUTPlJT TilE SC1U!T!,1!1 Fm IIEI!T:l'III\11 FUJII.
11 Ir IE liS I ()II [) E LT 1\ ( II f() I /\' :oo E ( ~ 1: I, 5 ) I E UTE T ( ':E , 5 ) j Q ( S1) J c E (t: K I I ) , T 0 r: L T ( I I+ 3 )
C I iiSER.T V.\LlJES OF UIIKIIOIJ!!S l llTO DELTA c
c
rn 1 o r -= 1 , ' : ~J F D [ u 'I ( l) =C E ( I I I )
1 0 Ul' I TI l·!lJF: 00 11 1=1 1 1·1N
C 0, 0. F o ! llJI!G ER S OF A ! I DOE. c
c
J 1 ::f\!<!OD E ( I I 3 ) J2=/1!10fll:( I,lf) J3=1\:noE( I, 5)
11 1 '~ ITE c r, I 1 ~) 1 I o E L r A ( .J 1 ) I n E L r /\ c J?.) , o E L r 1\ c .J3) 1 2 F m 1 lA r c 1 x, 15 I 1 ox I F 1 o. :; Is xI F 1 o. 5 , 5 x 1 F 1 o • s )
C FlJJ'.IRATE RF.r!UIRED. c
CALL FIJJRET(QtOELTi\,1\:!0fJE 1 IIK 1 iHJ) 1./R I H ( (-, , 1 3 )
13 Ff1\!1AT( iH0 1 'FLOI/RATE AT Vi\fUOUS SECT!nt-!S OF (r IF.' II6X 1 'SECTTOtl' ,8X, 1 I FLO\IRfl TE I) IIRITE(6,JI+)(I ,Q( I), I:::J 1 J)
14 FORi1i\T(8:<, I2 1 BX 1 F10.5) \/R I n: ( () , C) 0 )
SUBROUTINE CONVEG(DELTA,CE,NDF,KODE,NK) c c ******************** C TEST FOR CONVERGENCY c ******************** c
c
DIMENSION DELTA(NK) ,CE(NDF) KODE=1 DO 34 I=1, NDF Y=DELTA(I)
C VERY SMALL NODAL VALUES ARE NOT C TESTED FOR CONVERGENCY. c
IF(Y.LT.0.00001) GO TO 34 E=ABS(CE(I)-DELTA(I))-0.001*ABS(CE(I)) IF(E.GT.O.O)KODE=KODE+1
34 CONTINUE RETURN END
202
203
';·,,· .. -,':!JTI '.p·. '"I '''-l i'( .'-l'f\ Ll.-1 T .,, )"r· "' r·•·-1· ·., .. "" ·•r· "'"' "\.1 ITl- 1') ' ... . I I " \ .. ... v.. ~ l I '·. ~ ' \ • I \ I • } .:. , ...... t.. . • • I l I\ ' ' i J I 1 ... , ' . ' I , ' • ' • I ....
c c . ~************~***************************** c T'!I:-; :~-r:·'.TIT!;:r: :J:n:,r··.; T'l[ rr··:.t. '~:·:s~r:.r C ::.\C'I t·q:;:)t..\CE·.;:·:T 1 IS :ii'!~:-:.\T?.fl \!PT! !lY T!IF. C /\PPf\'JDrd.\TE VJ\IJJF. (1F TW FI.Cli/R;\1'E c ~****************************************** c
c
o r 1 E' 1s 1 'l: 1 :J E 1. T,\ < rn~) , .\ • 1o:J E ( u: 1, :>) I E t. Et lET< 11c Is:. , Q < 9) , r.: EL T ,, (1 1 K) R E,\1. lllfl0flf:: ( 81 ) 01 500 f::!IOF+l ,IIOF+!I
C K!Ja·/11 VI\LlJES OF TilE SnLUTr Otl. c·
c
CELT!\( 1)::1,0 500 DEI.TI\( I)::j .0
ll=iiOF+t-1 In. I E ( (j , 1 ) I TE R FflRII/\ T ( 1 I iO, 1 I TEP-1H I OWt·.'t~·.·~·t :lU! W ER ~·:·:"·:·:t ITER A TI n:J·::-::·::·:: l lUll l3ER'''''d:·::y TERA T I 0
T """·:,·,,·: r '' J: 1 n ER ,., ... ".,,., I r E·r:un r 'l' ,,."·"·,,·:r 1u • Fl ER =' , !51 1 T1X, ·~Jnf)E·:,,·,·!fl. ',:;x, 'X-Vt:L'!:: tTY'"'"''':-'.'!::LrJC ITY' 1 !;x, T' Y- VE Loc r yy,·:,·:~·,y- VE 1. nc I TY' , s x, 'PREs suc:;r·::·::~:~:·::·:"'' PRES suRE' 1 T18X: 1 PRF.SEilT1 ,SX, 1 PRt-:VTOI_JS 1 ,flX, 1 PRC:SE:Il"f ,SX, T'PREVIOU~' ,7:<, 1 PRESEtiT1 ,2X, 1 PREV IOUS 1 ).
DO 2 I=1 ,uti
C D.O.F. llliiiElERSS. OF 1\ tJODE c
c
J1=Ail0DE( !,3) J2=AIIODE (I 1 4) J 3 =A II ODE ( I , 5)
C OUTPUT TilE VAH TABLES OF TH/\T t!ODE. c
c
2 \-/R IT E ( 6 , 3 ) I , D E L T A ( J I ) , C E L TA ( J 1 ) , 0 E LT A ( .12 ) , C F. L T A ( J 2 ) , TDELTA(J3),CELTA(J3)
3 F m t 1AT ( L<, I 5 ~ I zx·, F 1 0. 5 , 2 X, F 1 0. 5 , 6 X, F I 0. 5 , I X, F 1 0. 5 ,3X , F 1 0. 5, F 1 0. 5)
C CALCU.LATE AilD PRI tiT THE Fl.OI-/RATE THROUGH THF. C DIE. c
c
99 CALL FLORET(Q 1 DELTA, AIIOOE, tiK,NN) \·IR I TE ( 6 , 1 3) ·
13 FClf.Uii\T(1110, 1 FLOI-/RATE AT VARIOUS SECTIQ:Js OF OIE 1 // T6X, 'SECTIOII' ,8X, 'FLOI-/RI\TE')
HR I TE ( 6 I JLf) ( I, Q ( I} I l= 1 '9) 14 FORt lA T ( 8 X, I 2 , 1 2X, F 1 0. 5)
204
C FiliAL SOLUTIOtl IS Wl\1 OflT/\TflEO. c c
c
I IR 1 T F. ( 6 I '1 ()()) 7CXJ FIYU11\T(JII0,27X, 1 Till~; r;, TltF: SCALED<;()!JITJOI) Fm~ i·lCHI-IIEI/TotiJi\tl FLO\-/
2Fnf:l\t\T 1 1SOX 1 1 SOLUTIITI Ill El?.,l1 t='flH1/\TI//1X 1 1 110DE·.':·:.,·:tl(), 1 1 ~iX 1 1 X··VEI.O :; c 1 TY I ' s X ' I y - v F UJC IT y I I 5 X ' I p 1\ E s s l )[{ E f I :< I) :< I f :< - \1 r: L 0 c 1 TY I I 5 X ' I y - 1/ E L 0 c I T 11 Y' I s :< , ' r n E s s u ~~ ~~ ' )
DO (,Q I::J,tlll J 1 ::J\IIDDE:( I, 3) J2::1'dl0 11[( I 1 l1) JJ::J\,;Jf)DE( I ,5) F8! ::o. 5
C THE 11001\L '/AI~!Al3LES 1\RE l!G\/ SCALED flY TilE C F LOI /k A TF.. c c c ~IULTIPL!CATim' BY zr-.:Hn IS tWOIDEO nY THE 11 IF STATEitF.IITS 11
c IF(Jl,GT .11) GO TO 20 0 E :_ T 1\ ( .J 1 ) ::DE I. T 1\ ( .Jl ) IQ ( 1 )
20 IF(J2,13T,11) Gfl TO 30 0 F.LTt1 ( J~·) ::DELTA ( J2) IQ ( 1)
30 IF(J3.GT.t1) GO TO ()0 · D E U A ( J 3 ) :: 1J f:: LT/\ ( .J 3 ) I ( Q ( l ) .,..,., F 8 I)
6 0 \ c I T E ( () ' 7 2 ) I I [) E LT A ( J 1 ) ' [) E LT 1\ ( J 2 ) I [) E LT /\ ( .I J ) ' [) F.l.T J\ ( .J 1 ) I [) E L T /\ ( .) 2 ) ' !DELTA( J3)
72 F m 11 exT ( 1 :<, Is , s :<, F 1 o. 5 , s x IF 1 o • 5 1 6 y, F 1 o .. L~ , 1 8 x, E 1 2 ,If , 1f x. E 1 7. ,11 , '1 x, E 1 2. Jll)
600 CCli ITI flUE o o n r: r ;: 1 tlE
7 2 5 EL UlE'r ( I , If~ ::: E L E I 1£ T ( I , If) I ( Q ( 1 ) ,·,·: ( F ll i - 1 • 0) ) c C VISCOSITY OF EACH IIODE CALCUU\TED !\tiD PRIIHED. c
C 1\ L L V I S r 0 ( l ·IU II 0 D E , E L Ell E T , I ill , t ·IE ) \IHITE(6,/50)
?SO FO:-::['\T(lli0,30X, 1 IIODE VISCOSITY') \ lit I H ( G , R 0 G ) ( ( I 1 IIU t I 0 D E ( r) ) , I:: 1 , I ,p 1)
110 0 FOR' 1/\ T ( 1 :<, I 6, 3 X , F 1 0 , 5 ) Rf.TIJRII EIID
S IJ!1 r; OUT I I IE V l SCi) ( t IU! lf)fl E, E LEt 1C T , tIt I 1 I IE ) c c ***************~********~****~**************************** C T'llS U\LCULIHES THE VISCOSITY llF Ei\C!l :HJDE IW /\!! t\VFJ~i\GI:H3
C Pf\OC E ;JI HU: c ***************************~******~*********************** c
c
llli!EIISIO!I r-:LEI!r-T(IIE 1 5) 1 LCrJU 11T(81) f{ El\ L r 1\J t I n !1 E( I l! I) DO 3 3 t I I::: I I tit I Len' J;n < rJI) =n t II n I OD H t II ) :: 0. 0
83 CmiTliiUE no 8 2 I= 1 1 liE
c I'IDDilL lltltlllms. c
c
I I =ELf:t1F.T( I 1 1) I 2 = EL [I\ ET ( I , 2) 13:: EL Ell ET ( I 1 3)
C t\lJi!ODE() STORES TilE V/\UJE Oi- THE ttOfll\L 'IISCOS ITY C /\I!Q LCOLHIT() I<EF.PS COlJI!T nr:- THE lllHif3F.R CJF C ELE:\EtiTS COi!TFU f-3UT I I!G TO pAT II ODE. c
TQ ( 9) , N3 c ( 1 2:5 I 9) , A( 3 > , 13 o ) ? co ) , 11 o E L r 1\ ( r.) , r 1 oF. u ( 9) , T 0 E L T u 4 3) I c F. LT .i\ (?. Lf 3 ) I c E ( /.I} 3 I 1 )
REAL K < 1 7 s , n > , 1\ r: < 9 I 9 > I 1z 1 < 6 I f1 ) , !< 2 < 6 , 3 > , r1u 11 Ofl E ( o 1 ) 1·/R l T E ( 6 I 99 8)
998 FORI 1•\ T { 1110 13 OX 1
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Ct\l.L COIITF\0( IIK,IIE 1 llli 1 11DF ,II, If3\/, DL, !1,1/, ALPHA) c C THE !\AT II StJElRrJUTI !IE REf1UIRED. c
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INITIAL NO. OF D.O.F. NK= 243 NO. OF ELEMENTS NE= 128 NO OF NODES NN= 81 UNKNOWN NO. OF D.O.F. NDF= 175 NON ZERO NO. OF D.O.F. N= 9 BANDWIDTH IBW= 29 DIE LENGTH DL= 1. 00000 HEIGHT OF DIE AT INLET H= 1. ooooo WIDTH OF DIE W= 1. 00000 ANGLE OF TAPER OF DIE ALPHA 0.00000 RADIAN
NODAL INFORMATION**********NODAL INFORMATION**********NODAL INFORMATION