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ReportingDate: July1974

Issued:January1975

4!3CIC-14 REPORT CWLECTW

FEPFK)DUCTIONCOPY

YOKIFER:A Two-Dimensional Hydrodynamics

and Radiation Transport Program

/alamos10scientific laboratory

of the University of CaliforniaLOS ALAMOS, NEW MEXICO 87544

4 \

Richard C. AndersonM. T. Sandford II

‘)

UNITED STATES

ATOMIC ENERGY COMMISSION

CONTRACT W-740 S-ENG. 36

—*- “ ‘L

An Equal OpportunityEmployer

0

.,-.

Work performed under thejointauspicesoftheUS Atomic EnergyCommission and the Defense Nuclear Agency.

s

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PrintedintbeUnitedStatesofAmerica.AvailablefromNationalTechnicalInformationService

US.DepartmentofCommerce5285PortRoyafRoadSpringfield,VA 221S1

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——

Y@KIFER: A TWO-DIMENSIONAL HYDRODYNAMICS

by

Richard C. Anderson and M., -. g$~ 10

-q-/7%qABSTRACT

AND RADIATION TRANSPORT PROGRAM

T. Sandford II

]7-4

q /9’~

The Y@KIFER program has been written to calculatethe coupled hydrodynamics and radiation transport prob-lem in two-dimensional (R-Z) cylindrical qeOmetrY. Thehydrodynamics are computed by %heradiation transport by either themethod. .—-

This report is a descriptionguide to its use.

ICED-ALE method, theMonte Carlo or Sn

of the program and a

INTRODUCTION

Y@KIFER solves the coupled hydrodyna-

mics and radiation transport problem in two-

dimensional cylindrical (R-Z) geometry. It

is written in Fortran IV and is run under

the CR@ system on the CDC-7600 computer.

A few subroutines are written in C@MPASS.

This report has been written for those

using the program and is limited to a de-

scription of the program and its operation.

Persons interested in details of either the

physics or the numerical methods involved

are directed to the following references.

The hydrodynamics problem is solved by

the ICED-ALE method of the YAQUI program.

Reference 1 is the basic reference for the

hydrodynamics. The radiation transport

problem is solved by either the Monte Carlo

or S~ method for which references 2 and 3,

respectively, are the basic references.

I. OVERVIEW OF THE Y@KIFER PROGIUIM AND

DESCRIPTION OF THE MAIN PROGRAM

This

gram as a

section describes ~he YdKIFER Pro-

whole and the main overlay. Also

described are subroutines and calculations

applicable to the entire program.

A. Program Organization

The program consists of the main pro-

gram, Y@KIFER (Overlay 0,0) and four prima-

ry overlays.

1,0 @FFWE@ Input and Problem Setup

2,0 Y@KKY Hydrodynamics

3,0 MCRT Radiation Transport (Nongrey,

Monte Carlo)

4,0 GREYSN Radiation Transport (Grey,

Sn)Appendix A contains a list of overlays

and subroutines, a list of the common blocks

and the overlays in which they are used, and

a list of the file sets that Y@KIFER uses.

B. Input

There are two different forms of input

to Y@KIFER. The input reading is controlled

by Sense Switch 1.

Bubble input Sense Switch 1 ON

Purd input Sense Switch I OFF(default)

Bubble input is produced by rotating the re-

sults of a one-dimensional starter calculation

1

(e.g. R.ADFL!3)through 90° to produce bubble

input cards. Appendix A lists the required

input cards. Purd input will be described

in a separate report.

c. General Order of Calculations

Calculations proceed in the order

shown schematically in Fig. 1:

IOFFWEGO

I

I1ormoreHYDRO cycles

&

1ormoreRADNTRNSPTcYcles

*

IOutputtoFileset7

The selection of Monte Carlo or Sn radiation

transport calculations is governed by Sense

Switch 2.

Monte Carlo Sense Switch 2 ON

Sn Sense Switch 2 OFF(default)

The calculation is terminated just before

the time limit on the J@B card is reached.

D. OVERLAY 0,0 - Y@KIFER

Y@KIFER is the main program, which

calls the primary overlays, writes data on

Fileset 7, and produces dump tapes at regu-

lar intervals.

1. Variables Computed by Y@KIFER

TTL

T1

T2

TCYCLE

TDUMP

NDUMP

Time limit on the job

card (s).

CP time at the beginning

of the cycle (s).

CP time at the end of the

cycle (s).

Length of a calculation

cycle (s). TCYCLE = T2-T1

Length of computing time

until the next release

(stage) of Fileset 7 to

tape (s).

Number of 60-bit words

written on Fileset 7.

2. Dump Procedure

Data are copied to Fileset 7 before

Y@KKY is entered whenever the problem time

is greater than the problem output time

T$3UT. (The determination of T@UT is de-

2

scribed in Sec. E.7.) Periodically, File-

set 7 is released from disk to magnetic

tape. A new, blank tape is used each time.

Initially, TDUMP is set to 900 CP sec-

onds, or TTL, whichever is smaller. After

each cycle is calculated, TDUMP is reduced

by TCYCLE. Fileset 7 is released to tape

whenever TDUMP < 2 x TCYCLE. After each

tape stage, TDUMP is reset to 900 s or

TTL-T2, whichever is smaller.

Initially NDUMP is set to O. When data

are copied to Fileset 7, NDUMP is incremented

by the approximate number of words dumped.

When NDUMP ? 106, Fileset 7 is dumped to

tape and NDUMP is reset to O.

Each dump tape contains all data copied

to Fileset 7 since the previous dump tape

was written.

E. General Topics

1. Y@KIFER Mesh

The Y$KIFER mesh is a two-dimensional

grid, in R-Z cylindrical geometry. (The hy-

drodynafnics program is written to handle X-Y

cartesian geometry also, but the radiation

transport programs cannot do this.) Each

cell is a volume of revolution about the z-

axis with a quadrilateral cross section.

(Initially, the cross sections are rectangu-

lar, but not necessarily of uniform size.)

Because no physical variables depend on a

third coordinate, the mesh can always be rep-

resented two-dimensionally by a grid in the

R-Z plane.

The mesh consists of IBAR cells in the

radial direction and JBAR cells in the axial

direction, and the left boundary is the cy-

lindrical axis. These cells are called the

“real mesh.” For computational purposes, a

single row of dummy cells is added at the

bottom, on the right, and at the top, for a

total of IP1 = IBAR + 1 cells radially and

JP2 = JBAR + 2 cells axially. The maximum

allowable value of IBAR and JBAR is 100. The

maximum number of cells (including dummy

cells) is 7200.

I is the radial cell index, 1 2 I ~ IP1; and

J is the axial cell index, 1 S J ~ JP2.

The single computationally equivalent index,

IJ, is frequently used:

IJ = (J-1) (IP1) + 1.

The cell index I, J, or IJ refers to the

lower left-hand vertex of the cell.

● Figure 2 shows the basic mesh conventions.

Because cell and vertex properties depend

on the properties of neighboring cells and*

vertices, a standard notation has been de-

veloped to describe the neighbor cells

(Fig. 3). The coordinate positions of the

vertices are given by

x Radial coordinate of vertex ijij

Y Axial coordinate of vertex ijij

R Geometry indicator. R. . = 1 forij 13

slab geometry, R. . = X. . for cylin-13 13

drical geometry.

The mechanics of setting up a mesh are de-

scribed in Sec. II.

v

I—--y’d+ ~

/ ///////////////////~////f/

Fig. 2. The basic mesh conventions.

/////////g/ +

l-l.J / I,J#

I+l,JIMJ / IJ IPJ

/ /////////L

I-I,J-l I I,J-l I I+I,J-l

Fig. 3. Neighbor cell notation.

2. Y@KIFER Mesh Variables

The principal mesh variables used

throughout the program are described below.

Other mesh variables are used locally with-

in the program and are described by comments

at the places where they are used.

SIE. .1]

TEMP. .1]

R@. .1]

Pij

uij

vij

RV@Lij

Specific internal energy at the

center of cell ij (J/mg).

Temperature at the center of cell

ij (eV).

Density at the center of cell ij

(mg/cm3).

Pressure at the center of cell

ij (MPa).

Radial fluid speed at vertex ij

(km/s).

Axial fluid speed at vertex ij

(km/s) .

2m/volume of cell ij, in cylin-2

drical geometry (l/km”).

3. Storage of Mesh Variables

Because SCM is not large enough to con-

tain all of the mesh data, most mesh vari-

ables are stored in LCM and periodically

read into SCM (usually) three rows at a time.

NQ words are provided for each cell, and the

mesh data needed for any given cell are

therefore stored in NQ adjacent locations.

At present, NQ = 18. Appendix A contains

a tabulation of the cell variables stored

in the NQ locations for different parts of

the program. Mesh data are written into

3

and out of LCM by subroutine L@@P (Sec. F.

1.) for one entire row of cells at each

call.

4. Y@KIFER Units

Unless otherwise indicated, the units

of all Y@KIFER variables are as follows.

Time sLength km (105cm)Volume km’ (lolscm’)

Velocity km/s (105cm/s)Acceleration bn/s2 (105cm/s2)

Density mg/cm3 (lo-’g/cm’)

Energy J (107ergs)

Specific energy J/mg (lOIOergs/g)

Energy density J/km3 (10-8ergs/cm3)

Pressure mg-km2/cm3-s2 (107dynes/cm2 = MPa)

Temperature eV (11 605.4 K)

Absorptioncoefficient km-’ (10-5cm)

Frequencys-l

The physical constants used are:

c = speed of light = 3.0 x 105 km/s,

a = radiation density constant = 137.214

x 108

J/km3-eV4.

5. Equation of State and Opacity Data

The equation of state and opacity data

are read from FileSet 6:

N@PT

N@PD

NFRQ

FRJ3Q(K)

@PTMP (I)

@pDEN(J)

@PSIG(K,I,J)

Number of temperatures for

which data are tabulated

(30).

Number of densities for

which data are tabulated

(9).

Number of wavelengths for

which absorption coeffi-

cients are tabulated (100).

Wavelengths (~). K = 1,

NFRQ, in order of decreas-

ing wavelengths.

Loglo temperatures (ev) for

which data are tabulated.

I = 1, N@T, in order of

ascending temperatures.

Loglo densities (g/cm3) for

which data are tabulated.

J = 1, N@PD, in order of

increasing densities.

Loglo absorption coeffi-

cients (l/cm). K=l,

NFRQ; I = 1, N@PT; J = 1,

N@PD .

SPTBL(I,J) Loglo Planck mean absorp-

tion coefficients (l/cm).

I = 1, N@PT; J = 1, N@PD.

SPTBL(I,J) Loglo Rosseland mean ab-

sorption coefficients

(l/cm). I = 1, N@PT;

J = 1, N@PD.

The mean absorption coefficients

Y@KIFER uses are

3.

Planck mean

Rosseland mean

PTAB(I,J)

ETAB(I,J)

BTBL(I,J)

controlled by Sense Switch

Sense Switch 3 ON

Sense Switch 3 OFF

(default)

Loglo pressures (dynes/cm3) .

I = 1, N$PT; J = 1, N@PD.

Loglo specific internal

energies (ergs/g) . 1=1,

N@PT; J = 1, N@PD.

Radiation derivatives, =

apr/aaT4. I = 1, N@PT;

J = 1, N@PD.

The data are arranged on the tape in a

single file. (NWL is the number of words

in the record, and it is not needed by the

program.)

Record 1

Record 2

Record 3

Record 4

Record 5

...

Record 272

Record 273

Record 274

Record 275

Record 276

Record 277

N@PD, N@PT, NFRQ

FREQ

NWL, @PTMP(l),

@PSIG(K,l,l)

NWL, @PTMP(2),

@P.51G(K,2,1)

NwL, @PTMP(3),

@P,51G(K,3,1)

...

@PDEN(l),

@PDEN(l),

@PDEN(l),

NwL, @PT~(N@pT), @pDEN(Nf5PD),

f3PSIG(K,N@PT,N@PD)

SPTBL (Planck)

SPTBL (Rosseland)

PTAB

ETAB

BTBL

6. Time and Time Interval Calculation

The program is permeated by the messy

calculation of problem times and time inter-

vals. Four main variables are involved:

TIME Radiation transport (R) time,

DTR Radiation transport (R) time inter-

val,

s

.-

4

T Hydrodynamic (H) time,

D’l! Hydrodynamic (H) time interval.

In the following discussion, this no-

tation is used:

TIMEm Time at the start of R cycle m,

DTRm Time interval of R cycle m,●

Tn Time at the start of H cycle n,

DTn Time interval of H cycle n.●

Hydrodynamic Time Calculations. The

hydrodynamic overlay (Y@KKY) is called if

and only if T = TIME. The status of the

time variables when Y@KKY is called is:

TIMEm Starting time of the next R

cycle,

DTRm Time interval of the next R

cycle,

Tn Starting time of the next H

cycle,

DTn_l Time interval of the last H

cycle (of no interest, now) .

At the start of PHASE1 within Y@KKY,

the new H cycle begins and the hydrodynamic

calculations occur. The calculated quanti-

ties are

DTn

Tn+l

DT~ is a

Time interval for the next H

cycle = min(DT& 5 x DTRm) .

Time at the end of the next H

cycle = Tn + DTn. When Tn = TIMEm,

Tn+l may exceed TIMEm+l.

time interval based on the hydro-

dynamic constraints described in Sec. III.

The 5 x DTRm limitation restricts the number

of R cycles between H cycles to about five.

At the end of Y@KKY, Tn+l is compared with

TIMEm+l = TIMEm + DTRm:

Tn+l < TIMEm+l Another H cycle is calcu-

lated. If another H cycle● is calculated, its DTn is

subject to the additional

* restriction that

DTn S TIMEm + DTRm-Tn.

T m+l Y@KKY is exited and a radia-Z TIMEn+l

tion transport overlay is

called.

Radiation Transport Time Calculations.

The radiation transport overlays (MCRT or

GREYSN) are called only if T ~ TIME + DTR.

The status of the four variables at the be-

ginning of an R cycle is:

TIMEm Starting time of the next R cycle

DTRm Time interval of the next R cycle

Tn+1 Time at the end of the last H cycle

DTn Time interval of the last H cycle

The time interval calculations for R cycle

m + 1 occur throughout the overlays during

the calculation of R cycle m:

DTRm+ 1

= min(DTR~+l, 10 x DTn) .

DTR~+l is a time interval based on radiation

transport constraints and described in Sees.

IV and V. The 10 x DTn limitation restricts

the number of H cycles between R cycles to

about 10. The problem time is advanced at

the end of each radiation cycle.

TIMEm+l = TIMEm + DTRm.

A number of checks and adjustment are made.

TIMEm+l < Tn. When this condition applies,

another R cycle is calculated.

TIMEm+2 = TIMEm+l + DTRm+ 1

is compared with Tn~ If

TIMEm+ ~ > Tn, the time inter-

val is reduced to DTRm+l =

Tn - TIMEm+l.

TIMEm+l = T When this condition applies,n.

the radiation transport over-

lay is exited and Y@KKY is

called.

Getting Started. The time calculations

are initialized as follows. TIME1 and DTRl

are the input numbers TIME and DTR.‘1 =

TIME and DTO = O are set by @FFWEG@. When

PHASE1 is first called (at the beginning of

H cycle 1), DT1 = DTR and T2 = T1 + DT1.

7. output

During every cycle there are several

short prints (also written on film) , and at

less frequent intervals more detailed, output

is written on film. These less frequent

times are called T@UT. T@lT is set to pro-

vide detailed output n times per decade of

elapsed problem-time= If t is the elapsed,-----Problem time at the start of the decade, out-

‘- - . _put occurs at elapsed times ft, f2t, .....

fnt = 10t; hence, f = ~’- The output

overlays (2,2;3,3; and 4,3) are called only

5

when T ~ T@UT. Initially (in @FFWEG@)

T@lT = TSTART, the problem starting time.

This causes Y@K@UT, the hydrodynamic output

program, to be called on cycle O. It is in

Y@K@UT that all subsequent

are made.

In cycle O T@JT = DTR.

In later cycles T@lT =

TSTART.

chansres in T@UT

f(T@lT-TSTART) +

The factor f is presently set at 1.15, which

corresponds to n - 16 outputs/decade.

8. Dumps and Restarts

At the output times, T@lT, data are

written on Fileset 7, and at less frequent

intervals, Fileset 7 is staged to tape.

The data written on Fileset 7 include all

data needed to restart the problem and other

data that are useful to analyze in detail

after the problem is run. Data are not

written on Fileset 7 after every cycle be-

cause of the enormous volume involved.

Fileset 7 is staged to tape when:

One reel of tape has accumulated

(NDUM.P> 106).

Fifteen CP minutes have elapsed since

the last tape was written.

The time limit from the job card is ap-

proaching.

Dump tape data are read and analyzed by the

program NEXTWAY, described in Appendix B.

The dump tapes contain as many problem cy-

cles as may happen to be written on them.

The mechanics of the dump procedure were

described in Sec. D; the structure of the

dump file is in Table A-VII in Appendix A.

F. Subroutines

1. Subroutine L@@P

L@@P, a highly efficient subroutine

originally written for YAQUI, is used to

transfer data between SCM and LCM. L@@P

maintains the NQ values for each cell in

three rows of mesh cells in SCM -- the row

for which calculations are being made and

the rows immediately above and below. To

aid in interpreting the source code listing,

the general form of a calculation using L@@P

is shown below.

CALL START

The bottom three rows of cell data are

read into small core. The indices of

the first cell in each row (IJM, IJ,

IJP) are set.

D@9J=2, J2

Each time through this loop, mesh data

are computed for row J. J2 = JP1 for

cell-centered quantities, J2 = JP2 for

vertex quantities.

D@ 8 I = 1, 12

Each time through this loop, mesh data

are computed for cell I h row J. 12 =

IBAR for cell-centered quantities, 12 =

IP1 for vertex quantities.

Set cell indices

Set indices for cells to the right and

left of I, as needed, IPJ = IJ + NQ,

IPJP = IJP + NQ, etc.

Calculate desired data

Increment cell indices

Set indices for the next cell in row I,

as needed,

8 C@NTINUE

CALL L@@P

Write data

indices IJ

IJ = T.PJ,IPJ = IPJP, etc.

for row IJM into LCM, reset

and IJP to IJM and

read data for IJP into SCM.

9 C@NTINUE

CALL D@NE

Compute data for two top rows

into LCM.

1.3,and

and write

.

●

2. SEARCH (XBAR, x, N, NDX, and MFLAG)

SEARCH is an extremely fast binary

search routine. Given a table of N values

of X, and a value XBAR, SEARCH finds NDX

such that X(NDX) S XBAR < X(NDX + 1) . MFLAG ●

is a returned error flag.

3. DBLINT (K, X, Y, XT, YT, TAB, Nl, M, *MC@LS, and NDIM

DBLINT performs double linear interpo-

lation of tabulated data. TAB(I,J) iS a

tabulated function of XT(I) and YT(J), I = 1,

MC@LS: J = 1, M. TAB(I,J) is dimensioned

for (MC@LS,M). NDIM is the actual number

of I values tabulated. DBLINT returns as

6

a function value the interpolated value of

TAB which corresponds to X and Y. Normally

K = N1 = O, but K, N # O allows use of tri-

ply subscripted tables.

4. GETEMP (XP, ZP, X, Y, Z, NX, and NY)

Z(I,J) is tabulated as a function of

X(I), I = 1, NX and Y(J), J= 1, NY. Given

the values of X, .Z,XP, and ZP, GETEMP com-

putes the corresponding value of Y by in-

verse interpolation. The subroutine is

used with equation of state data to compute

temperatures when densities and specific

internal energies are known.

5. PAKFN@ and UNPKFN

PAKFN@ packs three floating point

words into a single word. The packed words

have a seven bit exponent. UNPKI?N unpacks

the single word back into three words.

These subroutines save significant amounts

of space in return for decreasing the sig-

nificant figures to six.

II. @FFWEG@, THE INPUT AND SETUP OVERLAY

Overlay 1,0 (f5FFWEG@) is used to read

input data and to set up the initial mesh,

values of mesh variables, and marker parti-

cle distributions.

A. Overview of the Overlay

The setup overlay is @FFWEG@ (Overlay

1,0), which reads card and tape input data

and sets up the problem. There are no sec-

ondary overlays; the work is done by sub-

routines MESHMKR, PARTGEN, PARDEN, NSTART,

and FILMC@.

B. @FFWEG@

1. Equation of State and Opacity Data

Equation o.fstate and opacity data are

read from Fileset 6. The input wavelengths

are converted to frequencies (1/s), and,

where necessary, units are changed from

those of the tabulated data to Y@KIFER units

The frequency-dependent opacities, $PSIG,

are stored as a linear array, SIGA, in LCM.

SIGA(IJK) corresponds to @PSIG(I,J,K)

where the equivalent subscript is IJK = K +

(1-l)*NFRQ + (J-l)*NOPT*NFRQ. I, J, and K

are the temperature, densi~y, and frequency

indices, respectively.

.

2. Dump Tape Input

@FFWEG@ reads the dump file, Fileset

7. If the end-of-information is encounter-

ed on the first reading, one assumes that

there is no dump input and that initial

data are to be read from cards. If data

are found on Fileset 7, the file is read

until the end-of-information is encounter-

ed. When this occurs, the last dump on the

file has been stored in the computer and

is thus used to restart the problem.

3. @FFWEG@ Input Cards

@FFWEG@ reads the following data from

cards:

NAME Problem identification.

TIME Starting time of the problem (s).

DTR Initial radiation time interval

(s).

CYL Geometry parameter.

CYL = 0.0 for slab geometry,

CYL = 1.0 for cylindrical geom-

etry. It is not possible at

present to run radiation trans-

port calculations in slab geom-

metry.

GRDVEL Rezone parameter (Sec. III) .

ALPHA Radiation transport implicitness

parameter (Sees. ~ and V).

IBAR, JBAR, IUNF, JUNF, JMID, DR, DZ, and

FREZ Quantities that define the mesh

(See. C-1 and C-2).

AO, AOM, BO, XI, Mu, LAM, @M, EPS, ASQ,

GMl, GR, and GZ

Parameters used in the hydrody-

namic calculations (Sec. III) .

REZYO Axial coordinate (true altitude)

of the “center” of the mesh (Ion).

This value defines the mesh alti-

tude and, in practice, usually

corresponds to the coordinate at

the center of the bubble.

YBASE Axial coordinate of the bottom

of the real mesh (true altitude) .

YBASE is not independent of other

input quantities (Sec. C-2) .

The program operates on the assumption

that the altitude at REZYO = O. The input

value of REZYO is saved for reference,

7

but all other altitudes (Yij,YBASE) are

converted to true altitude-REZYO.

REZR@N Ambient density at REZYO

(mg/cm3).

REZSIE Ambient specific internal en-

ergy (excluding radiation) at

REZYO (J/mg).

4. Parameters Set and Computed by

@FFWEG@

@FFWEG@ sets initial values of some

parameters and precomputes others. These

parameters are defined and described in

the following sections, which describe the

parts of the program in which they are used.

5. Subroutine Calls

MESHMKR is called to read and compute

initial values for the mesh variables X,

R, Y, U, V, SIE, TEMP, R@, and RV@L.

PARTGEN or PARDEN is called by MESHMKR to

compute marker particle positions for Bub-

ble or Purd input problems, respectively.

FILMC@ is called to compute film-plotting

parameters.

6. Marker Particle Cells

ITAB(k) is the equivalent index of the

cell containing the kth marker particle:

ITAB = (J-l)*IP1 + I.

7. Mesh Variables

@FFWEG@ computes the mesh variables:

M Mass of cell ij (mg/km3/2m-cm3).i]

Eij Total specific material energy in

cell ij (internal + kinetic) (J/mg).

RM. . Reciprocal mass associated yith1-Jvertex ij (2n-cm3/mg-km3,). The mass

associated with a vertex is l/4 the

mass of the four adjacent cells.

8. @FFWEG@ output

@FFWEG@ prints the job number, the

date, all input data, and parameters whose

values are as set by @FFWEG@.

c. Subroutine MESHMKR

MESHMKR establishes

of the mesh variables X,

SIE, TEMP, and RV!3L.

1. Uniform Mesh

the initial values

Y, R, U, V, R@,

MESHMKR computes the coordinates X, Y,

and R for a uniform mesh of IBAR x JBAR

cells with specified cell dimensions DR

8

and DZ. The coordinate at the vertex,

J = JMID, is Y = REZYO; that at the bottom

of the mesh is Y = YBASE. The input value

of YBASE, for a uniform mesh, must be

REZYO-DZ*JMID .

2. Nonuniform Mesh

The nonuniform mesh is computed when

FREZ + 1.0. The nonuniform mesh contains

a total of IBAR X JBAR cells. There is an

inner, uniform region IUNF x JUNF cells,

for which the inner part of the previously

computed uniform mesh is used. In the

outer parts of the mesh, the cells grow

(or shrink) by amounts that depend on the

value specified for FREZ. At the bottom

of the mesh Y = YBASE. The FREZ input value

for a nonuniform mesh must be computed ac-

curately using the formula

REZYO = YPASE + JUNF x DZ

+ [1f x Dz x l-f‘JuNF/2-JMID])rl-f

where f = FREZ and MESHMKR sets Y2 = YBASE.

A nonuniform mesh is illustrated in Fig. 4.

The algorithms used to determine the co-

ordinates are

x.1. = ‘i-1 + ‘(xi-~-x~_2

where f = FREZ and

Yj = -t + ~ (l-f*j),

where f = FREZ, y<

,i= IUNF + 2, IP1,

x.1

=x...1-J

= 2, JB@T-1,

= Y:<, Az = DZ,

t=TJ=+ “(DZ),

Aj = JDB = lJ-JB@Tl, and

JB@T = JMID + 2 -JUNF/2.

‘j), j = JT@P + 1, JP2,‘j

.t+g(l-f

where f = FREZ, y. = Y *Z = DZ,3 ij8

t= TJ=F(Dz),

Aj = JDT = lJ-JT@Pl, and

JT@P = JMID + JUNF/2.

3. Background Mesh Variables

Ambient values of U, V, R@, and SIE are

placed in every mesh cell by one of two

methods

*

r

.

I-’=”l

MLLu-k+--i 3

Fig.

each

NB

NR

NT

NL

UI

VI

/////////////7z/////////////zzz’zz///h

4. A nonuniform mesh.

Uniform Regions. The data read for

uniform background region are

R@I

SIEI

Number of real cells below the

region

Number of real cells to the left

of the right boundary of the region

Number of real cells below the top

of the region

Number of real cells to the left

of the region

Input radial velocity in region

(km/s)

Input axial velocity in region

(km/s)

Input density in region (mg/cm3)

Input specific internal energy in

region (J/mg) (radiation not in-

cluded)

Figure 5 shows a uniform background re-

gion. Uij, Vij, R@ij, SIEij are set equalko

TNTd),

JTNB cdl,

P=-l’’.*$

7//////////////////////////////////

Fig. 5. Background mesh input.

UI, VI, R@, and SIEI, respectively, for

each cell in the region. TEMP. . = TEMP1 is13interpolated from the equation of state.

The total cell internal energies are found

from SIE + a(TEMP4)/R@. The input data and

the interpolated temperatures are printed.

Exponential Atmosphere. Densities that

decrease exponentially with increasing alti-

tude are computed for each row. The input

value is p = REZR@N, assumed appropriate

at y = REZYO. Temperatures corresponding

to the local density and the input ambient

energy, REZSIE, are interpolated from the

equation of state tables. The specific in-

ternal energies are found from REZSIE +

a(TEMP4)/R@r where R@ varies exponentially.

The density, specific internal energy, and

temperature are printed for each rov of

cells.

4. Bubble Input

Bubble input,

problems, consists

read only for Bubble input

of the specification~of

9

mesh variables in the upper right-hand quad-

rant of an R-Z plane. These values are re-

flected to the lower right quadrant, and if

required, the right semicircle is reflected

to form the left semicircle.

The variables used in the code are:

IBUB, JBUB

11, JJ

R@

SIEI

VI

UI

Indices of vertex correspond-

ing to the center of the bub-

ble

Temporary indices of cell

into which data are to be

placed. Typically, II and

JJ begin at 1.

Density in cell TI, JJ

Specific internal energy in

cell II, JJ

Axial velocity at vertex II,

JJ

Radial velocity at vertex 11,

JJ

The actual cell indices corresponding

to the bubble location in the mesh are cora-

puted from:

Quadrant Indices

Upper Right I=II+IBUB-1, J+JJ+JBUBY1

I.ower Right I=II+IBUB-1, J+JBUB-JJ

Upper Left I=IBUB-11, J=JJ+JBUB-1

Lower Left I=IBUB-11, J=JBUB-JJ

These mesh variables are assigned to the

appropriate cells and vertices, destroying

that part of the background mesh. Tempera-

tures are computed from the equation of

state as previously described, and all Bub-

ble input is printed. Bubble input is il-

lustrated in Fig. 6, but we note that one is

not restricted to spherical bubble data.

D. Subroutine NSTART

NSTART is called by MESHMKR and is used

only in Purd input problems.

E. Subroutine PARTGEN

PARTGEN generates marker particles for

Bubble input problems and is called by

MESHMKR . In addition to marking fluid po-

sitions, the marker particles are used to

define the “region of interest” in film

plots . For Bubble input problems, the par-

ticle regions generally coincide with the

bubbles.

1cellsIw.!.luchh,w,hut $sr,ad

vclm.IW8 J8U8

cut,,.,0

t.llld! LnJbblrilwlchrdlectui

_l-1-bi.–i—_i--l_Jl_J

Fig. 6. Bubble input.

Particles may be generated in one or more

regions of the mesh. The regions may be

either circular or rectangular. The data

that define the regions and the numbers of

particles

DRPAR

DZPAR

xc

Yc

XD

YD

in them are read from cards.

Radial spacing of particles in the

region (km).

Axial spacing of particles in the

region (km).

Radial coordinate. Center of cir-

cular region or left boundary of

rectangular region (km).

Axial coordinate. Center of cir-

cular region or bottom boundary

of a rectangular region (km).

Radius of a circular region or

right boundary of a rectangular

region (km).

Top boundary of a rectangular re-

gion (km). YD s O for a circular

region.

A maximum of 1000 particles can be gen-

erated, and for both information and plotting

purposes they should cover the area of the

bubbles. If DRPAR and DZPAR equal DR and DZ,

respectively, there will be one particle per

b

.

10

.

.

cell. The number of particles can be in-

creased by making DRPAR and DZPAR smaller.

Generally, the bubble is a semicircle along

the axis, and XC = O, YC = REZYO, XD = Bub-

ble radius, and YD = O.

Variables computed by PARTGEN are

XP~ Radial coordinate of thek’h

particle (km).

YPARk Axial coordinate of the kth

particle (km). XPAR and YPAR

are stored in LCM block YLC2.

NPT Total particles generated (1000

maximum) .

PYB Minimum value of YPAR (in all

particle regions) (km).

PY’r Maximum value of YPAR (in all

particle regions) (km).

PXR Maximum value of XPAR (in all

particle regions) (km).

All input data are printed.

F. Subroutine PARDEN

PARDEN is called by MESHMKR, and it

generates marker particles for PURD input

problems.

G. Subroutine FILMC@

FILMC@ computes certain parameters as-

sociated with film plots. It is called ini-

tially by @FFwEG@, and during each hydro-

dynamic cycle by REZ@NE. It resides with

the main overlay, Y@KIFER, so”the subroutine

and its results are available throughout the

program.

Variables computed by FILMC@ are:

XL Left boundary of the mesh, XL = O.

XR Right boundary of the mesh,

XR=max(X. .).1]

YB Bottom boundary of the mesh,

YB =min(Y. .).1-J

YT TOP boundary of the mesh,

YT=max(Y. .). The maxima and1]minima are over the boundaries of

real cells.

IXL 4020 coordinate of XL

IXR 4020 coordinate of XR

IYB 4020 coordinate of YB

IYT 4020 coordinate of YT

Also computed and stored are the

corresponding floating point val-

ues, FIXL, FIXR, FIYBI and FIYT,

Xc(mv

Yc@v

The

particle

PARDEN .

plotting

Factor for converting radial co-

ordinates to film coordinates,

xgZNv = (FIYT-FIxL)/(xR-XL).

Factor for converting axial co-

ordinates to film coordinates,

Yc@w = (FIYT-FIYB)/(YT-yB).

region of interest is defined by the

generator subroutine PARTGEN or

Region of interest plots eliminate

those parts of the mesh in which

nothing in particular is

corresponding quantities

interest are:

PXL, PXR, PYB, PYT

IPXL, IPXR, IPYB, IPYT

happening. The

for the region of

FIPXL, FIPXR, FIPYB, FIPYT

Pxc@V, PYc@V

The present region of interest defini-

tions are:

PYB = PYB - 3xPXR

PYT = PYT + 2xPXR

PXR = 3xPXR

PYB, PYT, and PXR on the right are values

computed by PARTGEN or PARDEN.

111. HYDRODYNAMICS CALCULATIONS

The hydrodynamics calculations are done

by Overlay 2,0 (Y@KKY), a modification of

YAQUI . The basic YAQUI hydrodynamic calcu-

lations are unchanged. The differences be-

tween Y@KKY and YAQUI (other than spelling)

are mostly associated with the input and

output, and with the fact that Y@KKY has

been divided into several overlays that com-

municate with the radiation programs. Pro-

perties of the original YAQUI program are

described in detail in Ref. 1.

A. Overview of the Overlay

The main overlay, Y@KKY, calls the five

secondary overlays and decides whether the

problem time is right for changing from hy-

drodynamic to radiation transport calcula-

tions.

1. Overlay 2,1 - PIiASEO

PHASEO performs the final calculations

for each cycle. It interpolates total pres-

sures and produces a short print of quanti-

ties of interest for each hydrodynamic cycle.

11

2. Overlay 2,2 - Y@K@T

Y@K@UT, the hydrodynamic output program,

is called only at output times, T@T. The

output is for the previously computed hydro-

dynamic cycle.

3. Overlay 2,3 - PHASE1

The hydrodynamic cycle begins in PHASE1,

which performs the explicit Lagrangian cal-

culations of YAQUI.

4. Overlay 2,4 - PHASE2

The implicit Lagrangran calculation

(pressure iteration) is done in PHASE2.

5. Overlay 2,5 - PHASE3

In PHASE3, the mesh is rezoned and the

Eulerian (transport) phase of YAQUI is solv-

ed to give final values of all mesh vari-

ables.

B. PHASEO

1. Variables Computed by PHASEO

Pij

Pressure in cell ij computed by in-

terpolation in the equation of state

table PTAB, (mg-km2/cm3-s = ma).

TIAME Total ambient internal energy in the

TI

TK

EP@T

TE

UM@M

VM@M

CIRC

TMAX

TGMX

TMDT

mesh (J).

Total internal energy (in excess of

ambient) , including radiation (J).

Total kinetic energy (J).

Total potential energy (J).

Total kinetic and internal (in excess

of ambient) energy (J).

Proportional to radial momentum of

the material in the mesh (mg/km4/

2m-cm-s).

Proportional to axial momentum of

the material in the mesh (mg-d4/

2n-cm-s) .

Line integral of the velocities

around the edge of the mesh.

Maximum specific internal energy in

the mesh (J/mg). ITM and JTM are

the indices of the cell containing

TMAX .

Maximum specific internal energy

gradient in the mesh (J/mg-km).

ITG and JTG are the indices of the

cell containing TGMX.

Time at the beginning of the hydro

cycle (s). ‘l?MDT= T - DT’.

output

The following data are printed and

written on film:

NCYC, TMDT, T, DT, NUMIT

TE, TI, TK, EP@T, TIAMB

UM@M, VM@M, CIRC

TMAX, ITM, JTM

TGMX, ITG, JTG

DTV, I.DTV,JDTV

1

Computed by Y@K@UT,DTC, IDTC, JDTC PHASE2, and PHASE3.

c. Y@K@JT

Y@K@JT is called only at output times,

T@UT, and it computes the value for the

next output time. Y@K@lT plots two zone

plots, two velocity vector plots, and one

velocity direction plot. Plotting is con-

trolled by the index NTHRU.

NTHRU = -1 zones in the entire mesh.

NTHRU = O Velocity vectors in the

entire mesh; zones in the

region of interest.

NTHRU = 1,2 Velocity vectors and di-

rections in the region of

interest.

The region of interest is defined in

Sec. II. Contour plots of density,

specific internal energy, vorticity, and

magnitude of velocity are plotted in the

region of interest.

The coding for a long print on film is

included, but this section of the program

is by-passed on all cycles except cycle O

to save film. The long print gives, for

each cell, 1, J, X, Y, U, V, SIE, R@,

l/Rvf5L, D, and P. D is V“; in l/s.

1. Subroutine PARPL@T

PARPL@T, called by Y@K@UT, plots the

marker particles in the region of interest.

D. PHASE1

1. Time Interval Calculation

PHASE1 calculates the hydrodynamic

time interval.

NCYC Hydrodynamic cycle number, for the

cycle to be started in PHASE1,

incremented to NCYC = NCYC + 1.

DT Hydrodynamic time interval for

the cycle to be calculated. For

the first cycle (NCYC=l), DT=DTR,

12

the input radiation transport time

interval.

In all subsequent cycles, the hydrody-

namic cycle time interval is DT = min (DTV,

DTC) , where DTV is the viscous stress time

interval computed in PHASE2 and DTC is

convective flux time interval computed

PHASE3 .

New maximum values of DTV and DTC

set in PHASE1. DTV = DTC = DT x DTFAC,

the

in

are

where DTFAC is a factor that causes the

time interval to change so as to hold the

number of pressure iterations in PHASE2

down to a small number (-5). DTFAC =20

15+N-UMIT where NUMIT is the number of

iterations required on the previous cycle.

DTFAC has a maximum value of 1.25.

The values of DTV and DTC are recom-

puted by PHASE2 and PHASE3.

T Initially in PHASE1 this is the time

at the end of the hydrodynamic cycle

just finishing, and it is incremented

(T = T + DT) to the time at the end

of the hydrodynamic cycle to be

started.

2. Mesh Variables

PHASE1 makes one pass through the mesh

loop and computes the following variables:

UTIL. .1-J

VTIL. .1].

GRIR. ,13

GRIZ. .1]

Eij

DELSM, .1-J

R@L. .~1

RCSQij

Explicit Lagrangian radial ve-

locity component

Explicit Lagrangian axial ve-

locity component

Radial velocity increment

Axial velocity increment

A geometric quantity

A geometric quantity

= R@..1-J

Reciprocal sound velocity

squared

.RCSQ. =

lj ASQ + GG:l*SIE:: ‘~J

where GGM1 = GM1*(l+GM1) (ambient cells)

and

PHASE1 utilizes the improved node

coupler that smooths vertex velocities by

the velocities of all eight surrounding

vertices.

3. Subroutine NADD

NADD is called by PHASE1 for Purd

problems only.

E. PHASE2

The variables computed by PHASE2 are:

PL.. Gas pressures obtained by itera-13

tion.

NUMIT Number of pressure iterations

required (500 maximum) .

ETIL. . Explicit Lagrangian internal1]

energy.

DTV Tentative value of DT based on

viscous stresses. It is the min-

imum of such values for all cells

and the value originally computed

in PHASE1. The eel

JDTV .

F. PHASE3

PHASE3 computes the final

the mesh variables X, Y, R, MP

is IDTV,

values of

RMP, EP =

SIE, U, V, RV@L, and R@. The temperature

is computed from SIE by the iterative

scheme described in Sec. IV. DTC , the con-

vective flux time interval is also computed

in PHASE3. DTC is computed for each cell,

and the final value is the minimum of the

value over all cells and the value pre-

viously computed in PHASE1. The cell is

IDTC, JDTC. PHASE3 calls subroutines

REZ@NE (to rezone the mesh), PARTM@V (to

move the marker particles) , and FILMC@ (to

modify the film plotting parameters) .

1. REZ@NE

REZ@NE is”called by PHASE3 when the

rezone parameter (an input number)

GRDV?3L = 2.0 or when the Sn radiation

transport calculation is being used.

(GRDVEL = 0.0 and GRDVEL = 1.0, respective-

ly, represent Eulerian or pure Lagrangian

rezones that are handled by PHASE3.) The

outside of the mesh is moved with velocities

FC3 (down), FCP2 (up), and FCX (tb the

(other cells).

13

right) .

rival of

the mesh

internal

the edge

These quantities depend on the ar-

velocities at the outer part of

and on the appearance of nonambient

energies (such as by radiation) at

of the mesh. The latter calcula-

tion is not presently activated. For Sn

radiation transport problems, the mesh

lines are moved, but they remain either

vertical or horizontal, thus retaining the

rectangular cells and representing continu-

ously rezoned Eulerian geometry. A mesh

that is not rectangular can be relaxed to

rectangularity by setting the variable mesh

“stiffener” parameter FSTF = 1.O*RI)T. The

rezone constants are printed.

2. PARTM@

PARTM@V moves marker particles to new

positions based on the velocity at the par-

ticle location. New values of PYB, PYT,

and PXR are computed to redefine the region

of interest in film plots.

IV. MONTE CARLO SOLUTION OF THE RADIATION

TRANSPORT PROBLEM

The Monte Carlo radiation transport

calculations are done by Overlay 3, 0 (MCRT).

A. Overview of the overlay

The main Monte Carlo radiation trans-

port program is MCRT (Overlay 3,0). MCRT

computes variables used throughout the ra-

diation calculation and calls the three

secondary overlays.

1. Overlay 3, 1 - REEFER*

REEFER performs the Monte Carlo solu-

tion to the radiation transport problem.

Its principal function is to generate and

follow statistical particles. In Subroutine

WALK (called by REEFER), the particles pur-

sue the random walk and meet their statisti-

cal fates. REEFER generates NSP (internally

set to 10) source particles in each cell in

which the temperature exceeds a specified

threshold value, TEMIT (internally set to

0.05 eV). REEFER sets values of the follow-

ing parameters for each particle.

*one who reefs (naut); a short coat orjacket of thick=th.

A random position in the cell,

A random direction of t~avel,

A random frequency (photon energy) ,

AI-Ienergy “weight” equal to the cell

emission energy divided NSP.

In WALK, the particles move from their

initial positions, with the above initial

properties, until one of the following ran-

domly selected events occurs:

The particle collides and is absorbed,

and the walk is terminated.

The particle collides and is scattered.

The particle energy is reduced to a neg-

ligible value (set by EDEATH).

The particle leaves the mesh, and the

walk terminates.

The radiation time interval ends.

When a particle is scattered, its random

variables are reset as follows.

Its position is the point where the scat-

tering collision occurred.

A new random direction of travel is sam-

pled (isotropic).

A new random frequency (photon energy) is

sampled.

The energy is equal to the energy of the

particle before the scattering.

If a particle has not died when the

time interval ends, it becomes a “census”

particle and its parameters are stored on

Fileset 1, so that its random walk can be

continued on the following radiation trans-

port cycle. The overlay REEFER reads File-

set 1 for census particles from the previous

cycle, before new (source) particles are

initialized.

When any particle (source or census)

undergoes NPCMAX scattering, its parameters.

are sent to the “bank.” The value of NPCMAX

is set by @FFWEG@ and is presently 50. Char- ●

acteristics of particles sent to the bank are

are also stored on Fileset 1 and are read

by REEFER after all census and source par-

ticles have been completed. REEFER splits

each bank particle into NBP particles. The

value of NBP

ently three.

acterized as

is set by @FFWEG@ and is pres-

Each sibling particle is char-

follows.

.

4

.

.

Its position is the point where the parent

particle was deposited in the bank.

The direction o% travel is the same as

that of the parent.

‘T-nefrequency (photon energy) is the same

as that of the parent.

The energy is the energy of the original

particle, divided by NBP.

These sibling particles, in turn, may

be deposited in the bank eventually during

their random walk, and it is not uncommon

to produce many particle progeny during

each cycle. As the particles move around

the mesh (in WALK), an exponential energy

loss is associated with each move, and the

energy is deposited (the weight scored)

along the path of the move. When a particle

is absorbed, dies from lack of energy, leaves

the mesh, or goes to census, its energy is

scored at the place where the event occurs,

by saving the coordinates of each sample.

The original particle energies, the energy

scores, and the energies of the terminated

particles (except census particles) are

stored on Fileset 3, for use by overlay

ESTEP . Also stored on Fileset 3 are the

particle frequencies. Fileset 3 is included

in the problem dump tapes so that.these data

can be analyzed.

2. Overlay 3.2 - ESTEP

ESTEP reads the particle production and

deposition energies, and coordinates, from

Fileset 3 and uses them to advance the inter-

nal energies, and hence temperatures, in

each mesh cell.

3. Overlay 3.3 - LISTING

LISTING is called at output times (T@UT)

and writes (on film) detailed mesh data as-

sociated with the Monte Carlo calculation.

B. MCRT 8

1. Mesh Variables

MCRT does one mesh loop calculation and

computes mesh variables and their spatial

integrals. These variables are stored in

LCM unless otherwise noted.

CENTX. .1]

Radial coordinate of the cent-

roid of cell ij (km).

CENTY . Axial coordinate of the cent-lj

roid of cell ij (km).

The centroids represent the positions of

cell-centered mesh variables, and they are

the arithmetic means of the radial and

axial coordinates of the vertices. CENTX

and CENTY are set to O when I = IP1, to

simlify subroutine CENTR@Y.

BETALC. .1-J

Radiation derivative in cell

ij .

SIGPLCij Mean absorption coefficient

in cell ij (l/km).

BETALC and SIGPLC are computed by double

logarithmic interpolation in the equation

of state tables BTBL and SPTBL, respectively,

FSCAT .. Absorption probability in cell1-J

ij (stored in SCM) .

FSN. . Identical to FSCAT.1-J Ij “

FSCAT is computed from

FSCAT =

l+c(ALPHA) (BETALCij)l (.SIGpLCij) (DT@LD) ‘

where ALPHA is an input quantity and DT@LD

is the radiation time interval. When

ALPHA = O, FSCAT = 1; there is no scatter-

ing, and the calculation that walks each

particle to absorption is explicit. When

ALPHA = 1, scattering is maximized and the

calculation is fully implicit, allowing

both absorption and scattering.

RZEDEN . Total energy to be radiatedIj

from cell ij during the ra-

diation transport cycle (J).

RZEDENij =aXcX(FScATij ) (BETALCij) (SIGPLCij)4

x ‘T~pij) ‘DT@LD) ‘O1weij’where Volume = 2T/RV@Lij.

SIEMIN Smallest amount of internal

energy (J) in any cell in which

the temperature exceeds TEMIT.

SIEMIN is used to determine the death energy

of statistical particles. The cell con-

taining SIEMIN is IJMIN, and the temperature

and density in the cell are TMIN (eV) and

DMIN (mg/cm3), respectively.

EINT Total internal energy (including

radiation) in the mesh (J).

EKIN Total kinetic energy in the mesh

(J).

15

EALL Total internal and kinetic energy

in the mesh (J).

URT@T Total radiation energy in the mesh

(J).

2. Time Interval

The time interval for the next radia-

tion transport cycle is calculated in MCRT.

TIME Initially, the problem time at the

beginning of the radiation trans-

port cycle (s).

At the end of MCRT, TIME is advanced to the

value at the end of the cycle (and the be-

ginning of the next cycle). Initially,

DTR is the time interval for the radiation

cycle, but throughout MCRT it is the time

interval of the next cycle. The original

value is retained in DT@LD.

T2 Time at the end of the cycle T2 =

TIME + DTR, where TIME and DTR

have their initial values.

DTR, the time interval for the next radiation

transport cycle is calculated by MCRT. The

inconsistencies introduced by using the time

interval for cycle m+l on values at the be-

ginning of cycle m are negligible. This

method is used to avoid recomputing quanti-

ties available during the MCRT calculation.

The energy radiated from’s cell during a

cycle is RZEDEN. .. We require that this not1]

exceed 15% of the total energy (E::) in any

cell where the

(Rz=ii)Let DTR in the

temperature exceed~JTEMIT.

X DTR S 0.15 X E..~1

denominator be the value for

the current cycle, (DT@LD). Also let the

value in the numerator be that for next cy-

cle, and solve for the latter;

(0.15 x Ei. X DT@LDDTR = min

R’EDEN. .13 )

At the end of MCRT, DTR may be reduced, if

necessary, to complete an even number of ra-

diation cycles per hydrodynamic cycle. Be-

fore the possible reduction, DT@LDER = DTR.

Later values of DTR are based on DT@LDER

rather than the reduced value of DTR, which

is usually very small.

3. output

MCRT prints

NCYC, TIME(at start of cycle), T2, DT@LD

URT@T, EINT, EKIN, EALL

IJMIN, SIEMIN, TMIN, DMIN

c. REEFER

1. Variables Computed by REEFER

JCEN Initially, the number of census

particles carried over from the

previous cycle.

At the end of REEFER, JCEN is the number of

census particles carried over to the follow-

ing cycle.

IBANK The number of particles to be

withdrawn from the bank on each

pass through the bank particle

calculation.

On the first pass, it is the number (after

splitting) of original source and census

particles sent to the bank. on each subse-

quent pass, it is the number (after splitting)

of particles from the previous pass deposited

in the bank.

ID Number of words of energy de-

position data currently stored

in the buffer array EBL@CK.

When ID = NBUF = 6000, EBL@CK is dumped to

Fileset 3 and ID is reset to O. NBUF is a

fixed parameter set by (3FFwEG@.

NFLUSH Total number of particles for

which deposition data are written

on Fileset 3.

The following indices are totaled sepa-

rately for census particles, source particles~

and for each pass through the bank calcula-

tion.

NGEN

NCEN

NBANK

NDIE

IESCAP

NM@VE

Number of particles

Number of particles

sus (for processing

radiation cycle) .

Number of particles

bank.

Number of particles

started.

sent to ten-

on the next

sent to the

that die from

either absorption, or loss of

energy (in WALK) .

Number of particles that leave

the mesh (in WALK).

The number of particle moves (in

“

f

16

WALK several moves may be re-

quired to effect a collision).

NC@L ‘I’henumber of particle collisions

(in WALK).

The random variables that define parti-

cles are listed below:

Statistical particle positions are defined

by three coordinates rather than the two co-

ordinates required by all other space-de-

pendent variables in the program. A(l),

A(2), and A(3), are the x, y, and z compo-

nents of the particle position (km). Before

WALK changes them, these are called XAl, XA2,

XA3, respectively. The corresponding R-Z

coordinates are:

[ 11/2Radial coordinate RH@P = A(l)2+A(2)2

Axial coordinate 2P = A(3).

The initial positions of census and

bank particles are the positions recorded

at census or deposited in the bank. The

initial positions of source particles are

randomly sampled in the cells in which they

are started. The random positions are cho-

sen by assigning a random weighting factor

to each cell vertex.

@MEGA(l), @MEGA(2), and @EGA(3), the x,

y, and z components of the particle di-

rection vector.

Before WALK changes them, these are called

X@4EGAl, X@MEGA2, and X@MEGA3, respectively.

‘Me initial directions of census and bank

particles are the same as those of the parti-

cles that arrived at census or the bank.

The initial directions of source particles

are randomly chosen polar and azimuthal

angles.

FREQP Particle frequency (1]s).

Before WALK changes it, FREQP is called

XFREQP . For census and bank particles, the

initial frequencies are the same as those

of the parent particles. For starting source

particles, FREQP is a random variable chosen

by subroutine PFREQ.

EPART Particle energy (J).

Before WALK changes it, EPART is called

XEPART . This is the weight that is used to

score the results of random particle walks.

The weight is given according to particle

type as follows.

Census Particles. Usually, EPART is

the energy with which the particle was sent

to census during the previous cycle. If

however, EPART for a census particle is sub-

sequently found greater than RZEDEN ij (the

energy to be radiated in cell ij, where the

census particle is located) , EPART is re-

duced to RZEDEN. Furthermore, no single

particle is allowed to carry more energy

than RZEDEN\lOm Thus, if the particle en-

ergy is larger than this, the census parti-

cle is split into NCP particles so that the

energy of each, EPART, is less than RZEDENI1O.

Whenever census particles are generated

in a cell ij, RZEDEN. , is reduced by the en-~J

ergy carried by the particles.

Source Particles. NSP = 10 source par-

ticles are started in each cell where the

temperature exceeds TEMIT = 0.05 eV. The

energy of each particle is EPART = RZEDEN\NSP.

Bank Particles. Each particle sent to

the bank (with energy EPART’) is split into

NBP = 3 daughter particles, each with ener-

gy EPART = EPART’/NBP.

I, J Indices of the cell in which the

particle is generated.

For source particles, I and J are known be-

cause particles are generated in particular

cells. I and J are known for bank particles

because WALK always knows in which cell a

particle lies and deposits this information

with the parent particle parameters. For

census particles, the cell in which the parti-

cle lay when it was sent to census is known,

but, because the mesh will generally have

been rezoned in the meantime, the particle

may be in a different cell. Subroutine

WHERE is called to find I and J for census

particles.

T1 The time when WALK is called (s).

For census and source particles, T1 = TIME.

For bank particles, T1 is the time the parti-

cle was sent to the bank.

EDEATH Particle death energy (J).

Particles whose energy is less than EDEATH

are terminated by WALK. For census and

17

source particles, EDEATH is 1% of EPART,

or 1% of SIEMIN, whichever is smaller. For

the first bank calculation, EDEATH = EDIE.

On subsequent bank calculations, EDEATH is

1% of SIEMIN.

EDIE Minimum value of EDEATH for all

source particles (J).

IDIE As input to WALK, this identifies

the particle type as:

IDIE = O Census or source particle,

IDIE = 1 Bank particle.

As output from WALK, IDIE identi-

fies the particle type as:

IDIE = O Escaped or dead particle,

IDIE = 1 Bank or census particle.

ERAD Total energy of all source par-

ticles (J).

ECEN1 Total energy of all input census

particles (J).

ECEN Total energy of all output census

particles (J).

EMC Total energy radiated (ERAD+ECEN1)

in the particle population.

2. REEFER Data Storage

Particle production and energy deposi-

tion data are stored on Fileset 3, two words

per particle, in floating point and integer

packed format.

Word 1 Bits 59-40 Radial coordinate

39-20 Axial coordinate

19-0 Energy

Word 2 Bits 59-18 Frequency

17-9 I

8-O J

The coordinates represent the position

where the particle was produced or the en-

ergy was deposited. The energy is either

the negative of the original source particle

energy (the emission”energy) , or the energy

deposited at a score. Word 1 is floating

point data packed by PAKFN@. The frequency

in Word 2 is truncated, and the integers I

and J are stored in the low-order bits.

Census and bank particle data are read from

Fileset 1 and written on Fileset 2, eight

words per particle.

Word Census Bank

1 A(l) A(1)

2 A(2) A(2)

3 A(3) A(3)

4 @MEGA(l) f5MEGA(1)

5 @EGA (2) @WEGA (2)

6 $NEGA(3) @MEGA(3)

7 EPART -EPART

8 FREQP(bits 59-18) FREQP(bits 59-40)

I(bits 17-9) Tl(bits 39-20)

J(bits 8-O) I(bits 17-9)

J(bits.8-0)

For census particles, FREQP is truncated

as described above, but not packed. For

bank particles, FREQP and T1 are packed by

Subroutine PAKFN@.

3. REEFER Output

The indices NGEN, NCEN, NBANK, NDIE,

IESCAP, NM@E, and NC@L are printed for

census particles, for source particles, and

for each pass through the bank particles.

Also printed are NFLUSH, EMC, ERAD, and

ECEN1 .

4. Subroutine WALK

Length of Particle Movements.

DM@E Distance the particle moves.

Generally WALK will move a particle several

times between its initial position and its

final position (collision, absorption, es-

cape, or census) . DM@VE is the minimum of

the following:

DCEN Collision Distance.

Initially, DCEN = c(T2-T1) , where T2-T1 is

the time remaining in the radiation trans-

port cycle, and c is the speed of light.

After each particle move, DC.EN is reduced

by DM@7E.

DC@L Collision Distance.

Initially, DC@L is the length of a random

number of mean free paths. The initial

value of DC@L is given by RMFP x DMFP.

RMFP is a randomly sampled number of mean

free paths. RMFP can vary from O to =, it

is usually less than 1 and it is sampled

from RMFP = Iln(y) 1, where y is a random

number.

18

——

After each particle move, RNFP is reduced

by an appropriate amount and DC@L is re-

computed.

DNFP = length of a mean free path at the

particle location; where DN.FP= l.O\SIGNU

and SIGNU is the absorption coefficient

dependent on the temperature and density

at the particle location, and on the par-

ticle frequency. The weighting factors

for computing density and temperature at

the particle position are found by sub-

routine CENTR@Y; the subscript of the

absorption coefficient, by subroutine

SUBSCR .

DCELL Nominal move distance.

Because mesh properties vary continuously,

DMFP is a continuous function of particle

position. To approximate this continuous

change, mesh properties are re-evaluated

whenever a particle moves. The nominal

move distance is the minimum dimension of

the cell in which the particle lies.

5. Energy Scores

When a particle moves, its energy is

reduced by

ESC@RE = (EPART) [ 1~_e(-FSP) (DM@VE)(SIGNU) ,

where FSP is the absorption probability at

the original position of the particle (the

weight factors for computing FSP are found

by CENTR@Y). The energy, ESC@RE, is de-

posited at a position RH@D, ZD, midway

between the initial and final particle po-

sitions. The new particle position and en-

ergy are then computed. Subroutine WHERE

is called to determine the indices I and J

of the cell in which the particle lies after

the move. Tests are next made to determine

whether the particle went to census, went to

collision, went out of the mesh, ran out of

energy, or merely moved while randomly seek-

ing one of the aforementioned fates. If

the particle went to census, its remaining

energy is deposited (scored) at its final

position and IDIE is set to 1 to tell REEFER

to store the particle parameters on File-

set 1. If the particle left the mesh, its

remaining energy and final position (outside

the mesh) are saved on Fileset 3, and IESCAP

is incremented. If the particle ran out of

energy, its remaining energy is deposited

(scored) at its final position and NDIE is

incremented. If the particle underwent a

collision, NPC@L is incremented. The parti-

cle will have been absorbed (with a proba-

bility FSP) , or scattered (with a probabil-

ity 1-FSP) at the collision. If the parti-

cle was absorbed, the remaining energy is

deposited (scored) at the point of collision

and NDIE is incremented. If the particle

was scattered (and NPC@LSNPCMAX) the parti-

cle is continued (reemitted) . Its random

frequency and direction after scattering

are computed by subroutines PFREQ and

P@MEGA, respectively. The local mesh pro-

perties are recomputed using subroutine

CENTR@Y. If the number of collisions ex-

ceeds NPCMAX, the particle is deposited in

the bank. If it is deposited, the particle

energy is made negative and IDIE is set to

1 to tell-REEFER to store the particle para-

meters on Fileset 1.

6. Subroutine FLUSH

FLUSH is a utility routine called by

WALK and REEFER. This routine writes parti-

cle production and energy deposition data —

(stored in the buffer EBL@CK) on Fileset 3.

FLUSH is called by WALK when the counter

ID = NBUF, and by REEFER after the last

deposition has been made. ID is reset to

O, and NFLUSH is incremented by the number

of particles (ID\2) for which data were

written.

7. Subroutine CENTR@Y

CENTR@Y is called by WALK, and it com-

putes interpolation factors for determining

the values of the cell-centered mesh vari-

ables at the position of a particle in a

cell whose indices are I and J.

ISC 1+1 if the particle is in the

right side of I, J.

I-1 if the particle is in the left

side of 1, J.

J+l if the particle is in the top

part of I, J.

J-1 if the particle is in the bot-

tom part of I, J.

JSC

19

CWGT1 Weight factor for cell ISC, J.

CWGT2 Weight factor for cell I, JSC.

CWGT3 Weight factor for cell I, J.

The value of mesh variable z at a posi-

tion x, y in cell i, j can be approximated

by ‘a linear combination of three known

values of z:

‘3= z(x3,y3) zl = Z(XIIY1) 22 = Z(X2JY2).

x3tY3 is the centroid of cell I, J.

xl 1Y1 is the centroid of cell ISC, J.

X.2rY2 is the centroid of cell I, JSC.

The three known values of z form a plane,

where

z = ‘lzl + ‘2Z2 + ‘3z3”

let

6X1 = xl - X3 ~Y1 = Y~ - Y3

6X2 = x2 - X3 ~Y2 = Y.2- Y~

6X =x-x3

6y ‘Y - Y3

.3x16y-6x6y1

‘2 = ~x~~Y2-~x.2~Y~ ‘

‘3=~-wl-~w2”

The weight factors WI, W2, and W3 are

represented by CWGTl, CWGT2, and CWGT3,

respectively, in the program. In a rec-

tangular mesh, 6yl = 6X2 = O and the

weights are

‘1 =&p2 ‘@tw3=1-wl-w2 “

Special provisions are made for the mesh

boundaries.

At the left boundary (i-1=0), or the right

boundary (i+l=IPl), set ax = 6x2 = 0“

‘1 = 0’ ‘2 =%, W3=1-W2 .

At the bottom (j-1=1), or the top

(j+l=JP2),lset &y = 6y1 = O.

‘1 ‘~’w2=0’w3=1 ‘Wl”

At the corners, both conditions apply,

and

‘1 =W=2 0 ‘ ‘3=1 “

8. Subroutine WHERE

WHERE is called by REEFER and its pur-

pose is to solve the general problem:

Given a position r, z, find the cell i,

j in which it is located.

Start with an initial guess i, j. In row

j, move to the left (west) until a cell is

found whose west boundary is west of r, z.

Let this cell be i-k, j (k=0,1,2, ...1)l).

If the r-values of both northwest and

southwest vertices are west of r, then

r is in the cell.

If the r-values of both northwest and

southwest vertices are east of r, then

r is in the next cell.

If one vertex is east of r and the other

is west of r, a test is made to determine

whether r is east or west of the line con-

necting the vertices.

If k > 0, the particle is in i-k, j, and

i is set to i-k.

If k = O (the original cell) , a similar

procedure is followed moving east.

When the value of i is determined, the

testing is done both south and north in

column i, to find j. If j is different

from the original j, the entire process

is repeated, because the i-value that is

correct for one value of j may be wrong

for another.

An improved version of WHERE is in prepara-

tion.

9. Subroutine SUBSCR

SUBSCR, called by WALK, is used to find

the frequency-dependent absorption coeffi-

cient at a particle position. The equation

of state variables L3PTMP(J), %PDEN(K) , and

FREQ(I) are tabulated. Analytic expressions

have been found for

J as a function of @PTMP,

K as a function of flPDEN,

I as a function of FREQ.

Given the particle frequency and the tempera-

ture and density at a position, I, J and K can

20

.

be computed. The combined subscript of the

frequency-dependent absorption coefficient

is

IJK = I + (J-l)*NFRQ + (K-l)+N@pT*NFRQo

Note that frequency-dependent absorption

coefficients are taken directly from the

tabulated data and are not interpolated,

and that this routine is data dependent.

10. Subroutine PFREQ

PFREQ, called

random frequencies

The frequency (eV)

distribution when

,’

by

of

is

WALK, is used to sample

statistical particles.

given by the Planckian

‘=-F(% In (Y1,Y2,Y31Y4)T I

where the y’s are uniform random numbers

on (0,1) and T is the temperature (eV).

C(k) = min lm,yc(m)] ,

where L(OY)= x — = 1.0823 and y isn=1 n;

a random number uniform on (0,1).

m is defined as the smallest integer for

which

m

~(m) = x — 2yc(@) .n=l n:

The factor 2.41814 x 1014

is the conversion

between hv (energy) and v(frequency) units,-1

expressed in eV/s .

11. Subroutine P@MEGA

P@lEGA samples random direction vectors

for statistical particles from the isotropic

density function.

D. ESTEP

The variables computed by ESTEP are:

‘pARTi jTotal energy deposited by parti-

cles in cell ij (J).

EMSN. .13Total energy emitted by particles

in cell ij (J).

ELfik5T Total energy of particles that

escape from the mesh (J).

EABS Total energy of particles ab-

sorbed in the mesh (J).

EEMIT Total energy of emitted parti-

cles (J).

RA Normalizing factor for absorp-

tion.

RE Normalizing factor for emissions.

RE and RA are the ratios of EMC

(the total energy of all parti-

cles, computed by REEFER when

the particles are generated) to

the total emission and absorp-

tion energies of the particles

actually retrieved by ESTEP.

The ratios differ very slightly

from 1 because of the loss of

significance caused by packing

the data. All energies are

normalized by RE or RA to con-

serve energy. The packing er-

SIE. .13

rors are random, so the solu-

tion accuracy is limited by

statistical error.

Specific internal energy in

cell ij (J/mg). SIE. . is basedI-J

on the original internal energy,

increased by the energy absorbed

in the cell and decreased by the

energy emitted in the cell:

EPARTi.-EMSN. .SIE. s = SIE.. + 1] X volume .

1] 13 R‘ij

TEMP. .1] Temperature in cell ij (eV).

The calculated total specific internal

energy of the cell contains the radiation

term, aT4. It is necessary to find a

SIE = I(P,T) + aT4 I

where I is the equation of state internal

energy for the material in the cell and

does not include the radiation term. An

iterative procedure is used. On the

same graph, (Fig. 7)

I. I = SIE-aT4 vs T ,

II. I(P,T) VS T .

Curve I decreases from SIE when T = O, to

O at some high value of T. Curve II has

a positive slope. The intersection of

the two curves corresponds to the T being

sought . The minimum value of the solution

is the ambient temperature TL@W = TAMB,

and the maximum value is that at which

the first curve goes to O, THIGH.

Guess a value of T midway between THIGH

21

Fig. 7. Calculation of temperature frominternal energy.

and TL@W and determine the value of both

curves. If curve I is above curve II,

the intersection is at a higher tempera-

ture value, so set TL@W = T. If curve II

is above curve I, the intersection is at

a lower value, so set THIGH = T. Repeat

the process until THIGH and TL@W are the

same.

TMAX Maximum temperature in the mesh

(eV).

DMAX Maximum density in the mesh

(mg\cm3).

DMIN Minimum density in the mesh

(mg\cm3).

ET@T Total energy deposited by particles

(J).

TAVG Midpoint of the time interval (s).

THY Total energy lost from the mesh,

summed over all cycles (J).

The following quantities are printed and

written on tape:

ET@T, EABS, EL@ST, EEMIT, ECEN, RE, RA

RAVG, TMAX, DMAX, DMIN, and THY.

E. LISTING

LISTING writes REEFER mesh data on film,

and it is called only at output times, T@UT.

The data written, for each cell, are

I, J Cell indices.

x, Y Radial and axial coordinates of the

[lower left-hand vertex) of the

cell.

EPART, EMSN, TEMP, FSN, SIGPLC, BETALC,

and RZEDEN.

v’. Sn SOLUTION OF THE RADIATION TRANSPORT

PROBLEM

Overlay 4, 0 (GREYSN) computes a grey

(one frequency group) solution to the ra-

diation transport problem, using the Sn

method.

A. Overview of the Overlay

The primary overlay for the Sn radia-

tion transport is GREYSN, Overlay 4.0,

which computes variables used in the calcu-

lation and calls the secondary overlays.

1. Overlay 4.1 CYLSN

CYLSN performs the Sn solution to the

radiation transport problem in RZ geometry.

2. Overlay 4.2 SNESTEP

SNESTEP uses the energy fluxes com-

puted by CYLSN to advance the internal

energies and temperatures in the mesh cells.

3. Overlay 4, 3 SN@UT

SN@UT is the output program for the Sn

overlay, and it is called only at output

times, T@UT.

B. Calculations of Overlay 4

1. GREYSN

GREYSN initializes parameters

ALPHA Implicitness parameter, originally

read as input by f5FFWEG@

(O S ALPHA S 1). ALPHA = O leads

to an explicit calculation (no

scattering) ; ALPHA = 1, to an im-

plicit calculation.

ISN Order of the Sn calculation, origi-

nally set to ISN = 4 by @FFWEG@.

Both parameters are stored in common block

CRIMSN by @FFWEG@. GREYSN does one mesh

loop calculation and computes mesh variables

and some totals of mesh variables.

CENTX. .1] Radial coordinate of the cen-

troid of cell ij (km).

CENTY. .1]

Axial coordinate of the cen-

troid of cell ij (km).

CENTX and CENTY are the arithmetic means

of the radial and axial coordinates, re-

spectively, of the cell vertices. They are

not used by GREYSN, but are required for the

analysis of dump tapes.

).

●

22

SIGPLC. . Planck or Rosseland mean ab-1]

sorption coefficient (l\km)

associated with the temperature

and density in cell ij, inter-

polated from the equation of

state table, SPTBL.

RZEDEN. . Explicit radiation source in~1

cell ij (J\km3-sr) .

RZEDEN. . = 3.2757 x 1014 (SIGPLCij)(TEMP 4,~1 ij

where the constant is%“

FSN. . Absorption probability for ra-1]

diation in cell ij.

FSN. = 1.0Ij

l+C(ALPHA (Bpij) (SIGPLC. .13)(DTR)‘

where BP. . is the radiation derivative as-1]

sociated with the temperature and density

in the cell and is interpolated from the

equation of state table, BTBL. When

ALPHA = O, FSN = 1 and there is no scatter-

ing.

AVINT. . The average intensity is set1]

to O in all cells before the

start of the iteration.

RSNi Radial coordinate of vertex

i = X(1,2) (km).

ZSN . Axial coordinate of vertexJ

j = y(l,J) (km).

ESN Total energy radiated (J).

ESN = 81T2 xij

(R2EDENij)/(RV@Lij) .

The principal time interval calculation

is in SNESTEP, but the time interval is

shortened, if necessary, at the “end of

GREYSN .

2. CYLSN

CYLSN computes the Sn constants,

SNC@N(I) I = 1, 181.

These constants are computed by sub-

routine SNGEN on the first cycle and on any

subsequent cycle when ISN is changed.

Bi Area of the top of cell i (for

all j) (km2), = IT(RSN~+l-RSN~) .

The rest of CYLSN is devoted to the Sn it-

eration.

AVflLDij Average intensities from the

previous iteration.

AVINT, ,Ig

Newly computed average inten-

sities.

In GREYSN, AVINT. , has been set to O for1]

all ij. On each iteration in CYLSN,

AV@LD. . = AVINT. ., and AVINTij

is recomputed

by su~~outine S&P.

The process is continued until AVINT. zl-j

AVflLDij for all ij.

ISTEP is the iteration counter. For explicit

calculations (ALPHA = sNC@N(182) = O.0)~only

one pass is made through SWEEP.

The calling arguments of the subroutines in

the CYLSN overlay are summarized in Table I.

SWEEP . Subroutine SWEEP is called by

CYLSN . SWEEP and its dependent subroutines

IN and @UT perform the Sn calculation.

The mesh variables computed by SWEEP are:

AVINT. .1-J

EM@lLC .lj

UI@M.Lcij

F@UTLCij

CYSSN

ZSN

SSN

B

SNC6N(LBST1)

sNcON LSET2 )

SNC@N [LO)

SNC!ZN(LS)

SNC@N (LW)

AL

BR

SB

ISAR

JSAR

NN

m

AVINT

AvOLn

Average intensity of radiation

in cell ij. (J\cm2-s-sr),

Vertical component of radiation

flux in cell ij (J\cm2-s),

Horizontal component of radia-

tion flux in cell ij (J\cm2-s) ,

Net rate of flow of energy out

of cell ij (J\s),

TABLE I

SUBROUTINE PASMISTSRS

xpgZSN

RSN R

s B

BET1 SETA

BET2

u u

E Ew wAL ALBR Bi3

BB Sv

IT IT

JT JT

NN NN

Nn

AVINT

AVOW

s s

CT CT

RNgill UN@!

E@N EM@

F@Y2 FOuT

1.0, ES

-1.0

DZP DZP

DZ DZ

,SM1 J

AVNEW AVNEN

K2

242

RsUARNS

Defined in text

Defined in text

Defined in text

sn constant

Sn constant

sn constant

Sn constant

Sn constant

Deffned in text

Defined i. text

Defined in text

Defined in ts%t

Defined in text

NN . lsN/2

24N - 1sw(Isw+2)/8

Defined in textDefined in text

De ffned in text

Defined in text

Oefined in text

Defined in text

Defined in text.

-1.0 in downward talc, 1.0 in

upward talc.

DZP - 2TDZ

Dz - ‘sNj+l-zsNjJN1 - inde= of sow being calcu-

lated - 1

Defined in text

N1=SSN+l

M2-2NM

23

SUM= ~ F@UTLCij .ij

SWEEP computes the constant M2 = 2*MM,

the total number of angles for which fluxes

are to be calculated. The maximum permissi-

ble value of M2 is 72.

SWEEP calculations are done one row at a

time, and they utilize row variables, (which

are the mesh variables for the row) , and

intensities.

F@UTi = F@JTLCij

EM@Mi = EM@MLC, .13

UM@Mi = UM@MLCij

AVNEWi = AVINT. .1-J

CTi = SIGPLCij .

.

.

.

.

Si = (R2EDENi~)(FSNij )

+ (AV@LDij) (l-FSNij) (SIGPLCij)

(radiation source).

BB . Vertical intensity for directionl,mm, m = 1, M2.

BR . Horizontal intensity for direc-3-l,mtion m, m = 1, M2.

‘Lk, i Angular flux k = 1, NN

The maximum dimensions of the variables are:

Row variables I = 100

BB I = 100, M = 72

1xM=7200

BR J= 101, M = 72

(J-1) X M = 7200

AL K=8, 1=100, KXI=8OO

SWEEP first computes the downward flux row

by row, starting with row JP1. The downward

intensities at the top of the mesh (BB) and

the inward intensities from the right (BR)

are set to O. The row variables, F@UTi,

EM@Mi, and UM@Mi are set to O, and AVNEWi,

cTi , and Si are computed.

Subroutine IN is called to compute row

variables and new intensities, from right

to left, and @UT is called to work from left

to right.

When the row has been calculated, the row

variables are stored as mesh variables.

When the bottom row is reached, the process

is reversed and the calculation is made from

bottom to top. The upward intensities at

bottom of the mesh are set to O. The row

24

the

variables F@UTi, ~@Mi, and ,UM@Mi are ini-

tialized to the corresponding values of the

mesh variables.

3. SNESTEP

SNESTEP advances the energies and tem-

peratures in the mesh cells. .

(-F@UTLCi.) (DT@LD) (RV@Li.)SIE.. =1-J + SIE.. ,

2T X 10-15(R@ij) 13 ,

where DT@LD is the time interval of the

radiation transport cycle. TEMP. .13

is found

‘rem ‘lEij by the iterative procedure de-

scribed in Sec. IV.

Other variables computed are:

SIET@T

URT@T

Pm

EL@ST

EABS

PWR2

Internal energy, in excess of

ambient, (including radiation)

in the mesh (J).

Radiation energy in the mesh (J).

Total radiation along the mesh

boundaries (J\s).

Total energy lost from mesh during

time Step (J).

Energy absorption rate during time

step (J/s).

Time rate of change of internal

energy (J/s).

The radiation transport time interval is

calculated by SWEEP.

DTR Initially, the radiation time in-

terval for the cycle being calcu-

lated, then the interval for the

following cycle.

DT@LD Interval for the cycle being

calculated.

During the cycle being calculated, let

ECELL. . be the total internal energy in

cell ;; and PMARK = lF@UTLCijl, the rate of .energy change in cell ij. The energy change

in cell ij during the cycle is (PMARK) (DTR).

DTR must be such that the energy change does.

not exceed 15% of ECELL. ., in any cell where~1

TEMP. . ~ 0.1 eV.13 Then,

XDTR. .1-J= 0“15 ‘EcELLij)’pMRKij’

DTR = min(XDTRij).

DTR may be modified further by GREYSN.

4. SN@UT

SN@UT plots the magnitude and direction

of radiation in each cell. These are two

plots, one of the entire mesh and another

of the region of interest around the ~ubble.

ACKNOWLEDGMENTS

The authors gratefully acknowledge

their indebtedness to the many people whose

assistance and support have made the Y@KIFER

program possible.

E. M. Jones for assistance in adapting the

hydrodynamics program YAQUI to Y@KIFER.

W. H. Reed, and H. G. Horak for writing

the Sn program and adapting it to Y@KIFER.

J. W. Kodis for subroutine PARTM@V and

for improved versions of other subroutines

to be incorporated into Y@KIFER in the

future.

LASL groups T-4 and J-15 for providing

the equation of state and opacity data.

L. W.

SEARCH ,

H. M.

agement

Fullerton for the subroutines

PAKFN@, and UNPKFN.

Peek and J. Zinn for their encour-

and support.

RIZFERENCES

1.

2.

3.

A. A. Amsden and C. W. Hirtr “YAQUI:An Arbitrary Lagrangian-Eulerian Com-puter Program for Fluid Flow at allSpeeds,” Los Alamos Scientific Labora-tory report LA-51OO (March 1973).

M. T. Sandford II and R. C. Anderson“Two Dimensional Implicit RadiationHydrodynamics,” J. Comput. Phys. ~,

130-157 (1973).

K. D. Lathrop and F. W. Bunklev“TW@TRAN-11 :- An Interfaced, E~portableVersion of the TW@TRAN Code for Two-Dimensional Transport,” Los AlamosScientific Laboratory report LA-4848-MS(July 1973].

APPENDIX A

INSTANT Y@KIFER

TABLE A-I

yOxlFER PROSRM ORGANIZATION

Qverlay O, 0

overlay 1, 0

Dverlay 2, 0

Qverlay 2, 1

Dverlay 2, 2

overlay 2, 3

Overlay 2, 4

Dwerlay 2, 5

Y!mxFEs

r4i!aP

F12Mcgi

GET2f4pDBLINT

SEAscn

OFFWSGO

MESKMKK

PASDEN

PASTG2N

NSTAST

YWSY

PHASEO

Y@KOUT

PARPLf7T

PKASE1

NADD

PKASE2

PSASE3

RIZONS

PA21TMw

Gverlay 3, 0 UCRT

PASINO

m4PH?4

Overlay 3, 1 AE2PER

FLUSK

P@EGA

PFSZQ

SUBSCR

CR@SS

CENTR@Y

WALK

WSEAZ

Overlay 3, 2 SSTEP

Overlay 3, 3 LIsTING

Overlay 4, 0 GREYSN

Overlay 4, 1 CYLSN

SWEEP

m

@lT

SNGEN

overlay 4, 2 SNSSTEP

overlay 4, 3 SN@UT

TAS3LE A-II

Y@KIFER CONt40N BLOCKS

Common @FFWEG@ Y@KKY NCRT GREYSN DUNF

STATE x xx x

RSD x Xxxx

PINK x xxx

@RANGE x x x

WHITE x x x

YELL@i x x

GREEN x x x

BLUE x

LAVNDER x

SILVER X x x x

CR2MSN x xx

SN@WITE x

NAWE x

SENSE x xxx

25

-2G‘x

-1-----

.J-x

“*ae-u

..-.m

a.?

--3

.:2A;3

.

..

+

ml-l(n

26

..

......

a.

......

Mdm<H

1-2..T,

x:In

>

-1

..

4-)-4

.-(-.ln

t-)O

-ZU

J<..M

CI

.Y*c

ln

“

.-l..Z

2Zz

>>Z

LO

azv

27

i-l

.“

APPENDIX B

THE NEXTWAY PROGRAM

NEXTWAY reads dump tapes from Y@KIFER and

plots the information on them. The program

generates a uniform mesh and interpolates

the values of the mesh variables to find

their values at the centers of the uniform

cells. These values are plotted. The plots

produced are:

a. Velocity vectors

b. Three-dimensional plot, rear view

c. Three-dimensional plot, rear view

d. Variable vs radius, through the bub-

ble center

e. Variable vs axial coordinate, along

the axis

Plots b, c, d, and e are plotted for each of

the following variables:

1.

2.

3.

4.

5.

6.

7.

8.

9.

TEMP

SIE

R@

SKE

RZEDEN

SIGPLC

P

l-FSN

AVINT

For Monte Carlo

Temperature (eV)

Specific internal energy

(J\mg)

Density (mg\cm3)

Specific kinetic energy

(J/mg)

Radiation source density

(J\cm3)

Mean absorption coefficient

(l/km)

Pressure (M-Pa)

Radiation scattering proba-

bility

Average radiation intensity

(Sn only)

calculations, the following

additional plots are made:

f. Spectrum of production particles

(W/eV vs eV)

9. Spectrum of energy depositions

(W/eV vs eV)

h. Spectrum of escaped particles

(W\eV vs eV)

i. Spectrum of census particles

(W/eV vs eV)

j. Map of production particles

k. Map of energy depositions

1. Map of census particles

Input Cards

Card 1

NXE

NYE

Card 2

XR

YT

YE

Cards 3-5

SCALEBi

SCALERi

Format 216

Number of cells, radially, in the

uniform mesh. Default: IBAR

Number of cells, axially, in the

uniform mesh. Default: JBAR

Format 3E12.4

Right boundary of the uniform

mesh (km). Default: X. ,I-J

,at IPl,2

!l!opboundary of the uniform mesh

(km).‘efault: ‘ij’ at 1’ ‘P2

Bottom boundary of the uniform

mesh (km). Default: Y. .1]

, at 1,2

Format 6E12.4

SCALEBi is the minimum ordinate

on graphs of mesh variable i.

SCALERi is the maximum ordinate

on graphs of mesh variable i.

i refers to the variable numbers

(1-9), above. The default values

are O and 1.5 x maximum value of

the variable, and default values

are used when SCALER, = O.1

Output Cards

NEXTWAY punches a complete set of input

cards that contain either the previous input

values or the default values. The cards may

be used in subsequent calculations to main-

tain uniform graph scales from one tape to

the next.

29

mal

2-:.lJus!

..

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..

WP

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zw:

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34

APPENDIX D.FLOW DIAGRAMS OF THE Y@KIFER

OVERLAYS AND SELECTED SUBROUTINES

0..

:=11

$w/m#)

O.?+ 1.0

ocallY+KKY

Ov.r13fZ,O

lb

c)STOP

b aCOmf.dc

7CUMP

!2+ UWMP -0

)

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no w+z dumpTHE LVD /?5+1-dmp

)Y’= Icy. I

—–—1~-1——

YOKIFIEROVERL4Y 0,0

OFFINEGO* ‘“m””’30

- m-*.,,:. M,E(,,I 4)

—--- . _____

-l-. —.-.m‘“-- G-nlul - .-

RM. .– -.–. _(almk)

.—__ ___ , ----

...——(5re{um _ . .

--

“-”---i~—---.—__ . ... . 1 —

Eoz ye~. . 7

? /t. d-— tmw-d cadsed d>~.— m -— ___ ._. /./duw Ikb.

+%3iZ7+ae

. 1I.wp –(F,d 7) )

A

rq+,rn

.— ...-—

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+--~.—. .

*

.. ..*C

XYR r“?-U.nG..-m- .VJ

.

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Rv@-aaCA

&+!za-.UVR$SZE7VAP

I

Lm- Pumpn46w

rPURD I)JPUT

1

*

h“m 2’1 WAY. n+.a

-M

f W!R&’p

qes - Pu.Qnpmu.m

I1 ..-OVERLAY 2,0

l=+t-

N:YC‘W TD-iNJICT

TE7zl-KEr’@mx,,~

l%i7- . ..-..—--

-.—“l-——.-.—53

..—...,7 rz 7— _-P$:,:(JI.’&.4

cm “!.TWXI TM JrM

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P

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— .

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.return +0cd. a“orkr

hqclmjdel%iR”-d ..-.

.—.

— /)——.-. return

lPNfAi!3E o0return OVERLAY c?,I

.. . . ... . .. .—-

~-E~.-...

T(!WT 551 -, y6N11w>2 I ? ,,,

0 z

i m -1,0--

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WRPL@T . . &ile Plat L&.43

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..— ..— —.

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2 . ... . ..

—-—.

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Cc.+ Nd-=

Cu@e GR1R,QRz2

urlL v7r L

P-SEOVERIAY 2,3

--’@10

-YOz{ c1cmwER!_4Y 2,?

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—-—— -—N1-ww Ci@rd“”-~-fA/ 20??.$ “.

0 /:’v>; ?jzrrs— .—. --- ——- .#ti/-” “-3~.r/

. T -------- -— J 70h W=ZWS

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cOmf.J2R@-E7R - bOUdM+S

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37

I

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1:8, Z?A .—. 2.TPI

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Rmnnx?CENSUS PART/CLE ‘

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39

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