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TRANSCRIPT
NASA CONTRACTOR
REPORT
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BOUNDARY-FITTED CURVILINEAR
COORDINATE SYSTEMS FOR SOLUTION
OF PARTIAL DIFFERENTIAL EQUATIONS
ON FIELDS CONTAINING ANY NUMBER
OF ARBITRARY TWO-DIMENSIONAL BODIES
Joe F. Thompson, Frank C. Thames,
and C. Wayne Mastin
Prepared by
MISSISSIPPI STATE UNIVERSITY
Mississippi State, Miss. 39762
]or Langley Research Center
NASA CR-2 7
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY
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1. Report No. 2. Government Acc_sion No. 3. R_ipient's Catal_ No.
NASA CR-2729
5, Repo_ Date
July 1977
• 4. Title and Subtitle
Boundary-Fitted Curvilinear Coordinate Systems For Solution ofPartial Differential Equations on Fields Containing Any Number ofArbitrary Two-Dimensional Bodies
7. Author(s)
Joe F. Thompson, Frank C. Thames, and C. Wayne Mastin
9. Performing Organization Name and Addre_
Mississippi State UniversityMississippi State, Mississippi 39762
12. Sponsoring Agency Name and Address
National Aeronautics & Space Administration
Washington, DC 2n546
6. Performing Organization Code
8. Performing Organization Report No.
10. Work Unit No.
11. Contract or Grant No.
Grant NGR 25-001-055
13, Type of Report and Period Covered
Contractor Report
14. Sponsoring Agency Code
3860
15, _pplementary Notes
Ruby Davis of the Theoretical Aerodynamics Branch of Langley Research Center converted the
UNIVAC 1106 code generated at Mississippi State University to the CDC 6000 Series code atLangley Research Center. Percy J. Bobbitt was the Langley Teclmical Monitor. Final report.
16. Abstract
A method for automatic numerical generation of a general curvilinear coordinate system withcoordinate lines coincident with all boundaries of a general multi-connected two-dimensional regioncontaining any number of arbitrarily shaped bodies is presented. No restrictions are placed onthe shape of the boundaries, which may even be time-dependent, and the approach is not restrictedin principle to two dimensions. With this procedure the numerical solution of a partialdifferential system may be done on a fixed rectangular field with a square mesh with no interpolationrequired regardless of the shape of the physical boundaries, regardless of the spacing of thecurvilinear coordinate lines in the physical field, and regardless of the movement of the coordinatesystem in the physical plane. A number of examples of coordinate systems and application thereofto the solution of partial differential equations are given. The FORTRAN computer program andinstructions for use are included.
17. Key Words (Sugg_ted by Authoris))
Coordinate transformationMultielement airfoilsNavier-Stokes
18. Distribution Statement
Unclassified - Unlimited
_.- Subject Category 02
19. S_urity Clauif. (of this report} 20. Security Classif, (of this _)
Unclassified Unclassified
21. No. of Pages 22. Price"
253 $9.00
Q
For saleby the National Technical Information Service, Springfield, Virginia 22161
TABLE OF CONTENTS
Ie
II.
III.
Introduction ...........................
Mathematical Development .....................
Preliminaries ..........................
Doubly-Connected Region ......................
Multiply-Connected Region .....................
Simply-Connected Region ......................
Coordinate System Control .....................
Time-Dependent Coordinate Systems ........ .........
Numerical Solution ........................
Difference Equations .......................
Multiple-Body Fields .......................
Multiple-Body Segment Arrangements ................
Initial Guess ..........................
Convergence Acceleration .....................
Optimum Acceleration Parameters ..................
IV. Examples of Coordinate Systems ..................
Coordinate System Control .....................
Various Body Shapes ........................
V. Examples of Application to Partial Differential Equations .....
Potential Flow ..........................
Viscous Flow ...........................
Velocity-Pressure Formulation ................
Pressure Distribution and Force Coefficients .........
Loaded Plate ...........................
_I. Instructions for Use - Coordinate Program ............
Coordinate System .........................
Dimensions ........................
Input Parameters .......................
Field Size ....................
Plotting ...........................
Initial Guess ........................
Body Contours ............ ............
Re-entrant Boundaries ....................
Acceleration Parameters ...................
Coordinate System Control ..................
Convergence of Very Large Fields ...............
Listing of Program TOMCAT .....................
Page
1
7
i0
13
15
15
19
21
21
23
24
26
30
33
36
36
37
38
38
42
44
45
46
47
47
48
48
48
48
49
49
50
51
51
54
56
iii
VII.
Vlll.
IX.
X.
XI.
TABLE OF CONTENTS, CONTINUED
Page
Sample Cases ............. •............. 93
i. Simply-Connected Region ................ 94
Input ....................... 94
Output ...................... 96
Plot ....................... i01
2. Single-Body Field ................... 102
Input ....................... 102
Output ...................... 103
Plot ....................... 109
3. Double-Body Field ................... ii0
Input ....................... ii0
Output ...................... 112
Plot ....................... 122
Instructions for Use - Factors Program ............. 123
Scale Factors .......................... 123
Listing of Program FATCAT .................... 124
Sample Case: Single-Body Field ................. 140
Input ........................... 140
Output ........................... 141
References ........................... 147
Tables ............................. 150
Figures ............................. 170
Appendices ........................... A-I
iv
BOUNDARY-FITTED CURVILINEAR COORDINATE SYSTEMS
FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONSON FIELDS CONTAINING ANY NUMBER OF ARBITRARY TWO-DIMENSIONAL BODIES
By Joe F. Thompson,t Frank C. Thames,* and C. Wayne Mastin §
Mississippi State University
Mississippi State, Mississippi 39762
Research Sponsored by NASA Langley Research Center, Grant NGR 25-001-055
SUMMARY
A method for automatic numerical generation of a general curvilinear
coordinate system with coordinate lines coincident with all boundaries of
a general multi-connected, two-dimensional region containing any number of
arbitrarily shaped bodies is presented. No restrictions are placed on the
shape of the boundaries, which may even be time-dependent, and the approach
is not restricted in principle to two dimensions. With this procedure the
numerical solution of a partial differential system may be done on a fixed
rectangular field with a square mesh with no interpolation required regard-
less of the shape of the physical boundaries, regardless of the spacing of
the curvilinear coordinate lines in the physical field, and regardless of
the movement of the coordinate system in the physical plane. A number of
examples of coordinate systems and application thereof to the solution of
partial differential equations are given. The FORTRAN computer program and
instructions for use are included.
I. INTRODUCTION
There arises in all fields concerned with the numerical solution of
partial differential equations the need for accurate numerical representa-
tion of boundary conditions. Such representation is best accomplished when
the boundary is such that it is coincident with some coordinate line, for
then the boundary can be made to pass through the points of a finite dif-
ference grid constructed on the coordinate lines; hence the choice of
cylindrical coordinates for circular boundaries, elliptic coordinates for
tprofessor of Aerophysics and Aerospace Engineering, Ph.D.
*Engineering Specialist, Ph.D., Present Affiliation:
Corporation, Dallas, Texas.
§Associate Professor of Mathematics, Ph.D.
LTV Aerospace
elliptical boundaries, etc. Finite difference expressions at, and adjacent
to, the boundary may then be applied using only grid points on the inter-
sections of coordinate lines, without the need for any interpolation between
points of the grid.
The avoidance of interpolation is particularly important for boundaries
with strong curvature or slope discontinuities, both of which are common in
physical applications. Likewise, interpolation between grid points not
coincident with _he boundaries is particularly inaccurate with differential
systems that produce large gradients in the vicinity of the boundaries, and
the character of the solution may be significantly altered in such cases.
In most partial differential systems the boundary conditions are the dominant
influence on the character of the solution, and the use of grid points not
coincident with the boundaries thus places the most inaccurate difference
representation in precisely the region of greatest sensitivity. The genera-
tion of a curvilinear coordinate system with coordinate lines coincident with
all boundaries (herein called a "boundary-fitted coordinate system" for pur-
poses of identification) is thus an essential part of a general numerical solu-
tion of a partial differential system.
A general method of generating boundary-fitted coordinate systems is to
let the curvilinear coordinates be solutions of an elliptic partial differen-
tial system in the physical plane, with Dirichlet boundary conditions on all
boundaries. One coordinate is specified to be constant on each of the bound-
aries, and a monotonic variation of the other coordinate around each boundary
is specified. Thus, there is a coordinate line coincident with each boundary.
The procedure is not restricted to two dimensions, allows the coordinate lines
to be concentrated as desired, and is applicable to all multi-connected regions
(and thus to fields containing any number of arbitrarily shaped bodies).
The coordinate system so generated is not necessarily orthogonal, but
orthogonality is not required, and its lack only requires that the partial
differential system to be solved on the coordinate system when generated
must be transformed directly through implicit partial differentiation rather
than by use of the scale factors and differential operators developed for
orthogonal curvilinear systems. An orthogonal system cannot be achieved
with aribtrary spacing of the coordinate lines, and the capability for such
concentration of coordinate lines is of more importance than orthogonality.
This general idea has been applied previously to two-dimensional regions
interior to a closed boundary (simply-connected regions) by Winslow [i],
Barfield [2], Chu [3], Amsden and Hirt [4], and Godunov and Prokopov [5].
Winslow [i] and Chu [3] took the transformed coordinates to be solutions of
Laplace's equation in the physical plane which, as is shown in the next sec-
tion, makes the physical cartesian coordinates solutions of a quasi-linear
elliptic system in the transformed plane. Barfield [2] and Amsden and Hirt
[4] reversed the procedure, taking the physical coordinates to be solutions in
the transformed plane of a linear elliptic system which consists of Laplace's
equation modified by a multiplicative constant on one term. This makes the
transformed coordinates solutions of a quasi-linear elliptic system in the
physical plane. Barfield also considered a hyperbolic system, but such a
system cannot be used to treat general closed boundaries, since only elliptic
systems allow specification of boundary conditions on the entirety of closed
boundaries. Stadius [6] also used a hyperbolic system to generate a coordinate
system for a doubly-connected region having parallel inner and outer boundaries.
With parallel boundaries it is only necessary to specify conditions on one of
the boundaries, the location of the other boundary being free. The elliptic
system, however, allows all boundaries to be specified as desired and thus has
much greater flexibility.
Amsden and Hirt [4] constructed the coordinate generation method by
iterative weighted averaging of the values of the physical coordinates at
fixed points in the transformed plane in terms of values at neighboring
points. Although not stated as such, this procedure is precisely equivalent
to solving Laplace's equation, or modification thereof of the form noted above
in Barfield [2], for the physical coordinates in the transformed plane by
Gauss-Seidel iteration. Amsden and Hirt also allowed the boundary to move at
each iteration, but this is simply equivalent to approaching the solution of
the boundary-value problem through a succession of boundary-value problems
converging to the problem of interest. In the approach of Godunov and Proko-
pov [5] the elliptic system is quasi-linear in both the physical and transformed
planes. These authors applied a second transformation to that used by Chu [3],
the transformation functions of this latter transformation being chosen a
priori to control the coordinate spacing. Though not stated as such, the over-
all transformation may be shown to be generated by taking the transformed coor-
dinates to be solutions in the physical plane of Laplace's equation modified
by the addition of a multiple of the square of the Jacobian, the multiplicative
factors being a priori chosen functions of the physical coordinates.
Meyder [7] generated an orthogonal curvilinear system by solving for the
potential and "force" lines in a simply-connected region and taking these as
the coordinate lines. This amounts to makin_ the curvilinear coordinates
5
solutions of Laplace equations in the physical plane with Dirichlet boundary
conditions (constant) on part of the boundary and Neumann boundary conditions
(vanishing normal derivative) on the remainder. The solution for the coordinates
was done, however, in the physical plane on a rectangular grid using interpola-
tion at the curved boundaries, rather than in the transformed plane.
Orthogonal curvilinear coordinates for multi-connected regions, including
regions with two bodies, have been generated by Ives [8] using conformal mapping.
Conformal mapping is a special case of the generation of coordinate systems
by solving an elliptic boundary value problem, but is not extendable to
three dimensions and is less flexible in the spacing of the coordinate lines.
There have also been a number of transformations developed directly for
special purposes without solving a partial differential system. One such
approach is that of Gal-Chen and Somerville [9] for the treatment of an
irregular boundary, such as mountaneous terrain, in a simply-connected region.
In the present research, the technique of generating the transformed coor-
dinates as solutions of an elliptic differential system in the physical plane
has been applied to multi-connected regions with any number of arbitrarily
shaped bodies (or holes). The elliptic equations for tile coordinates are
solved in finite difference approximation by SOR iteration. Procedures for
controlling the coordinate system so that coordinate lines can be concentrated
as desired have been developed. Present effort is confined to two dimensions
in the interest of computer economy, but the technique is extendable in
principle to three dimensions. The procedure is also applicable
6
to fields with time-dependent boundaries, one coordinate line remaining
fixed to the moving boundary. Here the equations for the coordinates must be
re-solved at each time step. The computational grid remains fixed in spite
of the movementof the physical grid.
Any partial differential system can be solved on the boundary-fitted
coordinate system by transforming the set of partial differential equations
of interest, and associated boundary conditions, to the curvilinear system.
(It is shownin Appendix B that the equations do not change type, i.e.,
elliptic, parabolic, hyperbolic, under the transformation.) Since the bound-
ary-fitted coordinate system has coordinate lines coincident with the surface
contours of all bodies present, all boundary conditions can be expressed at
grid points, and normal derivatives on the bodies can be represented using
only finite differences between grid points on coordinate lines, without
need of any interpolatio_ even though the coordinate system is not orthogo-
nal at the boundary. The transformed equations can then be approximated
using finite difference expressions and solved numerically in the transformed
plane. Thus, regardless of the shape of the physical boundaries, and regard-
less of the spacing of the finite grid in the physical field, all computa-
tions, both to generate the coordinate system an4 subsequently, to solve the
partial differential system of interest can be done on a rectangular field
with a square meshwith no interpolation required on the boundaries. More-
over, the physical boundaries mayeven be time-dependent without affecting
the grid in the transformed region.
The computer software utilized to generate the boundary-fitted coordi-
nate system is independent of the set of partial differential equations to
be solved on this system. For example, numerical solutions for inviscid
7and viscous fluid flows have been obtained using this system (Ref. 10-14).
The partial differential equations governing these phenomenadiffer drasti-
cally. However, for a given body geometry, the sameboundary-fitted system
generation program was used in both solutions. Another major advantage of
using boundary-fitted coordinates is that the computer software generated to
approximate the solution of a given set of partial differential equations is
completely independent of the physical geometry of the problem. The coordi-
nate systems for the wide variety of bodies included in this report, for
example, were all developed utilizing the samecomputer program. Finally,
it is shown in Appendix C that physical integral conservation relations need
not be lost in the transformed plane.
This report presents a detailed development of the method for generation
of boundary-fitted coordinate systems for general, multi-connected, two-
dimensional regions. The basic doubly-connected region transformation is
discussed in Section II in somedetail. Extensions of the basic transforma-
tion to multi-connected regions, contracted coordinate systems, and time-
dependent systems are also discussed. The numerical techniques used to
implement the method and a numberof specific examples are presented in the
Sections III and IV. Examplesof application to the solution of partial
differential equations are given in Section V. Finally, instructions for
use and a listing of the FORTRANprogram are given in Section VI. Various
derivative relations used in the transformation of partial differential systems
and program parameters used in the examples included are given in Appendix A.
II. MATHEMATICALDEVELOPMENT
Preliminaries
The general transformation from the physical plane [x,y] to the trans-
formed plane [_,n] is given by the vector-valued function
8
(x,y)J
(1)
The Jacobian matrix for this transformation is
F
J1
-- [_x qy
(2)
where the subscripts denote partial differentiation in the usual mannez.
The inverse function or transformation of (i) is, if it exists,
(3)
The Jacobian matrix of (3) will be denoted by J2 and is given by
(_)
The Jaeobian determinant, or Jacobian as it is normally called, is
then
j = det[J2] = x_y n - xqy$ (5)
The Jacobian matrices, (2) and (4), are related by
J1 = [J2 ]'I (6)
which implies the relations
_x = Yq/J' Ey = -xn/J' qx = -Y_/J' ny = x_/J (7a,b,c,d)
Partial derivatives are transformed as follows:
9
f = a(f,y) / _ (YT]f_ - Y_fn)x a(_,n) a($,n) = a (8)
f = _ / a(x,y) (-xT]f$ + x_fT])y _ (_,T]) a (_, T]) = J (9)
where f is some sufficiently differentiable function of x and y. Higher
derivatives are obtained by repeated application of (8) and (9). A com-
prehensive set of transformed derivatives, operators, unit vectors, and
other useful relations is given in Appendix A.
Sufficient conditions for the transformation described above to exist
are given by the inverse function theorem (Ref. 15). In particular, if
the component functions of (i) are continuously differentiable at a point,
say (Xo, yo) , and the Jacobian matrix (2) is nonsingular at (Xo, yo) then
there exists a disk N about (x yo ) such that the inverse function (3)o o'
exists and (6) holds for all [x,y] in N . It is readily apparent that theo
theorem guarantees existence only in a local fashion. For this reason com-
ponent functions of (i) which possess even more desirable properties than
those required by the inverse function theorem are sought.
Since the basic idea of the present transformation is to let the com-
ponent functions of (I) be solutions of an elliptic Dirichlet boundary value
problem, an obvious choice is to require that _(x,y) and n(x,y) be either
harmonic, subharmonic, or superharmonic. Harmonic functions have continuous
derivatives of all orders. Moreover, harmonic functions obey a maximum
principle,which states that the maximum and minimum values of the function
must occur on the boundaries of the region D. Thus, since no extrema occur
within D, the first derivatives of the function will not simultaneously
l0
vanishes in D, and hence the Jacobian J will not be zero due to the presence
of an extremum. (Note that this merely removes one condition which may cause
the Jacobian to vanish.) Further, the maximumprinciple guarantees uniqueness
of the coordinate functions _(x,y) and n(x,y), (Ref. 16), and thus ensures
that no overlapping of the boundaries will occur. Subharmonicand superhar-
monic functions are also continuously differentiable and obey a maximum
principle. (The maximumprinciple is not as strong for these functions as
it is for harmonic functions.) A more general discussion of the mathematical
properties of the transformation is given in [17].
Doubly-Connected Region
Consider the transformation of a two-dimensional, doubly-connected
region D boundedby two simple, closed, arbitrary contours onto a rectangular
region D* as shownin Figure i. (The basic transformation is discussed here
assuming that the body contour and outer boundary are transformed, respec-
tively, to the constant H-lines forming the bottom and top sides of the
transformed region. The more general case of segmentedbody contours trans-
forming to any side of the transformed region follows analogously and is
discussed in later sections. The computer program allows the body contour(s)
and outer boundary to be segmentedand placed around the sides of the trans-
formed plane in any mannerdesired.) Let FI maponto FI*, F2 maponto F2*,
F3 onto F3*, and F4 onto F4*" For identification purposes region D will be
referred to as the physical plane, D* as the transformed plane, and _ as the
body contour. Note that the transformed boundaries (FI* and F2*) are madecon-
stant coordinate lines (_-lines) in the transformed plane. The contours F3
and F4 which connect the contours FI and F2 are coincident in the physical
ii
plane and thus constitute re-entrant boundaries in the transformed plane.
In view of the closing remarks of the previous sectio_ consider taking
Laplace's equation as the generating elliptic system. That is, let _(x,y)
and n(x,y) be harmonic in D. Then
+ = 0 (10a)Sxx _yy
+ = 0 (10b)qxx qyy
with the Dirichlet boundary conditions
r! B 1[x,y] e rI (10e)
_2(x,Y)
n 2, [x,y] e r 2 (lOd)
where ql and q 2 are different constants (n2 > nl),and _l(x,y) and _2(x,y)
are specified monotonic functions on Pl and P2' respectively, varying over
the same range. The arbitrary curve joining F1 and P2 in the physical
plane, which transforms to the right and left sides of the transformed plane,
specifies a branch cut for the multiple-valued function $(x,y). Thus, the
values of the physical coordinate functions x(_,n) and Y(_,n) are the sale on
F3 as on P4' and these functions and their derivatives are continuous from F3
to F4. Therefore, boundary conditions are neither required nor allowed on
P3 and P4" (A graphic analog to the above ideas can be found in most com-
plex variable texts where Riemann surfaces are discussed. For example, see
Levinson and Redheffer, Ref. 16.).
12
Since it is desired to perform all numerical computations in the uni-
form rectangular transformed plane, the dependent and independent variables
must be interchanged in (i0). Use of equation:(A.18) of Appendix A yields
the coupled system
(lla)
(llb)
where
a - x 2 + yn2n
B -=xEx n + YEYn
_, - x 2 + y 2
with the transformed boundary conditions
fl(_,_l)
f2(_,n I)
(llc)
(lid)
(lle)
, [_,nl] e FI* (llf)
n2)]= , [_,n2]_
[g2(E,n2)j r2*
(llg)
The functions fl(_,nl), f2(_,nl ), gl(_,n2 ), and g2(_,n2) are specified by
the known shape of the contours F1 and F2 and the specified distribution of
thereon. As noted, boundary data are neither required nor allowed along
the re-entrant boundaries F3* and F4*.
The system given by the equations of (ii) is a quasi-linear elliptic
system for the physical coordinate functions, x(_,n) and _,n), in the
13
transformed plane. Although this system is considerably more complex than
that given by (i0), the boundary conditions (llf,g) are specified on straight
boundarie_ and the coordinate spaclng in the transformed plane is uniform.
The boundary-fitted coordinate system generated by the solution to (ii) has
a constant H-line coincident with each boundary in the physical plane. The
_=constant lines maybe spaced as desired around the boundaries since the
assignment of the _-values to the Ix,y] boundary points via the functions fl'
f2' gl' and g2 in (llf,g) is arbitrary. (Numerically the discrete boundary
values [xk,Yk] are transformed to equi-spaced discrete _k-points on both
boundaries.) Control of the radial spacing of the q=constant lines and of
the incidence angle of the _=constant lines at the boundaries is accomplished
by varying the generating elliptic system (i.e., the system of which _(x,y)
and q(x,y) are solutions) as will be demonstrated in Section IV. As illus-
trated in Figure i, the left and right boundaries of the transformed plane
are re-entrant boundaries, which implies that both solutions, x(_,q) and
y(_,_), are required to be periodic in the region {[_,n]I - _ < _ < _,
nI _<n _< n2}.
Multiply-Connected Region
The basic ideas and procedures introduced in the preceding section
can be extended to regions containing more than one body--that is, to general
multi-connected or multi-body regions. One transformation for two bodies is
illustrated in Figure 2. The bodies are connected with one arbitrary cut,
with an additional cut joining one of the body contours to the outer boundary.
The physical plane contours F1 - F8 map respectively onto the contours FI* -
FS* in the transformed plane. Note that the body defined by the union of
F7 and r 8 is split into two segments (F7* and F8*) , as is the cut joining
14
this body and the one defined by contour FI. The n-coordinate is the same
for both of the bodies and cut between them. Conversely, the cut defined by
F3 and F4 in the physical plane is taken as a _=constant line in the trans-
formed plane as before for the single body case. The outer boundary contour
F2 mapsonto the upper boundary in the [_,n] plane, becoming a constant
H-line in the manner of the one-body transformation. In contrast to the one-
body transformatio$ two re-entrant boundaries occur for this two-body trans-
formation. The left and right vertical boundaries (F4*, F3*) appear as
before. In addition a horizontal re-entrant segmentdue to the coincidence
of F5 and F6 in the physical plane arises. The coordinate functions and the
derivatives thereof are thus continuous across these re-entrant boundaries.
The boundary-fitted coordinates for the multi-body transformations are
again determined by the solution of the set of equations (Ii) with the added
boundary conditions
hl(_' I
nI)
h2(_,n I)
[$,_i ] c F7* (llh)
= , [$,nI] c F8*
2(_,Ul)]
(lli)
to define the additional body. Note that boundary conditions cannot be
specified along the re-entrant boundaries defined by F3*, F4* , F5* , and
F6*. As with the basic transformation all numerical computations both to
generate the system and subsequently to utilize the coordinates for
15
solving a set of partial differential equations,are executed on a rectangular
field with a uniform grid.
Simply-Connected Region
For a simply-connected region there are no bodies in the field and hence,
no cuts inthe physical plane and no re-entrant boundaries in the transformed
plane. The single continuous boundary surrounding the physical field trans-
forms to the entire rectangular boundary of the transformed field. The
manner in which the physical boundary is split into four segments for place-
ment on the four sides of the rectangular boundary in the transformed plane
is a matter of choice.
Coordinate SystemControl
Control of the spacing of the coordinate lines on the body is easily
accomplishe_ since the points on the body are input to the program. The
spacing of the coordinate lines in the field, however, must be controlled by
varying the elliptic generating system for the coordinates. Onemethod of
variation is to modify the Laplace equations (i0) by adding inhomogeneous
terms to the right sides so that the generating system becomes
Exx + Eyy = P(E,n) ' _xx + nyy = Q(_,n) (12)
In the transformed plane these equations become
+ + J2(pxE + Qxn) = 0- 2_x_ yxn_
_ - 2BY_n+ YYnn+ j2(py$ + Qyn) = 0
(13a)
(13b)
16
The effect of changing the functions P(_,n) and Q(_,n) on the coordinate
system can be seen by examining the system (12). If P _ Q _ 0, the system
reduces to the pair of Laplace equations used as the original generating
system, i.e., Eq. (i0). Let _', n' be the solutions of (i0) with boundary
values on D. Let _", n" be the solutions to system (12) with the same
boundary values and with positive functions P aLidQ on the right hand side
of the equations. Now_" and _" are subharmonic on D. Thus for any constant
k, the _" = k coordinate line would be closer to _" = Smaxthan would the 6' = k
line. The samerelation holds for the n" = k and n' = k coordinate lines.
Nowif all or part of the curve 6' = _ is a branch cut in D, this branchmax
cut will movein the direction of increasing _ to meet the curve 6" = max
if the right side of the system (12) is increased from 0 to a positive
function P. An analogous discussion can be madeon the effect of negative
functions P and Q on the generated curvilinear coordinate system.
Even though a change in P or Q in a subregion of D would change the
coordinate lines throughout the entire region, the effect would certainly be
more pronounced in the subregion. Also, the greater the change in P and Q,
the greater the movementof the coordinate lines. By varying the sign and
magnitude of P(_,n) and Q(E,n) for different values of _ and n, considerable
control can be exerted over the coordinate line spacing as will be seen from
the examples that follow in Section IV.
Oneparticularly effective procedure is to choose P and Q as exponential
terms so that the coordinates are generated as the solutions of
n
$xx + _yy = - Zi=l
a.i sgn(_ - _i)exp(-cil_ - _iJ)
17
m
- Z
j=lb.3 sgn(_ - _j)exp(-dj/(_ - _j )2 + (n - njYr-)
- P(_, n) (14a)
n
nxx + nyy i=l a.1 sgn(q - qi)exp(-ciJq - qil)
m
Z
j=lb.3 sgn(rl - qj)exp (-dj,/(_ -- _j)2 + (19 - rl.)2j )
Q(_, n) (14b)
where the positive amplitudes and decay factors are not necessarily the same
in the two equations. Here the first terms have the effect of attracting
the $ = constant lines to the $ = _i lines in Equation (14a),and attracting
n = constant lines to the q = qi lines in Equation (14b). The second terms
cause _ = constant lines to be attracted to the points (_j,qj) in (14a),
with similar effect on q = constant lines in (14b). No computational dif-
ficulties have been encountered because of the discontinuities in P and Q
caused by the sgn function, which is defined by setting sgn(x) to be i, 0,
or -i depending on whether x is positive, zero, or negative. Should pro-
blems arise in later applications, this function can be replaced by
2Arctan(nx), where n is a large positive integer. The sgn function was
chosen to give the maximum control. Several examples of the use of coordinate
system control are given in Section IV.
18
With the inclusion of the sgn function (or the arctan function) the
equations (14a & b) for the curvilinear coordinates are no longer subharmonic
or superharmonic, since the sgn function causes a sign change on the right
side when the attraction is to lines or points not on the boundaries. It is
possible, therefore, that too strong an attraction amplitude may cause the
system to overlap and therefore be unusable. Manysuccessful systems (all
those included in the examples given herein) have been generated, however,
using the above equations.
The use of the sign-changing sgn function is only necessary to cause
attraction to both sides of a line or point in the field. Elimination of this
function causes attraction on one side and repulsion on the other. If it is
only desired to concentrate coordinate lines near one boundary, such as the
body surface, then there is no need for the sign chang_ and the sgn function
can be eliminated. In this case the equations are subharmonic or superharmonic,
and a maximumprinciple is in effect to prevent overlap. Such a choice is
provided for in the computer program.
The subject of coordinate system control is still very muchunder investi-
gation, and other control functions are being evaluated. It is anticipated
that new coordinate control packageswill be madeavailable for inclusion in
the code whenwarranted. Of particular interest is the capability to cause a
specified numberof lines to fall within a certain physical region, such as
a boundary layer. Another area of further investigation is the coupling of
the coordinate equations with the partial differential system to be solved
thereon so that the coordinate lines concentrate automatically in regions of
high gradient.
Time-DependentCoordinate Systems
19
If the coordinate system changeswith time then the grid points move
in the physical plane. Ordinarily such movementof the physical grid
points would require interpolation amongthe grid points to produce values
of the dependent variables at the new locations of the grid points. With
the present method of coordinate system generation, however, it is possible
to perform all computation on the fixed rectangular grid in the transformed
plane without any interpolation no matter how the grid points move in the
physical plane as time progresses. This occurs as follows:
Recall that the coordinate system is generated as the solution
of someelliptic system with the values of the transformed coordinates
[_,_] specified on the boundaries in the physical plane, one of these
coordinates being specified to be constant on the boundaries and the other
being distributed as desired along the boundaries in order, perhaps, to
concentrate grid points in certain regions. The transformed coordinates
define a rectangular plane, the extent of which is determined by the range
of the values of E and n. Nowif the sameboundary values of _ and q are
redistributed in the physical plane, perhaps because the boundaries in the
physical plane have actually movedor maybejust to change the concentration
of grid points around the boundaries, and the elliptic system is re-solved for
the transformed coordinates with these new boundary conditions, new trans-
formation functions can be produced with still the samerange of values in
and n (provided the elliptic system used exhibits a maximumprinciple) and
hence to the samerectangular field in the transformed plane. The grid
points in the rectangular transformed plane thus remain stationary, and the
effect of the movementof the. coordinate system in the physical plane is just
to change the values of the physical coordinates [x,y] at the fixed grid
points in the rectangular transformed plane.
2O
Thus, although the position of a grid point changes on the physical
plane, its position in the transformed plane is fixed. The time
derivative transforms to the transformed plane as shownbelow:
3f _(x,y_f) / $(x,y,t)(_-t) = _ (_,n,t) _ (_,n,t)
x,y
= ft - xt(f_Y_ - fqY_)/J
+ yt(f_x n - f x_)/J(15)
Here all derivatives are expressed in the transformed plane, so that the
interpolation that would be necessary to supply values at grid points in
the physical plane that have moved is not required in the transformed plane.
(Note that in the transformed expression for the time derivative, all
derivatives are taken at the fixed grid points in the transformed plane.
The movement of the grid in the physical plane is reflected only through
the rates of change of x and y at the fixed grid points in the transformed
plane.)
The problem of solving N partial differential equations of any type in
a physical region with time dependent boundaries has been replaced by a new
problem consisting of N + 2 equations with fixed boundaries. The two
additional equations are, of course, those governing the transformation
(either (i0) or (12)). Thus, it is possible to construct numerical solutions
to physical problems with time dependent boundaries in a fixed rectangular
plane with a fixed square mesh with no interpolation required. Again the
above statements imply the independence of computer software from physical
geometry. Problems involving moving blast fronts, shocks, free surfaces,
and any other time-varying boundaries c_n be attacked successfully with these
21
procedures. (Someapplication to free surface flow is illustrated in Ref. 26.)
In addition this method allows time-dependent concentration of grid points as
desired in the physical plane.
The use of time-dependent coordinate systems requires that the difference
equations for the curvilinear coordinates be resolved at each time step,
of course. The coordinate values at the previous time step can serve as
the initial guess for the next, however, so that the iteration will converge
rapidly.
III. NUMERICALSOLUTION
Difference Equations
The discussion here assumesthe body contour and outer boundary trans-
form, respectively, to the n-lines forming the lower and upper sides of the
rectangular transformed plane as discussed in Section II. More general seg-
mentation of the body contour(s) and the outer boundary, and placement as
desired around the boundary of the transformed field, are provided for in the
computer program.
The finite difference grid for the single-body problem is illustrated
in Figure 3a. Circles denote the points at which the difference approxi-
mation to (lla,b) or (13a,b), are applied, while the triangles denote boundary
points. The left and right vertical boundaries are coincident in the physical
plane, and the values of x and y are thus equal along these lines. Such lines
are designated re-entrant boundaries as indicated before. If the number of
= constant lines is designated IMAXand the numberof n-lines by JMAX, the
computational field size is (JMAX-2)(IMAX-I). Boundary values are specified
22
on j=l and j=JMAXfor all I _ i j IMAX. The j=l line corresponds to
the contour F1 (the body contour) in the physical plane while j=JMAXis
associated with the remote boundary contour F2. Second-order central dif-
ference expressions (see Appendix D) are used to approximate all derivatives
in the transformed equations. The resulting equations are, for (lla,b),
+ Xi+l, j) - 8'i,j (x_n)i,j/2I
xl,j " [_i,j(Xi_l,j
+ T' + -1)]/[2(aii,j(xi,J+l xi,j ,J
!
+ Ti, j) ] (16a)
+ Yi+l,j) - $' ' ./2i,j (Y_n)i,3Yi,J " [_i,J(Yi-l,J
+ T'i,j(Yi,J+l + Yi,J-l)]/[2(sl,J + Ti,j' )] (16b)
where _'i,j' B'.l,],7i,j,. ' (x_)i,j, and (Y_)i,j are the difference ap-
proximations for e, 8, 7, and the cross derivatives respectively. These
expressions are developed in Appendix D. The re-entrant boundaries
occurring at i=l and i=IMAX are dealt with as follows. Since the values
of x and y are equal along these lines, iteration is necessary along only
one of them. Choosing i=l for convenience, the _-derivatives along this
line are approximated as exhibited below:
(x_)i, j = (x2, j - XIMAX_I,j)/2(17a)
(x ) = x 2 - 2x I +_ i,J ,j ,j XlMAX-I,J(17b)
(17c)(xEn)i, j -- (x2,j+ 1 - xz,j_ I + XlMAX_I,j- 1 - XIMAX_I,j+I)/4
for 2 _ j j JMAX-I. Similar expressions are used for the derivatives of
y. The set of non-linear simultaneous difference equations produced by
(16a,b) is solved by point SORiteration.
23
Multiple-Body Fields
A sample mesh for a two-body transformation is shownin Figure 3b.
As before, circles denote computational nodes and triangles the boundary
points. Values defining the outer boundary contour F2 (see Figure 2) are
required for 1 _ i _ IMAXalong the j-JMAX line. Boundary values specifying
the shape of the body contours FI, F7, and F8 in the physical plane are
required in segmentsalong the j=l line as follows:
rl: I2 <_i ! I3
F7: 1 2 1 2 Ii
]"8: I4 <__i < IMAX
The above requirements may be more clearly seen by comparing Figures 2 and
3b. The difference equations (16a,b) and the vertical re-entrant boundary
relations (17a,b,c) are also valid for multi-body transformations. The
primary difference in the two transformations results from the existence,
in the multi-body case, of the horizontal re-entrant boundaries along j=l.
The equivalence of x and y values along these segments is demonstrated in
Figure 3b. Again it is only required to calculate values along one re-
entrant segment. Choosing the leftmost (II < i < I2), the n-derivatives
are calculated using special procedures which are illustrated by the
expressions below for the point marked with an_ (i,j = II + i,i) in
Figure 3b:
24
(Xn)ll+l,l -_ (Xll÷l,2 - xi4 - 1,2)/2
(xnn)ii+l,l = Xll+l,2 - 2Xll+l,l + x14_l, 2
(18a)
(18b)
-_ _ + Xll+2, 2 _ Xli,2)/4 (18c)(x{n) II+i, i (x14,2 x14- 2,2
Note the existence of the horizontal re-entrant boundaries increases the
size of the computational field somewhat. In the two-body example given
the numberof additional equations to be solved is I2 - (Ii+l).
Multiple-Body Segment Arrangements
In the case of a single body it is logical to keep the body contour in
one segment, with a single cut connecting the single segment to the outer
boundary. This type of arrangement is illustrated in Figure 4a. (Figure 4b
showsan alternate single body arrangement. In these and all subsequent
figures the dotted lines on the segment arrangement diagrams identify the
two membersof a re-entrant pair.) In the case of multiple bodies there is
a wider choice of reasonable arrangements, someof which may be better than
others for certain applications. The boundaries in the physical plane may
be split into as manysegmentsas desired, and these segmentsmaybe arranged
around the rectangular boundary of the transformed plane in any way desired.
These segments are all connected by branch cuts in the physical plane and by
re-entrant boundaries in the transformed plane. Several of these arrangements
are illustrated in Figures 5-13. Illustrative values of the segment input
parameters are given in Table 1 for each of these arrangements, and input
instructions are given in Section VI.
25
In the arrangement of Figure 5, an n-llne encircles both bodies and
forms a cut between the bodies, the cut to the outer boundary being a G-llne.
The outer boundary is also a llne of constant n but at a different value.
Here one body is split into two segments, while the other body and the outer
boundary are each in single segments. Figure 6 shows an arrangement in
which each body is in a single segment, each body being a _-line of different
value. Here there is no cut between the bodies, but rather an n-line cut
between each body and the outer boundary, which is split into two segments,
each being an n-line of different value. (This produces a system similar
to a bi-polar coordinate system.) In Figure 7 each body is also in a single
segmentwith the outer boundary split into two segments, but here an
n-line encircles each body and forms the cut between that body and the outer
boundary. In Figure 8 one body is a single segment encircled by an n-llne
which forms a cut to the other body. The other body is split into two seg-
ments, each being a G-llne of different value, with each segment connected to
the outer boundary by a G-llne. The outer boundary is in a single segment
and is an n-line. Other arrangements are shownin Figures 9-13. All
these arrangements are shownwithout coordinate line attraction, and, conse-
quently, manyof the resulting systems exhibit wide spacing in concave areas.
This spacing can be improved by coordinate attraction as illustrated in the
examplesof Section IV. The results of the use of several of these multiple-
body segment arrangements are given for two-body potential flow in Section V.
Coordinate system control was used effectively in that case to improve the
spacing in the concave region. These concave regions occur when he cut and
body are on the samecoordinate line.
26Initial Guess
Since the difference equations are nonlinear, the initial guess must be
within a certain neighborhood of the solution if the iterative solution is to
converge. With somesegmentarrangements a logical choice of an initial
guess is difficult to perceive. Therefore several different types of initial
guesses have been inserted in the program, with the choice to be madeby the
user as guided by past experience. The rationale for someof these guesses
is more intuitive than analytical. The choices available are detailed below.
(In each case the initial values of x and y on all cuts are interpolated
linearly between the boundary values at the cut end points.) The choice is
controlled by the input parameter IGESas discussed in Section VI. The guess
type is identified by this number in the discussion below.
(a) Weighted Average of Four Boundary Points - Here the values of x
and y at each point in the field are set equal to the average of
the four boundary points having either the same_ index or the
same_ index, the average being weighted by the distance to the
boundary in the transformed plane. Thus
JMAX- " + j -i2 xij = (JMAX lJl)Xi,l (JM__X-l)Xi, JMAX
IMAX- i + i- 1+ (IMAX l)Xl,j (I_IAX- 1)XIMAX,j (19)
for i = 2, 3, ..., IMAX-I and j = 2, 3, ..., JMAX-I. An
analogous equation is used for y. (IGES = I). A variation of
this type (and also of types (b),(c), & (e)) is provided by IGED,
whereby the average maybe restricted to only a two-point average
in either the _ or q direction. The direction chosen should be
that which proceeds between the bodies and the outer boundary. This
variation is useful with an outer boundary located on three sides.
27
(b) Sameas (a), except zeroes replace the values on the cuts in the
formation of the average. This type of initial guess is particu-
larly effective with simply-connected regions, single-body fields,
and multiple-body segmentarrangements having the all body seg-
ments on one horizontal (vertical) side and the outer boundary a
single segment on the other horizontal (vertical) side. (IGES= 0)
(c) Weighted Average of Body SegmentBoundaryPoints Only - This type
guess is the sameas that of (a), except that boundary points on
cuts are not included in the formation of the average, which there-
fore maybe formed with fewer than four points. This type and the
exponential projection below are widely effective. (IGES = 2)
(d) MomentProjection - Here the initial value at each field point is
given by
(e)
(Z dijk)(Z X_k) - Z dijkX kk k k
xij N E dij k (20)k
with dij k = /(xij - Xk)Z + (Yij - Yk )2
Here xk is the value at a point on the boundary of the trans-
formed plane; dij k is the distnace to that boundary point in the
transformed plane; N is the total number of boundary points; and
the summations extend over the entire boundary in the transformed
plane (IGES = 3). A modification of this type omits the division
by N (IGES = 4).
Exponential Weighted Average of Body Segment Boundary Points -
This guess is similar to that of (c) except that the weight in the
average is exponential rather than linear. Increasing IGES causes
the points to contract nearer the boundary segments corresponding
28to the lowest values of n and _. This type is most effective
when strong coordinate attraction is used with a single-body
field having the body located on the bottom or left side of the
rectangular transformed field. (IGES > 4)
(f.) Exponential Projection - The initial value of x is determined at
each field point by
Exk exp(-llGESldijk/d o)k
xij E exp(-llGE--_ijk/d o) (21)k
where d is the diagonal length of the transformed field, theo
other quantities having the samedefinitions as in (d) above.
(IGES < 0)
Examplesof these initial guess types are shownin Figures 14-23 for
the segment arrangements given in Figures 4-13. The value of IGESfor each
is given in the upper left corner of each plot. Table 2 lists the types for
which convergence was obtained with each of these segmentarrangements.
The most widely effective initial guess in these cases was the expo-
nential projection. This type of guess produced convergence in the single-
body case and in six of the nine two-body cases. The optimum decay factor
for the exponential projection was 40 (IGES = -40) in most cases, with an
optimum of 20 (IGES= -20) in a few cases.
GuessType 2 (IGES= 2) also gave convergence in six of the nine two-
body cases and in the single-body case. However, the numberof iterations
required was a bit larger than with the exponential projection. Type 2 gave
convergence with one segmentarrangement (Figure 9) for which the exponential
projection gave divergence, but gave divergence for another arrangement (Figure
12) for which the exponential projection gave convergence.
Arrangements having the outer boundary in two segments tend to be the
more difficult to converge. Of the four arrangements of this type (Figures,
296, 7, 12, 13), GuessType 2 gave convergence for only one (Figure 13), while
the exponential projection failed for two of the four. The two arrangements
of Figures 6 and 7 proved to be particular recalcitrant, requiring a switch
to a different size field in order to get convergence for the arrangement of
Figure 6 and the introduction of a special guess for that of Figure 7.
The general suggestion for two-body cases is to use either exponential
projection with a decay factor of about 40 or GuessType 2. For simply-
connected regions, single-body cases, and the two-body segment arrangements
having both bodies on the samecoordinate line, GuessType 0 is generally
more efficient, although the exponential weighted average or projection and
GuessType 2 will also give convergence. Arrangements having two bodies in
single segmentson opposite sides of the transformed plane are particularly dif-
ficult, and GuessesType 1 and 4 may be used.
With somesegment arrangements with multiple bodies, convergence can
be achieved with a close outer boundary, but not with the outer boundary
farther out. Therefore, provision has been madefor initially converging
the solution with a circular outer boundary close in and then constructing
an initial guess for a field with a larger circular outer boundary from this
solution by linear projection to the larger field. This process can be
repeated as many times as desired with the outer boundary gradually being
movedout to its desired position. Two types of movementare provided: (a)
the outer boundary radius is doubled at each step, or (b) the outer boundary
radius is increased linearly at each step. This provision should not be used
unless necessary, and then the movementof (a) is to be preferred in general,
with as few steps as will produce convergence.
Finally, with strong coordinate line attraction it may not be possible
to achieve convergence from an initial guess that gives convergence without
attraction. Provision therefore has been madewhereby the attraction can be
30
added gradually, the converged solution for a small attraction becoming the
initial guess for a case of stronger attraction. Two types of increase
in attraction strength are provided: (a) the attraction amplitude is doubled
at each step, or (b) the attraction amplitude is increased linearly at each
step. This procedure is to be used only when necessary. In general, type
(a) is preferred, with as few steps as will provide convergence. Further dis-
cussion of the use of the various initial guesses is given in the instructions
of Section VI.
Convergence Acceleration
For a difference equation of the general form
al(fi+l, j + fi_l,j) + a2(fi,j+ I + fi,j_l ) + bl(fi+l, j - fi_l, j)
+ b2(fi j+l - f" ) + cf.. + d.. = 0, 1,j-i 13 13(22)
(i = i, 2, --, I ; j = i, 2, --, J)
with boundary values specified on i = j = O, i = I + i, and j = J + i, and
al, a2, bl, b2, c, and d constant, the optimum value of the SOR acceleration
parameter _ can be obtained in the case where a]2i b12 and a22 _ b22, and in
the case where alz _ b12 and a22 _ b22, (Ref. 18). The optimum parameters
in these two cases are as follows:
>_ b12Case #i: a12 and a22>_ b22
2= (over-relaxation, i < _ < 2) (23)
i + _--_- _
Case #2: a12 _ b12 and a22 _ b22
2w = (under-relaxation, 0 < w < i) (24)
1 + d-7-
31
where
0 = 2 - cos I + i
cos j +-----_ (25)
In the remaining case where al2 _ b12 and a22 _ b22 , no theoretical
determination of the optimum acceleration parameter exists as yet.
Since the difference equations for the coordinate system are nonlinear,
the above theory is not directly applicable. However, if the equations are
considered as locally linearized then a local optimum acceleration parameter
can be obtained which will vary over the field. It should be noted that
the local linearization is applied only to the determination of the accel-
eration parameters, not to the actual solution of the difference equations.
Following this approach and neglecting the effect of the cross deriva-
tives, the local constants in the above equations become
aI = _ a 2 = y
bl j2p j2Q= 2 b2 = 2
c = -2(_ + y)
so that locally optimum acceleration parameters are
Jij 2 IPij j > Jij2 IQij I
Case #i: _ij _ 2 and Yij - 2
_ij = (over-relaxation) (26)
i + /i - Pij 2
322 2
Jij JPij J Jij JQij 1
Case #2: _i-'j< 2 and Yij < 2
lJ i + /i + Pij z
(under-relaxation) (27)
where
2 J .P •
= I _ij - _ cos MAX - iPiJ _ij _ Yij
4212+ Yij - 4 (28)
j2JpJ j2jQJ
In the remaining case where a _ 2--and y < 2 , not even a local
optimum is available. The program allows a choice of strategy in this case:
over-relaxation, under-relaxation, or a weighted average as follows:
First Pl and P2 are calculated from
Pl = _ +-----_ 4 cos --_ _ 1(29a)
02= +y y2_ 4 - i(29b)
If over-relaxation is _pecified then m is calculated from Eq. (26) using
The same expression for p is used in Eq. (27) if under-relax-P = Pl + P2"
ation is specified. If the weighted average is to be used then, using Pl
in place of P' _I is calculated from Eq. (26) if _ _> _ or2 by Eq. (27)
J__P_i Similarly, using p 2 in place of p, m2 is calculated fromif _ _ 2 "
Eq. (26) if y > _ else by Eq. (27) The average is then formed by2 '
Plml + p 2m2= (30)
Pl + P2
OptimumAcceleration Parameters
In order to provide someguide to the selection of acceleration para-
meters for the most rapid convergence of the iterative solution, the
optimumvalues were determined in a number of representative cases by
computer experimentation.
The body for this study was a Karman-Trefftz airfoil, the contour of
which is shownin Figure 24. The points on the contour were spaced at
equal angular increments in the complex plane from which the airfoil was
generated, with three additional points added at half, fourth, and eighth
angular increments above and below the trailing edge to provide finer
resolution in that region.
A basic case was selected and four quantities, (the numberof points
on the body, the numberof coordinate lines surrounding the body, the
amplitude of the coordinate line attraction to the body, and the radius
of the circular outer boundary) were varied individually and in pairs
above and below the basic values. The initial guess type (Type 0) giving the
fastest convergence for the basic case was used for all. Each case was
run to convergence of 10-4. Additional cases were run with different
convergence criteria and different numbersof steps in the addition of the
final attraction amplitude. The basic case was also run with a circular
cylinder as the body for comparison of the effect of the body shape.
A series of two-body cases was also run with two of the sameKarman-
Trefftz airfoils positioned as an airfoil and flap system. Only one seg-
ment arrangement was considered, with both bodies on the samecurvilinear
coordinate line. The multiple-airfoil system and the segmentarrangement
are shownin Figure 25. The sameset of quantities varied in the single
body case was varied individually above and below the basic values (except
34
that variation of the numberof points on the bodies above the basic value
involved too muchcore and was omitted). The initial guess type was the
sameas that used for the single-body studies, which proved to give the
fastest convergence in the basic two-body case as well. The values of all
input parameters for the cases run are given in Tables 3-6.
The results of these studies are given in Tables 7-10. For the single-
body field, Table 7 shows the effect of individual variation of each of the
chosen quantities; Table 8 gives the effect of variation in pairs, and in
Table 9 other miscellaneous quantities are varied. Table i0 gives the results
for the double-body field. The numberof iterations required with the
variable acceleration parameter field and the average variable acceleration
parameter over the field are also included in these tables. (Under-relaxation
was used in the case of complex eigenvalues.) In a numberof cases the vari-
able acceleration parameters tend to be too large, and only in a few cases
was the variable field better than the uniform experimentally determined
optimum.
Plots of the numberof iterations required vs. the acceleration para-
meter are given for a few cases in Figure 26. In a numberof cases the
optimumparameter was only 0.i below the divergence limit for the particular
case. A typical exampleof this appears in Figure 26b. The effect of the
general field size is evident in Figure 26c and d, where the larger field
(d) gives a muchsharper minimum. Strong attraction with small fields makes
the convergence more difficult as can be seen in Figures 26e and f. With
strong attraction and a small field convergencewas obtained only in a very
narrow band of acceleration parameters.
35
A few trends are evident in these results:
(i) The optimum acceleration parameter, m*, and the number of iterations,
I*, increase with the numberof grid points.
(2) w* increases slightly as the outer boundary movesoutward, this effect
being more pronounced at small outer boundary radius. I* experiences a
minimumas the outer boundary movesoutward. Both of these effects are stronger
with two bodies than with one.
(3) _* is little affected by attraction amplitude with one body, but tends
to decrease with increasing attraction amplitude with two bodies. (This dif-
ference is probably due to the fact that in the two-body case there wasattraction
to a line in the field, i. e., the cut between the bodies, as well as the body
contours.) I* tends to increase with increasing attraction.
(4) There is an optimumnumber of steps for addition of the attraction,
with increasing I* occurring on both sides of this optimum. Too few steps
mayproduce divergence. This optimum is less apparent with two bodies than
with one.
(5) The numberof attraction lines had only a small effect on either
m* or I* in the cases considered.
(6) w* increases as convergence tolerance is tightened.
(7) The intermediate convergence tolerance value that should be
used when the attraction is added gradually should be 0.01. A tighter
tolerance requires more iterations, while a looser tolerance maynot produce
a sufficiently close initial guess for the next addition of attraction.
(8) The use of the variable acceleration parameter field is not generally
recon_nendedin cases where the optimum can be estimated from experience. This
feature can be useful, however, in the absenceof a good estimate. In that
36
case it is probably best to use the variable field once, and then to use a
factor about 3%less than the average over the variable for subsequent runs.
IV. EXAMPLESOFCOORDINATESYSTEMS
A variety of results utilizing the theory and numerical procedures
outlined in previous sections are now presented. The results selected for
presentation here were chosen to exemplify the generality of the method, and
somewere used for the numerical fluids studies in Ref. ii. In most of the
plots only a portion of the physical field is shownin order that the coord-
inate lines maybe distinguishable near the bodies. The actual fields were
generally extended someten chord_ _, radit_ from the hodip_.
Coordinate SystemControl
As discussed in Section II, the curvilinear coordinate lines maybe
concentrated by attracting the lines to other lines or points in the field.
The control of the coordinate system in this manner is illustrated in Figures
27-28. Input parameters involved are given in Table ii. In Figure 27, the
basic system generated by the Laplace equations (zero right hand sides) is
shownin (a). In (b) the n-lines have been attracted to the body. In (c)
the attraction to the body has been madestronger on two sides, while in (d)
the lines are more strongly attracted over a small portion of the body. In
(e) and (f) the angle of intersection of the lines with the body has been
controlled, over the entire body in (e) and over only a portion of the body
in (f).
Figure 28 illustrates the use of control to pull the coordinate lines
into a concave portion of the body contour, (a) being the result of the
Laplace equations and (b) having the lines attracted to the slope disconti-
nuity on the lower surface.
37
Various Body Shapes
Coordinate systems for a circular cylinder and a camberedJoukowski air-
foil are pictured in Figure 29. The contours are of equi-spaced _ = constant
and n = constant lines in the physical plane. (In the interest of clarity
only a portion of the grid is shownin this and subsequent figures. The
outer boundary for these and all succeeding results is circular and has a
radius of ten body-lengths.) Note that the two systems given in Figure 29
are orthogonal (The transformations in these instances are conformal.).
Slightly more general bodies are shownin Figure 30. These include a cambered
and an integral flap Karman-Trefftz airfoil. Although systems for each of
these could be generated by conformal transformations, the ones shownwere
obviously not. The effect of the coordinate system control is demonstrated
for both airfoils in Figure 31. The figures exhibit the effect of contraction
to the n-line coincident with the body profile. Note that the grid spacing is
significantly collapsed in the contracted transformation. The contracted
meshspacing near the solid body boundary allows the solution of viscous
flows (Ref. ii).
Contracted coordinate systems for more general airfoils are exhibited
in Figures 32-34. These are the Liebeck laminar airfoil, the G_ttingen
625 airfoil and a NACA0018 profile. Contraction to the single-body n-line
is demonstrated for the Liebeck profile and to the initial 15 n-lines for the
G_ttingen 625 and NACA0018 airfoils. The effects of multiple-H-line con-
traction are seen to be quite dramatic.
Coordinate systems for a multiple-airfoil system are shown in Figure 35,
with coordinate line concentration to the bodies and into the concave region
formed by the airfoils and the cut between. The coordinate line attraction is
to the first ten H-lines surrounding the bodies with an amplitude of i0,000
38and a decay factor of 1.0 on all but the tenth line, where 0.5 was used. The
coordinate system for simply connected regions are shownin Figures 36 and 37.
Finally, to demonstrate the applicability of the transformation method
to quite arbitrary bodies, a system in contracted form (15 line) for a rather
odd looking body--denoted the camberedrock--is given in Figure 38.
V. EXAMPLESOFAPPLICATIONTOPARTIALDIFFERENTIALEQUATIONS
As noted above,any set of partial differential equations may be solved
on the boundary-fitted coordinate system by transforming the equations and
associated boundary conditions and solving the transformed equations numeri-
cally in the transformed plane. All computation can be done on the fixed
square grid in the rectangular transformed region regardless of the shape
of the physical boundaries. Several examplesof such application are given
below.
Potential Flow (Ref.ll, 12, 19)
The two-dimensional irrotational flow about any numberof bodies may
be described by the Laplace equation for the stream function _:
_xx + _yy = 0 (31)
with boundary conditions
_(x,y) = _k on the surface of the kth body.(32a)
_(x,y) = y cose - x sine at infinity. (32b)
where e is the angle of attack of the free stream relative to the positive
x-axis. Here the stream function is nondimensionalized relative to the air-
foil chord and the free stream velocity.
coordinate system this equation becomes
39
Whentransformed to the curvilinear
_$ - 2B_n + Y_nn + °_n + _ = 0 (33)
The transformed boundary conditions are (see Figure I).
qJ(_'"11) = )o on n = nI (i.e., on I'I*) (34a)
,(_.,n2) = y(_.,rl2) cos0 - x(Y.,,12) si,10 ou n = 112 (i.e., ou F_) (34b)
The uniqueness is implied by insisting that the solution be periodic in
-_ < _ < _' ql i n _ _2" The coefficients _, B, y, o, and T are calculated
during the generation of the coordinate system (see Appendix A). Equation (33) was
approximated using second-order, central differences for all derivatives,
and the resulting difference equation was solved by accelerated Gauss-
Seidel (SOR) iteration on the rectangular transformed field. The value of
the boundary values of _ on the bodies were determined by imposing the Kutta
condition on each body.
The pressure coefficient at any point in the field may be obtained
from the velocities via the Bernoulli equation, which in the present non-
dimensional variables is
Cp = 1- IvI 2 (35)
On the body surface this becomes
Cp = 1 __I_Yj2 _D 2
(36)
40
with the derivative evaluated by a second-order one-sided difference
expression. The nondimensional force on the body is given by
_"= - _ Cp lj ds (37)
where n is the unit outward normal to the surface, and ds is an increment
of arc length along the surface. The lift and drag coefficients are
CL = _ Cp (-x¢cos0 - yEsin0)d¢(38a)
CD = _ Cp (y¢cos0 + xi,sin0)d_j(38b)
These integrals were evaluated by numerical quadrature using the
trapezoidal rule.
The coordinate system for a Karman-Trefftz airfoil having an integral
flap is shownin Fig. 30b, and the streamlines and pressure distribution for
this airfoil are comparedwith the analytic solution (Ref. 20) in Figure 39.
Similar excellent comparisons have been obtained with other Karman-Trefftz
airfoils. Fig. 52 shows the coordinate system for a Liebeck laminar air-
foil, the solution for which is comparedwith experimental results (Ref.
21) for the pressure distribution in Fig. 40. Finally the coordinate
system for a multi-element airfoil is shown in Fig. 35, with the stream-
lines and pressure distributions shownin Fig. 41. Here coordinate system
control was employed as discussed above to attract the coordinate lines
into the concave region formed by the intersections of the cut between the
airfoils, as well as to the bodies.
Figure 42 shows a coordinate system for a pair of circular cylinders,
the coordinate lines being attracted to the intersections of the cut between
the bodies with their surfaces. (The accompanyingdiagram shows
the arrangement of the body and re-entrant segments in the transformed
plane.) The streamlines for potential flow obtained on this coordinate
system are shownin Figure 43a, and the surface pressure distribution is
comparedwith the analytic solution (Ref. 22) in Figure 43b. By contrast,
the uncontracted coordinate system, generated by Eq. (ii), and resulting
pressure distribution are shownin Figure 44. The effectiveness of the
coordinate system control is clear in the comparison of these results with
those in Figure 43b. To illustrate the use of different segment arrangements,
Figures 45 and 46 show another coordinate system and pressure distribution
for the sametwo cylinders.
The effects of the various numerical parameters involved for the potential
flow solution were investigated in somedetail, and the results are reported
in Ref. 19. These results should serve as a guide to reasonable choices of
such parameters as field size, convergence criteria, meshspacing, etc. for
the use of the body-fitted coordinate systems in other applications as well.
Ref. 19 also serves to illustrate in somedetail the procedure of application
of the body-fitted coordinate system to the solution of a partial differentialsystem.
42
Viscous Flow
The time-dependent, two-dimensional viscous incompressible flow about
any numberof bodies maybe described by the Navier-Stokes equations in
various formulations, two of which are illustrated below.
Vorticity-Stream Function Formulation. (Ref. Ii -14) with the vorticity and
stream function as dependent variables the transformed Navier-Stokes
equations are
_t + (_ - _$_n )IJ = (am_ - 2Bm_n
+ om + _m_)/j2 R (39)
- + Y_nn + + = _j2_ (40)a_ 2B_n °_ n _
with boundary conditions:
= constant A _g = 0 on body surface (41a), j
= y cose - x sine, m = 0 on remote boundary (41b)
All quantities are non-dimensionalized with respect to the free stream
velocity and the airfoil chord. All space derivatives in the field were
represented by second-order, central difference expressions. The time
derivatives were represented by two-point backward difference expressions.
The H-derivatives on the body surface were represented by second-order one-
sided difference expressions. The solution was implicit in time, all the
difference equations being solved simultaneously by SOR iteration at each
time step.
The boundary conditions were implemented directly except for the second
of (41a), which was satisfied by adjusting the value of the vorticity on the
body by a false-position iteration procedure until the second-order, one-
sided difference representation of the tangential velocity, -_ _n was belowj
some tolerance:
43
(k) (k-l) I _I (k)(k+l) (k) mil - mil Fyy
Here (k) is the iteration count, K an adjustable parameter, and (i,l) refers
to a point on the body surface.
The surface pressure is calculated from the line integral of the Navier-Stokes
equations on the surface:
P2 - Pl = R f[2 (V x _). drr ~ ~~I
2 ¢
= R-_ f 2 (B_ - ym )d_ (43)
¢I
The body force components are then obtained from the integration of the
pressure and shear forces around the body surface:
F = + _ py_ d_ - 2x _ _ mx$ d_ (44a)
= 2 d_ (44b)Fy - _ px_ d_ - _ _ _y_
Finally, the lift and drag coefficients are given by
CL = F cos0 - F sin0 (45a)y x
CD = F sin0 + F cos0y x
where O is the angle of attack.
(45b)
The coordinate system for a GDttingen 625 airfoil shown in Fig. 33 was
used in this solution. The high density of constant n-lines near the airfoil
is the result of contraction to the first 15 n-lines. Streamline contours are
shown in Fig. 47, and velocity profiles are shown in Fig. 48. Pressure and
force coefficients are illustrated in Fig. 49.
To show that the boundary-fitted coordinate system can be used with
arbitrary shaped bodies, the viscous flow about a cambered rock at a Reynolds
number of 500 was developed. The contracted coordinate system used in
the solution is given in Fig. 38. _ and _ contours are shown in
44
Figure 50, and velocity profiles are shown in Figure 51.
In order that the pressure be single-valued, it is necessary that the
value of the stream function on each body of a multiple-body system be such
that the line integral of the Navier-Stokes equation on each body vanishes:
2
!vl0 = _ V p • dr =- _ [_t + V(---_2-)
i
- v x m +_ V x a] dr
= _ 1 _ (V x a) • drR (46)
Thus in the transformed plane it must be that
(yan - B_)d_ = 0 (47)
on each body. Since this requires a double iteration, i.e., for both
and _ on each body, it appears that this formulation is not as well suited
for two-body calculations as is the primitive variable formulation that
follows.
Velocity-Pressure Formulation (Ref. 23, 14). With the velocity and pressure
(primitive variables) as the dependent variables the transformed Navier-
Stokes equations are
ut + [y (u 2)_ - y_(u2)]/Jq + [x_(UV)n - x_(uv_]/J
+ (yQp_ - y_pn)/J = (_u_ - 2Burn + YUnn
+ OuR + Tu_)/Rj2 (48)
v t + [yn(uv) - y[(uv)n ]/J + [x[(V2)n- x_(v2)[ ]/J
+ (x_p n - xnp_)/J = (av_ - 2Bvsn + yvnn
+ gv + Tv[)/RJ 2n
(49)
45
aP_ 28P_n + YPnn + + = - (YnU_- oph Tp_ - y_un) 2
- 2(x_u - xnu_)(ynv _ - y_vn)
where(x_v n - xDv_) 2 - J2D t (50)
D _ (ynu_ - ygu + x_v - xnv$)/J
Equation (50) is the transformed Poisson equation for the pressure,
obtained by taking the divergence of the Navier-Stokes equations.
The boundary conditions are
(51)
u = v = 0 on body surface (52a)
u = cose, v = sin0, p = 0 on remote boundary (52b)
The pressure at each point on the body was adjusted at each iteration by
an amount proportional to the velocity divergence evaluated using second-
order one-sided differences for the H-derivatives on the body.
Pressure Distribution and Force Coefficients. The surface pressure distri-
bution is calculated in the vorticity-stream function formulation from
the line integral of the Navier-Stokes equation around the body surface.
In the velocity-pressure formulation the surface pressure, is, of course,
obtained directly. In the velocity-pressure formulation it is necessary to
calculate the body vorticity before applying (44) from
1
w = - _(y_v n + x_u ) (53)
Figure 35 shows the coordinate system for a multiple airfoil consisting
of two Karman-Trefftz airfoils, one simulating a separated flap. Coordinate
system control was used to attract the coordinate lines strongly to the first
ten lines around the bodies and to the intersections of the cut between the
bodies with the trailing edge of the fore body and the leading edge of the
aft body. Velocity vectors and pressure distributions for the viscous flow
46
solution at Reynolds number i000 are shownin Figure 52.
Loaded Plate (Ref. 24)
Figure 53 showsa comparison between the numerical and analytic solution
(Ref. 25) for deflection contours for a simply supported uniformly loaded
triangular flat plate. This problem involves the solution of the biharmonic
equation by splitting into two Poisson equations. The transformed Poisson
equations appear as in Eq. (40) above. Again all computation was done in the
rectangular transformed plane.
47
Vl. INSTRUCTIONSFORUSE- COORDINATESYSTEMS
The coordinate system is generated by the program TOMCATwith the
subroutines BNDRY,CORPLOT,LINWT,ERROR,GUESSA,MAXMIN,PARA,RHS,SOR,
PLOT,and SYMBOL.Subroutines PLOTand SYMBOLwere added for compatibility
between the GOULDand CALCOMPplotters. Each facility will probably have
to makeminor changes for plotting. A complete set of instructions for the
input is included in the listing of TOMCAT.
Core must be set to zero at load time.
Files. The program uses two essential files with internal namesi0 and ii.
File ii is used to store a partially converged solution so that the itera-
tion can be continued by a subsequent run. This file need not be retained
once the solution has converged.
The converged coordinate system is written on file i0. This should be
saved to use as input for PROGRAMFATCAT.
48
Certain files and additional control statements will also be necessary
for the operation of the plotter, and these must be added to fit the user's
installation. The program is compatible with both the GOULDand the CALCOMP
plotters.
Dimensions. The standard program allows a maximum field size of 70 $ lines
and 60 _ lines and requires a core size of 131,000 words for the Langley
Research Center's CDC 6000 Series Computer System. Error signals and instruc-
tions for modification will be given if these limits are exceeded. The three
statements requiring modification for larger fields are separated from the
rest of the dimension and data statements of the main program for convenience.
In_n_ Parameters. Most of the input is self-explanatory in the instruc-
tions given in the listing of TOMCAT. However, a few additional comments
may be in order.
Field Size. The parameter IDISK controls the storage of the converged
system on the disk file and also signals the restart of a partially con-
verged solution. The format of the storage of the coordinate system on
the disk file is given following IDISK in the instructions.
Plotting. Plotting may be by-passed by setting IPLOT to zero and eliminating
certain control cards. The selection of the GOULD, CALCOMP, or other plotter
is made by the parameter IPLTR. Recall that the user must also add certain
site-dependent control statements to the run stream appropriate to the
particular plotter installation. The parameters NUMBR and NUMBRI allow the
49plotted field to be confined to a portion of the actual field by restricting
the numberof curvilinear coordinatelines plotted. Coordinate lines maybe
skipped in the plot by adjusting ISKIPI and ISKIP2. The parameters XBI,
XB2, YBI, and YB2 also allow the plotted field to be restricted to the
portion of the actual field between limits in the cartesian coordinates.
Initial Guess. The parameter IGES controls the initial guess for the
iterative solution of the difference equations. Since these equations are
nonlinear, convergence can be obtained only from an initial guess within
some neighborhood of the solution. The same initial guess will not in
general give convergence for all segment arrangements. The type 0 is
suitable for single-body and simply-connected systems, however, as well as
for multi-body systems having all body segments on the same curvilinear
coordinate line. Types 2 and -40 are widely applicable to multiple-body
fields, except those having two bodies in single segments on opposite sides
of the transformed field, where types 1 or 4 may be effective. Very
strong coordinate attraction near a boundary having a sharp convex corner
requires an initial guess having sufficiently closely-spaced lines in the
region of line attraction, else the iterates may overlap the boundary. In
such a situation the exponential weighted average guess (type IGES > 4) should
be used. The lines in the guess will be more strongly contracted as IGES
is increased. Note that the use of this type of guess requires that the
boundary to which the lines are attracted be located on the bottom or left side
of the rectangular transformed field. Gradual movement of the outer boundary
may also help(see INFAC). See Section III for more information.
Body Contours. Points on the body and outer boundary contours may be placed
as desired around the contours, and the cartesian coordinates of these points
are input in order from cards, one card per point, or from a file with one
image per point.
Some of the contours of a multiple-body system may be split into several
50
segmentswhich maybe located on the rectangular boundary of the trans-
formed field in manydifferent ways as noted above in Section III. For
single bodies the body and outer boundary contours are simply cut and
opened into single segments. The points will be placed on the rectangular
boundary from the first index, LBI, to the second index, LB2, even when
LBI exceeds LB2. The contour segmentsmust be arranged so that the segment
ends are connected by coordinate lines that do not cross. This is simply a
matter of arranging the segmentson the rectangular transformed field
boundary such that a continuous path is traversed over the contour seg-
ments and connecting cuts in the physical field as a closed circuit is
madeo_ the rectangular boundary of the transformed field. The order in
which these sets are input is immaterial, except that the outer boundary
contour must be last. There is no relation between the order of input of
the sets and the order of their appearance on the circuit of the rectangu-
lar boundary.
Note that no points are repeated in the input; the closure of each
body contour is accomplished internally by the program. (The total number
of _ and n lines, IMAXand JMAX,however, will include the repeated points
that close the contours. See, for example , the test cases given following
the program listings in this section.
If a circular outer boundary is desired, this contour maybe calculated
internally rather than being input. In this case the radius and origin
of the circle, and the angle of its initial point counter-clockwise with
respect to the positive x-axis, are input. The points on the outer boundary
contour will then be placed at equal angular increments clockwise from this
initial angle. This outer boundary may then be located on the rectangular
boundary in one or more segmentsin the samemanner described above.
Re-entrant Boundaries. The re-entrant segments pairs are specified by
their end points and the sides on which they lie, but no points are input
51
thereon, since these are actually cuts rather than boundaries in the phy-
sical plane. The order in which the re-entrant segments are input bears no
relation either to the order in which these segments occur on the circuit
of the rectangular boundary or to the order in which the body contours are
input.
Acceleration Paramaters. If a non-zero value is input for R(1), then this
value will be used as a uniform SOR acceleration parameter. Typical values
for a number of cases have been given above in the Section III. The
program also has the capability of calculating a field of variable local
acceleration parameters which are updated at each iteration until the
maximum absolute change of acceleration parameter over the field is less
than the input value R(10), after which the acceleration parameter field is
frozen. Since these local parameters are calculated from linear theory, they
are not true optimum values. Furthermore, in certain local situations, not
even the linear optimum is known. A choice is given, via IEV, for these
situations, but under-relaxation is generally the safest course. As noted
in Section III, in some cases these calculated variable acceleration
parameters tend to be too high and may not give convergence. The use of
the variable acceleration parameters also requires extra computer time for
their calculation, of course, and this calculation involves a square root.
Therefore, the constant input acceleration parameter is usually to be
preferred, provided this value is selected with some care with attention to
the results in Section III.
Coordinate System Control. The curvilinear coordinate lines may be concen-
trated by attracting the lines to the body contours or other lines or to
points in the field. Generally an e_fective way for concentration in the
52
vicinity of a body contour is to use attraction to the contour and also to
the first several lines off the contour, with decreasing attraction amplitude
on each line outward and a decay factor of 0.5 or less on all lines.
On a field that is I0 chords is radius with 40 lines surrounding the
body, an attraction amplitude of i000 with attraction to i0 lines gives
moderate concentration, while i00,000 gives very strong concentration.
Amplitudes of i00 or below give only slight changes from the concentration
that is inherent in the basic homogeneous equations.
Some attention must be paid to the rapidity of the change of coordinate
line spacing with strong attraction else truncation error in the form of
artificial diffusion may be introduced as follows: Consider the finite
difference approximation of a first derivative with variation only in the
x-direction to which the _-lines are normal.
Ynf_ f_f -
x x_y n x_
The difference approximation then would be
fi+l- fi-1f = + T.
x xi÷ 1 - xi_ 1 i
where T. is the local truncation error.l
Taylor series expansions of fi+l and fi-i
Then by (8)
about f. then yield, after1
some algebraic rearrangement,
T = 1 - 2x.)i - 2 (fxx)i (Xi+l + xi-i 1
But the last factor is simply the difference approximations of x so that
1
T = - _ x_ fxx
This truncation error thus introduces a numerical diffusive effect in
the difference approximation of first derivatives. Care must therefore be
taken that the second derivatives of the physical coordinates (i.e., the rate
of change of the physical spacing between curvilinear coordinate lines) are not
too large in regions where the dependent variables have significant second
derivatives in the direction normal to the closely spaced coordinate lines.
_3
Just what is a permissible upper limit to the rate of change of the line
spacing is problem dependent. Consider, for instance, viscous flow past a
finite flat plate parallel to the x-direction. Here the velocity parallel
to the wall changes rapidly from zero at the wall to its free stream value
over a small distance that is of the order of-_--i where R is the Reynolds
number, R = _U_ , based on freestream velocity, U_ the distance from the
leading edge of the plate, x, and the kinematic viscosity, v.
The equation for the time rate of change of the velocity parallel to the
wall is
1
ut = -uu x - VUy + _(Uxx + Uyy)
Recalling that the large spacial variation in velocity occurs in the y-direc-
tion, coordinate lines would be contracted near the plate. The truncation
error introduced by this contraction would be
v
-v(-T) = (- _ yn_)Uyy
v
This introduces a negative numerical viscosity (- _ ynn), since v and Y_n
are both positive.
The effective viscosity is thus reduced (effective Reynolds number
increased),so that the velocity gradient near the wall is steepened. There-
fore care should be taken that y_ is limited so that the numerical viscosity
v
(- _ ynn) not significant in comparison with the physical viscosity (_).
The situation is mitigated somewhat of the fact that the numerical
viscosity is proportioned to the small velocity normal to the wall, this
velocity being of order 1 Actually this limit is conservative, since the/f
normal velocity drops to zero at the wall and only attains the order 1
£fin the outer portion of the region of large gradient of velocity parallel
to the wall where u is very small.YY
Sufficiently close spacing of lines can be obtained even subject to
such limits on the rate of change of the spacing by using decay factors in
54
the tenths range for the coordinate attraction.
The use of coordinate system control tends to slow the convergence
of the iterative solution, and it is necessary to add the attraction grad-
ually for strong concentrations. Convergencecan be achieved even with
very strong attraction amplitude by successively partially converging the
field with a weaker amplitude and then using this result as the initial
guess for the iteration with a stronger amplitude. This can all be done in
one run by inputing the numberof steps to be used for addition of the full
amplitude (IFAC) and the multiple of the final convergence criterion to
be used as the criterion for the partial convergence of each succeeding
amplitude (EFAC). Generally the lowest or perhaps the next-to-the-lowest
numberof steps that will produce convergence is the most economical. In
typical single-body fields, an amplitude of i000 has required three steps,
while i0,000 has required six steps. A value of i00.0 is typical for EFAC.
Whenvery strong coordinate attraction is used to a boundary having
a sharp convex corner the lines may tend to overlap the corner unless the
SORiterative sweepis toward this boundary. Since the sweepis done to-
ward lower ¢ and n values, such a boundary should be located on the bottom or
left side of the rectangular transformed plane if strong attraction thereto
is to be used. In such a case the initial guess should be IGES> 4
as mentioned above, the stronger the attraction, the larger IGES. When
IGESis large enough it should not be necessary to use gradual addition
of the attraction. Movementof the outer boundary mayalso help (INFAC).
Convergence of Very Large Fields. With some segment arrangements for
multiple-body fields convergence problems have occured with large fields
(20 chords or so). This problem arises since with some arrangements, fewer
lines pass between the bodies and the outer boundary in some directions
than in others. Therefore, provision has been made for approaching con-
55
vergence on the final field by successively partially converging smaller
fields and using each succeeding result to produce an initial guess by
linear projection for the next larger field. This can be done in a single
run by specifying INFACand INFACO. A choice is given between doubling
the field radius at each step and increasing it linearly. In the former
case the initial size is completely determined by the numberof steps
specified, while in the latter case it is necessary also to specify the
initial point in the linear increase from zero at which the radius is to
start. Care should be exercised that the initial outer boundary does not
intersect the bodies.
56
Coordinate System
Program TOMCAT
57
PROGRA_ TO_CAT(I_RUToOUTPUToTAPESIINp_ITwTAPEblOUTPljTeITAPEIO,TAPEI|)
E ******** MT$$T$$rPPT STATE _-0 80DY-_ITTED CnOR_TNAT_ _YSTE_ ********C * ($I_PLY nR MULTIPLy CON_ECTE_ REGIONS)E *
e * {DEPARTMENT OF AEROPMY$IC5 a_ AEROSPACE E_INEERI_G )E * ( _I$$1$STPP! 8TATE !J_TVERBITY tg?5 }C * {DEVELOPmEnT @PO_@nREO BY _ABA,L*_GLEV RESEA_C_ CENTER)C *
C * DIRECT IN_UTR_F@ TO _R, JOE F, THUMP@ONc • _A_ER A
C * _I_BISBIPPI _TATEw _S ]976_C *
C , PHnNEI bnl_3_S-]_5C *
C *********************************************************************C
DI,EN$10_ X(7np60), Y(TO,60], RETA(?O,_), RxT(70,bO), _ACC(70,bO)I, T*CC(?O,bO), XPLOT67_), YPLOT(7_)
l), LBI(6)_ LR_(6), LBDY(6], LR8_D(6)_ LRI_6), LRE(6) e L?$ID(b)_ LI_1(6], tIE(_)_ LTYPE(6), LSE_(6), IwER(_}
INTEGER TACC
DATA NbIM,NDIV! /70,60/OATA RA_/S?,EOST?OS1308/_ATA _NBBEG,_N_BEG /6,61_ATA ZERO /I,OE-08/
C
C *
C *** CARDS(S) , CIICE/BnY - FOR_AT{BAt_)C i (NAy RE BLANK)C *
C * C_ AND C_ • BO CHARACTER A/h ARRAYS ._ICH ARE PRINTED ATC w TH_ TOP OF EACH OUTPUT P'_GE AhD DN ANY PLDT$_C *
C * BOY • NA_E OF BODY BEING TRAN$FOR_ED (BO CNARACTER$ NAX),C *
C _*l CARD I IMAX,JMAXeNBDYeITER_IGEB,IDISK,t_R_I_INTL_I_FIN,IGED
C
C * I_AX = NUVBER OF XI•LI_EB,
C * J_AX . NUMBER OF ETa-LINES,C *
C * NBDY • NUMBER OF BODIES _N THE FIELD.C * (ZERO FOR B_NPLYeCONNECTEO REGION)C *
C * _TER = _AXI_U_ NUMBER OF ITERATIONB ALLOWED *C *
C * IGE$ • INITIAL GUE8B TYPE I (BEE IGEO AL80]C * (IOME BEGHENT ARRANGEMENT8 _ILL NOT.CONVERGEC * WITH THE |TANDARD _NIT_AL _UEBB, THEREPOREC * SEVERAL ALTERNATIVES ARC PROVI_ED,]C * (IGE$mO I8 TYPICAL _OR BINGLE•BOOY FIELOI ORC * MULTiPLE=BODY FIELOB HAVIN_ ALL BODIF8 ON THEC * SAME 81DE OF THE TRAN|FOR_ED FIELD. _GE$sE DR -_0
tE3U
S6?
o1o11t213
15
;Y
EOElE22x2,E5EbE7iB29_03132333.3S_6]T$8
taO
.5
.7
.e505t5_53S,5S_6
E85,bO
58
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cccCCccc
ccc
cCKCCCcCcccCCC
cCCcCCCCCcCcCc
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I$ IYPICAL FOR OTHER MULTIPLF-_O_Y FIELDSe EXCEPTF_R THOSE HAVING TWO @_DIE@ IN SINGLE SEGMENTS0_' (}PP_$ITE SIDE@ OF T_E TRAN@ FORME_ FIELD, IN THE
LATTFR CASE TRY IGE§II O_ A,)(IGED.NE,O T@ MOR_ FFFECTI_F FOR CASE@ ,ITH TWEriOTER BOUNDARY O_ THREE @IOE$,)
mt - wEIGWTED AVERAGE OF FDL!R PRPJECTE_
BOUNDARY VALUE@,mO - SAME kS I EXCEPT ZERO I $EO I N PLACE OF VALL_E@
O_ CUT@,I_ - _AM_ i_ % EXCE pT _ OUNDARY VALUES ON CUTS
OMITTE_ IN AVERAGE,I_ - MOMENT PROJECTIONI
WI(@U_D,@UMXmSUMXD)/@UMD, DIVIDED BY IOTALNUMBER OF BOI_k$OARY POINTSew_ERE $UMWmSUV OF ROU_ARY VALUES,
@UMDISUP OF DISTANCE@ TO HOUNDARY POINTS,@U_XDmSUw OF RRO_UCT@ OF ABOVE,
m_ - SAME k@ ] EXCEPT NO _IVISlON BY TOTALNUMBER OF BOLINDARY ROIMT@,
)a - _AME A_ _ EXCEPT EXPONFNTIAL _EIGMT RATHER TMANLINEAR= CONCF_TRAT_ON I@ T_wARD LUwER VALUESOF XT AND/OR ETA WITH DECAY FACTOR OFO,I_(IGESea) ,
¢0 - EXPONENTI AL PROJECTIONIXISUMXE/@UME; wiTH EXPONENTIAL DECAYFACTOR EOUAL 10 IAMS(IGES),*WERE StJMEm@UM OF EXP{._ECAY*OISTANCE),
SUMWEISUM OF PRODUCT OF BOUNDARY VALUEAND ABOVE EXP,
(DISTANCE IS NDNDIMENSIONALIZEO RELATIVETO DIAGONAL OF RECTANGULAR TRAN$F _RMED FIELD)
IUISK - DISK READ/WRITE CDNTROLImO START ITERATION pROM INITTAL GUESS,
DON_T STORE C_ORDINATE $VSTEM ON DISK,m_ START ITERATION FROM INITIAL GUESS,
@TORE COORDINATE SYSTEM ON DISK,mE CONTINUE ITERATION OF k PARTIALLY
CONVERGED _OLUTION READ FROM RESTARTFILEON DISK, STOKE COORDINATE SYSTEM ON DISK,
m_ A@ #E EXCEPT _ONeT @TORE COORDINATE SYSTE M,
NOTEI IDI$K OF 1 _R Z CAUSES THF COOROINATE SYSTE _ TO BE _RITTEN
, TO DISK IN TWE FOLLOWING FORMkTI
,RITE(tOet) C1,RITE(IO,1] C2*RITE(tOet) IMA.X,jMAX,NBSEG,NRSEG,LISEGeNBDY*RITE(tO,I} (LBSID(L]_LBL(L)_L@_(L)_L@OY(L)*LSEN[L)_
I Lmt,NBSEG)_RITE(10_I) (LRSIO(L)eLRt(L),LR_(L).LISIO(L}eLIt(L}eLIZ{L]_
1LTYPE(L)_LmieNRSEG),RITE(IO_I) ((X(I_J), I=I_IMAW)_JII'JMAX}_
1 {(y(ImJ)_III_IMAX)_JI_JMAX)
MERE LISEG IS THE TOTAL NUMBER OF BODY @EGMENTS(EXCLUSIVEOF OUTER BOUNDARY SEGMENTS) PLUB 11 LBEN(L) IS et IF
61_Ze_@g
6Sb_elbB()@70T1?Z
7aTS
7'77_7';SO@1@Z@]5S_8'38bST@B@9_0919E93ea9596
_899
tO0101
tO$tOU1.OSlobtot10810@11011ttti_IL$
It@tL?118119tEO
59
cCCCt"CcCCcCc.Ct'ccCccCCCCCCC
Cc¢CCC
CCCcCCcCCCCCCCCCCCCCCC
CCC
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LP2CL_>LR1(I } ANn IS -I {_T_E_ISE(LRI a_D LR2 AREI_TE_CHA_GEO I_TERNALLY AFTF_ LSE_J TS _ET IF NECFSSA_YS_ T.AT LB2>LR1}I LTYPE IR Im2,],_,_,OR _, _ESPFCTZVELY,IF THE TWO SEGMENT8 Or A :E-E_T_a_T PaIR k_F tl) n_F ONmOTTO_ AND O_E _ T_P, (_) RPT_ _ ROTTO_. _) ROTH O_ TOP.{_) ONE ON LEFT AND O_E 0_' RIGHT, (_) B_TH nN LEFT_ ORfb) _OT_ 0_ RIGHT! X _D Y IRF THF _ARTEST_N COP_DIN_TF$.
IF CONVERGENCE IS NOT aTTAINED TN TWE ALLOWED N(_BE_ OFITERATInNS T_E PARTIALLY CONVERGE_ S_LLJTIO_ I$ _Tf_E_ ON
* DISK FOR RESTART, T_E ITE_ATInN _AY THEN HE C_NTT_UE_ Ry***** SETTING IOISK TO 20_ 3 AN_ INC_EASTN G ITEm,
T*I_ - _0 nO_T PRINT EAC_ ITERATION ERROR NOR_,_I PRINT EACH ITERATION ER_n_ NO_,
T*TNTL - m_ OON_T PRINT _NTTIaL _UF$$,11 PRINT INITIAL GtJE_S,
I.FI_ * *O PRINT C_ORDIkaTE SYSTEM,
II _ONeT PRINT CDORDINATF SYSTEM,
IGED - CONTROLS OIRECT]O_J OF wF]_TED AVERAGES FO_ INITIALGUESS TYPES 0,1,2, AND ,GT,U I
• 0 . AVERAGE IN ROTH Xl AND ETA _IRECTTONS(FOU_ POINT AVEragE)
• ! * AVERAGE IN nNLY ETA DIRECTION
(T._ ROINT AVERAGE)• E - AVERAGE _N _NLY Xy nI_ECTTON
(T*O _UI_T AVE_AGF}
**t CARD I IPLOT_IPLTR_NCOPY,LIN*T_,LIN_TE.NUHBR_NU_RR|_*** 18KIPI,ISKIP2, - FORHAT(qI_)
I
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IPLOT - PLOT OPTIONS1
mO NO PLOTS (INPUT OF REST OF CARD NOT REQUIRED),xl PLOT COORDINATE SYSTEM,l_ PLOT INITIAL GUE88 aND COORDINATE SYSTEm,
IPLTR - I1 PLOT _ITH OOULO _800,B2 PLOT kITH GERRER
m_ OTHER ( CALL P_EUO0 - DEVICE INDEPENDENT .VARIAN AND CALCOMP DO NOT HAVE LINE*TCAPABILITY]
NCOPY - NUMBER OF COPIES O_ PLOT DE8IRED,
LINwT1 . PLOT LINE wEIGHT DESIRED POR PLOT TITLES,(=_ "I_ O_ I_ _ RESPECTIVELY FOR TRIPLE_ DOUBLE_
NORMAL_ HALF_ THIRD WEIGHT LINES)(THIS APPLIE| ONLY TO THE GOULD _800)
LIN_T2 * PLOT LINE WEIGHT DESIRED FOR COORDINATE 8YSTEH,{|[E NOTE BELOW LINWTI]
(TH_8 APPL_F8 ONLY TO THE GOULD 4800)
NUHBR . NUMBER OF ETAmLINE8 DESIRED FOR PLOT,[ZERO VALUE PLOT$ ALL]
121
125I2(J
127
150131
135
13b157I_
I_I
I_
I_?
150
IS$
15S
157158
It)OI_
165
17o17I
177
60
Cr
cFcccc
cccc
t
t
t
_.LJMmPl .. _ll,_E_ OF XT..LI_ER r_E._Tt_E r_ Ircl_ PLC1T,
(ZF.R{] VALLIE PLUT$ ALL)
TSKIPt - SKIP PARAMETER FOR XI-LT_E$,(1PL_TS EACH LINE, 2 PLOTS EVEPY SECO_ LT_E,ETC,
]SKIP2 - SKIP PkRA_ETE_ FOR ETk-LI_ES,
(REE NOTE BfLnW ]SKIP1)
t
*** EACW B_OY A_D TMF OUTFR _0 IjN_A_¥ IS nIVI_E_ I_T0 ONE OR
*** M_PF SEGMENTS PLACE_ ON TM_ BI_ES nF X_F RECTAN_ULA_ T_ANIFO _EO
*** FIWLn, TWESE 5EGME_IT$ _RF CONNECTED PY RE-E_TRA_;T SEGMENTS,
C *** TMF BFG_E_T CONFIG U_ATIo_ INPUT IS AS _OLLO_Sl
C **** CARD I N_SEG.NRSEG - FORMAT(2IS)
, NBBEG - T_TAL NUMBER OF _O_Y A_O O_TER BOUNDARY SEGMENTS,
, NRBEG - TOTAL NUMBER 8F PAIRS OF WE-ENTRA_;T SEGMENTS.
, (ZENO FnR $1MPLY-CON_ECTED REGION)
*** C&RD$(N_$EG) I LbSIDeLR%eLR2#LRDY " FOP_AT(_IS)
cCccCrCr
C *C *C *C *C *C *C *C *
C *C *(- ,C *C *
C *C *C *C *C *C *C *C *C *C *CCCCCCCCCCCCCC
L_$1D - BIDE OF RECTANGULAR TRAkSFOHMED FIELD nN wHICHR_DY _EGwE NT OR OUTER BO(IK, nARY SEGMENT I8 LOCATED,
(BOTTOM IS 1_ LEFT IB _, T_P I8 ], RIG wT IS _,)
LHI_Lm? - FIRS T AND LAST INDICES OF REGMENT,
(Lm} MAY EXCEED LB2)
LBDY -RQDY NUMBER (BUDIES ARE NIJMRERED CONSECUTIVELY
FROM | TO NBDY
OUTER BOUNDARY IS BODY NUMBER 0 OR Nb_Y_%,
NUMBER 0 CAUSES OUTER BOU_DIRY TO BE CALCULATED
INTERNALLY IS A CIRCLE, P_STTIVE nR NFGATIVENBDYe% CAUSE8 OUTE R BOUNDARY TO RE READ AS OTHER
BODIES,)NOTE I EACH BODY IS RFA_ VIA A SINGLE CADD OR SET
OF CARD8 wiTH _0 RFPEATED POINTS_ NOT EVENTME FIRST POINT, BODY SEGMENTS MUST BECONSECUTIVE ACCORDIN_ TO LBDY, WITH THEOUTER BOUNOARY LAST REGARDLESS OF ITSLBDY, _OINTS IN FACM SEGMENT ARE READCONSECUTIVELY AND PLACED ON BOUNDARYFROM INDEX LBI T_ INDEX LB_,
*** CAROS(NR_EG) I LRSID,LRI_LRE,LISIO,LII,LIE " FORMATCbIS}, (OMIT FOR 8IMPLY-CONNFCTEB REGION)
LI
R
LR$ID,LISI_ - SIDES OF RECTANGULAR TRANSFORMED FIELD ONWHICH IEOMENTB OF A RE-ENTRANT pAIR ARELOCATED, BEE NOTE BELOW LBSIB FOR 8IDES,(LIIIb MUST EQUAL OR EXCEED LRBIO)
LRteLREeLIIeLIE = FIRST AND LAST INDICES OF EACM SEGMENTOF A REtENTRANT PAIR, (I_ LISID II EQUALTO LRBID THEN LII MUST EXCEED LRE, IFLIiID T_ NOT EQUAL TO LRBI_ THEN LitMUST EQUAL LR%. IN ANY CASE LRE MUST
$ba18S
187188
Tel
E00
CO2_0520_E05EO__0?ZOb_09
2%2
ElS
_EO
E22
2_5Z2_
228
_0
!]?
6i
CCcCccCcCCcCr¢ccCCC
cC
CCCCf,CC
CCCccCCC
CCCCCCCcCCCCC¢CC¢CCC
* EXCEED L_I, ANn LI_ MU_T EXCEED LII.]t
*** CARD I _(1)eR(2),R(3)eYINFI_oAIN_IN,XOT_.fjYOINF,_JI_F -FORMAT(?KIn.0,15)
R(I) - GAt,$$-$EIDEL ACCELE_ATIn_ PArAMETEr.ZERn VALUE CAUSES VARTARLF ACCELERATION PARAMETER
FIELD TO RE CALCULATE_ tNTEPNALLY,(TYPICALLY 1,85)
R(2) - CQNVERGENCE CRITERION Fn_ X ITE_ATInN EPROk NOR_,(TYPICALLY 0,0000|}
R(3] - CONVERGENCE CRITERION FO_ Y ITEWATIO_ ERPn_ NO_M,(TYPICALLY 0,00001}
YINFIh = RADIUS Or CIRCULAR OUIER _ULINOARY,
AINFIn - ANGLE OF FIRST PnINT nN OUTEP _O_NDARv, (nEGQEE$)(ANGLE t_ POSITIVE COIJNTER-CLOCK_ISE FWnM POSITIVE
X-AXIS, PC)INT8 ARE CLOCKWISE FROM T_T$ ANGLE.)
XOINF,YOINF - CENTEN OF CIRCilLAR O_ITER BOUNDARY,
k,INF= NUMBER _F UNIQUE POI6T$ ON DUTE_ ROUNDARY e
HOTEl YINFIN,AINFIN,XOINF_ & VnlNF _AV BE RLA_K IFOUTER BOUNDARY I_ TO _E READ,
*** CARD I IffvmIAIT,R(10] - FORMAT(@IS,FIO.01(BLANK CARD MAY e_ INPUT IF COh_TANT ACCELERATIOkPARAMETER I_ USE_}
IEV - CONTROLS COMPLEX JACOBI FIGENVALLJE PROCEnURE.[OPTIMUM ACCELERATION WITH R_AL _IGENVALUE_ ISOVER-RELAXATION. OPTIWIJw _ITW IMAGINARY I_UNDER-RELAXATION, NO THFORETICAL OPTIMUM 18 KNOWNFOR COMPLEX EYGENVALUE$,)-l I UNDERmRELAX=
0 I WEIGWTED AVERAGE,el I OVEHuRELAX,
IAIT - NONmZERO VALUE CAUSES VARIARLE ACCELERATIONPARAMETER FIELD TO _E PRINTffD,
R(|O) - CONVERGENCE CRITERION FOR VARIARLE ACCELERATIO hPARAMETER FIELDm FIELD I$ FROZEN wWE_ _AXlMUMABSOLUTE CHANGE ON FIELD I_ LESS THAN RCtO].(TYPICALLY O,OI)
*** CARD I INFAC,INFACO - FORMAT(EI_](CAN RE USED TO AID CONVERGENCE BY CONVERGING ASMALLER FIELO FIRST AND USING TwI! REiULT TO PRODUCE&N INITIAL GUE$! FOR k LARGER FIELD, BLANK CARDMAY RE INPUT If ?HI8 FEATURE I_ NOT TO BE II!ED,!TANDARD I! TO NOT U!E TWI! FEATURE,|
INFAC = NUMBER OF !TEP! IN ATTAINMENT OF OUTER BOUNDARY.POSITIVE DOUBLE! OUTER BOUNDARY AT EACH 8TEP_
NEGATIVE MOVE! OUTER BOUNDARY LINEARLY,
62
cc
rCCr
CcrccC
ccccr
cccCcCcc
cccCCcCCCcccCcCccCCCCCCcCc
CC
CC
CCCcC
t {INCREASE MAGNITUDE IF DIVEWGENCE _CCUR$)t
* INFACn - INITIAL STEP IN ATTAINMENT OF OiITER BDU_DAWY,
, (LINEAW ATTAIk_E_T ONLY)
*** CArD I $1ZE,RATIO,_IST,XBIw_B_YRIeYm_ - FORmAT(IF)0,0)
(C)MIT IF NO PLOTS DESIRED= I,Fo= IF IPLOT IS ZFwO)
SIZE = PLOT SlZE IN Y=DIWECTION, (TNCH_B)[TYPICALLY 8,0)
RAIl0 = I0 X AND Y PLOT _CaLES ARE E_UAL,>0 X A_D Y PLOT SCALES ARE AnJIJSTE_ 80 THAT THE
PLOT I$ SQUARE,(IYPICALLY 0)
nlST - GOULD PAGES REOUEST(X=DTRECTIDN)p EITHER 10 OR 20,(TYPICALLY tO,O)
_BI,XB_ - _INIWUM A_D MAXIMUM W-VALUE8 TO BE PLOTTEO_
(ZERO8 PLOT ALL)
YBIeYB2 = MINIMU_ AND uAXI_UM Y=VALUE$ TO BE PLnTIED,(ZERO8 PLOT ALL)
*** AFTER THIS INPUT, READ IN BODY COORDTNATF$ = FORWATCEFtOmO)
_et
*** TF NO COORDINATE SYSTEM CONTROL 18 TO RE USED, FnLLOW TMESE CARDS
*** *ITH T_REE BLINK CARDS, TF CONTNnL I$ TO BE USED, *lSF THE
*** FOLL0.1NG I_PUT RATwFR THAN THF BLANK CARDSl
*** INPUT FOR COORDINATE SYSTE M CONTROL= USE TWO 8ETSe ONE FOR
*** XI-LINE CONTROL AND ONE FOR ETA=LINE CnKTROL,
*** (THIS DATA 18 RFA_ I_ SUBROUTINE SOR.) I#
*** CARD I ITYPwITYP,NLN=NPTeDECwAMPFAC = FDQMAT(Ab=I_I_F|O=0)
ATYP = TYPE OF ATTRACTION e (Xl FON XI'LINE ATTRICTION_
* ET_ FOR ETA-LINE ATTRACTION,) LEFT JUSTIFIED,
* ITYP - ZERO GIVE8 ATTRACTION ON BOTH $1DES,
* NON=ZERO GIVES ATTRACTInN ON CONVEX SIDE AND
, REPULSION ON CONCAVE 81_E,
e NLN = NUMBER OF ATTRACTION LINES,
l
• NPT = NUMBER OF ATTRACTION POINTS,
* OEC - NON-ZERO DEC U_E8 DEC FOR BECAY FACTOR,
• AMPFAC= NON=ZERO AMPFAC WULTIPLIE$ ALL AMPLITUDES
• BY AMPFACI
**• CARO$(NLN) I JLN_ALN_DLN - FDRMAT[§W=IS_F[0,0)• (OMIT IF NLN I8 ZERn)
• JLN = ATTRACTION LINE INhEW,
* ALN = AMPLITUDE (NEGATIVE REPELS) FOR LINE ATTRACTION,
e
30t30_
_05306$0130B30e
313
316
3IS3193_n
3_3
329330
33335.
336337_3833q
3.3
3_6
3_83wg3S0
3S$
35_
63
* nL_ = DECAY FACT_ FnR LT_E XTT_AC?IO_J, 3hiC * 362
*** CA_S(NPT) I IPT,JPT,APTe_PT - FnWMAT(_I_,2FI_.Q) ]_C * (O_TT IF _PT T$ ZEkn) 3b_
C * TPTwJPT - ATT_ACTTOh POINT INDICES. 36_C * 367C * APT - A_PLTTU_F (NFGATIVF REPEL$_ WOQ POINT ATTrACTIOn. 368C * 36q
* f)PT - _ECAY FACTO_ FO_ P_INT ATT_ACTI_. 370C * 371C *** FOLLOW TMF COORDINATE SYSTE_ C_TROL CAR_S wITH THE 37_C *** FOLLOWING CADOt _T_C * 37.
C *** CArD I IFaC.TRTTeEFAC m FO_MAT{_ISeKIO.O) ]75C * (CAN BE USED T_ AID CONVERGENCE HY CO_vERGING ;IELn 37k
* _IT_ LESS ATTRACTION FIRST A_O D_TNG TWIS RESULT 377C * A_ TH_ INITIAL GHE_S F_ ST_N_E_ ATTRACTION, ]T_
* BLANK C_RD -LIST RE INPUT TF TwTS FEATURE I_ _OT USED. 3ToC * STAN_AR_ IS TO NOT USE THIS FEATI_IRE . _UT TT8 LISE MAY 380
* _E _ECESSARY _TTH STRONG ATTRACTION,) 381
C * _FAC - NUWRER OF STEP_ IN ADOITI_h 0F INMOMOGENEOUB T_M, _3
C * POSITIVE OOUflLES INHOM0_ENEOUS TERM AT EACW STEP, 3_Q
* NE_ATIVF INCWFASE_ INMO_OGE_tEOU$ TER_ LTNEARLY, 38_
C * (INCREASE MAGNITUnE IF _IVERGENCE OCCURS,) 386C * 38T
C * IRIT - NONmZERO VALUE CAUSES INWO_nGENFOUS TERM TO BE $_C * PRINTED, 38qC * 390
C * EFIC - MULTIPLE OF CONVERGenCE C_ITERIO_ TO RE USE_ FOR ]gtC * INTERMEDIATE CONVERCENCF BETWEEN AOnITI_NS OF ]9_C * INNOM_ENEOU8 TERM, (TYPICALLY |00,0 _ _C * $9_
C ********************************************************************* SqSC $qbC READ INPUT PARAMETERS ]97
C 3qO*RITE(_,6_O} Sqq
C _00
C A CALL TD PSEUDO INITIALIZES THE GRAPHIC OUTPUT SYSTEM, THIS MUST _OtC BE THE FIRST CALL FOR GRAPHICS IN ORDER TO GENERATE PLDT DATA Q02
C IN THE FORM DF A DEVICE INDEPENDENT PLQT VECTOR FILE, _03C k CALL TO LEROY _LOW8 TMF 8PFED FO_ GERBER F_R LTOUIO INK PEN, _0aC IT I$ OK TO LEAVE CALL LEROY IN FOR T_ OTHER DEVICES, _OSC _Ob
CALL PSEUDO _OTCALL LEROY _08RE_O (§_bSO) C|,C_BDY QO_
READ iS,k20) IMAW.JMAXeNBDYeITEReIGE8.1DISK,I*IR,TWINTLeIWFINeIGEO _10IF (IMAW,GT,NOIM) wRITE (6e700) IMAX_NDIM _11IF (JWAW,GT,NOIM|) WRITE (6,?_0} JMAX.NDI_[ U12IF (IMAX,GT,NDIM,OR,JMAW,GT,NDIM[) STOP [ _15READ (Se6_O) IPLOT_IPLTR,NCOPY_LINWTI,LINWT_NUMBReNUNBRI_I _1_
tIKIPI_IIKIP_ _[_REAO (§_q60) NBSEG.NRSEG _16IF (NB|EG,OT,MNBBEG) WRITE (_,qtO) NBSEG,MNBOEG _|7IF (NR|EG,GT,MNRDEG) WRITE (6#9_0) NRBEGeMNRBEG _18IF (NB|EG.GT,MNB|EG.OR.NRSE_,GT_MNR8EQ) 8TOP Z _19READ (_,880} (LROID(L]_LBI(L}.LRI(L).LRDY(L)eLBt_NBB[_) 0_0
64
tL)wLIE(L)wLSloNQ$_G)RFA_ (_eO00) (_(1)_Tule$),YT_FI"_eAIKFT_,XnI4FjYOI_F,_I_'F
_An (S,_20) IEV,IAIT,_(In)RFAb (_b20) I_FAC_I_rACn
IF (IRLOT.hE.O] _FAP(_M30) !TZFogkTIOenI!T,X_leX_2wvR1wYR2
_WITF T_PLIT PARAMETFR$
*RITE (6,1n60)
_rRITE (6,990}
%,IGEO
_RITE (_,I010)
wWI_E (_,Ih205_RITP (a,tO3h]
*RITE (6,101o)
I"_W_J"_X,N_DY,ITKR,I_ES,T{)ISw_I*IR,IWINTL,IWFI _
IPLOT,TPLTW,_COPY_I. IN-T_,I I_WTP,NU_BP_U'_R|
I_V_I_IT,R(IO}
$1ZE,_ATTC},I)T!T_XBI,XB_,YHI_YB_
Ir (NRS[G.EQ.e) _RIT[ (b,IOMO)
_PITE (6_glO) (L,LR$I_{L)eL_I(L)eLBE(L),L_DY(L),LmI,NR!EG)
III(L),LI)(L)eLmI,NNSEG)
IF (INFACO.EG.O) INFACOmlIF (INFIC.EQ.O) INFACm|
wRTTF (6_9_0) YIkFT_INFTNeWOI_r_YOI_@,NTNF,INr_C,INr_CD
IF (R(1).GT.O.O) wRIT( (6,8?0)
SET {JP P_RAMETERS
I0
NPLT!m!
_INVINe_INFI_/R_D
IF (NUMBR,EO,0) _ILI_BRBJMAX
IF (NUMBR|.EQ.O) NUMBR[ZI_AX
IF (I$KI_I.E_.O} 18KIP!BI
Ir (I$KIPE.EO.O} 18KIP2utIF (NBDYoLE.O) NRDYm|
IF (I_LOT,EQ,O) G_ TO 10
IF (kB$(XBI),LT,ZERO) XBIB=t,0EEOIF (AB$(XBE).LT.ZERO) XS_le!lO[_O
I_ (AB$(YRE).LT.ZERO) YB_meI.0[_O
IF (DIST,LE,O,O) DTSTmlO,OCONTINU_
RE_D POINT8 ON BOOIE_ AND
READ OR CALCULATE _OINT5 ON OUTER ROUNDARY
DO 20 LmI_NB$E$E0 If (IAB_[LBDYtL]},EQ.NBOYeI.DR.LBDV(L).EG.O)
LISEGmNBOEG_IGO TO _0
30 LISEGmL
_0 DO SO luleZM_X
DO _0 Jmt,JMkXX(Z,J)uO.O
SO VCI,J)u0,Ok_OYOmO
GO TO $0
_EEu23
UE/__27(_28uEq
u3!
(_3q_W0
UWQ
Q_0
_SB_sq/4b0
/4e/_
/4bq
_0
LBDYCNB8EG+I)me|O0 _81nO _EO LmSmNBSEG .82
XImLBI(L) _8_
T_mL82(L] _8_
ISENml _B5TF (Tt.GT.I_) TBENB-I ,86
LB[N(L)mTBEN _87
KImMTNO(II_T2) _8
K_mMAXO(II.I2] ,89LBI(L)mKI U90
LBZ(L)mK2 _91
Ie (LBnY(L).EQ.LBnYO) IIIIIeISEN _2IF (L_DY(L).NE.LBDY(LeI]] I2mT2-TSE_J _93IF (LBDYtL)mNEmLBOYO) I_mTI ,q,
KImMTN0(TI,12} .9_
K2mMAX0(IIwI2} _q6IGOT0mLBBIO(L) ,q?
GO TO (_Owl&O._Oe160). TGOIO ,98
C**** BOTTOM OR TOP .99
60 IF (LBBID(L),EQ,l} j=! SO0
IF (LBBIOCL),EQ,3) JmJMAX S0I
IF (NRBEGmEGmO) GO T0 gO _O_
IF (L,LT,LISEG] Gn TO 90 S0]IF (LBDY(L]) Q0o?0eQ0 S0Q
70 CJLL BNORY (XmY,JIIImI2mLB$10(L}pYI_FI_eAINFINmXOINF#YOIN_wNINF_ _0_INDIM} 50b
GO TO 130 S07
90 DO _00 XmKl,X_ 50BIIII÷(KmKI)*IBEN S09
100 RE_ (S,850) W(T,J],Y(I,J} 51_1_0 I_ (LBOV(L),EQ,LBDYO) GO TO I_0 SII
I_mj 51Zx._mXCla,la) 513
Y_6tY(I_,Ia) 51Q
_0 TO t50 SISIQO CONTINUE 51b
XCII-18EN.J]mX]5 517
Y(II-IBEN,J)uY$_ 518150 ISwJ 51q
ISmI_ 520
X$SmX(l$_IS) 5_1Y$SIY{I$115) 5_
IF (LBOY(L),EQ,LROY(Lel]) GO 70 2bO 5_3IF (NR_EGmEOm0eAND_NBOYmGTmI) GO TO _0 5_
X(II+IBEN_J)mW.6 5_YfI2+IBEN_J)aYQa 526GO TO _0 5_?
C*** LEFT OR RIGMT S_8160 IF (LBSIDCL],EQ,Z) Iml 5_9
I_ (LBBIDCL),EQ,#) TmIM_X 530IF (NROEG,EQ,O) GO 70 190 S]lIF (L,LT,LTSEG) GO TO 190 S$ZIF (LBDY(L)) 190_t70_190 533
170 C_LL BNDRY (X.YeI,I|_I_LBBIO(L),YINFIN,_TNFINeXOINFmYOINF_NINF_ 5}_INDIM) 5$5
GO TO _$0 _)b
190 DO _00 KIKI_K_ S]?JmIItCK-Wt)*ISEN S38
I00 READ ($,830)-X(I,J),y(I,j] 5$q
66
2aO
_50
2eO
11)
it)
_Oo
CALL WAXMIN (X,T_AX,JMaXpNDI_,XB_AX,XRVTN,IDLwM,InlIM,IDUM,IDUMel,
)9O
Cn_TINUELROYOuLB_Y(L)cONTINU_
DO _90 LI|aNMSEGL_DYfL)mI_B$(LBnY(L))
CLASSIFY RE-ENTRANT SEG_MENT_
IF (NRSEG.EO.O} GO Tn 3_0no ),0 LmI,NRSEG
IGOTOmLR$10(L)GO TO ($00,$t0,_)0,330), IGOTn
100 LTYPE(L)m!Ir (LISID(L).EQ.LRSI_(L)) LTYPE(L)m_GO ?0 $_0
31o LTYPE(L)m_Ir (LISIO(L),EO,LRSID(L)) LTYPE(L)m_GO TO 3_0
_0 LTYPE(L)miIF (LIBID(L),EU,LRSTD(L)) LTYPE(L)m)GO TO 3_0
)30 LTYPE(L)m_I_ (LI$10(L).EQ.LR$1D(L)) LTYPE(L)m6
_aO CONTINUE3_0 CONTINUE
PULL OUTEP BOUNDARY IN FOR INITIIL GUE$_ Eft IN@_C NOT ZERO
$60
Ir (IN_C.GT.0)IF (INFAC.LT.0)IF (LISEG.GT.N68_) GO TODO QO0 LmLISEG_NB_EG
ItmLBt(L)I)mLS)(L)IGOTOmLBSlO(L)_0 TO (_60_380_60,_80))IF (IGOTO,EQ,I) JmlIF (I_OTO,IQ,)) JmJW&XDO 370 Imll,IE
ClN_Cm!.0/FLO_T(E**(INF_{-!))CIN_A_m_LOAT(IN_ACO)/AB_(FLOAT(TNFA_))
IGOTO
67
370 Y(I,J)mY(I,J)*CINFACGO T_ _00
380 IW (IGOTO,_Om2) Iml
IF (IGOTn,_Q,_) ImI_AXDO ]_0 JmIleI2
w(I,J)mX[I,J)*CI_FAC
3QO Y(I,J)nY(I,J)*CINFAC
QO0 CONTINUE
INITIAL GUESS
_lO CALL GUE$$A (I_AX,JMAX,NDI_wXpY.XBMAX,WBMINpyRMAX,y_MINeYINFI_w_.NBIEGeLRIpLR2,LIIeLI2,LRSID,LIBIDeIGE$oIGEn)
QO_ IW (IPLOT.EUe_) CALL CORPLT {XpYjNDI_,NIJ'_WIw_UM_RtCIwC2wRJTIOp$1Z
IEe_COPY,LI_wTI,LINWT2eI$_IPI,ITKIP2,XPL_ToYPLnT,DTSTelPLTRe_LT$,LB$1D*LBIeLM2*LBDY,LR$1D_LI$1D,LRImLW_,LTIeLI2_LTYPE,_BTEG,N
TR_EG_LI$_G,J_AX_IM_X,XBI_XB_eYBI,YB_]
PRINT INITIAL DATA
wRITE (6#6_0) C1
wRITE (6ea6n} C2
wRITE(6_6TO) NDY_IMAX_jWAXmR(|)eITER_R(_)_R(])_RITE (_,A_O) NBDY
IF (NBDYeEQml) GO TO _aO
_0 WRITE (_0)60 TO .TO
_0 wRITE (km_go) NCO_¥,LINWTE,$1ZE_RATIOQT0 CONTINUE
IF (IWINTL.EQ.O) GO TO SO0wRIT( (_?_0)
WRITE (SmTQO)
DO _00 Jmi_jMAX
wRITE (_TSO) d
QAO wRITE (_?I0) (X(I,J)_ImI.IMA_)wRITE (_.760}
DO _0 JmlmJMAX
wRITE (a_?60) J
_o0 CONTINUE
ITERATIVE SOLUTION
C_LL 80R (RmW,YmlWAXBJMAX,ITER_IXER_IYER_IEND,NOI_IWEReNBOY_WACIC,RETA_RWI_LRI_LR_LII_LI_LTYPE_NR$EG,TWYR_IAIT,INFAC_INFACO_NO$_
_G_LIS_G_LBI_LB_L_SID,TACCmIEVmIDITK)
PRINT VINAL VALUES
I_ (IWFIN.NE.O) GO TO STOWRITE (5_6_0)
WRITE(6_6_O} CIwRITE(_.b_O) C_
W_ITE (_770)
O0 TO _0
wRIT[ (keT_O) ITER
68
530 ,_ITE {6,8nnl
_QITE (6,_50) TXERCI),TXFR(2)eIYF_(1),IVF_(_)*mITE (_,7_0)
_0 550 JII,JMiX
,mITE (_,75o) J
550 _ITE (6,71n) Cx(Y,J)oI¢IoIMaX)
*_ITE (6e7_0)
Dn 560 JsleJ_X_RITE (b_75_) J
S_O ,_ITE (6,71e) (Y(I.J).ImI.I_AX)
_RITE (6,6Q0)570 TK (ZDISK.EU.O.n_.TDISK.EQ._) _P TO 6_
_RITE (10,q80) C1
wRITE (lO,gSO) C2
TF (IDISK.EQ.,) GO TO $80
*RITE(IO_620) I_AX,J_X,_aSEG,_'_$£G,L?SEG,_BDY_RITE(IO.qFI)TLSBT_CL),L_](L),LR2(L),LBnY_L_.LSE_rL),Lmt,N_SEG)
wRITE(IO.ga_)(LRST_(L),L_I(L),L_?(L),LIST_(L),LIIfLI,LI2(1),
tLTYPE(L),LBI,N@$EG)GO TO _0
580 _RITE(IO,_I) IMAX,J"_X5_ *_I?E(10eg83)[(XC_,l),Im1,I_AX),Jm1,J_iX),(CY(led)_lileI_al)_Jlle
tj_X)
600 COnTInUE
PLOT
IF (IPLOT.GT.O) _0 Tn 61nSTOP
610 CALL CORPLT (X,Y,NDI_,4tJ_RR1,NU"BR,C1,C_,RATIO,$IZE,NCOPY_LINWT1,LIIN_T_,ISKIPI,IB_IP_eXPLOTeYPLOT,DTST_TPLT_NPLT$,LBSIDeLBle_LB_,LSOYeL_$ID,LISIO,L_t,LH_,LII,LI_eLTYPEe_BBEG,NRSEGeLI$EG,JMAXe31_AX_X_t_XR_,Y_I_YR_)IF (_PLOT.GT._) CALL PLOT (O..O..g09)
STOP 010t
6ZO FORMAT (1615)
6]0 FORMAl {8F10.0)b_O FORMAT _IH1)
6S0 FORMAT(BA|O)660 FORMAT(_tXeSAtO)&TO FORMAT {/]TXe*BODY_FITTED COORDINATE SY_T_M*tt_IX,*TRA_FOR_ED 80D
IY. *,8AtOIt_IX_*FtELD PARAMETERS, NU_RER OF XI=LINE8 sw,Igt3RXe* N
_UMBER OR ETA-LINES • *,I_t/ItqX,* ITF_ATION PARAMETEmS! $OR ACCEL]ERATION PARAMETER • *,FS,St_ZX,* MAXI_u_ NUMBER OF ITERATIONS ALLO_wEO u *,I=/3_X,* ALLOWABLE ITERATIO_ ERROR NORMSl Xl *,E10,_/75SX_* Yl *,E|O,S)
bBO FORMAT {I_IX_*NO PLOTS DESIRED*)
650 FORMAT (/EIX,*PLOT PARAMETERS1 COPIES DESIRED • *I3/_?x,* LINE*,1*wEIGHT _ES_RED • *I3t_qxe*PLOT SIZE TN YoDIRECTIDN • *F§,]/3qX_*R|ATlO I *_F8,$)
?00 FORMAT (*0--- ERROR ---- IMAX TO0 LARGE*,IOX,*IMAXs*,TS_SXn*MAXI_UtM IB*,IS/t6X,*INCREASF NDIM IN OATA STATE.ENT AND*,* FIR8T DI_ENSl
_ON OP X,V_RETA,RXI_ACC,TACC_*/16X,eALSO*,* INCREASE nIMENSTON OF
SXPLOT AND YPLOT TO MAXIMUM OF*,* NDIM AND NDIMI PtUS _,*)710 FORMAT (_Xe10F11.S)?_0 FORMAT (*Omo- ERROR ..o= JMAX TOO LARGEteIOXetJMAXi*e_eSXeeMAXtMU
IM IS,,I_/t6X,*INCREASE NDIMt TN DATA STATEMENT AND*,,' SECOND DIMEN
b_1
662t665
b6_
6e7
67n_71
e73
675
67867_
68U
bBb68T668
68_
69t69_
6q3
69_b_66_7698
bq97O0
701
TO_?Oh_OT
?08
710T11
Tt|Tl$
TiT710715
69
2SION OF X,Y,WETA,RXIewACCoTACC**IIbX,,ALSn,** INCREASE _IMENSION O T213F WPLOT AND YPLOT TO MAXIMUM OF*,m NDIM A_O NDIM| PLUS 2**) T22
?]0 FORMAT (/21We*INITIAL GUESSES FDW X AND Ye) 7237UO FORMAT (/_Xe]3(JM,)wq_p,XmARRAymSw_](|Ht)} 72QTSO FORMAT (SXw*JmeI_} 725T60 FORMAT (/_Xw35(tH,)egxetY.ARRAY,BXw_](IH_}) 726?TO FORMAT (/StW,*FINAL VALUES./) 727
T80 F_RMAT (2|Ww*ITERATION DIVERGE8,,) 728?gO F_RMAT (2tXw*ITERATION 18 CnNVERGING RLIT DOE8 _OT REACM ERROR TOLE 729
IRJNC[8 IN *I3e* ITERATIONS.*) 730
800 FORMAT (2|We*ITERATION CONVERGE$,,) 731
8tO FORMAT (/21Xw*INItIAL ITERATION ERROR _,_RM$1 Xl *wE|O.Se* YI _eE 7_2
tlO,Sw* AT ITERATE i |*/2IX**FINAL ITERATInN ERRQP,,, NORMS1 Xl 7_2*wElO,Sw* YI *eE|O,So* AT ITERATE #*,I_) 73Q
820 FORMAT (215,F10.0)830 FORMAT (2F|O.O) 7_5
736850 FORMAT (21Xo*LOCkTION OF MAXIMUM TTE_ATIO_I ERRORI Xl I=*I_p*,Jm*Z5 737
l/SOXe*Yl II*I_we J=*IS) T]_860 FORMAT (/21XweNUMBER OF BODIES TN FIELD 1.I5) 739870 FORMAT (*OUNIFORM ACCELERATION PARJMETER USED**) ?gO880 FORMAT (_15)890 FORMAT (61S) 7WI
7_2qO0 FORMAT (TFIO,Op2IS) 7_3q|o FORMAT (*O_w. ERROR ._._ NBSEG TOO LARGF*,IOX_*NBSEGIe_I_eSXe*MAXI 7_
IMUM IS*,I3/16X,*INCREASE MNRSEG IN _aTA 8TATEMEN T AND,o, DIMENSIO N TQ528 OF LB8IO_LBt,L_2,L_OY_L8EN,,) 7_6
q20 FOWMAT (*O*=w ERROR .-.= NRSEG TOO LARGEI,IOX_*NRSEGle_I$_§W_*MAXI ?atIMUM 18*_I3/IbX**INCREASE MNR8EG IN. DATA STATEMENT AND*_* DIMENSION TQ8
2S OF LRBID_LRI_LR2_LISIO_LII_LI2,LTYPE,,) TWqq30 FDRMAT (cO--BODY 8E_MENTS''*//$W,*L*_2W,*LB8IDe_QX_,LBI._X_.LB2,_ TSO
I$W_*LBOY*//(Ia_I?)) ?St9_0 FORMAT (*O--RE-EnTRANT 8EGMENTS''*I/$Xe*L*_2X,,L_SIO,,_X_,LR|,_X_ T52
I*LR2**2X_*LISIO*,_X,*LII*_X,*LI2,1/(I_,bI?)) 753qSO FORMAT (*OeeOUTER BOLINOARYw.,//3X_,RA_IU s I,_FI2.B_IOWe,|NITII L AN TSg
|GLE s**Ft_,813W_*ORIGIN AT X I*,FI! 8;; Y s*_Ftt B//]Xe*NUMBER TS_ZOF _OINT$ =*,IS,lOW,IS,* STEPS IN*,_ tA_ OF ?S6NMENT _NFINITY,,IOW_,]INITIAL STEP (LINEAR CASE) m*_I_) ?ST
_60 FORMAT (2IS_2F10,0) 7SBqTO FORMAT (*OINITIAL GUESS TVPEl*eI_) 7YqqO0 FORMAT( OAlO)q81FORMAT(SI_} 760982 FORMAT(TIg) 76!q83 FORMAT(SEI_,_) Yb2
Tb$9qo FORMAT (,OIMAXeJMAX_NBDYeITEReI_ESeIDISKelWIR_IWINTI__I_FIN_IGED I* 76_
I,lOIS)
1000 FORMAT (,OIPLOTelPLTReNCOPY_LINWTI_LINWT2eNUMSR=NUM_RIe, 766|*I|KIPI,ISKIP_ 1*,9IS) ?_T
|0|0 FORMAT {*ONBSE_eNRSE_ I*_I_) ?b81010 FORMAT (*OR(t)_R(2)_R(])_YINFIN,AINFIN,XOINF,YOINF_NINF I*/?F15.8, 76qtlS)
?70|030 FORMAT (*OIEV,IAIT_R(IO) 1*_2IS.FIS,8) f?!tOaO FORM.AT (*OINFAC,INFACO I*,_IS) T7210§0 FORMAT (*08IZE_RATIO_DIST_XB|_XB2_yBI_yB_,_TFt2,8) 77_1060 FORMAT (IXetT(IM*),* INPUT *e]O(1M*)) ?7_1070 FORMAT (tX,t]O(tH,))
7751080 FORMAT (*OSIMPLYwCONNECTED REGION*)
C IT6
END 77T?T8TTe780
7O
8UHROUTTNE BNORY {X,y,IJ,TI,12,LBI_,R,A,XO_YO,_:TNF,N]
CC ********************* CIRCULAR OUTER BOUNDARY ***********************
C tt $ THIS ROLITINE CALCULATE8 NINF XeY COORDTNATE8 AROUND A CIRCLE OFC • RADIUS R AT EQUALLY 8PACED ANGULAR INCREHENT8 STARTING AT ANGLE AC * (POSITIVE COUNTERoCLOCKWISE FRO_ POSITIVE x-AXIS] ANO PROCEEDING
CLOCLOCK_ISE FROM THI8 &NGLEe
C *C i_,il**ililllli.i.,lllllll.li_i_lilil*_lil*illl'li**'lllli_ll**illll*
C_IMENSION X(Nol)w Y(N,1)DATA PI/_.tU1_9_65_91
KIsMIN0[II,I_)
KEmMAX0(II,12_ISENelIV (II.GT.IE) ISENI=I
IXMIININFDm.E.OtPI/FLOAT(IXMI}GO TO (I0,30,I0,$0], LSID
Ct*** BOTTOM _R TOP10 00 EO KIKteKE
ImIle(K-Kt]*IBENXtI,IJ)mR*COBtA)+XOy(I.IJ)IRI$1N(A}eYO
EO AskedGO TO SO
_0 O0 _0 KmKIoKE
C***t LEFT OR RIGHTJmlle(KmKl)ilBENX[IJ,J)mR*COS(A)+XOy(IJeJ]BRIBIN(A]eYO
_0 ksA÷OSO CONTINUE
RETURN
CEND
SUBROUTINE CORPLT (%,YwNDIM,NUNBR|P NUMBR'CI'C_pRATIOpBIZEwNCOPYwLIINWTI,LINWTEwlSKIPI,18KIPE,XPLOT,YPLOT.DISTeIPLTR'NPLT$tLBBIEO.LBI,LB2,LBOY_LRSID,LISID_LRt_L R_'LII_LI_'LTYP_'NBSEG_NRSEG_LISEG
$.JJMAX,IIMAX.XBIeXB_YBI_YB_]
CC **_ PLOT ROUTINE • PLOT8 COORDINATE 8YBTBM AND SEGMENT DIAGRAM e_mweC *C *********************************************************************
C DIMENSION LBBIO(1), LBt(1)e LBE(I]e LBOY(t), LRBIDCt]. LZ8IO(i]e LIRI(I), LRE(I), LIICI), LIE(t)e LTYPE(I]
DIMENSION X(NDIMel], Y(NDIMeI]_ XPLOT(I), YPLOT(I]* CI(I], C_(l)e
lkXISL(_)e XA(6), YA(6]OATA 8CAL /O.IIDATA 81 I0,08?S/DATA 8_ tO,l?S/
781?BE
'_8S786?ST?88789?eOTgt?BE793
79_T96?R??98TR980080180E80_BO_80S80_SOT80B809810811
817818819BEO
BEE8E$
B_SBEbBETSEB81$050
85785S
71
CCC
DATA HA 11,01DATA Hg /Oe5 /
C*****ISYm2T
JMAWmNUMBRIMAXINUW_R|
HlmO.5*81M2si.S*81_3m0.S,8_H_ml.S,S2HSmO.2S,8!H6m2.0,81H?ml.$,8)IMAXMImIMAX.I
CC JXl8 M_NIMuM 8
C
I0
2o
AN_ $CkLE FACTON$
CALL MAXMTN _XeIMAX,JMAX,NOIMoXMAX,X_IN,IXMXe3XMXeIX_NejXMNelSKIp |I,I8KIPE)
CALL MAXM)N (y. IMAX,J.AX,NDT_,yMAxeyMTN.IX.X,JXMX,IXHN,JXMN,IBKIPIIoISKIP_)
WMAXmAMINI(W_AX,XR2)XMINIA_AX|(XMIN,XR|)
YMAXIA_IN|(YMAX,yR2)
YMINIAMAX|(YM_N,YR|)
X||XMIN
Y|mYMIN
AXIBL(2)mBIZ[IF (RATIO) 10,10,20
Y2m(YMAX-YMIN)IAXIBL(2)X2mY)
AWISL(I)m(XMAX-XMIN)IX2GO TO 30AXISLCI)mAXISL(2)W2m(xMAx-XMIN)IAWISL(I)Y2I(YMAX-YMIN)IAXISL(2)
SET UP PLOTTER
30 CONTINUE
NPLT|I2
LAB_L$
IF(IPLTR,E@,I.OR,IPLTR,EG,_) CALL LTNwT@IPLTR,LINWTI)CALL PLOT (,5,,_,-3)
CALL 8YMBOL(O,,.OOIw,08?S_CI,90_,80)CALL 8YWBOL(,3e,OOI,,OS?Sec_,qo,eSO)CALL PLOT (1..0.e-$)CALL PLOT (0..9'99.|)CALL PLOT (.Se.Se-))
PLOT LINES OF CONSTANT ETA
IFCIPLTR,EQ,[,O_,IPLTR,EG,2) CALL LINWT(IPL_R_LINWy2)DO TO Jm)_MAX_ISKIP)
WmO
O0 _0 Iml,IMAX,18KIPl
IF (Y(I_J),OT,YSI,OR,Y(I_J),LT,YSI) GO TO (0
8_3
8_5
8_n
_53
85_
869_70
_728_3
87S
87q
88288388_8aS886
8_8889
8q]eq_8q_8q_8qT
_qqqO0
72
KIK÷IXPL_T(K)WX(I,J)YPLOT(K)aY(I,J)
_ CONTI_OExPLOT(Ke|)IX_XPLOTrK+E),X2yPL_TCKel)mYIyPLOT(KeE_sY2
50 CaLL LINE (XPLnTwYPLOT_KwI*O_Oe.OT}
PLOT LINE8 OF COhSTANT XI
b_
?0
On TO II],IMAXjISKIP]KmOOO 60 JmlwJMkX,lSKl p2
I; (X(I_J).GT,XBE,OR,W(IwJ},LT.WBI)IF (y(IeJ).GT,YBE,OR,Y(IpJ),LT,YBI)
xmKeIWPLDT(K)mX(I,J)yPLOT(K)mY(IpJ)CONTINUE
XPLOT(K_|}BX[XPLOT(Ke2}BX@yPLOT(K+_)mvlYPLOT(KeE)mY2
CaLL LINE (XPLOTpYPLOTpKRIpOeOp,O?)
CC SEGMENT CONFIGURATION nIAGRAMC
CALL PLOT (AWISL(1)*E,Ow[,O,-3)JMAXmJj_AXIMAXBIIMAX
CC****C
6nbO
BODY SEGMENT8 ****
00 100 LslwNBBEGOlBLBI(L)*SCALDEBLB2(L)i$CALIGOTOBLBBID(L)GO TO (&OegO_BOigO). IGOTOIf (LBBID(L),EQ,%} DSmBCAL
_OTTOM OR TOPIF (LBSlD(L},EQ,$) D]mJMAX*BCALLwTm-_I_ (L,GE,LIIEG) LWTI'I
IP(IPLTR.EQ.t.OR.IPLTR.EG.E}CALL PLOT (DtwO_,$)CALL PLOT (OioD]el)
IF(IPLIR.EQ,I,DR,IPLTR,EQ,E)IF (L88ID(L},EQ,I) MIwM_IF (LBBID(L),EQ,$) MuM208!MMIBIGN(MgwM)-_2t(I.O_BIGNHMM|BIGN(M]eM)CALL NUMBER (D[eMt_OSeM_B|
CAL.L LINWT(IPLTRwLWT}
CALL LIN_T{IPLTR_O)
(t,OeM)}*O.$
_FLOAT(LBt(L)},O.O_'t}_LOAT[LB_[L})eO,O_=t)
CALL LINWT(IPLTRe-I)CALL NUMBER (O|-Wt_O|eMell
IF[IPLTR,EQ'.|.OR.IPLTR.EQ.|}IF (L,LT.LIIE;] CALL NUMBER ((DteDE),O.S=M3,DSeMM.IEeFLOAT(LBDY(
IL)),O,O_e[)IF(LeGE,LIBEG)CALL BYMBOL((DteDI)_O_S-M$_DSeHM_B_IBY_OeO_=I]IF(IPLTR.EQ.i.OR.IPLTR.E_.2) CALL LINWT(IPLTR_O}
90190E905
905_0690?90P,q09OtO
915
915
911OrB9199Z.O
9E29;t$9Z_9ZS9Zb0/.?9Ze9Zq9ZO9],1
9],$
93SO$6
9]IB
9_0
9_Z
9_8
9St9'JE955
R_'J_Sb9S?eSe_)IH)_)bO
73 '_
CALL PLOT (_].D3-HHH_2) 96]
CALL PLOT (D2pD$+Hww,]) 962CALL PLOT (D2_D].Hw_w2 }" 963GO TO £00 ¢b.
C**** LEFT OR RIGHT 9bS
90 IF (LBBID(L).EQ.2) O]w$CAL oh6
IF (LRBTDCL).EQ._) _]II_AX,BCIL gb7LwTx-2 9_B
¢b9IF (L.GE.LIBEG) LWTm.[ g70
IF(IPLT_.EQ,I,OR,IPLTR.EQ,2) CJll LIN_T(IPLTRoL_T) g71CALL PLUT (D].Dle_) gT_CALL PLOT (_3_npw2) g75
IF(IPLTRoEg. I.OR,IPLT_mEQ,2) CALL LINwT(IPLT_a_ ) Q7QIF (LBBTD(L),EQ.2) HI-H6 Q75IF (LB$1D(L)mEQ,_) waN6-8! g7b
wMmBIGN(HTww)'B2*(|'O+$TGN(l'_'w))*n'l q77HMHmBIGN(MS,H) g7_CALL _IJMBE_ (D_W_DtwBIeFLDATCLBI(L_),O,O,.I) 979CALL NUMBEP (D)+MeDP#SIjFLDAT(LR_(L)),O,Op.)) q_O
IF(IPLTR,EO,lo0_eIPLT_mEQ,@) CALt. LIN*TtTPLT_.I) 081IF {L.LTmLIBEG ) CALL NUMBER (_3÷HH_tDI÷O_)IOm_$2,FI..OAI(LRDY(I.)) q_2
l,O.O_-l)
IF(L,_E.LIBE_)CALL 8Y_BOL(O_÷WM,{OI÷O_)*O.S_82elSy,O._.I) g_I_(IPLTR,EQ,I.O_,TPL, TP,EO._) C_LL LTN_T{IPLTR,O) Q8_
CALL PL_T (D]÷HHH.OI,_) gB&CILL PLOT (_]-HHH_|e2) getCALL PLOT (n]+wwH.02,_) _8_CALL PL_T (D]-wHM,D2_2)
100 CONTINUE 9BgC ¢90
Cw*** RE_ENT_NT 8EGMENT_ ,*** gg|C _92
IF (NRBEGoEQ,O] GD TO 260 g9_DO 2§0 LBI_NRBEG 995
DIILR_(L)*BCAL gg6D_mLR?(L)*BCAL 997
IF(IPLTR,EQ, I,OR,IPLTR,EQ,)) C_LL LIN*T(IPLTR_O) gIA
IGOTOmLRBIO(L) ggg
GO TO (liO,l_O,llOel_O), IGOT_ 1000C**** BOTTOM OR TOP - FIRST OF PAIR 1001
lln IF (LRBIDLL),EQ,I) n3mBCAL _00_
IF (LRBID(L),E_.]) DBmJM_Xe$CAL 100_CALL PLOT (01_D]_$) [OOQCALL PLOT (D2_D3_2) [OOSGO TO 13o
C_*_* LEFT OR RIGHT - FIRST OF PAIR 1006[007
120 IF (LMBIO(L),EQ,_) D3o$CAL 1008IF (LRBIO(L),EQ,_) D3BIMAXm$C_L 100g
CALL PLOT (D$_DI,)) 1010C_LL PLOT (0],02,2)
150 D_mLII(L),BCAL 1011101_
D_mLI_(L)_BCAL 1015IGOTOeLIBIDtL)
_0 TO (laOelSO,lOO,l_O)_ IGOTO 101_C,m** BOTTOM OR TOP - BECOND OF PAIR 1015
laO IF (LIBID(L).EG.I) D6mBC_L lOlb1017
IF (LI81OtL),EQ,3) D6BJM_X*BCAL 101B
CALL PLOT.(D_D_$) 101gCALL PLOT COS_O6_) 10_0
74
GO T_ lb0LEFT O_ RTG_T - _ECO_n OF PAI_
IF (LI$ID(L).EQ._) _blIMAX*SCALCILL PLQT (Dbm_Uw31CALL PLDT (nb,DS#2)
]hA TF(|PLTg,Eq,],OR,TPLTR,EQ,2) CALL L;N*T(IPLT_,_ 1
IGOTOILTYPE{L)GO TO (170_l_OpiQO,200e_lOw22_), IGOTe
C**** ONE 0_ BOTTOm, O_F _ TOPi?n XA(I)m(_t+_2)*O.S
yA(t)mn3Xi(2)mx*(t)yIc_)uVk(1)-H_XA(_)ITMAX*SCAL+HB
IF (2,0*D_,LT, FL_TfTM_x)*$CAL) XA(3)m-H8
¥A(_)NYA(_)
yA(Q)IJMAX_$CAL÷_8XAC_)uCD_÷D_)*O.5
xA(_)mX4(5)Yi(b)lO6
NAmbGO TO 23_
C**** _OT_ ON HOTTO_180 XA(l)m(nl*O2)*O.5
Yk(I)ID]
yA(_)IYAC1)-HQ
y_(3)lYk(_)
_kmQGO TO 2_0
C**** BOT_ ON TOPt_0 XA(1)m(nl+O_)*O,5
VA(1).O$XA(2)mXA([)Y_CZ)|VA(I)_HgXk(3)m(O_eO_)/O,_YI(_)IYA(_)
YA(_)mDb
N_N_
GO TO 2)0
C**** O_E ON LEFT_ ONE ON _IGHT200 X_(I)103
YA(|)I(D]_D2)*O,_
XA(E)mXA(t)-H8Y_(2)mYA(I)XA(])IXA(2)y_($)mJMAX*BCAL*H8IF (2.0,D2.LT.FLOAT(jH4X)*BC_L) YA())mm#8
XI({)mlMAXISCAL+_8
YA(_)mYA())
_PTTED LINES CON_EC TI_'G HE-E_THANT
lO)t
I0_$
I0_
IO_S
I0_61027
1028
I0_
I0$0
lO$tI0_
Io31
I05_
1015ln_6
10_7
1058tO)_
I0_0
I0_2
I0_1
lo_a
1o_6lO_T1o_8IOU_loso
1051I05_
IOS1105_1o_$
I056
I0_?1058
lOng1060lOet106_1061lOb_
lOb_1066
rob?
lOb_
lOb_
I070
I011107_1073107_
107_1076107?1018
10?_1080
75
CCCC
C
XA(S)mXA(_)
YiCS)mCO,tO_),n.SXACb)aDb
Y*(6)mvA(S)_Am6
GO TO E30
_OTH O_ LEFT
Xa(l)mO3
YA(1)I@OI÷_E),n.SXA(E)mXA(1).M9Y*C2)mYACI)Xi(_)mXA(2)
YAC$)m(O,÷DS),n._Xa(_)mO6
YA(_)mYA(3)
GO TO E$O
80Tw ON RIGHTXA(1)mO$
YaC1)m(_leO2)eO.5Xi_2)uXA(l)÷M9Va(E)uY,(1)XaC3)mXaT2)
YAC$)m(Do+D_).O.5XA(_)m06YAC,)mva($)NAN_
PLOT DOTTED LINES
230 CALL PLOT CXACI)wYA(1).))DO EaO NIE,NA
E,O CALL PLOT (XA(N),Va(N),E)250 CONTINUE
TERMINATION SEQUENCE
E_,nCALL PLOT ($CALiKL_JT(IMAX)_/.Oj.2.Se.3_CALL NFRAMERETUR_
END
SUBROUTINE LINWT(IPLTR,IPEN)
******************** SETS PLOT LINE wEIGHT **************************t
;0 TO (IO.EO). IPLTR10 CaLL LI_EWT(IPEN)
RETURN
EO CILL PENSL (IPEN)R_TURNEND
1081
I0_
1083I08.
In_b
lOeb
I087
1o88
108qlOgO
I0@IIOq_
I09,
Inq5log6
10qT
IOqM
I0991100
II01
IIoE1103
110_
1105
1106
1107
1108
11o_
1110
1111
111_
111_
Ilia1115
1117
111_
1119
112011E1
11EElIE3
lIESllEb
lIE?
11_
11]0
11311112
1133
11_
1156
11371138
11$0
11_0
76
_W_.CTIO_ E_ROW (v_IF,.vOLD.E_W_ILr. elFLDIJ FLmwlvER)
DTME_$IO_, TVEP(1)
_AXT_I_M Nnk _ OF ITEkIT_ C_A_GE ,**************m***
10
TIA_S{V_EW-V_Lb)
IF (T.LF.ERROL_I
IVER(1)uIFLn
IVE_(_)IJFLO
ERRO_IT
RETuR_F_WOwIEPWOL_
RETUW_
_o ?0 10
E_c)
C_
IE(IGE_.EQ.2.0N.IGE_.GE.51J,XwIIJWAX-|
I_XMIII_IX=I
C
C PROJECTION FROM BODY SEGMENTS
C
TGEBuO
Ie CIGES,NE,O) GO 10 )0TIIO,I*tIGESS-"]_RG_ITI*JMWM1
T_II,0/(EXP(ARG2)'I,O)ARG)ITI*IMWMI
TSII.O/(EX_(&RG$)-I.0)
DFACII,0neACCIl,OIK(IGED.EQ.$) DFACCIO.O
DO _0 JI_,JMWwtIF(IGES$,GE,5) GO TOFACIFLO&TCJ-I)/FLOATCjMXMt)
GO TO ?
_RGtITt*(J-t)FACI(EXP(ARGI]-I,O)=TZCONTINU_
DO 10 II_IMXMIT_(IGES$,GE,S) GO TO 8FkCCIFLOATCI-I}/FLOkT(INXMI)
II_7
115n
1151I15_
1153
1155
115_
tlS?
1159
1160
11_1
11_2
11_
1166
1167
11bq
1170
1171
117_1175
117_
117511761177
117q118011811182
1185
118=
118611BT118B
llBq
11901191
1195
11qS
11qq1_00
77
10ZO
GO TO 98 ARGImTI*(I=I)
FACCI(EXP(kRGI)-I.O].T3g CONTINUE
X(ZeJ)m(X(Zel)e(XCZejMAX)=X(Zel))*FAC)*_FACe(CX(ZM&XeJ).X(Iej)l)*FJCCeX(teJ))wOFJCC
Y(ZwJ)m(Y(]et)etY(ZeJMAX)'Y(Ie]))*FACI*DFkCe(/y(]MAXej)=y(|wj)11*FACC+Y(|wJ])*OFkCC
CF (XGE$8,EO,R,OR.ZGESS,GE,S) GO TO 1_]F(DFACzEGeO*OmORmDFk_CmEO.O,O) GO TO 1_
X(%.J)mO.S.X(r.j)Y(IeJ)aO.S*Y(ZwJ)CONT|NUE
COkTZNUE
RE-ENTRkNT SEGMENT8
30 ZF (hR$EG.EO.O) GO TO t,OO0 15n LBL._RSEG
ZloLRI(L)tZmLRZ[L)ZlisZt+iZZE=Z2-iZGOTOaLRSZD(L)GO TO (_OebOe_OebO)w IGOTO
"0 re (LRSID(L).EO.t) JzlIF (LRSYD(L),EO,3) JzJ_kXOXmX(IE,J]=X(I1,J)OVlYtlZeJ]-V(IteJ)OXm_X/CIZ-%[]OYNDY/(12*II)
DO SO ]mll1,11ZDDXm(I-II],DXODYm(I*II)*DY
X(leJ)mX(llwJ]+ODXSO Y(I,J)IY(II,J),ODY
GO TO 80
60 IF (LRSXD(L).EO.2] ZmlIF (LRSID(L)mEQ._] TmZHAXDWmX(I,IZ)-W(I,II)DYmY(I,Ii)mY(I,II)OXIDX/(II-II)DYIDYI(_-X[)O0 70 JslIt,lt2
DDXm(J-II),OXDDYm(J=Zt)*DYXtI,J)mX(Z_It)÷O_X
70 Y(leJ)IV(Iolt)÷OOV80 IImLII(L)
I)mLII(L)lllmll+!IIi=IE-IIGOTOeLI|ID(L)GO TO (90oltOjgOwllO)e IGOTO
90 IF (L_$IDCL).Eg.I) JetIF (LI$_DCL).EG.3] JIJ_X
OX|DX#(_-]I)OYIDY#(]_||)
1_0112o2lZn3120_12051206lEO?12081209121n121112121213
1215121_1217121812191220
12221223122_12Z_122a12_7
1_291250121112321233123_1215
12371238123_12,012=112_E
12_5lZa_12_?
12So12S11_§2IZS$
12SS
1_$8I_So1260
78
LINEAN PROJECTTO_ o IGE881
%F (IGES.NE.I) GO TO 170OFiCl|,OOFACCIi.O%e[%GEO,Eg.t] OekCCmO.O]V(TGEO,EQ,2) DFACIO.ODO lbO JxEeJMXM!
VACxVLO[T(j-i)/FLOkT(JHXM1]DO 150 %mE,%MXM!
FACCIFLOATC]-|)tFLOAT(]MX_I)X(I,j)x(XIT_tlefXIT,JMAX_=Xfl,I)),KkCI*DFAC*(_X(IHAXJJI=X(I,J]
I),FACC_X(twJ))*OFkCCy([eJIBIY(Iel)e(Y(IeJM&XI=Y¢I_I_] *FAC)*DVAC+((YflHAX°JI.Y(IeJ)
t],VACCeY(IpJ]]*DFACC%V(DVACoOT,O,OoOR,DFACC,GT,O.O) GO TO 1_0
x(%,J)mO,5*X(T,J]Y(%eJ]lO,_*Y(TwJ)
150 CONTTNUE160 CONTINUE
RETURN
NOMENT OR EXPONENTTAL PROJECTTON
1?o BNII, CIMAX÷JMAX)-_DNm$ORT(FLOAT((%_AX-I]**E*(J_AX-I]**2]5OMmtTABB(%GE$])/OMDO ZIO Jm|,JMXHt
O0 _|0 II2o_XH|$XBO_O|YmO,O|OlO.O8XOlO,OIYDIO,OIXNIO_OIf_lO,OtE_O_ODO 180 TIml,IMkX
|XllX+X(%Iel)eX[%_eJ"kX)|YISYeY(Zlel]eY(l_.J"AX)
lee!126212b3IEb_126_1_6612671268IR_q1270I_71I_721273IE7_1275127_127T1E7BIE7e1E801281128_1283IE8U1285128_128T128812891290
12_212g_12q_129S
l_gT
I$00
15o_13o3I]0_130513061_071308150_1$10I$II
1312I]I_
13151316
1318131_13_0
79
180
lqO
DmIAB$CTI=I)I)ImTAB$(1-J)DEmlAB8CJM&X.J)OmD**ZOInSORTCD+DI**2)D2mS_RT(O_D_**2)_OnSDeDI÷D_SXDsSXDeX{TIet).DIeX(II_JMAX)._2SYOaSYOeYtIIe|)*()|÷Y(TIeJMjX).D_EImEXP(-DIeDM]EEuEXP[-DE*OM)SXMISXM÷X{II,I)*EIeXCIIpj_AW).E28YMBSYMeY(IIel}*EI÷Y(IIwjMAW),E?9EBSEeEtsE_CONTINUE
D0 190 JJn2eJMX_!8XmSXeX(teJJ)÷X(IMAXeJj)SYeSYeY(toJJ),YCIMkX_JJ)DmIABSCJJ=J)DIuIABS(I-I)D2eIIO$(IMAX-I}OLD**201mSQRTtOeDI,mE)O_mSQRT(Oe_2**2]SDISDeDIeO_
$X0mSXDeX[lejJ]*0teX(IMJX,jJ}.D28YOmBYDeY(I.JJ)*OI÷Y(IMAX.JJ),O?EIBEXP(eDI,DM)E2|EXP(wD2,0M)
SXMISXMeW(IoJJ)eEIeW(IMAXeJJ)tE)8YMISYMeY(|oJJ),EIeY(IMAX.JJ),E_SEB$EeEI÷E2CONTINUE
IF (IGESoEQ.3) GO TO ZOO
EXPONENTIAL PROJECTION - IGE$cO
TImI.O/$Ew(I.J)mTI*SxMY(IoJ)oTI*gYM
GO TO 210200 CONTINUE
ZIO
MOMENT PROJECTION - IGES= ] OR .
TIIt.O/BDIF (IGES.EG.$) TImTI/8N
X(I.J)u($D*SW-BXO)*TIY(IeJ]O(BO*SY=BYD]tTI
CONTINUERETURN
END
|UBROUTINE MAXMIN (XBIMAX.JMAXeNOIW.XMIWeXMIN.INXoJMXeIMNeJMNwIIKIteleIlwIPl)
132Z1323132.
13E613_7
13291330
1332
1533
1_3513_613371358133_13.0
13.313,,13.S13.613.713_813.9
135O135!13621353135"13551356135?13§8135913e01361
1363136_13bS13b_I$b7
13b_13701171137_1375137,I]?S
1377I$?|l]?e1300
80
c **** TwI$ SU_ROoTIN_ CALCSJLATES TM_ _XI_;_ A_ MINIMUM VALUFS ******C **** (IF A T*O-_IWENSTnNAL DATA A_RAY. ****_*C *C , T_PLI? DATAI
C *C * Xm2-D _TA ARPAY _OSF MAXIMUM & _TNIMU_ IS T_ BE O£TERwINEnC • I_AXmLARG_SI VALIJF or ;IRST SUBSCRIPT nF X TO RE SCANNED
£ • JMAXsLAR_ES? VALUE _r SECO_JO SUR$CRIPT _F X 70 BF SCANNEDC * NDIMmIST DIME_$IOK' OK X
C * ISKIPIaSKIP PARAMETFR FOR |ST INDEX OF X (?ME l INDEX)
C * ISKIP2mSKIP PARAMETER FOR _ND I_OFX OF X (THE J IN_[X)
C .• OUTPUT OATAI
C *• _AXmMAXI_UM OF X ARRAY
C * I_X,J_XmI,J L(}CATIP_ OF W-AXC * W_TNmMINI_U, OF X ARRAY
C * I_N_J_kmI.J LOCATT_ OF X.INC *C *********************************************************************
10
X,AXsX(I.t)XMINIX(_I)I_XI|J_XI|
J_NBIDO 20 II|_IMAX_IBKIP|
O_ 20 jm]_JMAX,ISKI_I
IF (W(I_J).LT.XMAW) GO TO I0X"_XmXCI.J)IMXm|
JMXmJIF (X(I.J).GT.XMIN) GO TO 20X,INmW(I_J)IMNmlJMNmJCONTINUE
RETURN
ENO
SUBROUTINE _ARA (WWI_YXI,XETA_YETA,LACC,CFAC_RETA,RWI,CBI_CSJ,_ACCI.R_TeACCL.ACCLI.NDIM.I.J.I"ER.TACC_IEV)
**** C_EFFICIENT| AND VARIABLE ACCELERATION PARA_ETERB ON ************** RE-ENTRANT 8.EGMENT8 ********_*
DIMENSION RETA(NOIM,I)_ RWI(NDIW,I)_ *ACC(NDIM_I}_ R(|)_ T(|)DIWENSION TACC(NDIM_t)DIMEN810N IWER(1)REAL JACBS,JBAGINTEGER TACt
l_SE
1_861387118RI_891590
15_3I$9_119S1596
I$9_15_91_00
I_02l_O_l.Oa
I_06I_O?I_O8ISO9I_I0l_ll
I_I_I_I_I.IS
I_EO
1_1_$1_
l_mI_?la_8
la$ola$ll_1_33
1_$8I_
I_0
BI
LOGICAL LACCCtt**t
UACC[Rj)nE.OICI.O,RqRT(I.0÷RJ**E))OJCC(RJ)m2.01(I.0÷$QRT(I.neRj**E]]ALFAsXETA**E+YETA**_GA_ImXX_**2*Y_**EAGI].O/(ALFA÷GAMA)JAC_8=(XXI*YETA-XETA*YXI)*tE*CFACJSAGmJAC_8*AG*O.O_2S
VARIABLE ACCELERATIO_ PARA,ETER
YF (LACC] GO TU t_OTE_IaJACBS*O.12581|-TEMI*RXICYwJ)_BeTEMI*RETI[Zej)81siSB(Bl]REmABB(R2]TIBALFA*QE-BI**2T2mGAHA**2-B2**2*TIm*B$(Tt)ATZmABS(T2]IF (TI,_EaOIO.AND.T_.GE,O.O) _0 TO BO_F (Tt,LT,O,O.AND.T_,LT._,O] GO Tn 90RJtBSGRTCAT1)*CSI*kGIF (TI,GE,O,O] GO TO IOTEMPwlmUkCC(RJI)TACC(I,J)IIOGO TO 20
10 TENP_lsOACC(RJ1)TACC(I.J)a20
EO RJEBSQRT(ATE)*CSJ_AGIF (TE.GE.O._) GO TO SnT_MPwESUACC(RJ2]TACC(I,J)BTACC(I,J),IGO TO _0
_0 T_MCwEBOkCC(RJE)TACC(IwJ)mTACC(I,J)÷E
O0 IF (IEV) $0,60e?0SO TEMPwmUACC($QRT(RjI**E_RJE_wE])
GO TO 10060 TEH_m(Rj|eTEMPW_e_J_eTEHP_E)I(RJ_eRJ2)
GO TO 100?0 TEMPWmO_CC($ORT(RJI**EeRjE**E))
GO TO 1o080 RJB(8_RT(ATI),CS_,SORT(kT_)_C_J_wk_
TEHP_BOACC[Rj)T_CCCI_J)m_GO TO 100
90 RJ_(8_RT(ATI)*CST,8_RT(AT_)*CSJ]*AGTEMPwmUaCC(RJ)T_CCCI_J)_!
I00 R(I|)mERROR(TEMP**w_CC(I.J].R(I_).I.J.IWER)wACC(_#j]_TEMP_
|10 CONTINUE
COEFFICIENTS
T(S]wJSkG*RETA(Z,J]T(O]mJ8kG*RXI(I,J]
I..1I..2
I_
la.9
Ia521.53
1.551.561.57l_Sm
l_eO
I.62I.65Ia6.
1.6bl.b7
1_69
I_711,72
l_T?I_78
I_80
1.821.8]
10851_86
1.881_89laeO1.91lace
l._O
l,e?10981O991500
82
T(L)B,5*ALFA*AGT(2)B,5*_AMA*AGT_3)I,25*(XXI*XETA+YXT*YETA)*AGA_CLB*ACC(ItJ)*CCLImI,0oACCLR(12)RR(12)+ACCLRFTURN
I DPT(IOO)w IPT{|O0], JRT(IO0)INTEGER XIwETA,BLKoATYPDATA ZE_O/I,_E-tn/DkTA XI,ETAeBLKI6HXI o6HET&
C*****IF _NLN÷NPT,EQ,O) RETURN*RITE (6,200)IF {ITYP,EQ,O) *RITE (bw2_O)
IF (ITYP,N[,O) wRITE (6e270)IF (ATYPoEQ,ETA) wRITE (6o280]IF (ATYP,EQ,XI) *RITE (6_Z90)
SET (_P JMPLITUDE AND OECAY FACTOR
IF (NLN,NE,O) REaD (5,210)IF (NPT,NE,O] R_AO (S,220]IF (DEC,LT,ZERO) GO TO _0IF (NLN,EO,O) GO TO _0OO |O LBIwNLN
10 OLN(L)IOEC
20 IF (NPT,_O,O) GO TO _0DO 30 LBI,NPT
]O D_T(L)BDEC_0 CONTINU_
IF {NLN,EQ,O) GO TO 60DO 50 LmI_NLN
50 _LN(L)B_LN(L)*IMPF_C
_0 I_ (NPT,[Q,O) ;0 TO _0DO TO LBI_NPT
70 a_T(L]ma_TCL),aMPFAC80 CONTINUE
IF (NLN,N[,O) _RITE (6,_0)IF (NPT,NE,O) _RITE (6,_0)
GO TO _o
C_LCUL_T[ INHOMOGENEOU_ TEEM
DO 190 I|I_IMAXDO 190 JmI,JMAX
(JLN(L}_kLN(.L),OLN(L)_LB%,NL N)(I_TtL},JPT(L),_PT{L)e_T(L)_LB|,NPT)
(jLN(L),kLN(L},OLN(L),LBI_NLN)
150115021_u315o_15o5150615o?150815091510151115121513151_151515161517
1519I_O1521IS_15_$
1525152615_71528IS_153015$11532IS_$
153a153515_6IS)?15}8155_1_015_1
15_$
15w515_615_?
15501551
155_1556
155_155_1560
83
9OC****
1uo150
POINTIFDO
160I70l_n190
CC PRINTC
IFC*****
RETURNC
ATT_ACTINNIF (ATYP,EQ,ETA] IjIJIF (ATYP,EQ,xI) IJmlT2m_mOIF (NLNmEOmO _ GO TO 100O0 90 LBIaNLN
TmAL_(L.]*EXP(-_LN(L)*IaHSfIJ-JLN(I ]5)IF (ITYP.NE._) G_ TO 90TmT*ISIG_(I,IJ-JL_J(L))
IF (IJ.EQ.JL_(L)) Tmh,O
T_mT_-TATTRACTION(NPTmEQ.O) GO TO I_0170 LsI,_JPT
ItmIPT{L)IF (ATYPaN_oETA) G[} TD t=nIlaJ-JPT(L)GO TO 150IImI-IRT(L)18NuISIG_'(I,II)IF (II.EQ.O) I$_.mOTe(I-IPT(L))**2÷(J-JPT(L))**2TxAPT(L}*EXP(-OPT(L)mSQRT(T))IF (ITYPaNEoh) GO TO 160TmT*IBN
IF (I._QoIPTCL)oANDoJaE_.JPT(L)} TxO.OT2mT2-TCONTINUE
RETA(IjJ)eT_CONTINUE
INHOUOGENEOUS TERM
(IRIT.NE.O) WRIT[ (_,250) {(I,J,RETA{I,j),TBI,IMAX),JmI,jMAX)
_00 FORMAT (*OEXPONENTTAL DECAY RMS.]210 FORMAT (I10,2FIOofi)220 FORMAT (2IS,IFIO.O)2]0 FORMAT (IOATTRACTION LINES*IIQW,*J*oITX,*AMP*olSW,*DECAYmt(ISe2F2O
1.8]]
2_0 FORMJT (tOATTRACTION POINTS*t/QXe*I*_X,*J*,I?XwtAMP*_ISX,*DE_AY*e1/(215,_F20,8))
_SO FORMAT (*O_HS*//b(]X,*I*,3X,*J*.6X,*R[TA -*)/(6(2I_,1E10,_* -*)))_&O FORMAT (* - ATTRACTION -*)
_?0 FORMAT (* - ATTRACTION TO CONVEY _IOE, RE_UL$10N TO CONCAVE -*)_80 FORMAT (*0--. ETa EQUATION RH8 e-at)290 FORMAT (*0--- Xl E_UATION RH8 -.-i)
END
SUBROUTINE 80R (ReX,Y,IMAWeJMAX.ITERelXER,IYEReIENO,NOIMe$wEReNBIDY_ACC,RETA,RXI,LRI,LR_,LIt,LI_,LTYP[,NRIEG_IWIR,IAIT,INRAC_INF_C_O_NBSEG_LIIEG,LBI,LB2,LBSIDeTACC_IEV_IOISK)
15o]1562
15o315h_150_15e6I_b7156_156915701571IS72157_157"15751576IS77IS7_157_15_0158115_215_3158_
15861S8T15_158915RO
159115_21595159.15_5159615RT159_159016001601160_1605160_160_16061607Ib00160_1610161116121615161_16151616161TI&18I&1916_0
84
* LOGICAL COe, T_(_I_S I LACC - VArIAbLE ACCELFWAIIn_ PaRAUETER rilL n
. CO_veRG_ (TWRFLEVA_T IF LCONmT_LIE)
.
* LCrAC - AnDITTO_ eF I_Fn_nGE_r_i!$ TER_ COMPLETEDw
, L_C_ _ - CO_ST_T ACCELFkATIO_ PARAMETER.
* ACCEL_WATI_ _ PA_AmETE_ TYPE l I - ROT_ _V T_AGINARY {U_J_R_RFLAX)
* 2 - @OT_ EV REAL (OVER-RELAX)* 12 - XI EV T_AGINAWYw ETA EV REAL
* 21 - Xl Ev REAL, ETA _v I_AGINARY
nTwENSIO_ -ACC(NDT_,I), RETA(_DIM_I),DT_SIDN LRI(%]_ LR_(1)_ LBSID(1)
DOUBLE PRECISION AG
I_TEG_R ATYP,_TA,XI,TICCREAL JSAG,J_CR_
LOGICAL LACC,LCFAC,LCON_ATA XI,_?A/_Xl ,_wETA .'
n,TA PII$,I.ISq?_53591
IXFR(1), IY_R(11. T(5), LRI(
RXIt_DI_,i), TACC(_nIM,I)
_ACK(X,T,I,jeTI,12,JI,JE,A_AI_TE,T_]mAI*W(I,J)÷**(T(1)*(X(12,.I)ex(
111,J))+T(_)*(X(I,J_)÷X(T,JI))-T{3_*(X{I_,32)-X(12.JI)+X{II,JI)-X(I
21,J2))÷T(S)*T2÷T{_)*T1)
M*CKECW,T,I,J,II.I_,JI,J_,A,AI,T_.TI)mA_*X(T,J)÷A*(T(1)*CX(12,J)_XI(II,J))_T(?)*tW(I,J_)÷X(JI_JE))-Tt$1*tW(I;,JE)-X(JI-I,JE)÷X(JI+I,J
MACW3(X_T_I,J,II,T2,jI,j2_A,AI_T2.TI)mAI*X(I,J)_A*(T(q)*(X(12,J)÷WI{I$,J))*T(E)*(XfJ_,J_)÷X(I,JE)}-T(3)*(WtJI-I,J_)-X(I_,J2)*X(It,J_)
_-X(Jt*I,J2))+TCS)*T_+T(_)*T_)
MACK,(X,T,I,J,II,I_,JI,J2,A,AI_TE_TI)mAI*X(I,J)÷A*CT(1)*(X(12_J)÷X
t(I_,II))÷T(2)*(W(I,J2)+X(T,JI))-T(])*(W(I_,JE)-W(IE,JI)_X(I_,II.I)
I-X(I_,II-I))*T(S]*T2eT(_)*TI)
_ACKSCX,T,I,J,II,IE,JI,JE,A,At_TE,TI}mA_*X(I,J)÷A*(T(1)*(X(IE,II)÷
IX(12,J))÷T(2)*(X(I,J2)÷XtI,JI))-TCI)*(xfII,TI-I)-X(12,11*I)_X(IE_JEI)-X(12,J2))_TCS)*TE÷T(_)*TI)
UACC(RJ)mE.O/(I.0*SQRT(I.0÷Rj**2))
OACC(RJ)wE,O/(I,0÷SGRT(1,0-RJ**E))C*****
WRITE (6,630)
C
C READ COORDINATE ATTkACTIDN PARAMETER8 AND CALCULATEC INkOMOGENEOU8 TERMC
KOBIREA_ (§,610) ATYP,ITYP_NLNeN_T#DEC,AMPFAC
IF (ATY_.EG.ETA) CALL RH8 (R_IMAX_jMAW_NDTM_NLN_N¢T_ATYP_ITYP_D[CeIAMPFAC,_ETA]I_ (ATYPeEQ.XI) CALL RH8 (R_IMAXeJMAX,NOIMeNLNeNPTeATYPeITYPeDECeA
IM_FACeRXI)
READ (5_610) ATYP,ITYPeNLN_NPT_EC_AMPFACIF (ATYP,EU,ETA] CALL RH8 (R,IMAX_JMAX_N_M_NLW_NPT_ATYP_ITYP_O_C_
IAMPFAC,RETA)
IF (ATYP.EO.XI) CALL RHa (ReIMAXeJMAX,NOIMeNLN#NPTeATYP_ITYPeDECeA
16E2
16E3
IbES
16E6
16_?162816_9
16_0
163316_
1635
1636
163?
I_3_
16_o
16QI
1a_2
IM_
16_516_616.7
16_8Ib_q
16501651
165E1653
165a1655
165616571658
165.
1660
1661
166_
1665166_
1665
1666
1667
1668
166916701671
167E
1673167_1675I676
16T?1678
16791680
CCC
1MPFICwBXI)
_EA_ (_p_qO) TFACpI_ITeEFAC
TNITXAL SETUP
IF(R(I).GToO,O) GO T_ _0IF (I_V) lOe_nl30
10 _RITE (6_6_0)GO TO _0
20 _RITE (6,650)GO TO _0
$0 _RITE (6,660)_0 CONTINUE
IF tIDIRK,EQ.2,0R,IDISK.EQ,3) GO TO 50;0 70 6O
Se REA_ (116720) LkEC,LCONwCFAC,LCFAC,IWCC.I!_,INCOONeCINFACp_eIEV,IFkC
IeEeAC,jWXMI,IMXMt,JMXPI,CSImCSJ,NPT,I_FkE,_INF,IENDREAD (11,730) tR(I)el=l,t])
REkD (It,7]O)((Xtloj),V(I,J),RVI(IpJ),_[Ik(I,J],wACC(Xpj),ymt,iMAlX),Jlt,JM&X)RE*IND tl
KOIK_|
IF (IENDeNE,3) G_ TO 100KOstGO ?0 80
60 le (IFACoEO.O) IFACIIWRITE (6ekO0) IFACeEFACEFACmAMINI(R(2),_($)),EFAC
IF {IFJC.GT.O) C_AC=I.O/_LOAT(2**(IFAC.i}}Ie (IKAC.LT.O) CwACmI.O/A_8(WLOAI('IW,C))W_ITE (6,670) CF_CIRCOUN_I
IF (IWR.GT.O) *RITE (66570)JMXMIIJM_X.IIMXMIIIMAXB_
JMXPIBJM_X÷I
CSJBCOS(PI/FLO_T(JMXM1))CSImCO$(PIIFLOkT(IMXMt))NPTm(IMAX_)wCJM_W._}
IF (NRSEG.EQ.O) GO ?0 7_
O0 70 LIIeNR_Ee70 NPTBNPTsLR_(L)-LRt(L).!7S IF (INFAC.Ee,O) INFACII
IF (INF&C.GT.O) CINFACBIeO/FLOAT(2mI(yNFAC.|))
IF (INFAC.LT.O) CINFACmFLOAT(INKACO)IAS_(FLOkT(INF,C))wRITE {6,710} CINeACINCOUNIt
IF (INFIC.L7.O} INCOUNmINFACOEINPmEFIC
SET UP ITERkTION
LCFACI,FALSE_I_ (IABS(IF_C).LE.I) LCFACI.TRUE.DO 90 ImI_IMAX
DO _0 Jlt,JM_X_0 *_CC(IeJ)=l,0
w_lTE (6,6_0)
1b8116821683168a168S166616871688168Q16901691
1_921693169U
1696
1697169_
169917001701
17021705
170_170S
1706
1707
170817UQ
1710171!
171_
1713
171_
171S
171617171718
1719
17_0
17_1
17_2
17_
17_
17_51726
17_7
17_8
17_9
1710
17]_
175_
173_175S
17]_17371738
1759
17#0
86
80m YTERATZO_
100 IEh_sODO _50 XlXOwIT{R
XF (K.LE,2) LACCm.TRUE,IF {KmNEm_} GO TO I_0
LICC1,FALSE,7F (LCON) LACCtmTRLJE,
TF_.R(1)DO 11o yul,I_aX
O0 110 jm1,J_aX
I10 *aCC(I,J)=TE_120 R(_)m_,O
R(5)a_,OR(11)m0,0
R(1_)m0,0
C
C**** VTFLOC
C##WW
O0 E60 JJIE.jMXM!
JIJMXHIwjJ+2J_lIJ=tJPlmJ+lDO 260 ITII_,_HXMI
III_XHI-III+2
IPlmI+1
IF [I,G7,1) GO TO 130%_IzTMXMI
GO TO I_0
130 I_III=!
I_0 XXIIX(IPI,J)=X(T_I,J)YxIIY(IPI,J)-Y(IMIoJ)
XETAmX(IpJPl)=W(I,J"I)
Y[TAmY(IwJPI)°Y(I,JMI)
ALWAmXETA**)+Y[TI**)GJ_AmXXI**E+YII**E
AGIi,Ot(ALFAeGAMA)JACBSm(XXI,YETA=XETA*YXT)**Z*CFACjSAGmJACBB*AG*O,O6ZS
VARIABLE ACCELERATION PARAHETERIF (LACC] GO ?0 _)0TEHIIJACB8*O,I_SBIImT_I*RXI(_J)BEI-TEHI*RETk(I.J)BI_ABI(BI)
B_mABS(BE)Tlm_LFA¢*2-BI**2T)IGAM&i*2-BEiIEkTIBABB(Tf)ATIIAB$(TE)
IF (TI,GE,O,O,_ND,T_,G[,O_O] GO TqIF (TI.LT.O.OmAND.T_..LT.O,O) GO T_RJIII_RT(AT1)*C$%*AGIF (TI.GE.O.O) GO TO 1SOTE_P_I_UACC(RJI]TACC(I,J)_IOGO TO lbO
150 TE_PWtIOACC(RJi)7kCC[l,J)_ZO
1_0 RJEIiQRTCAT_)*CSJ*AG
EZO
17_I17_E17_$
17_517_617_711"817_91750175117_E1755175_
17_S
17S6
1757
1758I?S_1760
17611762_763176_
17e5
1766
1767
176817691770
1771
177_1773
177_177S17761777
177g
1779
17801781
178E1783178_17aS17861787
17_817891790
1791179_
1793
179_179S1791
17_7179817991800
87
270
180lqO
EO0
EIO
E3O
_ao
_so
Z60¢C***,C
IF [TE.GE.n.O) _0 TO i?OTEMP*Emt.ACC(R,YE)TaCC(XoJ)uTACC(T,J).!GO TO 180TEMPw2uOACCCRJ2)TACC(IwJ)ITACCCItJ)+E
XF (X[V) 190.EOOpElOTF_PwmUACC($ORTCRJI**2+RJ2**_I)GO TO E_O
TE_Pwm(RJL*TEMPw1+RJ_ITE_P_)I(_jI+Rj_)GO TO E_OTEMPwmOACC(8_RT(RJ|**_eRj_**_))GO TO 2_0
RJm(SORT(ATI)*CSX$$ORT(ATP),CSJI,AGTEMP_IOACCCRJ)
TACC(I,J)u_GO TO E_ORJB(SORTCAT|)tESZeSQNT(AT_),C_J),AG
TE_P_uUACC(RJ]TACCCX,J)ml
R(I|)aERROR(TE_Pwe_ACCCIeJ)eRCI|)._.J,XwER)WACC[XeJ)mTE_P_CONTINUF
ITERATET(S)mJBAGtRETACTpj)T(_)wJ8AG*RXI(_,J)T(I]m,S*_LFA*AGT(_)mtStG_k.AGTf_)te_(XXZwXETAeYXTwYETA)wAG
ACCLmXACC(I.J)_CCLImI,O-ACCLR(I))mR(IE)+ACCL
TEMPXNHACK(X_T.I_J._MI.IPi,J_I.JPt,ACCL.ACCLI_XFTA.XXI)TEHPYIHACK(Y_T.I_J.IHt. IPieJ_i,JPI.ACCL.ACCL_.VETA.YXI)R(_)IERROR(TEMPX_X(Z_J)_Rf_)_I_J_TXER)R(S)mEHROR(T[MPY_Y(ZeJ),R())_T#J,)Y(R)X(I,J)mTEM_XY(Z.J)ITEMPY
IF (XtGT.I) GO TO 2_0X(_MAXej]mTEMPXY(IMAX,J)mTEMPYCONTINUE
REENTRANT SEGMENT8 ****
ONE
IF (NRSEGs[O.O) GO TO _00DO SqO LNIeNR|EG
IIoLRI(L)÷tX_|LRZ[L)-[ISmLXt(L}etXamLX=(L}=tIGOTO_LTYPE(L)GO TO (_?O.EqO,$tO.$30,3$O.$TO)_ON BOTTOM. ONE ON TOP00 180 JJ_Xt_IJZaX_-JJ+X!
XXIIX(IeI.I)*X(I-I_|)YXZtY(I#|et)eY(]e|,|)XETk_X(_e|]*X(Z.JMXMI]YETAEY(_)oY(|,JMXM|)
XGOTO
l_ot18o_1803180_1805180b18071_08tSOg18101811
1813
181S181b18171_18181q1_dO
18E218E3
18_S18E6teE?18E8182q18301831183E18_$183_t8$S183618371838183g18_0
18_E18_3184_18_S18_b18_T18_818_q1850
185_1853185_IBS5
18S?18S818Sg1860
88
28o
290
300
C****3t0
SEo
c**** ONE
330
CALL PAPA (XXT,YXI,XETA,YETA,LACC,CFACp_FTA,_XIoCSI,CSjo_ACC 10b1t_.T.ACCL.ACCLt._IM.T.I._FReTaCCeIEV) 1092
tE_PXm_ACK(XaT,I_I,YoI,I÷I,J_XMlmPpiCCL,i_CLImXFTA,XXI) 1063TE_PYm_ACK(Y,ToTptwT-t;T_Iwj_X_|_wACCL,ACCLIJYETapVX_) 186_
_(S}xE_RO_tTEMPVeY(_,I)e_f_)eT;I,TYF_) 1866xtI_I)mTE_PX 10_TY(I_I)zTE_PY IRbO
X(_,J_AX)mIF_Px I_b9Y(I.J_AX)BTE_PV 1070CO_TINuE 1071
GO ?_ }90 1072ROT_ _ _nTTO_ 1_7_
_ 300 JJmIlo_2 107__mI_-JJ÷I! 107S
ITmT_-(_-II) 187b
XXIaX(?+I.t)-X(I-t.t] 1077YXIIY{_÷I,1)-Y(?-Iet) 107_XETAmX(I,2)-xtI_2) 107_
YETAuY(_,_)-Y{II,2) 1800
CILL PAPA CXX_,YX|,XETA,YETA,LACC,C_aC,_ETA,_XI,C$I,CSJvwA_C 10811._.TeACCL_ACCLI._DIM.TeteT_F_eTICC_EV) 1082
TE_PXIHACK2_X.T.YeleZ-IeI÷teI_.2.ACCL°aCCLI.XFT_eXXI) 108]TE_PYI_ACK_CY,T,I,I,I'IeI÷I,II_,ACCL,ACCLI,YETA_¥XI) 188__(_)zE_RO_CTEMPX.XCZel)_Rf_).I.I.ZX_) 180_
_(S)*E_ROR(TEHP¥,Y(T,1),_tS)e_.t,TYER1 1086X(IeI)wTE_PX 1887Y(Z.I)xTE_PY 1000X(II_I)mT[_PX 188qY(II,I)NTE_PY 1090CO_TI_U[ 1891
GO TO _90 189__OT_ ON TOP 1093
oO $20 JJsZt,12 189_
IsI2=JJ÷It 1095
IIBI_-(Z-II] 1896XXIIX(_.I,JMAX)-X(I-I,JMAX) 1897YXIxY(I*I,J_AX)-Y(I-I,J_AX) 1896XET/IX(II_J_XNI)-X(_.J_X_t) 1099YETAIY(II.jHXMt)=Y(I,JMX_I) 1900
CALL _ARA (XXI,YXIeXETAeYETA,LACC.CFAC,qETA_RXIeCBI,CSJewACC 19011,_,7,1CCL,ACCLI,NOIN,I,J_AX.IWE_eTACC,IEV} 190Z
TEMPXIHACK$(X,T,|,IMAX_I_IeY_t,_,JqXMt_ACCL,ACCLI_XETA,XXI) 1905TE_PYI_ACK$tY,T,I,I_AX,I*{,I*I,T_,J_XMI,ACCL,ACCLt_YETA_YXI) 190__(_)IERROR(TEMPX,X(IeJ_AX).R(_).I.JMAX,ZX_R) 190S
R(S)IERROR(TEHPY,Y(I,jNAX_,RtS),I,J_AX,_YER] 1906XCI_JMAX]ITEHPX 1907Y(I,J_AX)mTE_PY 1900X(I_J_AX]mTEHPX 1909¥(%%,jMAX]xTEMPY 1910CONTZNUE 1911
GO TO ]gO IgIZ
ON LEFT, ONE ON RI_T 191$
O0 I_O JJmIl,I_ 191_
JzIE-JJ¢Ii 1915xXIwX(_,J)-X(I_X_t,J) 191_
YXIIY(_,J)-Y{I.XM1,J) 1917XETA_X(l_J÷t]-X(l,J-l) 1918
YETAIV(t_Jel)-Y(I,J-I) 1919CALL PARA (XXIeYXI,XETA_YETAeLACC.CFAC,RETA,RXI,C$I,CSJ_WACC 19|0
_9
3_0
$S0
360
370
];8039O
I'RwT'ACCL'ACCLI'_IMel'JP_*E_wTACC'IEV) 19_!TEMPXBHICK(XwTeI°JP_XMle_JeIP'T_IeACCLwACCLIwXETAeXXI) 1921TE_PYs_ACK{V'Te1'J_XM[_2PJ'IeJ_I'ACCL'ACCLI'YETAeYX_) 1923_(_)mERROR(TE_PXeX{IeJ)eRt_].I,JeTXER] 192_R(5)IE_ROR(TEMPYPY(|mJ)eR(_)°IeJeTYFR) 192SX(I,J]sTEMpx 1926Y{]ej)mTEMpY 19_?X(IMAX,J)sTEMPx 1928YfIMAXwJ}ITEMPy 1929CONTINUE |930
GO TO $90ROT_ ON LEFT |9$1
I912DO 360 JJxIl,I_ 1935JmI2-JJ*I! 193_
I_sr_'(J=It) 195SXXI=X(_.J)-X(2,I%) 1956YXIIY(E.J)mY{2jTI) 195?XETAmXCI,J÷t)-X(teJ.I) 19$8_[TAsYC1,Jet]-Y(leJ.l) 19$_CALL _ARA (XX_*YXIeXETAeYETAeLACC,CFACeRETA_RXIeC_%eCSJewACC [9_0
!,R,T,ACCL,ACCLIeNOI_teJeI_EReTACC_]EV) |9_|TEHPXIHACKU(XeT,IeJ_|_e2eJ-|eJ÷IeACCL_ACCLIeXETAeXXZ) 19_TEHPYmHACK_(YeT'teJeIIe2eJ'1eJe]'ACCLeACCLi*Y_TAeYXZ] 19_3R(_]sERROR(TE_PX_X(|_J)_{_],t,j,_XER] IRUa
X(I,J)ITE_PX 1R_6Y(I,J)ITEHPY 19_7X([_II)sTE_PX 19_8Y_I_ZI]mTE_PY 19_C_NTINUE 1_0
GO TO 390 _95!ROTH ON RIGHT [9S_
DO $80 JJsli,I2 tRS$JmTZ-JJ+I! 195_
Ilit_'[J'II) 19§SXXImX(I_XMI,II)-X(I_XM!_j] 1_§6YWIIY(IMXM|eII)eYCIMXMteJ) |_S?XETAxX(I_AX_JeI].X[_AX,J.!] leSSYETABY(ZMAX_Je!)=Y(IHAX_J.I] 1_9CALL _ARA (XXI;YXI_XETAeY_TA_LACC,C_AC_RETA_RXI,CS|_CSJ_WACC [_/0
I_R,T,ACCL.ACCL[_NDIN_INAX_J_IkER._TACC_ZEV) Ieb!TEMPX|HACKSCX_T_IMAX_JelI*IHXHI'J'IeJe|eACCLeACCL|eXETA_XXI) lgb_TEHPYIHACKS(Y_T'IMAX_JeII*IMX_I_J't_JeI_ACCL_ACCLIeYETA_YXZ) |96_Rf_)IERROR(TEMPXpX(_AXeJ]eR(_)'|H_eJe]XER} 1_6_R(S)IERROR(TENPY_V(IMAX_J)_R(S)e_NAXej_yER) [qiSX(I_AX,J)BTEMPX 1966Y(ZMAX,J)mTE_PV 1e6?XCI_AX,II)mTE_PX 1q66Y(I_AX_ZZ)sTE_PY l_b9CONTINUE 1970
CONTINUE 1_?|CONTINUE 1_11
STORE INITIAL ITERATION ERROR HORN|
R(IZ]PR(II)/FLOAT(NPT)ZV (K,GTs|) GO TO _10R(6)eR(_)R(?)IR(§)
1975le?_IST5197619171978[97S1$B0
9O
C _AITE ITERATION ERROR NORM8 19811982
CUIO IF (IWIR.GT O) WRITE (6.§_0) W.R(_)._(S).R(II}.RCI2}.LACE.LCFIC_ 1983• 19@_
ILCON 198S
C 1986C CHECK TO SEE IF ITERATION I8 COMPLETE 1987CC,Wt* CHECK VARIABLE ACCELERATION PARAMETE_ FTELD C_VERGEhCE 1988
1980LACCmR(tl},LT,R(tO) 1990IF CLEON} LACCm,TRUE, 1991IF (LACE) R(II)eO,O
C*,** CHECK INTERMEOIA'TE FIELD CONVERGENCE REFO_E INEREABIN_ 19921993
C**** COORDINATE ATTRACTION 199_IF (LCF&C} GO T_ _0IF (R[_],GT,EVAC,OR,R(_),GT,EKAC) GO TO _20 1905
1996C**** INCREASE COORDINATE ATTRiCTION 10qT
R(U)xI,O 1998
LACCI,FAL$[, 1999IF (LEON} LACCm,TRUE,IV (IFAC.GT.O_ CFICuE,OICFAC 2000IF (IFACoLT,O) CFACICFACtVLOAT(IRCOUNel)/FLOAT(IWCOU_} _OOi
2002IRCOUNmlRCOUN÷! E00$XF (CFACmGT,I,0) CFkCml,0LCFACIABB(CPAC.1,0),LT.O.O00001 200__RITE (6e67_) CFAC 200_
U20 CONTINUE 2006C**** CHECK INTERMEDIATE FIELO CONVERGENCE REVORE _OVING 200T
Cew*m OUTER BOUNDARY 2008IF (INCOUNeGEeIABB(INFAC)) GO TO _30 2009
IF (R(_},LToEINFoAND, R(S)*LT*EINE) _ Tn _90 _OIO
• C**** CHECK F_ELO CONVERGENCE E011_30 IF (R(_},LT,R(2),ANOtR(_), LT,R(3)] _0 TO _9_ 201_
C**** PRI NT VARIABLE ACCELERATION PARAMETER FTELD AT 8TART OF ITERATION |01_201_
IF (IAIT.EQ.O) GO TO _SO 201_
IF (K.N[._} GO TO _SO _OZ6WRITE (69680) _OITDO _0 JSl,JMAW
wRITE (6,690) J 2018wRITE (6_700} (TACC(I_J)eWACC(I,J)* I_I_IMAx) 2019
20_0_0 CONTINUE 20_1_0 CONTINUE 2022
C 20_3C ITERATION 00E8 NOT CONVERGE 202.CC_*_* WRITE PARTIALLY CONVERGED 80LUT_ON TO DIBK _0_
i_ _RITE (1|_7|0) LACCeLCONeCFAC_LCFAC_IRCOUNeINCOUN_CINFAC_KeIEV*IFA 202&ZCeEFAC_JHXMI_IMXMI.JMXPteC|I_CBJ_NPT_INFAC_EINF_IEN O _027WRITE (II.?]0} (R(1).IBI.I3) 2028
WRITE (11_?)0) ((W(I_j)_y(I_J)_RXI(I.J).RETA(I.J)_wACC(I.J}.III.IM iO@q_030
IkX).Jn1_J_AX}
C**** PRINT VARIABLE ACCELERATION PARAMETER FIELD _0$1Z032
IF (IAIT.EG,O} GO TO _70 _033WRITE (6,680} 203uO0 _60 j_I_JMAW 205S
wRITE (6_690) JWRIT[ (6_700) (TACC(I_J)_WACC(I_J)_I|I_ IMAX_ _036
_60 CONTINUE 2037_70 CONTINUE _0)8
IF (R(6],GT,R(_),AND,R(T),GT,R(S)] GO TO _80 E039IENDIt E0_0
91
RETUPNOAO IENDm2
WETU_N
ITERATION CONVE_GE8
Qgn IFNDI]
C**** MOVE OUTER BOUNDARYOINFACmCINFAC
IF (INCn(IN.GE,IAR$(INFAC)) GO Tn 550IF (INFAC.GT.O) CINFACn2.0*CINFAC
IF (INFJC.LT.O) CINFACmCI_FAC*FLOAT(I_COuN+I)/FLDAT(I_COUN}DINFACmCINFAC/OINFAC*RITE (6_710) CINFACI%COtJNmTNCOUN+_
00 S_O LmLISEG,NBSEGIImLBI(L)12mLB2(L)IGOTOmLBSID(L}
Gn TO (_00o520_S00_5_0)p IGOT_50n Ir (IGOTO,[Q,I) Jml
IW (I_OTO.[Q._) J|J_AXDO SIO Imlt,12
X(I,J)mX(I,j)*DINFACSI0 Y(IoJ)mY(IeJ}*DINFAC
GO TO SUO
$20 IF (IGOTO.EQ.2) Im!IF (IGOTO.EQ._) In|MAX00 S$0 JsIl,12
W(IeJ)IW(IaJ)*DINFACS$O Y(IoJ)uY(I_J)*OI_FAC_UO CONTINUE
IF(LCFAC) IFACmlKOml
IR(K.EO.ITER) GO TO _Y5GO TO Oo
C**** FINAL CONVERGENCE550 CONTINUE
ITERIK
C,i** PRINT VARIABLE ACCELERATION PARAMETER FIELDIF (IAIT.EQ.O) RETURN_RITE (6e6_0)DO _0 JI|$JMIX
_RIT[ (&_&90) J
wRITE (&,TO0} (TACC(I_J),_ACCfI,J%,IBI,IMAX}$60 CONTINUE
RETURN
S?O FORWAT t*|*#|SX,mITERATION ERROR NORMS*//}SO0 FORMAl (I_)[IS.S_F_O.S_SL|O)S_O FORMAT {IIS,IFlO.O,I_)
600 FORMAT {eO*#I_#* STEP8 IN ADDITION OF INMOMOGENEOU8 TERM. INTERMIEOI&TE CONVERGENCE FACTOR I*_F|_,_)
&20 FORMAT (*0----- MAXIMUM NORM8 OF ITERATE CHANGESI._OX.eLOGICAL CON|TROL_m//eOITER_TEs_$W_,X_NORMm_gX_,Y_NORMm#?X_eACCmNORM_?X,_AVG,A_CCmPARA**_bX_*LACC*_SX_ILCFACI_X_wLCONI )
&]O FOR.IT (*OLOGICAL CONTROLS I LACC - VARIABLE ACCELERATION*_, PARAMIETER FIELD CONVERGED (IRRELEVANT IF LCONITRUE)I//_OWtiLCFA C . AODI_TION OF INMOMOGENEOU_ TERM COMPLETEDe//_OW_,LCO N B UNIFORM ACCELER
20_!20_20_20_20_S_0_6Zoa?20.820_9_0502051_052Eos_20SU_05$_05__057
_05920602061
2o_E_o_$20b_20bS20b_EO_?EO_8EOb9_n?OZ071ZOT__o7__o?aE07S_O?&_077Ro/#_079EO#OE081_08Z2083_08_ZOeSZ086_08?_088_ose2090
_oel_o_1o93_o9_io_s_oeb_Oq?_058
ilO0
92
65O
6?068O_qoTO0710?En730
ATION PA_A_ETER*//EnX)FORMAT (*OCOMPLEX EIGENVALUE P_CEDIJR¢ ! UN_EW°_ELAX*_FORMAT (*OC_MPLFX EZGE_VACUE DR_CEDU_F I _ETGWTE_ AVERAGE*}FOR.AT (,OCn-PLEx _TG_NVALUE PR_CEnURF I _VE_=_ELAX*)FORWAT {*Oi,FiS,R_* (IF I_HOMOGENEOU$ TEPK*)FORMAT (,OACCELERATIO_z PARA_ETER$*//}FORMAT (*Oj m*,I}/)FORMAT (IO(IX,*(t,I_,*]t,FT,_,IX))FORMJT (*0,.FI_.8.* OF OUTER BOUNDARY*}FURMAT(_LIOpEI6,B.LIO,21_,EI6,8/3T_,E_,B,_IS._I_,_/_I_,E1_,8'IS)
FDRMaT(_EI_._)
END
BUBROUTTNE PLOT(X,Y,I}CALL CALPLT(X,Y,I)RETUNNEND
BUBROUTINE _YMBOL(X*Y_M_TFXT, ANGLE_NC_p}CALL NOTATE(X_Y,M_TEXT_ANGLf_NCHAR)
RETURNEND
EIUEEIO_E1o_
EIO_
aI07EIO_EIo__11oa111EIIE
E_5E116E11?
EIIB
EI_O_IEI_I_
EI_EI_S
EIE8EIE9EI$0E131
_133EI__135EI_6EI_?
E_38Z13e
EI_I_I_E
93
SampleCases
96
Sample Case Output" Simply-Connected Region
ZMAX,J_AA,N_QY,ITE_#I_LS,10I_,InlR,L*INTL,I*PlN,IGE0 I dl
IPLOT,IPLTR,NCUPYeLI_*II,LI_T_,NUM_,N_MB_t,I$_I_I,I_KIP_ I
NB_EGeNH$EG I U Q
_(I},_(_},N(_),¥INFIN,AINFI_,XU1NF,YUI_FeNINF S
l,BO000OO0 .0_010000 .00O]O0_0 0,00000000
IEVpIAII,_(10) I o 0 0.0DOQ0000
I_FACeINFACQ 8 0 0
21 0 100 O L L 0 0 0
1 ] I 0 0 0 0 1 1
OmOOOOO000 O,O00OOOOO O,O0000000 0
SIZEFRArI0,OISTeXBI,XB_,YBL,YB_ B,0OO00000 0,0VO00000 10,00000000 0,00000000 0,OOO00O00 0,00000O0o ODD0000000
L_DY
INITIAL STEP {L|NEAW CA_[) • 1
$1MPLV-C0_NECTED NEGIUN
INITIAL GoES$ T_PEI 0
--BODY _EGMENT$--
L LB$1U Ldl L_
I I 1 _1
_ l _t
_ _I 1
=-OUTEN BUU_qDIRY= _
R4OIU@ • g=O0000000
0_IGl_ AT X • 0,0000000,)
N_BEH OF POINTS • 0
-1
-I
-1
ol
INITIAL ANGLE • 0o00g0000O
, Y • g.000O0000
I _TEPS I_ aTTAINmENT OF INFINITY
UNIFO_ _ ACCELERAI_U_ PArASiTE w U$_.
TEST CASE * BUOY-FITTED COORDINATE SYSTEM
SImPLY-CONNECTED REGION
BODY-FITTEO EOOR_INATE SYSTE_
T_&NSFOWMEQ UOOYt KEY _E&T _MAFX
F_ELU PARAMETERS, NUMBER UF XIJLINE_ • _I
N_MBEH OF ETa-LINkS • _I
ITERATION PANAMkTER$! SU_ ACCELEHAIION PARAMETER • l.O0gO0
MAXIMUM NUMBER OF ITE_ATION_ _LLO_ED • tOO
ALLO*ABLE ITERATION [NROR NON_I X| t|DOoOE*0J
Vl olOO00_-O$
NUMBER UF BODIES IN FIELQ : 1
PLOT PARd,ETC.5! CUPIE_ QL$1_EQ • 1
LINE_EIGMT OE$1_ED • O,
PLOT S_ZE _N YeOINECTION • _,000
RATIO • O.OO0
LOGICAL CUNTNU_S S LACC - VARIABLE ACCELERATION P_N_METER FIEL_ CUNVEWGEO (IMHELEVINT IF L{ONmTRUEJ
LCFAC - AUDITION UF _NM0_O_ENEOUS TER_ CU_PLETED
LC0_ - UNIFUH_ &CCELEHATION PARAMETER
93
Sample Cases
94 Sample Case Input: Simply-Connected Region
TEST CASE - BODY=FITTEOSIMpLy=CONNECTED REGIONKEY SEAT 5HAFT
CUOWDINATE SYSTEM
21 21 0 100
01 1 21 =1
1 21 o13 21 1 "l2 21 1 °1
le8 O,O00l 0,00010 0 0.00 0
8,0 0,0 I0,0
eel,S ,9g215
=,JIQ6! ,9@718
°o6_932 ,77715=,8@gOU ,55919
°,9563 ,@9257°I,
=,9563 -,29237=,8290_ =,b5919
=,b2932 =.77715°,37_61 ".92718
°|I
,37_bl =,92718o_@932 °,77715,8290_ =,55919
,9563 -,29_37
I,
,9563 o29237o8290_ .55919
,b@932 ,77715
,J7_61 ,92718o125 o99215
,125 ,98b,125 .978.125 .97.125 .962,125 ,95_
,I_5 o9Q6
,125 ,91b
,t25 .932,125 .9_eo125 ,92,125 ,91_,t25 .908o125 .903,1_5 .898
0 1 1 0 00 0 0 1 t
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0
95
.125,125,125
,122
,125
,12,115011,1,09
cO0,Oh
,02
-,02=,0_=.Oh-,08=,09
",11=,115",12°,125°.125°,125=,125
=,125-,125
",125".1_5",125=.125
=,125
=,125".125".125
",125
".125°,125",I_5
"o125".125ETA
XI
1
,8q3,888.88U.88,877
,875,872,d7S,872.875,875
,875,8750675,875,875,075.872,875,872.875.875,875,875,872.875
m877.88m88_
,888
,89_.898
.90J,908
,92
t92_
,9_2,936.9_b.95_
,962
.97
,978.98b
l
10.00
100,0
0 0,00,00!
0 0,0
OoO
OoO
96
Sample Case Output" Simply-Connected Region
t**t*t**t****m*mm INPUI ******************************
IMAX,JMAX,N_OY,ITER,IGES,IOISa,I_IR,I_INTL,I*FINelGEO ! _I
IPLOT,IPLTReNCUPVeLIN_IteLI_T_,N_M_,NUMBRt,IS_IPI,ISKIP_ I
I,_0000000 ,00010000 ,00010000 OmO000000O
IEv,laIT,R(IO) | 0 0 0,00000000
I_FAC,INFAC0 1 0 0
_I 0 IO0 0 1 ! 0 0 0
l ] 1 0 0 0 0 l 1
0,00000000 0.00000000 Q,00000000
$_ZEeRATLO#OIST_XBIf_B_rYBI,Y_2 8,OUO00000 O,OUOOOODO lO,O0000000 OmO0000000 O.OUOOO_O0 OtOOOOOOOO OmO0000000
L_OY
INITIAL STEP ILINEA_ CASE) •
SIMPLv-CONNECTE0 REGION
IN|TI&L GUESS TYPEt 0
*-BODY S_GMENT$--
L L_SIU LBI u_
1 I l _l
_ 21 I
--OU_ER _OU,WOaRY-°
RaOlU_ • O,O00000OO
OR_IN _T X • 0,00000000
NUMBER OF ROINTS • 0
-I
-I
-I
-I
INITIAL aN_Lk • 0e00gO0000
_ • Om00000000
I STEPS IN alT&IN_LNT OF INFINITY
unIFORM A_CELERATIU_ P&_AM_TE w U]_O,
TEST CASE - _OOY-_ITIE0 COUROINaTE STITEM
$1_PLY-C0NNECTED _EGION
UODY-FITT_ C00_INalE $YSTE_
TRANSFO_E_ UOOY, KEY SEAT S_FT
FIELO PAWA_ETEH$, NUMBER UF XI-L_N[$ • _|
NUMBER O_ ETa-LINES • _1
ITER_TION PaRAeETER$I SO_ ACCE_LMAIION PARAMETER • I,B0OO0
MAXIMUM NUMH_W 0F ITEWATION$ &LLO_LD • I00
ALLO_BLL ITLRaTION ERROR h0N_@l Xl ,I0000E-0_
YI mlOO00_'O_
nU_E_ oF UOOIE$ IN FIEUD I I
PLOT PARaMETeRS| COPIES _ESIREQ • 1
LINEmEIGMT DESIWED • O.
PLOT SIZE In Y-DIW_CTION • _00O
RaIIU • 0,O0Q
LOGICa& CONTRUL$ I LACC - VaRIaBLE aCCELERATiON PARAMETER FIELQ CONVEWG[D (I_R[LEVINT IF LCONITRUEI
LCFAC - A_DITION UF INMO_OGENEUUS TERM CU_GETE0
LCON - UNIFURM AC_EL_RaTION PaRkM[IER
97
EXPONENTIAL UECAY _MS
o ATTrACTiON •
-°- Ela EGUATION RM$ ---
ATTRACTION L|NE$
J AM_ O_CAY
1 1O_0O000000 .U0100000
I 5TLP3 |N AOOITION UP ZNMOMOG[NEUU8 T[_M e
1,000U0000 OF INHOM0_ENLOU$ TE_
I,0000U000 UF 0UTb_ _OUNDJ_Y
----- MAXIMUM NOHM$ OF ITERkTE CHANGES
ITEHATEI
$
5b7
910Ll
13
1017
1819_021
2_
272_
_Q3O313235
3b37
3_39_U
_5_o
.77jb7bo01
.755_3E-01
._ESBbE_00:192_2E*00ml_070E_o0
_I09J_E÷00.81_BdE-01
75859_-01_7_32E-01W75@0E-01
299_bE-012700_E-01
.17_5_E-0119b_oE-ol
_70_bE-01
_3gE=o2
99U_-U210,0dE-Ol
.11177b-01
.710bbb-OdeSbbb_E-U2._1eSIE-02._}512E°02,265|_E_02
.21gl_E-02
mlg00W_02_l_0bSb-0_,|021_E-02
.Tbb_2E-0$
.b_99_E-O_
.2858iE=0J
,36811_-OJ
.270_gb-03
.18_I,E-03
.IwO_UE-0J
,gBbBB{-U_
,II059b-0_
,5507_E-0.
Y-NORM
.187_0E_00 0
.179_5b_00 0
.52_70E+OU 0
._2021E+00 0
.3e712L+0_ 0
.2177_+00 0
,_3q7_E+OO 0
.I,885E+00 0
.|,OSbE.00 0,gb_23E_Ol 0._8bTbEo01 0.55B_1E°01 0.57719_-01 0
.bIZ_Ob-Ol 0
.,52_E-01 0
.29517E-01 0m327._E-01 0._0b_0Eo01 0
._bT55Eo01 0
,22_90E-01 0.20057L-01 0_15_bqE_Ol 0
.1297bE'01 0
.10755E'01 0
._085bg-0_ 0
,_b27E-02 0
.328b/E-0_ 0,255_3L-02 0
.ITq,SE-02 0
.1055_E°0_ 0
.89375_-03 0
.B3blgb-03 0
.5_767_-0] 0
,.2585E°OJ 0
,361_7E-0_ Om
.3g2O_-0J 0
.3072bE-0J 0
._I)0oE=U_ 0
._bb87L-O_ 0,289dSE-OJ 0_Ig891E-0_ 0
.I_S/E-0_ 0,IU_b_E-O_ 0
.78_OOEo0_ 0
INTL_MEO|AI[ CONVENGENCE PAGTQR • 100,00000000
_VG,_CC
LOGIck_ CONTW06I
PA_A. _CC L_FAC _CONOOO00 T T T
00000 T T T8000O 1 l T8OOOO T T T
80000 T T T800QO T l T80000 T 1 T
80000 T T T80000 T l T80000 I T T80000 T T T
80000 T T T80000 T T T
80000 f i T80000 T T T8O0OO T T T80000 T T T
80000 T T TBOO00 T T T
80000 T T T80O00 T T T,80000 T T" T
80000 T T T80000 I T T80000 T T TBOO00 T T T
80000 T T T80000 T l T
_0000 T T T80000 T 1 TBOO00 T T TBOO00 T T T
1.80000 T T T
1.80000 I I T1_80000 T T Tl.BO000 T T T
1,80000 T T T1,80000 T T T
lm80000 T T TlmB0000 T T T1.80000 T T T
IMP0000 i T T1,80000 T T TImSO000 T T TlmSO000 1 I Tlm_O000 T l T
98
TEST CA$_ - _O_YeF|ITED [QURO|NATk JYIIEM_IMPLv-CO_ECTEO REGION
FZNJL v*_uE|
%TE_ATZON CONV[NGESe
I_{T_AL [TE_AT_0N [RROR NUKMSI Xl ,77167E-0| Yl e|8730[_00 AT _TERATE IFINAL ITk#ATION [RHOR NURNSI Xl ,5507gE-0_ YI 178160[-0g AT _TERAT[ # 4bLuCATIO_ UF .AXI_U_ |Tb_ATION [HRORI Xl |x 5_JI 4
YI |x 17 Jm 5
*l#_**t****_ttQt*_tt_*Qt*t**l_**l XeAR_y *_Rttttatt*le*i/ttt_ttt/tttltl/tllt/ttlRl//tt//llllt¢Js !
_l_5_E_*]7_6iE_*b29_2E_u_82_t_*95b_E+_'_Et_|_9_b]_Et_e_829_att_mI6_]_{t_=*_74_|_t_O, ,37_1E*00 .b2932E÷O0 ,8290_E_00 _9_b30E_O0 .lO00OE+Ol m95650_00 ts_QO4[tOO i_l_[tO0 ,_7_k|[900
.IESOOE÷O0
-_1_+_-_e_5E*_-*_5_E*_u°_j_E*_-*7_8_5E*_8]59_*_*B_8_E_?_49_$_5_$_E_-_89E_-,_1931E-05 ,3128BE*00 .55130[*00 .Y25aTL+O0 ,82281E*00 .835_0E*00 ,7685ZE*00 ,63b61[*00 ,_tSSIE*O0 =ll71_[*O0
_i_500b+O0Jm $
-._970E-UQ ._EeO0 .qb_9_+O0 .bOTQdEeO0 mbT_gT_eO0 .bb_Sb_tO0 m_g_J|ke00 .Q_Q_EeO0 e_b||b(#O0 e_Q_||eOOm1_5OOb÷O0
-_leq2oE=U_ ,_Ob_gE*O0 ,_7l_bE÷O0 ,_7867EtOU ,5_1_8E.00 .508_E+00 mQSSb_E$O0 t$8|tSEfO0 _i9905_+Q0 ,_lbll|$O0,12500E_OOJ= 5
-,eU85_E-05 ,|55_5E*00 ,_781_E_0o ,3590JE_00 ,$9_3_E+00 ,_lb_E÷O0 ,36_0|L+00 ,$1592[_00 ,_07|[_00 ,|9933[t00
.12500E_00J= b
-,179J_E-05 ,11165E÷O0 ._057LE÷O0 ,_7151E+00 ,_ObbSEgUO ,31_71E÷00 ,]OIS_E÷O0 ,2752b[*00 e_$_SbE_O0 ,|870$[_00.I_bOOE+OO
Jm ?
_'_5_E÷_7_q3E+_-'_5_5E_-`_9_E+_-_2_2_E+_°_b_91_+_z_|_E*_*z1_9_Et_[_B_|
=_J_bEoU5 .855_qE-01 mlbOO_E*O0 ._1589E*00 ._50tSE*OO ._6_OE*O0 ._602_E*00 ,2Q_88E+O0 ._1S0_£_00 .1774|[*00
=l_500E+OO
-_5_E_1e95_E__E+_°_9_8_*_°'_97_E_'_775E÷_2_1_7_*_*_789S_13_8E_=_e9_y1Ee_1m_6_qSE-05 .bqOoSE-o! .IJOSBE*O0 .1789bE$00 ._11J7E_O0 ._775E900 ._97_E*00 .2tgb/E*O0 .1995bE+00 .16q5$[*00
J=
-_5_E*_ej_+_°_6_9E*_E_'_59_E_]_Et_°_8_7_E*_E_e_1_99_*_*5_e_Ee_|._9007E-05 .57097E-01 .1099JE*00 ._5_aJE*O0 .18_15E*00 .ZOO3aE*O0 ._0597E*00 .Zo_ogE_O0 .tete?[*oo .|t=sl[,oo._500E÷O0J= t_
_*_-`_5_j_E+_°'17_1_E_57bE_1_eB_+_°_78?_E*_6_*_t34$3_9_$_SE_1_e_|._,bgaE-o5 .qgJtq[-01 .9;SJSE-O! .[_233[*00 .10065E*00 .1787i[*00 .1868ZE*00 .tSS?;[*O0 .tTiiS[*O0 .;57J1[*00
.l_bOOb*O0J= 11
.=705_L-05 ._8_7r-0! ._25eJE-Ol .iibSbE*OO .laJOOE*O0 .IbII_E+O0 .1709aL*O0 .11_79E+00 .|bbgiEtO0 .l§_3eE900
.I_bOOE_O0J= 1_
°=_E*_t=79SE_`t588UE*_t6|e_E_°_t$7S_E*_e_eE_-_1_8_E_=*_|_E_-=73_e$Ee_|=e]77$_|ee_,7e007_-05 ,$77SIE*01 ,71082E-01 ,lOIBb[*O0 ,1_66JE*00 ,|_iSgE*O0 ,t_?S?|*O0 D|I|/?|90O ,|Slli[*_O ,J4794|_1t,I_SOOE*O0
99
j= L_
-g_$8|2EeO5 .3]bb7E-O! ,6S]qbE=01 .93512E*0 .1167bE*O0 ,|5_40EtOO oleb|tEtO0 j|S|QO[*O0 ,|S|baE*O0 ,|4]q4|tOI
,i_OOE*O0J= I_
.l_bOO_*OOJ= l_
.t_bOOE_O0J= te
,7105_Lo05 ,_l_E-Ot ,50005L=01 ,7_a73E=O ,9_OeaEoo! ,10817E÷00 ,12063E_00 ".1_34E_00 ,t]aO6E*O0 ,15180E+00.t_5OOE*O0J# t7
.b_a7E-o_ ._JOObE-O! ._67b3b*_ .08010E-O ._eegTE*01 .IO_17E*O0 .tta_?E*O0 .lZ$S6E+O0 .1_9_6E*00 .1300_[+00
.t_5OOb*O0J= lO
_5_E_'1_E*_'_a_E_11_9E*_-_1_93_*_973a9E_1°_8_8_E_1*_59_-_aa_a_E_a6_E_
.a7_79E-05 ,_aeSE*oi ,ua_SqE-g] ,baotaE-0 ,8_egbE=01 ,_7379E*01 ,1093bE*00 ,ttOb_E*O0 ,_gZE+O0 ,IZ6OSE*O0
.1_500E*00
J= 19_+_Sj7_÷_1_$_*_*t_a_-'_E*_'9$_9E-_°_9_S_e_t°_6_9Ee_1-*_3_6E=_t=_a_5E=_|
.110]Ot-Ob ,_tqSeE-_i ._9_-0_ .b_EteE-Ot .7_86_E-01 .957bSE=O| .tO5EOE*O0 .ItiS_E_O0 .t_tOaE_O0 .1_5$8Ee00,l_bOOE_O0J= _0
-.11050[-Ob ._Ob_OE-O! ._lO,1E-Ot .bOTbOE=Ot .7el$_E-Ot .9133bE-0! .IO_E+O0 .ttt§_EgO0 .1176BE*00 ._Tk[*O0
,l_bOOE÷O0J= _
-_b_+_|_E+_*1_5_L÷_*|1_E*_-_1_+_e_9_E*_1_8_G_=_1_6_Ee_1_[e_1e*_
O, ,_O000Eo01 ,aOO_OE*01 ,eOOOOE-01 ,80000E-U! ,_O000E=01 ,tOOOOE_O0 ,IIO00E*O0 .11500E+00 ._O00E*O0.|_500E÷O0
,_9_15E*00 ,9_710E+0_ ,77715E*_0 ,55919E+OU ,_q_$7[÷000, _.29_$TE+OO-.sSqtqE+OOe,7171SE+OO-e_7|_E_O0..IO000E_OI..9_71_E÷OO-.77715E+OO-.55919E+OO-o_9_37E+O00. ._9_7E+00 .5591qE+00 .777_SE_00 .9_71_E_00
._9_|5[*00J.
._oOOE+O0 .9_7_oE+00 .8_b_gE÷O0 .bSO?OE+Og ._SbE+O0 ._9_lE+OOee|_O_SE+OO*e_90_E+OO°ebO_9_+O0_*7530_900-.St_OOE*OO*.7550_E+OO-.bO_90_+OO-.3_O]gEeOO*e_JO_E+O0 ._9_5[e00 ._18bt_eO0 .6_O?a[+O0 .8_b_[÷O0 .93707E900
,_860UE*O0J= 3
.97800E+00 ,95897EegO .85767E*O0 .7_515E_OU .55959[+00 ._Ug/_EeO0-.b_b75E+OO*._9_E*OO=.$_qJ_E*OO-.tgO_E+O0 .Sb511E-Ot .30989E+00
.q7800E÷O0
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102
Sample Case Input" Single-Body Field
T[$T CASE - BUOY-FITTED CUO_DINATE SYSTEM8INGLE _ODY I KA_A_-T_EFFTZ AIRFOILe _b PU_NT8
K-T AZRFOZL #127 20 1 200 0 1 1 0 0
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109Sample Case Plot: Single-Body Field
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Sample Case Input" Double-Body Field
TEST CASE - _UDY-FITTEO COU_DINATE SYSTEM
OOU_Lb BUOY I TwU KARMAN-IR_FFIZ AI_FUXL@, _b POINTS ON LACH
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Sample Case Output: Double-Body Field
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Sample Case Plot: Double-Body Field
123
VII. INSTRUCTIONS FOR USE - SCALE FACTORS
After a coordinate system has been generated, the "scale factors" for
use in the solution in any partial differential equation transformed to the
rectangular transformed plane are generated and written on a disk file by
the main program FATCAT with its subroutine ABG, MXM , PARAI, PARA2, PARA3,
and WRDATA. Instructions for use are included in the listing of FATCAT.
Core must be set to zero at load time.
Dimensions. The standard program allows a maximum field size of 70 _ lines
and 60 n lines and requires a core size of 201,000 words for the Langley
Research Center's CDC 6000 Series Computer System.
Input Parameters. Explanation found with the sample test data, and program
listing.
Files. This program requires 2 essential files:
TAPE i - input tape - generated as TAPE10 by Program TOMCAT.
TAPEIO - disk on which the factors are to be written.
124
Scale Factors
Program FATCAT
125
PROGRAM FATCAT(I_PUT.OUTPUT,TARFSmINPUT,TAPF6m_LITPUT,IIAREIOwTkPEt)
r *******,** _I$EI$SlPBI _TATE 2=D COORDINATE TRANSFORMATION *******_**
C * CDnRDI_aTE SYSTE M "SCALE FACTDRE*I ALPwA,B_TA,RAMAeSIGMA,TAUeC * JAC_HTAN,DXIDWI,[)XlbETA,C * I)YIOXI, AND DY/DETAC *
C *********************************************************************C
DIMENSIO_ CI(8)wC2(8).LBSID(b),LMI(6),Ln_f6],LEDY(6}, LRS_DI(6), LRI(6), LRZ(6), LISID(6). LI](_), II2(6), LTYPE(_), LSEN(6)_ATA NDIM,NDIMteNDIMX 170e_0,70/
_*****C
C
C *C *C *C *C *C *C *C *C *C *
C *C *
C *C *C *
C *
C *C *C *C *C *
C *C *C *C *C *C *C *C *C *C *C *C *C *C *C *
THE COORDINATE @VSTEw IS REAO _R_M nTSKCTAPEt] IN THE $_[• DRW*T USEO BY TO_CAT TO *RITE ON _ISK. THE SCALE F_CTC.R8kRF WRITTFN ON DISK(TAPE|O] IN ONE OF THF FOLLOWING FONM_T8,SP_CIFIEO BY IFORM,
*** FORMAT _1 (IFORMmt) ***
_RIT[(tO,t} c]
_RITE(tO,t) IM_X,J_W.N_SEG,NWSEG,LISE_,NBDY.RITE(IO,_) (LnSID(L],LB_(L].L_2(L)_L_Yt' },LEEN(L} ,
I LmI,NBEEG}w_ITE(tO,I) (L_BID(L),L_I(L).LRE(L),I_I_ID(L).LIt(L),LIZ(L),
I LTYPE(L),LmI,NR8EG)DO 2 jBIIJMAX-2
2 *RITE(tO,I) ((CtN,T,J),Nml.IO).IB_.IM_X-2)DO _ Jmt_
$ WRITE(IO,I) ((CB(N,I_j)_NII_|O)_m_wM_XO(IMAX,JM_X}
THE _RR_Y C CONTAINS THE F_CTOR$ FROM THE _ECOND POINT TO THEPENULTIMATE POINT_ ON SECOND ROW TO THE PENULTIMATE ROW,THE *RRkY C_ CONTAIN8 THE FACTORS ON THE RECT_NGULkR B_UNDkRY,JmI,2,3,_ IN CS CORRE8RONDINS, RESPECTIVELY, TO THE SOTTOM,
LEFT, TOPe RIGHT SIDES, ON EiCH SIDE THE POINT8 RUN FROM 1 TOEITHER IM_X OR jMAXe A_ _PPROPRI_TE,
*** FORMkT _E (IFOWMI|) ***
*RITE(tO,l) C1WRITE(IO,I] C2_RITE(tOwt) IM&X,JMAX,NBEEG,NREEG,LIEEG,NBDV*RITE(tO,l) (LB$ID(L),LBt(L)_LB_(L}eLBOYtL)_LSEN(L]e
I LBI,NBOEO).RITE¢tO_t) tLR8ID(L),LRt(L].LRZ(L)_LIEID(L]eLIt(L)eLIE(L),
| LTYPE(L]_LlleNREEO}DO _ JmI,jMkWwRITE(IOn1) ((C(N_I_J)_NBI,IO)_IBI_IMAX)
THE JRRAY C CONTAIN8 THE FACTORS FROM THE FIR|T POINT TO TMELAST POINT_ ON THE FIR*T.ROW TO THE L_ET RO_,
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C
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CC
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C
CC
C
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t
* qll I JACORIAN
* NO2 I ALPHA* Nm3 t B_TA, Nm_ I GAMMA, Nm_ t SIGMA
* NB_ | TAU* NmT I DXIDXI
* Nm8 l DXlDETA
, Nm9 l DYIDXI
t Nm10 I.DYIDETA
t
tit w_
T_E FACTORS CORREBRON_ TO TMF TNnEx _ AS FOLLL)*SI
*** CARD l IwRTIjI_RTE,IFOR_ - FORmAT(SIS]
IWRTI - I0 DONeT PRINT COORDINATE SYSTE M FRO_ _HICH
, FACTORS ARE CALCULATFD,
t )0 PRINT COnRDINATE SYSTEM,t* T_T2 - m0 _DNWT PRINT FACTOW$,
* >0 PRINT FACTORS.e
t IFORM - FILE STORAGE FORMAT CONTWOL, {SEE AMOVE)t
READ INPUT DATA
_RITEC6eISO)
N|m|REWINO NI
READ C5,110] I_RTt,T_RTE,IFOR __RITE (_ISO) IWRT%,IWRTE,IFDRM
IF (IFORM.LT.I,OR.IFORM.GT.2) _RITE (6,%70}
IF (IFORM,LT,|,OR,IFORM,GT*_} STOP |
READ COORDINATE SYSTEM
REAO (NI,IOI] C|
READ (NI_IOi) C_REAn (Ni,IIO) IMAX,JMAX,NRSEG,NR8_G,LISEG,NRDY
ITEST|IMAX-_
JTE$TmjMAX*2IF (IFORM,EQ,|) ITESTmlMAX
IF (IFORM,EU,_) JTE$TBJMAXIF (ITEBT.GT.NDIM) WRITE tbeL]O]
IF (JTEST,GT,NDIM1) wRITE (6_IQO)IF (ITEST.GT,NDIM.OR.JTE8T,GT,NDIMt) STOP
IF (MAXOCIMAX,JMAX).GT,NDIMX) WRITE (b_160)
IF (MAXO(IM&X,JMAX).GTtNDIMX) 8TOP ]READ (N|_IO2]fLBSID(L),LRI[L]_LB_(L),LBDYfL}_LSENfL)_t. mt_NB$EG)
READfNI_tO])(LRBID(L),LRI(L)_LR_(L).LT8IDtL),LII(L),LIE(L),tLTYPE(L)_LIt,NRSEG)READ(NI,IO_)((X(I,J}_ImleIMAX)e.TmI_JMAX)_f(Y(IeJ),ImI_IMAW]_
1Jmt.JMAX)
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127
PRINT LAREL
_ITE (b,lO0) Cl_ITE (B,lnn) C2_ITE f6,120) IM_XeJ_JX
CALCt,LATE FACTORS
jMxMImj_AX.I
M_IHWm-AXn(I-Axpj_AW)I_X_2mT_AX-)j_XM2IjMAX-p
CALL ABG (C,_,Yp*D_,I_AW,JUAX,?_X_le.IpY_I,T_,_T2,_$E_rLISEGwCB,LBIST_,LH_.L_,LB_V,LR_T_,LQI_LP2,L.I$1_,LII.LI2,LTYPF,LBF_NDI_X,NR_E_G,_iDIMI)
PWIK,T X A_r_ V FIELr)$
IF (I_RTI.EQ.O) GO T{} 10CALL *_T* (X,I_aw,J_AX_I,2,_DI_}
*WITE F_CTOW$ T_ nISK
*** FORMal It ***
_R
WR
ILT
3n _ROO
_0
ITE(IO,IOt) C!ITE(IO_IOi) C_ITE[IO.110] I_AX,J_AX,NBBEG,NRSEG,LISEG.NRDY
ITEttO_IO2)(LB$1_{L},LRt(L)_LR_(L},LBnYtL_LSEN(L),Lm_NB_EG )ITE(tO_IOI}tLR$1DtL),LRt(L),LRE(L),LI_I_(L),LII(L},LI2(L},YPE(L)_LIt.NRSEG}_0 Jmt_JMXM_
ITE(IO_IOQ} ((C(N,I,J)_NII_IO)_Imt_IwWM_}_0 Jm1._
_ITE(IO_IOQ)((CB(N,I.J)_mI_IO)_III;MDIMX]STO_ 01o1
*** FORMAT _E ***
So _RITE(tO,t01) C1*RITE(IO,tOI] CE
_RITE(t_,110) IMkXeJMAX,NBSEG,NRBEG,LIIEG,NBDY
wRITE(IOe_O2](LB$1D(L)eLRI(L),LRE(L)eLH_Y(L)eLSEN(L),LmI,NBBEG]_RITE(IOetO_](LR$1_(L}eLR_(L)eLR_(L)eLI$I_(L]eLII(L)eLIE(L),
ILTYPE(L)_LmI#NRSEG)DO 60 JJ|I,JMXM_
DO 60 IIat,I_XM2DO 60 NIt,tO
JJJIJ_XMI-JJIIISlMWMI-IIJIJMAX-JJIBIMAX-II
bO C(N_I_J)mC(N,III,JJJ]O0 70 IBI,IM_X
DO 70 NII_IOC(N,I,I)ICB(N,I_I}
l_t
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70 C(N.I.jMAX)uCB(_.I.$)00 80 Jz2.JMXMI
DO FO _uI.10C(N,I,J)mC_(_',J,2)
BO C(N,IMAX,J)mCB(N,Jp_)DO gO JmlpjMA_
90 *RITE(IO,IO_) ((CCNeT,J)w_II_IOIeIBIeT _AXI
STOP OlO_
100 F_RMAT(lXeBAIO)lOl FORMAT(BAIO)102 FORMAT(SIS)10] FORMAT(TIS)
10_ FnRMAT(_[16,8)110 FORMAT (1hiS)
120 FORMAT (*OFTELD I IMAX meuIUeSXe*J MAX meet_)
130 FORMAT (*O=-e.- ERROR m'--'*o|OW_*IwAX Ton LARGE. I_CREASE*o* _DI"
I AND FIRST OIMENBION nF XeY A_JD SECOhn DI_E_JSTON OFew* C .*}
|QO FORMAT (*O.--o- ERROR --.--*,IOX,*JMAX TO_ LARGE. INCREASE*ee N_I M
11 AND SECOND DI_[_BION OF XeY A_O THIRD*w* OIMENSTO_ OF C. ,)ISO FORMAT {*OINPU] I Iw_TIu*wI2,SW,*IwRT_B*,I2,SX,*IFORMu*pI_)|hO FORMAT (*O-o-w- _RROR --.--*w|OXe*MAXTMU_ OF IMAX AND JMAX MUSTew*
I NOT BE GREATER T_AN SECOND DIM_N$10N OF _B. TNC_FASE THIS*,* DIME
2NBION*/_ ANO NDIMX IN DATA BTAT_M_NT**)
170 FORMAT (*Om_w-- _RRO_ -----*._OX_*IFOR_ _U_T BE 1 _R 2.*)I_0 FORMAT(tM|/I)
END
_UBAOUTIN[ ABG (CF,X_Y_NDIM,IMAX_jMAX_I_XWI,JMXWI_IWNT2.NRS[G_
ILIgEG_C,LB_ID,LBI,LB2,LROY_LRSI_LRI,IR_,LI$10,LII,LI_,LTYPE_2LBEN_NOIMX_NB_EG_NDIM|)
DIMENSION W(NDIM_I), Y(NDIM_I)o CF(IO._IW,NDIMI)_ C(tO,NDIMX_)_IMENBION LBBID(1)_ LBI(1)_ LB_([), LBDY(1)_ LRBIO(I)_ LRt(1)o LR2
1(I), iIBID(t}, LIt(1)_ LIZ(I}_ LTYPE(1), L§EN(1)DIMENSION FAC(IO}_ 81D[(_)INTEGER FAC_SID[
DATA FAC /6HJACOBN_bNALPNk ,&MB_TA ,6MGAMMA ,6NSIGMA ,bMTAU
ItNX,Xl ebHX,ETA _6NY.XI _bHY,ETA /DATA 8|DE/bNBOTTOM_bHLEFT ,6HTOP #6MRIGHT t
C*****O0 TO _60
CC**_* BODY 8EGMENTB **_*C
tO DO 100 L_i_NBBEGIIzH,.BI(L)I|BLB_(L)I$lIitil_el|-l
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ZGOTOmLRSID(L)Gn TO (2n,_,6n,Bo), IGOTO
C**w* BOTTOM
PO CALL PARA_ (X,Y,II,I,NDIM,XXI,YXT,IIpI,TIeI,IeII÷_,I,XETA,yFTA,III,I.It.2,II,3,1,),B[TA,GAMA,IGOTC,JCb,JLP_AI
JmlImllJCx!TeaT1C(I,IC,JC)mJCBC(2,IC,JC)xALP_C(_,ICe,|C)BBETAC(_,IC,JC)mGAMAC(5,IC,JC]xSIGC(_,IC,JC)BTAUC(7,1CIJC)BWXIC(_,ICpJC)mX_TAC(9,1C,JC}mYXIC(IO,IC,JC)mYETACALL PARA3 (X,Y,12,I,NDTM,XXI,YXT,IE,I,T2-I,I,12=2,I,XETk,YFTA,I
I_,I,I_,E*IEw3w'I.IeRETAeGAMAeIG_TO,JCReALPMA)
ICmIEC(l,IC,JC]wJCSC(2,1C,JC]mALPMAC(),IC,JC)mBETAC(_,IC,JC)mGAMACt5,TC,JC)mSIGC(6,IC,JC)ITAUC(7,IC,JC]IWXIC(8,IC,JC)mXFTAC(9,IC ,JC)BVxIC(IOwIC,JC)BYET*DO $0 Iml),I_
CALL _R_$ (W,Y,I,I,N_IM,WXT,YWI,I-I,I,O,O,I÷I,I,W(T_,YETA,I,II,Y,2,1,),O,I,BETA,GAM_,IGOTO,JCR,_LPH_)
C(t,IC,JC)=JCSC(),IC,JC)m_L)M*C($,IC,JC)mBET_C(_IC_JC)mG_MAC(),IC_JC)m$1GC(b,IC,JC)mYAUC(?,IC,JC)mXwIC(i,IC,JC)mXETAC(I,IC,JC)mYwIC(IO,IC,JC)mYETA
)0 CONTINUE
;0 TO 1ooC**** LEVT
_0 CALL PARk3 (X,Y,I,II,NDTW,XXI.YXI_I,II,),II.],II,XET_.YETI_I_II,tl,II+I,I,II,_,I,I,BETA,gaMa,IGOTO,JCB,aL_a)
JmllJCm_
ICml!C(I,IC,JC)mJCBC(E,IC,JC)mAL_MAC($,IC,JC)mBEYk
130
C(5,1CeJC)mBIGC(6,1CjJC)mTAL!C(T,ICeJC)mWXlC(8,ICeJC)mXETAc(geIc,JC)IYXTC(iO,IC,JC)mYEYACALL PARA) (XeY,I,12,_OTM,XWI,v_I,I_T?,)wT2,),IP,WETA,YET*,I,12,
II,I_-I,IaI_-_,Io-I,_ETI,GA_A,IGOTO,JC_,ALPHA)Jml2ICml2C(I,IC,JC)mJCBCCZ,IC_JC)mAL?_AC($_ICeJC)mRETAC(_,IC,JC)mGA"AC(),ICeJC)mSIGC(6,1C,JC)mTAUC(?_ICwJC)mXXIC(i,IC,JC)mXETIC(IelCeJC)mYXlC(iOolC_JC)mYETADO 50 Iml],l_
CALL ;ARA3 CX,YwI.Iw_DI"_XXT,YXI,I,T,),Y,3_I,XETA,YFTA,I,I-I,nI,O,I,_I,L,O,flETI,GA_A,TGOTOeJCB./LP_A)
ICm!C(I,ICwJC)mJCRC(),IC,JC)IALP.AC(),ICfJC)mBETAC(_,ICeJC)mGAMAC(l,IC,JC)m$1GC(b,ICeJC)mTAUC(7,1CoJC)mXXlC(8,1C,JC)mXiTAC(I,IC,JC)mYXl
C(IO,IC,JC)mYEYA50 COnTInUE
GO ?o lOOC**** TOP
bO CALL PARk) fX,Y,II,JHAX_OIHeXXT,YX),TI,JMAX,ZI+IoJ_AW,IIe2,JMAXIeX[TA,YETA,ZlwjMAX,TIeJMAX-I,II,jMAX-_eI,-IpBETA,GA_A,IGO?OejCBeAL_PHA)
JmdMAXImllJCm$ICmllC(I_IC,JC)IJCBC(),IC,JC)mALP_C(3,1C_JC)mBETAC(_,IC_JC)mGA_A
C(),IC_JC)mSlGC(b,IC_JC)mTAUC(?_IC_JC)mXXIC(8_IC,JC)nXETAC(_IC_JC)mYXIC(iO,IC,JC)mYETA
CALL PARA) (X_Y_I2_j"AXeNDI_XXY_YXI.y2_jMAX,I_ml,J.AX_I_.2eJ_AXI_XETA_YETA,I2,JMAX,I_,JMAXeI,T2wJMAX-_,-I,-I_RETA,GAHA_TGOTO,JCB_A2LPHA)
l_Z2ICml)C(I,IC,JC)mJCO
3ol)0230)
$ OS
)&o7)08];0(;_10311)12
$16_&l'r
$21
$Z(43Z5
_0
)&_(¢)(;5
_50_St
)STSSe
Sbo
131
CC2,1_,JC)nALP_A 361
CC],T_,JC)n_ETA ]62
C(S,IrwJC)mSIG ]b_C(6,1CojC)mTAU 365
_C7,1CwJC)mXx! 366CC_,IC,JC)mWETA 367CCl. IC.JC)mYxl ]&AC(IO,IC,JC)mYFTA ]6g
nO ?o ImI3,T_ 370
CALL PAPA3 CW.Y.I.J_AX.NDIk'.XXImYWT.I-I.j_AWeO.O.I+L.J_AXmXETA 371I'YFTA.I.JHAX.I..Y"AW=I.I.J_AX').O.-I._FTA.CA,A.IGO_OmJC_.ALpHA) 37)
ICm! 373C(I,IC,JC)mJC_ 37.C(2,!C,JC)xALPHA ]75C(],ICwJC)mRETA )76C(,.IC.JC)mGAMA _77C(5'IC'JClm$1G 378
C(6.1C.JC)miAiJ 37Q
[(7.1C.JC)mXWI 3_0
CCA,IC,JC)mX_T¢ ]81CCQ,IC,JC)mYwI 3_2CCiAelCwJC)mYEIA 38)
?0 CONTINUE 38.GO TO 100 3_5
C**** WIGwT _8680 CALL PA_A$ (W,y_IMAX,IIw_DIM,WXI,YWT,I.AW,II,IMAx. ImlI,IMAX._I) 387
I'XETA'Y_TA'I_IX'II'I_AX'II+|'IMAX'II÷P,'I'I,B_TA,_AMA,IGOTO_JCB_IL )882_A_ 38_
ImlWAX )gO
Jml! Sgl
jCm. 39_
ICmll ]93C(I.IC.JC)mJC_ 3_
C().IC.JC)m_LPWA Sg)C(3.1C.JC)mBETA Sg_
C(QmlCmJC)IGAM* 311C(S.IC.JC)m$1G ]9_C(6,1C,JC)mT_U 399C(?,IC,JC)mWXI _00
C(8_IC.JC)mWET_ "01
C(9,IC_JC)mYXI _02
CaLL PaRA3 (WmY_IMAW_I2mND|M_XXI_YXI_IMAX_I)_IMAX_ImI2_IMAX._#I_ _00
I'XETA,YETA,IMAX_I?,IMAX,12. L,IMAX.I_._,.I,.I,RETA,GAMA,IGOTO_JC_,_ _OS2L_M_) _06
JmI_ _0_
ICml_ ,08
C(I.ICmJC)mJCB _0_
C(|_IC,JC)IALPMA _10
C($,IC,JC)mBE?_ _II
C(_,IC,JC)mGAMA _1)
CtS.IC_JC)mBIG _1)
C(6,IC,JC)m_AU _1_
C(?,IC_JC)mXWI UIS
C(8,IC,JC)mXET_ _16
C(9_IC_JC)mYX! _17C(IOmlC_JC)mYEIA _IM
CALL PARA] (X,YeIMAXeleND_MmXXIeyxIeIMAX_ImIMAX_IeleIMkX_eIeX U_O
132
|ETA,yE?AeIMAW,I.Io0,0,1wAW,T*tI-I,0,HFTA,_A_A,IGOTn,JC_,AL pwA)
ICmY
C(I,ICIJC)mJCR
C(_IC_JC)mALPHA
C($,ICeJC)mMFTA
C(_,ICeJC)mGA"A
C(5wlCpJC)m$1G
C(aw|CpJC)mTAU
C(?elC_JC)mXXlC(_elCoJC)mX_TA
C(qwlC_JC)mYWI
C(IOwlCwJC)mYFTA
CONTINU_
CONTINUEQ0
I00
CC**** REENTRANT SEGMENTS ****
C
IF (NR$_G,EQ,0) GO TO 2"5DO 2_0 LmI,N_SEG
I_mLRI(L)÷I
16mL_ZCL)-II)mLII(L)_I
l_mLl)(t)-IIGUTOmLTYRE(L)
GO TO CllOwl)OillO_l?OelqOi210), IGOTO
C_*_ ONE ON BOTTOMp ONE 0_ TOP
tlO O0 I)0 lell,l&jml
XXll(X(l+l,l)-X(l-t_t))*O,5
YXlm(Y(I_I,I)-Y(I-I,I))*O,5XETAm(X(I,))-X(I,jwX"I))*O,)
YETAm(Y(I,2)-Y(IeJMX'I))*O,5
IlmI-II_sI÷l
JImjMXM!
J292CALL PaRAI (XXI$,VXISelE,J,TIeJ,XEIA),YETIB,I,J2,T,JI*XXlETI,Y
IXIETA,I_IJ2,12,JI,II,JI,II,J2,X,Y,I,j°NnlM}CALL PaRI2 (wxIwYWI_WETa,YETa,WXIS,YxI$,xETaS,Y_TIS_XXI_Ta,Yx!
IETa,ALPHA_@AMA,BFTA,IIG_TAU,jCB,I,J_N_I_)JCglICm!
C(I,IC,JC}mJCB
C(2,IC_JC)IAL)HAC(3,%C_JC)mBET*
C(§,IC_JC)I$IG
C(tmlC,JC)mTAU
C(?,I{,JC)mWWI
C(i_lC_JC)mXET_C(q_IC_JC)mYX!
120 CONTINUEGO TO _10
C**** BOTH 0N BOTTOM130 00 1_0 Iml$,l_
jol
II_li-(I-I_)
XXle(l(I*l,l)-l(I-l,l})*O,5Yllm{V(l$1,l}-V(I-l,l))_O,_
u21
"35_3b
_5(1
_SE_53
_55
_Sb_57
_5_q_sq
_b2
_e I]_bb
_bq_70
_7'__Te_7q
133
C**** BOTw150 DO
160
Ct*e_
170
XETAm(x(T,E)-X(TI,E))*O,5
Y_TAu(v{I,E)=Y(II,2))*O,5IlmI-I
12ml*i
JIull
J2=ECJLL PARAI {XXTS,YXI8,IE,J,TI,J,X_TAS,VETA$,I,J2,J1,J2,XxIETA,
IYWIETA,12,J2,JI"I,J2,JI+I,J2wII,J2,X,Y,T,JaNDI_)
CaLL Dt_a2 (XXI,YXI,XETA,YETa,wXI$,YXI$,XFTAS_YFTa$,XXIETA,YxI
IETA,ALPWJ,GAMA,BETA,_IG,TAU,JC_,I,JeN_I_)
JCB1IC,I
C(I,IC,JC)IJC8
CC_eIC,JC}sALRMAC(3,1C,JC}xBETA
C(_,IC_JC)BGAMA
C(5,1C,JC)mSIG
C(_jIC,JC)BTAU
C(7,IC,JC)wXxI
C(8,IC,JC)mWETAC(9pICtJC)BYXIC(IOwICeJC)mY[TACONTINUE
GO TO 2_0ON TOP
I_0 Tml_,I6JBJMAX
IImI_-(I-15)
XXIm(X(I÷!wJMAX)-X(I-IeJMAX))*O,5
YXIm(Y(I÷ItJMAX)-Y(I-IpJMAX)}*O,5
W_TAm(X(IIwJMXMI)-X(I.jMXMI))*O,5
YETAI(Y(ITeJWXM%}-Y(IwjMXW|))*O,5IlmI-I
12mI*!
JIwll
J2IJMXMI
CALL PARAI (XWI_eYXI_fI_wJwII,JwX[TASeYETA$1JteJ2,1pJ2,XXIETA,
CaLL PARA_ (XXI_YXI,XFTA,YETa_XXI8,YXT$_XETA$,YETI$,XWIETA,YXI
IET_ALPW_,GAMA,BET_w_IG_TAU_JCB,I_J,NDIM)JC=]ICuI
C(1,1C_JC)_JC_C(E,ICeJC}mALP_AC(3,IC_JC)mBET*C(_,IC_JC}wG_MIC(5,ICwJC]_SIGC(6_IC,JC}uTAU
C(?,IC_JC)mXXI
C(8_IC_JC}BXETA
C(q,IC,JC)wYX!C(IO,IC,JC)mVETA
CONTINU[
GO TO E30ONE ON LE_T_ ONE ON RIGMT
DO I_0 JwlS,I_
Iml
XXIm(XC_,J)-X(IMXMIwJ))*O,S
YXII(Y(_J)-Y(IMXMI_J))*O,S
XETAw(X(I,J*I)-X(I,J-I))*O,5
_S5
/_B¢;
Q92_'{;3
_gq50050150250_SOU50550_507
50q$10511512
515
515
511
S_O
S_5
5_?5ZSS_q5}0
531
SWO
134
180
1oo
2OO
C**t*ZlO
YETAm(Y(t,J*I)-Y(IjJ-i})*O.5I1mIMXW1IEmEJtmJ-%J2mJ*!CALL PARAI (XXISeYXIS,IE,J,I1,j,XFTAS,YETAS,I,j2,I,JIeXXIFTA,V
lXIETA,I2eJ2_I2eJl,ItwJ1PIlwJ21X,YeI,J,NnI')CALL PA_A2 (XXIsYXIeXETAwYETAoXXI_eYXTS,XETASjYFTASeXXIETAeYXI
IETAeALP_A,GA_A,BETAo$1G,TAU,jCB,I.J,_nI")JCm_ICmJC(IeICpJC)mJCBC(2eICwJC)eALP_AC($oIC,jC)m_ETAC(_elCwJC)mGA_A
C(S_ICpJC)m$1GC(6eICeJC)mTAUC(?eICfJC)sXXIC(8,IC,JC)mWET*C(,.IC.JC)mYXIC(IO_IC.JC)mYETACONTINUE
GO TO 2]0BOTM ON LEFT
DO _00 JoIS,16In!IIoIW-(J-15)
wXIu(X(_,J)-x(2,1T))*O.SYXID(Y(2pJ)-Y(_,II))*O.SXETkmfX(lwJ+1)-X(leJ-l))*O.qYETAmCY(I.J+I)-Y(I_J-I))*OeSIImll12m2
JlmJ-IJZtJ*iCALL PARA1 (WXI8,YXlS,I2,J,12,11_WET_9_VETAS,I,J2,I,JI,XXIETA,
tYXIETkeI2eJ2,1_,JIel)_II+leI2elIoleX,V_I,J,NDIM)CALL PIRA_ (XXI,YXI_XETA,YETA_XXI_,YWIS,X_TA$.YETA$_XWIETA,YXI
1ETA,AL_MA_GAMA_BETA.$IG_TAU_JCB_I_J,NDIM)JCm_ICmJC(I.IC.JC)mJCBC().IC_JC)mAL)HAC(),IC,JC)mBETAC(_.IC.JC)BGAMaC(S.IC_JC)mSIGC(_.IC_JC)ITAUC(?.IC.JC)mXWIC(O,IC,JC)mXETAc(g_Ic,JC)oYXIC(IO_IC_JC)BYETACONTINUE
60 TO _$0BOTW ON RIGHT
DO 210 JmIS,%6ImIMAXIImIQ-(J-IS)
XXII(X(IMXMI,II}-X(IMXMI_J)),O.SYXII(Y(IMXMI_II)-Y(IMXMt_j)),O._XET_m(X(IMAX,J.I)-X(I_AX_J.t)),OmS
5u25_3
5_55_6
5_c_55055155_55355_555556557
559SbO
563SbtacS6S1_6b56756856c_57O571572
575576S7T$7857O'580
50358_58S5865675_8S_959O$91Sgi_$93$9_595$9_
59_$9Ob00
135
vFTAs_Y(I_AX'J÷I)=Y(I_AX*J'I))eA,5 601It=IT _OETEsT_X_I _03
JlsJ=1 bU_
j2sj+! b05
CALL PA_AI (WXIS,VXI$,I_,II.12,j,yETAA,YETAS,I,JE,I,JI,WXIETA. 606IYXIFTA'IE'II'I*I2'II+I'I2'JI'I_'J2'W'Y'T"TI_I_) bO?
CALL P_Rj_ (XXI,YWI,XETJ,YE?A,XXIS_YWT$,WFTA$_yETA_oXXIETA,YXI b08IETAeALPWAeBAMAw_TAwSTGeTAUwJC_'I'JeN_I_) 60q
JCmQ blOICmJ ell
C(IelC.JC)mJCB 61ECCE,IC,JC)mALPMA 61)CC3,1C,JC)mBETA elQC(_,IC,JK)mGa,a 615
C(5,1C,JC)mBIG bibC(b,lC,JC)mTAU bit
C(TeICeJC)sXXI bl_C(R,TC,jC)mX_TA blg
C(Q,IC,JC)mYXT 620C(IOpICpJC)mY_TA 621
2_0 C_TINLJE 6E2
E]n CONTINUE 6E32_0 C_TINUE _
_a5 DO _6 _ml,lO 6_TF_ • C(_,I_I) * _(_,I,_) 626C(N,I_I) m TEM 6_7
C(N.Io_) • TE_ 628
?_M I C(NmJMAX,_) ÷ Cf_.I_3) _q
C(N,I,]) m TE _ 6}1
TE _ • C(_IWAW,]) ÷ C(N*JMAXeU) 63EC(N,IWIW_$) • T_M b33C(_JM_X,_) • TE_ 63_
C(_.l,_) • TE_ b}6
_A CONTINUE 6_
I_ (I_RT_,EQ,O) GO Th 2QO 63RwRITE (b,330) 6WOnn E_O Jml,_ 6_1
_RITE (6,3_0] SIDE(J) _E
I_ (J,EO.1) IM_XWi • IM_X b_$IF (JmEQ,)) I_aXWI s IMAX 6_IFCJ.EQ.E) IMAXXI • JMkX 6_IF(J.EQ._] l_axx! • JMaX 6W&
25h _RITE(b,IEO) FkC(N),(C(N,I,J).ImI_I_AWWI] b_8
GO TO 290 b_eC 6S0
C**** FIELD **** 6§1C 65_
_bO CONTINUE 6§)O0 _TO Jm_IJMXM| 6S_
JCmJ-I 655JMlmJ-t 6§6JPlmJ+[ 6)?DO E?O ImE,IWXMI &S8
ICml-l b§eIPlml+I bbO
136
_70C
*_ITEC
_BO
EgO
IPlmI-1WETAICXCIpJPl}-XCT,JMI))*Oo_
VfTAmCYCIojPl)mYCIpJ_I))t_),5
XXlmCX(I_1,J)=X(IM1_J))*O,5YXIm(Y(IPIeJ)-Y(IMIeJ))*O,5
ALP_AmXETA**2eYETA**2HETAmXXI*XETA+YXI*YFTA
GA_&mXXI*m2+vWI**2JCHmX_ImYETA-_ETA*YXIXXI_mX(IP1,J)-2,0*X(l,J)÷X(i_l,J)
yXISmY(IPI_j)-2,0*Y(IwJ)+Y(TMI,J)
WETA$mw(I,JPl}-2,0*X(TwJ)÷XfI,J_I_YETASmY(I,JPl)-_,O*Y(TpJ)+YtI,JMII
XXIETAmO,ES,CXCIPIPjpl)-X(IWI,JPII-WCTPI,J_I)+x(I_oJMI))
yWIETAmO,_,(Y(IPI,JPl)-Y(IMIwJPl)-Y(TP|pJMI)+Y_I"I*JU%))
DXmALPHA,XXI$.E,O,FFTA*XWIETA+GA_a*_ETA$
DYmALPWA,yxI$.2,0,@ETA*YxIETA+G_*YFTAB
8IGm(DX*VXI=DY*WXI)/JCBTAUm(DY*X_TA-DW*y_TA)IJ_R
ABSJmAHS(JC8)
CF(I,IC_JC)mjCBCFCE_IC,JC)BAL.PHA
CF($_IC_JC]mBET_
CF(_IC_JC)mGAMA
CF(I_IC,JC)m81C
CF(_IC_JC)mTAu
CF(7elC_JC)mXWI
CF(M,IC,JC)mXETA
CF(I_IC,JC)mYXlCF(IO_IC_JC)mYETA
CONTINUE
KACTOR$ TO PRINTER
I_ (IWRT_,EQ,O) GO TO I0
wRITE [b,)O0)
JWxwEmJMAX-2
IWXM2mIWkX-2
00 280 jmlmJMXM2JJmJ÷l
wRITE (b,$10) JJDO 280 Nml_lO
wRITE (_$20) FAC(N),(CW(N_I,J),ImI,I"XMEI
GO TO tOCONTINUERETURN
C$00 FORMkT (*0---- SCALE FACTORS ---=*)
_10 FORMAT (// * J m*_I)/)$20 FORMAT (*O*mA_t/(SEt_,8))$]0 FORMAT (// * BOUNOkRYm)]aO FORMAT (//2H *_A6)
CEND
fUNCTION _MWMN (NO_Te_,AWeI,J,IXeJX)
_b5_e,h¢",b ?e_bP_hoqb70
672_73_7_
e,75_7_hT?e,78_7q/_8t)
e_BE
(',B_
_8L&
e_8?
_.SR
bee
ag_
bgT
6(]B
?On
?0%
7O2
?03
?0/4
707
708
TlO
?11
?I$Tl(&
?I?
Tie
'V_O
137
G_ T0 {|0_20)e NOPT
10 IF _A,LT,AX) GO TO _n
GO TO 3hEO IF (A.GT.AX) GO Tn ._30 AMX_N=A
TXu!jXBj_ETUR_
.0 A_X_NIAX
RETIJRN
E_'O
SUBROUTTNE PARAI (XI,Yt,lt,J1,IP,j2,X_,Y_,T3,J3,1",J",X12jYt2,15,JIS*I_,J6,I?,J7,I8,J_,X,Y,Iwj,NDI_
*4*44,44********444 _ECONP OEWlV_T_V_S ,4,44,4*****4***4,44**44,4****
*********************************************************************
_IMEN$_ON X(NDIM_I)_ Y(ND_M_I)
XIBX{It,JI}-2.0*Xt_,j)÷X(_E,jE)vluY{II.JI)=E.O*Y(I.j)÷Y(TZ_J2)X_BX(T3_J3)-2.0*X{I.J)÷X(T,.J_)Y2BY(I3,J3)-_.O*Y{T,j)eY(;_,J_]XIEmO.2S*(X(TS,J_5-X(T_j6)+X(I?,JT)-X(TS,j8]]Y_2IO.ES4(YCIS_J_)-Y(T_J_)eY(I?_JT_-Y(TS_J8))RETURN
ENO
BUBROUTTNE PARA_ (XXI_YXI_XETA_YETA_XXIS,VXZS,XETAS_YETAS_XX_ETA_YIX_ETA_ALPWA_GAMA_BETA_SIG_TAU_jCB_j,NDZM}
C
C *C *********************************************************************C
RE_L JCB
ALPHAIXETA*I_eYETA**EBETA_XX_*XETA_YXI*¥ETAGAMAmXXI**_eYX%**_JCBIXXZ*YETAeXETA*YXZDXIALPHA*XX_8_,O*RETA*XX_ETAeG_MA_XETA$
?E!12E723
?E6
7Eq73073173E
7_6
?38"/)9
?#3
7_6
751
?$3
?557_6
?58759
76t?eET63
?iS
T68
?70771??l
??e
780
138
DYmALP_A,YXI$=2,0*R[?A*YXIETA$GA_A*YETA_$1Gi(DX*YWI-DY*WWT)/JCBTAUm(_YeX_TA-DW*YETA)IJC_HETU_N
END
Ct_t_
IF (KI.EQ.O) GO TO 10xXIm.O,5*(X(I],J_)-_,O*X(I2,J_)*],O*X(II,JI_)*KIYXII-O,S*(Y(I$_J])-_.O*Y(I2,J2)÷_,O*Y(II,,II))*KIGO TO _o
to XXlmO.S*(x(13,J])-x(It,Jt)}vxImO,S*(v(1),J$)-Y(If,Jt))
_0 I_ (_.EQ,O) _0 TO _0WETAm-O,S*(W(I_,j_)-_.O*X(IS,JS}÷3,0*x(I_,J_))*K_YETAm.O,5*(Y(16,J_)-_,O*Y(IS,JS)*$,O*v(T,,J_))*K2GO TO _0
$0 XETAmO,S*(X(Ib,Jb)-W(I_,J_))YETAmOeS_(Y(16,Jb)-Y(IQ,JQ))
QO CONTINUE
ALPMAmXETA**2$YETA**_
BETAmWXI*XETA_YXI*Y[T_
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END
*_t6H** YAR#_MRAY **/
OIMENSION CODE(_]
OIMENSION X(NDI_,I)DATA COOE/bH*_ XAR,&HRk v
C_
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SAMPLE CASE INPUT: SINGLE-BODY FIELD
Input for Program FATCAT requires only one card and is described beginningat line number 76 in FATCAT listing. For the sample case computed:
1 1 2
Disk file 1 needed as input is TAPE I0 saved from the TOMCAT program.
Sample Case Output: Single-Body Field
141
INPUT I ]M_TIm 1 ZxHT2m ] 1FO_Ma 2
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O_
Oe
O,
Oe
O,
Oe
Oo
Oe
O,
Oe
Oo
O|
Oo
Oe
O,
Oe
Oo
Oe
OI
Oe
O,
Oe
O,
Oe
O,
Oe
0,,
Oo
O_
147REFERENCES
i. Winslow, A. J., "Numerical Solution of the Quasi-Linear Poisson
Equation in a Non-Uniform Triangular Mesh," Journal of Computational
Physics, 2, 149 (1966).
2. Barfield, W. D., "An Optimal Mesh Generator for Lagrangian Hydrodynamic
Calculations in Two Space Dimensions," Journal of Computational Physics,
6, 417 (1970).
3. Chu, W. H., "Development of a General Finite Difference Approximation
for a General Domain, Part I: Machine Transformation," Journal of
Computational Physics, 8, 392 (1971).
4. Amsden, A. A. and Hirt, C. W., "A Simple Scheme for Generating General
Curvilinear Grids," Journal of Computational Physics, ii, 348 (1973).
5. Godunov, S. K. and Prokopov, G. P., "The Use of Moving Meshes in Gas
Dynamics Computations," USSR Computational Mathematics and Mathematical
Physics, 12, 182 (1972).
6. Stadius, G., "Construction of Orthogonal Curvilinear Meshes by Solving
Initial Value Problems," Dept. of Computer Sciences Report No. 53,
Uppsala University, Sweden (1974).
7. Meyder, R., "Solving the Conservation Equations in Fuel Rod Bundles
Exposed to Parallel Flow by Means of Curvilinear-Orthogonal Coor-
dinates," Journal of Computational Physics, 17, 53 (1975).
8. Ives, D. C., "A Modern Look at Conformal Mapping, Including Doubly
Connected Regions," AIAA Paper No. 75-842, AIAA 8th Fluid and Plasma
Dynamics Conference, Hartford, (1975).
9. Gal-Chen, T. and Somerville, R. C. J., "On the Use of a Coordinate Trans-
formation for the Solution of the Navier-Stokes Equations," Journal of
Computational Physics, 17, 209 (1975).
148
i0. Thompson, J. F., Thames, F. C., and Mastin, C. W., "Automatic Numerical
Generation of Body-Fitted Curvilinear Coordinate System for Field
Containing Any Number of Arbitrary Two-Dimensional Bodies," Journal of
Computational Physics, 15, 299, (1974).
ii. Thames, Frank C., "Numerical Solution of the Incompressible Navier-
Stokes Equations About Arbitrary Two-Dimensional Bodies," Ph.D. Disser-
tation, Mississippi State University, (1975).
12. Thames, F. C., Thompson, J. F., and Mastin, C. W., "Numerical Solution
of the Navier-Stokes Equations for Arbitrary Two-Dimensional Airfoils,"
Proceedings of NASA Conference on Aerodynamic Analyses Requiring
Advanced Computers, Langley Research Center, NASA SP-347, (1975).
13. Thompson, J. F., Thames, F. C., Mastin, C. W., and Shanks, S. P., "Numer-
ical Solutions of the Unsteady Navier-Stokes Equations for Arbitrary
Bodies Using Boundary-Fitted Curvilinear Coordinates," Proceedings of
Arizona/AFOSR Symposium on Unsteady Aerodynamics, Univ. of Arizona,
Tucson, Arizona, (1975).
14. Thompson, J. F., Thames, F. C., and Shanks, S. P., "Use of Numerically
Generated Body-Fitted Coordinate Systems for Solution of the Navier-
Stokes Equations," Proceedings of AIAA 2nd Computational Fluid Dynamics
Conference, Hartford, Connecticut, (1975).
15. Kolman, B., and Trench, W. F., Elementary Multivariable Calculus,
Academic Press, New York, (1970).
16. Levinson, N., and Redheffer, R. M., Co mi_ Variables, Holden Day
Nan Francisco, (1970).
17. Mastln, C. W., and Thompson, J. F., "Elliptic Systems and Numerical
Transformations," ICASEReport 76-14, NASA Langley Research Center (1976).
18.
19.
14N
Varga, R. L., Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood
Cliffs, New Jersey, (1962).
Thompson, J. F., and Thames, F. C., "Numerical Solution for Potential
Flow about Arbitrary Two-Dimensional Bodies Using Body-Fitted Coordinate
Systems," NASA CR to be released 1976.
20. Karamacheti, K., Principles of Ideal-Fluid Aerodynamics, John Wiley &
Sons, Inc., (1966).
21. Liebeck, R. H., Wind Tunnel Tests of Two Airfoils Designed for High
Lift Without Separation in Incompressible Flow, McDonnel Douglas
Corporation, Report No. MDC-J5667/OI, (1972). See also, Liebeck, R. H.,
"A Class of Airfoils Designed for High Lift in Incompressible Flow,"
AIAA Paper 73-86, AIAA llth Aerospace Sciences Meeting, Washington, 1973.
22. Morse, P. M. K., and Feshbach, H., Methods of Theoretical Physics, Vol.
II, McGraw-Hill, (1953).
23. Hodge, J. K., "Numerical Solution in Natural Curvilinear Coordinates of
Incompressible Laminar Flow Through Arbitrary Two-Dimensional and
Axisymmetric Bodies," Ph.D. Dissertation, Mississippi State University,
(1975).
24. McWhorter, J. C., Unpublished Research, Department of Aerophsycis and
Aerospace Engineering, Mississippi State University, (1975).
25. Timoshenko, S., Theory of Plates and Shells, McGraw-Hill Book Company,
Inc., 1940.
26. Thompson, J. F., Thames, F. C., Shanks, S. P., Reddy, R. N., and Mastin,
C. W., "Solutions of the Navier-Stokes Equations in Various Flow Regimes
on Fields Containing Any Number of Arbitrary Bodies Using Boundary-Fitted
Coordinate Systems," Proceeding of V International Conference on Numerical
Methods in Fluid Dynamics, to be published in Lecture Notes in Physics,
Springer Verlag, (1976).
150
0
0
,-q
or_
_ ocq
_J
_ _o
0
L_
_3 _q
!0
r_
0=0
_o,-qu_
_o
IM
_g
!0
e,3
I0
0t_
,-4 ,--I
_oo
!0
!o
_-4_
& &
I I0 0
o _
0 0
g
Iop..
_.--t o4 e-i o
,-4
,-,4
"O
OL3,-4
O
¢',1
t-,I
H
t.,I
o'J
I-i
!O
oh
Io
O
O
IO
!Of-.
oI--.
o
Ot_
O
¢',1
O
O
OC0 _D
_NOO
,4
o
_OO
,-IO
t,h
,.-I
_. _,_ "__ _ I0 _ N
0
_OM
_.._
0,.,_
• lJ ,_ 0
• N __ m 0
_ ° _
•_ O -N
151
•,'4 O00 l_00000000 00 _ ....... 0 .00
t_,l ,--I > ,--I O O _'_ O O O O
!
0
o
152
TABLE 2
Effect of Initial Guess on Convergence
Configuration
(Fig. 4a)
(Fig. 5)
(Fig. 6)
(Fig. 7)
(Fig. 8)
(Fig. 9)
(Fig. i0)
(Fig. ii)
(Fig. 12)
(Fig. 13)
IGES :
0 i 2 3 -i -4 -i0 -20 -40 -i00
68
75
NC
NC
i00+
i00+
99
NC
NC
NC
NC 80 NC NC
NC 64 NC i00+
NC 1 NC NC NC
NC NC NC 2 NC
i00+ i00+ NC NC
I00+ i00+ NC NC
89 99 NC NC
i00+ i00+ NC NC
NC NC NC NC
NC I00+ NC NC
NC
i00+
NC
NC
i00+
i00+
NC -- 69 71
70 63 67 68
NC NC NC NC
NC NC NC NC
i00+ 87 84 86
NC NC NC NC
i00+ 91 84 82 86
i00+ i00 93 91 94
i00+ i00+ i00+ I00+ i00+
i00+ i00+ i00+ i00+ 99
Legend:
Notes:
Numbers given are the number of iterations required for convergence
to 0.0001.
NC indicates no tendency toward convergence in I00 iterations.
i00+ indicates a converging solution at i00 iterations.
IIGES = 1 gave convergence for the field shown in Figure 6.
2IGES = 4 (Guess 3 without division by total number of boundary
points) gave convergence for the field shown in Figure 7.
TABLE3
Input Parameters for Basic Single-Body Field Used inOptimumAcceleration Parameter Studies
Case 1
153
IMAX 67
JMAX 40
NBDY 1
IGES 0
NBSEG 2
NRSEG 1
LBSID 1
LBI 1
LB2 67
LBDY 1
LRSID 2
LRI 1
LR2 40
LISID 4
LII 1
LI2 40
R(1) varied
R(2) 0.0001
R(3) 0.0001
YINFIN i0.0
AINFIN 0.0
XOINF 0.0
YOINF 0.0
NINF 66
3
1
67
0
154
TABLE3 Continued
IEV 0
R(10) 0.0
INFAC 0
INFACO 0
ATYP ETA
ITYP 0
NLN i
NPT 0
DEC 0.0
AMPFAC 0.0
JLN i
ALN i000.0
DLN i. 0
ATYP 0
ITYP 0
NLN 0
NPT 0
DEC 0.0
AMPFAC 0.0
IFAC 0
EFAC i00.0
Case
2
3
4
5
6
7
8
9
i0
ii
12
13
14
15
16
17
18
19
20
21
TABLE4
Input Parameters Varied for Single-Body Field Used inOptimumAcceleration Parameter Studies
IMAX
37
97
37
37
97
97
JMAX LB2 LR2 LI2
37
37
97
97
20 20 20
60 60 60
37
37
37
37
97
97
97
97
20 20 20
20 20 20
60 60 60
60 60 60
YINFIN NINF ALN
I00.0
i0000.0
36
96
5.0
20.0
36 I00.0
36 i0000.0
96 i00.0
96 i0000.0
i00.0
i0000.0
i00.0
i0000.0
5.0 i00.0
5.0 i0000.0
20.0 i00.0
20.0 i0000.0
IFAC
4
4
4
4
155
156
Case IMAX JMAX
22
23
24
25
26
27
28
29
3O
31
32
33
37 20
97 20
37 60
97 60
37
97
37
97
LB2
3737
9797
3737
9797
3737
9797
3737
9797
TABLE
LR2
20
20
60
60
4 Continued
LI2
20
20
6O
60
20 20 20
60 60 60
20 20 20
60 60 60
YINFIN NINF ALN
36
96
36
96
5.0 36
5.0 96
20.0 36
20.0 96
5.0
5.0
20.0
20.0
IFAC
Case
34
35
36
37
38
39
NLN ALN DLN IFAC EFAC
i0000.0 i
i0000.0 2
i0000.0 3
i0000.0 5
i0000.0 4 i000.0
i0000.0 4 i0.0
157
TABLE4 Continued
Case NLN
40 2
41 3
ALN DLN IFAC EFAC
i000.0, i000.0 1.0, 1.0
i000.0, i000.0, i000.0 1.0, 1.0, 1.0
NOTE: Blanks indicate the basic value (Case i, see Table 3).
158
TABLE5
Input Parameters for Basic Double-Body FieldUsed in OptimumAcceleration Parameter Studies
Case42
IMAX 93
JMAX 40
NBDY 2
IGES 0
NBSEG 4
NRSEG 2
LBSID i i i 3
LBI 1 75 29 1
LB2 19 93 65 93
LBDY I i 2 0
LRSID i 2
LRI 19 i
LR2 29 40
LISID i 4
LII 65 i
LI2 75 40
R(1) varied
R(2) 0.0001
R(3) 0.0001
YINFIN i0.0
AINFIN -30.0
XOINF 0.0
YOINF 0.0
NINF 92
159
TABLE5 Continued
IEV
R(i0)
INFAC
INFACO
ATYP
ITYP
NLN
NPT
DEC
AMPFAC
JLN
ALN
DLN
ATYP
ITYP
NLN
NPT
DEC
AMPFAC
I FAC
EFAC
0
0.0
0
0
ETA
0
i
0
0.0
0.0
i
i000.0
1.0
0
0
0
0
0.0
0.0
3
i00.0
160
TABLE6
Input Parameters Varied for Double-Body Field Used in OptimumAcceleration Parameter Studies
Case IMAX JMAX LBI LB2
45
44
47
46
49
48
60
43
20
LRI LR2 LII LI2
19 29 65 751 60 i 60
19 29 65 751 20 1 20
i 33 33 43 107 117149 117 149 i 40 i 40
43 107i 149
YINFIN NINF ALN IFAC
i0000.0 6
i00.0 1
20.0
5.0
148
Case NLN ALN DLN IFAC EFAC
54 I0000.0 6 i0.0
53 i0000.0 5 i0.0
52 i0000.0 5
51 lO000.0 4
50 i0000.0 3
Note: Blanks indicate the basic value (Case 42, see Table 5).
161
O"
,-4_0
oD
O
O,H_._
.H
>
4;
.,--I
r_ O
.r..I
O
O
(D
U
O
r/l
O
"_ Z ¢1_ 4-1
ffl
O
•,_ Z
0
_J
z0
•el 4-
r/)
z
"HO
O_ OO r--OO O_ o_
u_ r-. Ooo _O o-,
oo-..i-I'--.,.o r-. oh
,-I
r-- -._- ooP-. oo oo
oool_O', O_ _-..I
o'1 r-- Ooo oo O'_
O -._ '.OO_ I'-. cxl
,..-I
oo oo oo
,4,4,4
r-- oo ._r
oo _.. ,._oo oo oo
,-I ,-_ ,-I
oo i-.. ,.._
Lth ..-r _oo oo oo
l> ooi-i o'_C_
,-I r-.. ooo_ oo oo
,-I ,-4 ,-I
O r--. oo,-_
O -.T -._"oo oo oo
I_ "_ oo :n
4-;
•r..I _
O O _1 _ O _ _ .;.-I
•_ m m
_ _ .-_._
CO _
•..T _,D
z v_
._ II II IItY
OOOO c'q _T ',.O
.U II II II
1-14-1
u_O
_.1
II II II
nnl "_ nn+
_.J _.J 4J
,-_ ,-_ ,_
_ vO O O
O
0OJ
_ _) _-_
O_t:D
O
Lth ,_ ¢',,Ik_
II U II O,-W
_ _ _ o o_•,_-r_ ._ _ _._ _J
O0 _-_ 0"_ +.--I 0
162
TABLE 8
Optimum Acceleration Parameters for Single-Body
Field : Variation of Pairs of Quantities
(12) IMAX Up
Amplitude Down
1.89 : 122
(1.90 : >122)
(2) Amplitude Down
1.85 : 81
(i0) IMAX Down
Amplitude Down
1.79 : 63
(1.85 : 92)
(5) IMAX Up
1.88 : 137
(i) Basic
1.84 : 74
(4) IMAX Down
1.77 : 68
(13) IMAX Up
Amplitude Up
1.86 : 183
(1.89 : >183)
(3) Amplitude Up
1.86 : 114
(ii) IMAX Down
Amplitude Up
1.83 : 86
(1.84 : 83)
(16) JMAX Up
Amplitude Down1.86 : 129
(1.90 : DIV)
(2) Amplitude Down
1.85 : 81
(14) JMAX Down
Amplitude Down
1.84 : 89
(1.84 : 94)
(7) JMAX Up
1.85 : 126
(I) Basic
1.84 : 74
(6) JMAX Down
i .81 : 90
(17) JMAX Up
Amplitude Up1.87 : 147
(1.90 : DIV)
(3) Amplitude Up
1.86 : 114
(15) JMAX Down
Amplitude Up
1.81 : 139
(1.81 : >139)
(20) Radius Up
Amplitude Down1.89 : 131
(1.88 : >131)
(2) Amplitude Down
1.85 : 81
(18) Radius Down
Amplitude Down
1.80 : 185
(1.91 : DIV)
(9) Radius Up
1.84 : 205
(i) Basic
1.84 : 74
(8) Radius Down
1.80 : 184
(21) Radius Up
Amplitude Up1.81 : 243
(1.88 : DIV)
(3) Amplitude Up1.86 : 114
(19) Radius Down
Amplitude Up
1.81 : 176
(1.91 : DIV)
163
TABLE8 Continued
(24) JMAXUpIMAXDown1.83 : 77
(1.89 : >i00)
(4) IMAXDown1.77 : 68
(22) JMAXDownIMAXDown1.71 : 48
(1.76 : 62)
(7) JMAXUp
1.85 : 126
(i) Basici. 84 : 74
(6) JMAXDown
1.81 : 90
(25) JMAXUpI_IAXUp1.89 : 128
(1.91 : DIV)
(5) IMAXUp1.88 : 137
(23) JMAXDownIMAXUp1.84 : 142
(1.87 : >142)
(28) Radius UpIMAXDown1.81 : 77
(1.84 : 92)
(4) IMAXDown1.77 : 68
(26) Radius DownIMAXDown1.78 : 102
(1.84 : >102)
(9) Radius Up
1.84 : 205
(29) Radius UpIMAXUp1.87 : 219
(1.90 : >219)
(i) Basic1.84 : 74
(8) Radius Down
1.80 : 184
(5) IMAXUp1.88 : 137
(27) Radius DownIMAXUp1.82 : 285
(1.93 : DIV)
(32) Radius UpJMAXDown1.82 : 117
(1.84 : >117)
(6) JMAXDown1.81 : 90
(30) Radius DownJMAXDown1.80 : 176
(1.91 : DIV)
(9) Radius Up
1.84 : 205
(33) Radius UpJMAXUp1.89 : 103
(1.90 : >103)
(i) Basic1.84 : 74
(8) Radius Down
1.80 : 184
(7) JMAXUp1.85 : 126
(31) Radius DownJMAXUp1.58 : 218
(1.91 : DIV)
164
TABLE8 CoNtinued
Format: (Case Number)CaseOptimumAcceleration Parameter: Numberof Iterations(Average Variable Acceleration Parameter: Numberof Iterations)
SeeTables 3-4 for values of the quantities varied.
165
TABLE9
OptimumAcceleration Parameterfor Single-Body Field : Miscellaneous Variations
CaseNumber
3435363
37
383
39
i4O41
Case
Experimental Optimum
OptimumAcceleration
Parameter
Numberof
Iterations
Effect of Numberof Steps in Addition of InhomogeneousTerm (Attraction Up)
One StepTwo StepThree StepFour StepFive Step
1.351.611.851.861.85
561317119114125
Effect of Intermediate ConvergenceCriterion (Attraction Up)
Convergence : -01Convergence : -02Convergence : -03
1.861.861.85
i01114167
Effect of Numberof Attraction Lines (Basic)i
One Line 1.84
Two Lines 1.83
Three Lines 1.83
I
Effect of Convergence Criterion (Basic)
Convergence : -03
Convergence : -04
Convergence : -05
Convergence : -06
1.84
1.84
1.84
1.89
74
76
79
49
74
117
175
166
(U-r-I
O
IO
0
m
0.N
,-.-t
<
E
.el
0
.-4
0
ia
(Q
0
,.Q u_ .I-II_ 0 _
Z,LIH
0
o_<
_DIn
c.D
Z
.r_Ou_
<
O
(n4-J
.r4O_U
O
Z
O
"OO
O_ .-Too O,.4 c_
-,1" _OoO oo
,.-I ,-4
-r_
m m__1 4.1
•r't ._0 0
r_ u_
_.1 o'3
0
u,-4
c'_ O', oOO oo c,'_eq ,-.I c..I
o0 -.T u%r-... 00 O0
II II II
-<t" c.l u'_
_ oo u'3
,--t r-t ,--t
°_ O0
4-1 ,--_ ,--t ,--t4-1"_ II II II
O "_D "_D '_
4J .Ira 4-_ 4-1(0 -r4 -,q .,-4(D _-I ,--4,-4
_4D oq r----.T -.._-.T
tOm.,-4
O
Q)4-J
O
,4-iO
J.J
O
q_q_
u_ Ch c_oO oO _Dc-q ,--Ieq
,-4 -.I"_D_._ cO oo
0 c_ _:_,
•r-I -_ .,-4
_ c-4
II U II
"_ "tJ "_
O _-I eq r--u-% u-.lu_ ..-i-
167
J_J
0
O
4.J
O
OCD ._
_ O _
Z4.J
O
0
r/l
O
O
4.J
O
(DO
0r_
.r..I
0
4_1
0
_. u'_ _- _
0'_ o'_ O_ o_
0000I I I I
_.) r,..) 0 0
g-
> > N N
168
TABLE ii
Input Parameters for Coordinate System Control Demonstration of Figure 27
IMAX = 31, JMAX = 30, YINFIN = I0, AINFIN = 0.0
(a.) No Attraction
(b.) ETA Attraction: JLN = 20, ALN = I0.0, DLN = 1.0
(c.) ETA Attraction: IPT = i, JPT = 20, APT = i0.0, DPT = 1.0
IPT = 16, JPT = 20, APT = i0.0, DPT = 1.0
(d.) ETA Attraction: IPT = 5, JPT = 2, APT = i000.0, DPT = 1.0
IPT = 6, JPT = 2, APT = i000.0, DPT = i00.0
(e.) XI Attraction: JLN = 2, ALN = i00.0, DLN = 1.0
(f.) XI Attraction: IPT = 5, JPT = i, APT = i000.0, DPT = 1.0
169
_D
0U.4
C8
0
0
4..1 _
0og
,-4
,-4
0 _ 0 _"-t
-,-i
_ "_ 0
0 _ _
•N _ _ O
_ .H ,-4
oo_
0 .I-I 0 ,-I_._
_o-__o
"O,-4 _-H-H O
O_-_O _
•,_ _ _:_ O,-4-,-4 _
_-_,_
. O _
0
0•,'4 I_
0
0•_ 0
•,-I "_
0 _
,-4 0
0 ",-4O
O _ O¢J O
O
O "_
O ,-40 °
_ O',_ _O m O
4_ O O _ _
4-_ O_O _ O_0 _ _,--I•I-_ 0 '4-1
¢xl _ 0 .,-I
I"-,.
o
,-_ _ _
._
u_ _ O
o4
(D
I--I _ r--
•_ O
_ _._ _ _
0 -I_ ._ _O,_U_ t_ 0
(D
m
170
Physical Plane
q=q2
_=N 1
n
i"2
I ' I
Region D*
II il
rl
Transformed Plane
F 3
Figure I. Field Transformation - Single Body
171
Physical Plane
N=D2
F4
_=n I
_2
I ' I I
Region D*
I
J
F7 F 5 FI F 6 F 8
F3
Transformed Plane
Figure 2. Field Transformation - Multiple Bodies
172
j=JMAX
j=l
i=l
I
] ..... i _ ..... T
t
] (
,""I I " I I I I I"_ ,, ,,[III _,_'
L
i=IMAX
a) Single Body
j =JMAX
j=l
" " 7 ........... _ "" ¶ ....... T
I
I
• J.... , , ," , ...... , ,
.... r III,,II
i Ii=l I1 I2 I3
I
I
J,
I
b) Two Body
Figure 3. Computational Grids - Single and Two Body Regions
173
37
1 37
Figure 4a. Single-Body Configuration
174
g
g
4-1
oo .,-Io
4 ,I
1I
i
o-1-t.,_
_0.1-1
=o
o
I_J
_J
175
8!
2 16 _ ! s_ 86 2 81
!
Figure 5. Double-Body Segment Configuration #i
176
31 2L _L31
2
\
Figure 6. Double-Body Segment Configuration #2
177
,oo
11 2 _I
11 i _I
I
Figure 7. Double-Body Segment Configuration #3
178
16
.......................................... i
51
16
2
! !
/
\ /\
Figure 8. Double-Body Segment Configuration #4
179
11l
I
L1
_03!
1 31
26
11
//
\
Figure 9. Double-Body Segment Configuration #5
180
2_ 51 26
2
i 71
2
l',ilk
\
2
281
2
01
i
Figure ii. Double-Body Segment Configuration #7
182
21
51
Figure 12. Double-Body Segment Configuration #8
183
16 76 2
_6 86 !Ol
Figure 13. Double-Body Segment Configuration #9
184
0
L
, ++ _" - I,
. _ _----,.. --
z _
7
-4O
" 7-_- I-
<
Figure 14. Initial Guess Types - Single Body
185
2
_'liI//II//M,I ! ! i ! _t//,
ili_,/iI,,'//_/(,Ait!,i,O7,'//,
_////." ;&<./.K7_',/ /// 4.'/.."_
!1
Figure 15. Initial Guess Types - Double-Body Segment Configuration #i
186
\
1
/
/
/
-4
'\_/ \ \/" // " ,\....,- X \, /
<,: '" 1 \ X
Figure 16. Initial Guess Types - Double-Body Segment Configuration #2
187
\ ,,, / /
' // i/
2
¢'" "_" 4, ', ,, ,,
\¢i . ,....... ',
x - /
"" i' / ,, t ,\ -4
\ ,/" / i j\d / ,_
Figure 17. Initial Guess Types - Double-Body Segment Configuration #3
188
.J
/ ,
j_
t-..
. -2
7,
/
2
1 i
i!'.-/// //
./
3
Figure 18. Initial Guess Types - Double-Body Segment Configuration #4
189
0
i _*t_ / '
-4O
Figure 19. Initial Guess Types - Double-Body Segment Configuration #5
190
i"" _"
\.
A,
\
.%/]
-40
Figure 20. Initial Guess Types - Double-Body Segment Configuration #6
191
3
/-
.fJ
\
\_///Y'
Figure 21. Initial Guess Types - Double-Body Segment Configuration #7
192
0
/ , .,/x\
.-/
/
• / /
/-"
_k
j_'TC-
," // /
,, / /' _x
J
_. y ,t
\\,_ \
Figure 22. Initial Guess Types - Double-Body Segment Configuration #8
193
o< i/
196
50_ 0 BASIC, ONE LINE GTTRRCTION
_SD'_
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.oL
100_-
50._-
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RCCELERRT ION Pq_RMETER
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IOPTIMUM - 1.80O. I 1 I I I I 1 I I!.50 1.55 1.60 1.65 1.70 1.75 1.80 1,85 1.90 1._5 2,]0
ACCELERATION PARAMETER
500
t°_50
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= 2soL.
._ ZOO F
150 I
fOOL
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50" _OPTI_UM . 1.8tO. I I I I t I I 1
1.50 1.55 1.60 1.65 1.70 1.75 1,80 1.85 I.gD 1.95 2.00ACCELERATION PARAMETER
(b)
ioo 160 r
l_ok
£
ioo_ i i= eo.b
i80. m i
20. _
Io-I_O,- _.-O. I I I I I l I
1.50 i.55 1.60 1.65 1,70 1.75 1.80 1.85 1.90 1.95 2.00ACCELERATION PARAMETER
500
_50 t
_O0_-
Is5oL
I
i::i= 2ooL
(d)
,o.LO, _ I 1 I t 1 I I
1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 ?.00_CCELERRT ION PARAMETER
(e) (f)
Figure 26. Effect of Acceleration Parameter on Convergence Rate
197
(e)
Figure 27.
(f)
/
\
Examples of Coordinate System Control
CSee Table ii for control parameters.)
198
, , P?_/_./,:-" . ..,,/..J_/_.X3J_ L !,'_, ,_v' _ ..... ' -_,
"-'_'I., <"X.<k"'v_ '
(a)
__L_._., __,.,_. , _,._ , , ¢ - ,,? ,,:',,, ,_ ,,
• _,_, ,_ ,I, ,
(b)
Figure 28. Example of Attraction into Concave Region
199
Figure 29. Coordinate Systems- Circular Cylinder And JoukowskiAirfoil
200
(a)
J
\
(b)
\
Figure 30. Coordinate Systems - Karman-Trefftz Airfoils, No Coordinate Line
Contraction
201
Figure 31. Coordinate Systems - Karman-Trefftz Airfoils, With Coordinate Line
Contraction
202
\
, .÷f_
4
1l
/
/
,/
_f
l;7-j-7.7/
J
1-
/
ou_
.H
I
o)4J
oo
4-1
4-1
o_)
_)°H
203
"'4
/
/
g.._
t-""
/
"\
:7::7/--i1
c,I
,--I,.Q
00
0u,-i
M°N
0
4..i
I
m
m
4.1
°_..t
00
1.1
or_
204
i
/
i
\
J
,.Q
[._
0
oo
00
<
Z
I
00
4J0
._J
0
205
| m : _ 2 sin' i .T ! _ 1_
Figure 35. Contracted Coordinate System - Multiple Airfoil
Coordinate line attraction to the first i0 lines around the airfoils
with amplitude i0,000 on the body. Decay factor of 1.0 for all except
last line, where 0.5 was used. 37 points on airfoils, 40 lines around
airfoil. Circular boundary of radius i0 fore body chords.
206
Figure 36. Coordinate System - Triangular Simply-Connected Region.
¢O
! !! i
i ii ii Id
i 21
207
Figure 37. Coordinate System - Key-seat Shaft, Simply-Connected Region
(See Table iZ)
2O8
\
o
0o
o
I
4Jm
m
4.1
.N
I-ioo
4_1
4-)
o
t-¢3
.1-4
2O9
Figure 39.
LIFT COEFFICIENT I 2. 535Ii9t_
ORRG C3EFFIC!ENT - 0.0039¼6
L:=T C_;EFFICIENT EBR_ I --0.00S_0_
I. 00qa
-°-°° r
b r_
\ :0I
-2.00 L
I
I _ Numert¢il Solu¢lo
-3.00 II -- Analytic Solution I-I,O0 -0.00 I.OO
X
Comparison of Numerical and Analytic Potential Flow Solutions for
Flapped Karman-Trefftz Airfoil. No coordinate attraction.
210
c_c_
....i i.........i.....!.........i. " " _. '_' ._.i i _ ! i LIEBECK LAMINAR AIRFOIL
o i :: ! i ! -- NUMERICAL0 : : ; ;
d re TEST DATA [2L]' i
• . . .
] ..... i.....!.....i ....i....
: i : i$ .... i .... ,_..... ;
i....!
.....i.....i....i.....)
I
I ..................... i
ooi....i....._....i....i....._....i....i.....!....i.........._.-_..-.I .....i....i....i.....i....!• ! :, i _ i ! i _ ! ! _ihl_-.,l:/_,i ! _ i I
° ....i....i.....ii.............i.....i....i!.........i....i.....i..-......i.._ " il
,0.00 0.25 0.50 O.?5 :.DO
X/C - DIMEN$1ONLESB CHORD
Figure 40. Comparison of Experimental ,_nd Numerical
Pressure DistributLon - l,ieback Laminar
Airfoil
211
I_j_y.F_/_-_._ _ _ ._-_------_._.._.__I__--_ _ _-_..._
-O.t_L.
-$._7 b
e. "|$'_-II.I
-:S._
-ll.I,
-LT.q
-10. ill
-IO.I
-ZI.I
-H.I
-il.q
"M.I
-K.I
"N.I
-N.Ig.M
-0.0_
-t.O_
-$._
-q. O(
$U° .
I II ; ._1. OdO.fHD I.N -I.M -LID O.N
II meql,. IlI
Figure 41. Potential Flow Streamlines and Pressure Distributions
- Double Airfoil
212
co
Figure 42. Coordinate System - Double Cylinders, Coordinate Attraction to Cut
Intersections
Attraction to intersections of body contours with cut between at
amplitude i000 and decay factor 0.5. 31 points on each obdy. 40
lines around the bodies. Outer boundary at 20 diameters.
(a) 213
f.
(b) I-! NUMERICAL- ANALYTIC
-5.00
Figure 43.
-0.00X
Potential Flow Streamlines and
Pressure Distribution - Double Cylinder
214
BSOT
-,.,%-_.90_-
-5.90_.
-6.90L-
90L _r-
L
-9.90_-
- I O. 9(],.
-_z.*_- [] NUMERICAL
-_3.9_- -- ANALYTIC-lll. g_.
-15.9_-16.9
,,,L _oo-0,50
%
0.50
Figure 44. Potential Flow Pressure Distribution for Double Cylinder without
Coordinate Attraction
215
qD
Figure 45. Alternate Segment Arrangement - Double Cylinders
No coordinate attraction.
216
-o,ooI
-1.ooI
-2-°°I
-3.00
-_,. 00
-5.00 [] NUMERICAL
-- ANALYTIC
-6.00 I-1.00 -O,O0 1.00
X
Figure 46. Potential Flow Pressure Distribution for Double Cylinders - Alternate
Segment Arrangement
217
t=3.13
t=3.33
Figure 47. Streamlines - G_ttingen 625 Airfoil, R = 2000, e = 5 °
218
Figure 48. Velocity Profiles, Gottingen 625 Airfoil, R = 2000, _ = 5 ° t = 1 83
t = 0.118 C, =0._15916
O
CD=O.c-_a),u CDp=O- CO ='-'l.l]7Ei31S
I ]i
Z
(-)
C.
cr_
i
=P
?
C_
0
! ...... :
i _ ! _ i . i
iiV_____ ___-----------
l
I
i
"i_N'j. O0
r
0.25 O. 50 O, 75 1.00
X/C - DIMENSIONLESS CHORD
219
W
O_
L_
t = 3.33 CL=0"83051i
CD:0.258096 CDp=0.090_55 CDF:O.1677u, I(::)
"T'l _ !
I
• t _ t t_
!If- ! - , ,...... E0
II
- i I
i I '
_ioo o125 olso o'• ?5
X/C - DIMENSIONLESS CHORD
.00
Figure 49. Pressure Distribution - G_ttingen 625 Airfoil,
R = 2000, _ = 5 °, t = 0.118 and 3.33
220
Figure 50. Streamlines and Vorticity Contours - Rock, R = 500, t = 0.5
221
Figure 51. Velocity Profiles - Rock, R - 500, t - 1.2
222
1.00 1.00
:. -O.CO_- -o.oaL
-1.00L ] -I.00 I i0.50 0.80 ;.00 -0.60 -1 ) -0.20 -,l,OC 9._0
X
Figure 52a. Pressure Distribution and Velocity Vectors - Double Airfoil,
R = 1000, t = 0.i
223
2,00
F)i
;.00 _-
_. -o.ooL
-,.ooL
\\
i
'/
B_OY z1.00
LI
,._-0.00_
2.001 I -_.ooi I l ; ' ;0.60 0.80 1.00 -0.'$0 -0.'¢0 -0.20 -0.00 ')._0 O.WO
I X
Figure 52b. Pressure Distribution and Velocity Vectors - Double Airfoil,
R = i000, t = 1.2
224
BOD_ !1.00
-O,OC
-2.00
t
i
-3.00 I0.60 0.80 1.00
X
1.00
-0"00 1
-I.O¢ L 1 1 A I
-0.60 -0,_0 -0.20 -0._ 0.20 0._0x
Figure 52c. Pressure Distribution and Velocity Vectors - Double Airfoil,
R = i000, t = 3.02
225
Fig. 53. Comparisonof Numerical and Analytic Solution for DeflectionContours for a Simply-Supported Uniformly Loaded Plate.
APPENDIX A
DERIVATIVES AND VECTORS IN THE TRANSFORMED PLANE
This appendix contains a comprehensive set of relations in the trans-
formed [_,q] plane. A few relations involving x and y derivatives of the
coordinate functions _(x,y) and n(x,y) are also included. Since the intent
here is to provide a quick reference only, most of the algebraic development
is omitted. The following function definitions are applicable throughout this
appendix:
f(x,y,t) - a twice continuously differentiable scalar function of x, y,
and t.
F(x,y) = i Fl(X,y) + j F2(x,y) - a continuously differentiable vector-
valued function of x and y.
coordinate unit vectors.
J = x_y n - xny _
= xn2 + yn2
= x_x n + Y_Y_
y = x2 + y2
Dx = _x_ --2BXEn + yxnn
Dy = _y_ - 2By_ + YYnn
= (y_Dx - x_Dy)/J
= (xnDy - ynDx)/J
i and j are the conventional cartesian
A-I
A-2
Derivative Transformations
fx _ (_f) = (Yqf_ - Y_fq)/J_x y,t(A.1)
fy _ (_f) = xqf_)/J_y x,t (x_fq -(A. 2)
_f = _f 1ft -- (_-)x,y (_t)_,q - 7 (Yqf_
_x
- y_fq) (-_-)_, q
1 (x_fq (3_-_t)- _ - xqf_) _, q(A.3)
._2f. = (yq2f_ + /j2fxx E (_x2)Y,t - 2ysyqfsq y_2fqn)
+ [(yq2y_ _ 2ysynysq + y 2yqq)(xqf_ _ x_fq)
+ (yq2x_ - 2y_yqx_q + y_2Xnq)(y_f n - yqf_)]/j3 (A.4)
(_2f) = - 2x_xqf_q + Ij2fYY _y2 x,t (xB2f_ x_2fqq )
+ [(xq2y_ - 2x_xqy_q + x_2yqq)(Xnf _ - x_f )n
+ (xn2x_ - 2x_xqx_q + x_2xqq)(y_fq - yqf_)]/j3 (A.5)
fxy = [(x_yq + xqy_)f_q - x_y_fnq - xnyqf_]/J2
+ [xqyqx_ - (x_yq + xqyE)x_q + x_y_xqq](yqf_ - y_fq)/j3
+ [xqyqy$_ - (x_yq + xqy_)ysq + x_y_yqq](x_fq - xqf_)/J 3 (A.6)
A-3
Derivatives of _(x,y) and n(x,y)
_x = Yn/J (A.7)
_y -xn/J (A.8)
nx = -y_/J (A.9)
ny = x_/J (A. 10)
Cxx = (¢xYcn + nxYnn)/J - (_x2J¢ + CxnxJn) IJ (A.II)
Cyy - - (_yX + CyXcq)IJ - (¢ynyJ B + Sy2jE)/j (A.12)
_xy " (nyYnn + _yYcn )/J - (_xCyJ¢ + _xnxJn )IJ (A.13)
nxx = _ (¢xy¢¢ + nxYcn)IJ - (¢xnxJ_ + nx2Jn)/J (A. 14)
nyy = (nyXcq + CyX¢_)/J - (_ynyJ¢ + ny2j )/J (A.15)
nxy = - (nyYE_ + _yYE¢)/J - (nxnyJ n + CynxJ$)/J (A.16)
A-4
Vector Derivative Transformations
• Laplacian:
V2f = (af_$ - 28f_n + yfnn)/J 2 + [(_x_ - 28x_B + yxqn)(ysf n - ynf_)
+ (_y¢$ _ 2BY_n + XYnn ) (Xnf $ _ X_fn )]/j3 (A.17)
or3
V2f = (af_ - 28f_n + Yfnn + °fn + Tf_ )/J2(A.18)
Gradient:
Vf = [(y f_ - y_fn) _ + (xsf n - x f_)j]/J (A.19)
Divergence:
V • F = [yB(FI) _ - y_(Fl) _ + x_(F2) n - x (F2)_]/J (A.20)
Curl:
V x F = k[yn(F2) _ y_(F2) n n~ . - - x_(F I) + xn(Fl)_]/J (A.21)
Unit Tangent and Unlt Normal Vectors in the _,q Plane
A-5
In many appllcatlons components of vector valued functions
either normal or tangent to a line of constant _ or n are required.
Similarly, dlrectlonal derlvatlves in these dlrections are often
needed to evaluate boundary conditions. These quantities may be
obtained by trivial calculations if unit vectors tangent and normal
to the _ and q-llnes are available. These vectors are developed
below.
It is well known that the unit noKmal to the graph of f(_,y).
= constant is given by
Vf
n (f) = -
Associating the coordinate function q(x,y) with f(x,y), we have
Utilizing equation (A.19) this reduces to
= (-ycl+ xcj)ICv (A.22)
which is the unit vector normal to a llne of constant q. In a
similar manner the unit vector normal to a llne of constant _ is
given by
n(t)= _ = (yqi - xqJ)//_ (A.23)
These vectors are illustrated as they appear in the physical plane
A-6
in Figure A.I below.
Y
_=K2 _=K3
_=K_ __ n=C3
n(n) n c 2t(EJ)
q=C I
.r- x
i
Figure A.I. Unit Tangent and Normal Vectors
The unit tangent vectors are then given by
t (D) = n (n) x k = (x$i + y_j)/_ (A.24)
t($). = n(_)~ x k. = - (xn ~i+ ynJ)/v_a. (A.25)
Vector Components Tangent and Normal to Lines of Constant _ and n
Fn (n) : n(n) " F : (-ysF 1 + x_F2)//Y'Y (A.26)
Ft (n) : t(n) " F~ = (x_F I + Y_F2)/V_y (A.27)
A-7
Fn(_) - n(_) • F = (YnFl - xnF2)/_ (A.28)
Ft ($) = t (_) • F = - (xnF 1 + YuF2)/_a (A.29)
Directional Derivatives
_--f = n (n) • Vf = .(Tfa - 8f_)/J_ __ (A.30)_n(N) ~ -
af (n>= .t • Vf = f¢l_Y (A.31)_--_(n)
Bf <_)B---_(_)"= n • V f = (sf_- 8fT])/J/_ (A.32)
_--f = t ($) • Vf = - f /4 (A.33)_t(¢) - - n
Integral Transform
Scalar Function:
/ f(x,y)dxdy = / f(x(_,D) , y(_,_))IJId_dn
D D*
Vector Function:
Consider a vector integral in the physical plane of the form
I = / f(x,y) n(x,y) as- S ~
where S is the closed cylindrical surface of unit depth whose
perimeter is specified by the contour F1 in the physical plane
(see Figure A.2) and whose outward unit normal at any point is
given by n(x,y).
(A.34)
(A.35)
A-8
n(x,y)
rI
x
Figure A.2. Integration Around Contour r I
If r = r(x,y) is the position vector describing rI, then
dS = (i.0) Idrl
which implies that (A.35) becomes
I - ¢ f(x,y) n(x,y) Idrl
rI
We now wish to transform (A.36) to the (_,n) plane.
(A.36)
Consider Idrl :
Idrl = _(dx) 2 + (dy) 2 = _(d_) 2 + Bd_dh + a(dn) 2
" /_'7d_ (A.37)
since rI transforms to rl*, a constant R-line (n = nl). Noting that
(nI)n(x,y) = n - (-y_i~ +x_J)/_ (Equation (A.22)), Equation (A.37)
becomes
A-9
_mAX
I = I f(x(_,n I) , y(_,nl)) (x_J - y_i)d_" _min
_max
-/_mln
f([,n I) (x_J - y_i)d_(A.38)
where _min and _max are the minimum and maximum values respectively of
on rl*. Note that all quantities in (A.38) are evaluated along
B = nI. If the vector n(x,y)~ is incorporated into the function f(x,y),
we can define the vector function f(x,y) as
f(x,y) _ f(x,y) n(x,y)
Equation (A.3_ now becomes
_x
I = f f(_,_l ) _y d_ (A.39)&rain
which is merely an alternate form of (A.38).
APPENDIXB
PRESERVATIONOFEOUATIONTYPE
Whencoordinate transformations are utilized as an aid in
solving a given partial differential equation, it is imperative
that the equation not change type under the transformation. Such
an invariance will now be demonstrated for the transformation
discussed in Section II. Consider the general, second order,
quasi-linear partial differential equation
A(x,y,f)fxx + B(x,y,f)fxy + C(x,y,f)fyy + E(x,y,f)f x
+ F(x,y,f)fy + G(x,y,f) = 0 (B.I)
where f = f(x,y) is a twice continuously differentiable scalar
function and A, B, C, E, F, and G are continuous functions. Recall
that the equation type is determined by the coefficient functions
A, B, and C as follows:
Elliptic if B 2 - 4AC < 0
Parabolic if B2 - 4AC = 0
Hyperbolic if B 2 - 4AC > 0
Utilizing equations (A.I), (A.2), (A.4) - (A.6), and (A. 7) -
(A.I_) of Appendix A, equation (B.I) transforms to
+ E*f_ + F*f + G* = 0 (B.2)A*f_ + B*fsn + C*fnn n
B-1
B-2
where: A* E A_x2 + B_x_y + C_y2
B* E 2A_xnx + B(_xny + Synx) + 2C_yny
C* E A_x2 + Bnxny+ Cny2
E* _ A_xx + B_xy + C_yy + E_x + F_y
F* _ Anxx + Bnxy + Cnyy+ E_x + Fqy
F* -F
Now, consider (B*) 2 - 4A'C*:
(B*) 2 - 4A'C* = [2A_xn x + B(_xny + nx_y) + 2C_yny] 2
- 4(A_x2 + B_x_y + C_y2)(Anx2 + Bnxny + Cny 2)
= (B 2 - 4AC)(_xny - _y_x )2
= (B2 - 4AC)/J 2
Since j2 > O, B 2 - 4AC and (B*) 2 - 4A'C* are either both positive,
both negative, or both zero. This implies that (B.I) and (B.2)
have the same type.
APPENDIXC
PRESERVATION OF INTEGRAL CONSERVATION RELATIONS
IN THE TRANSFORMED PLANE
It is now shown that the divergence property is not lost in
the transformed plane. Let F(x,y) be a vector valued function
defined on the physical plane (Region D in Figure I) whose component
functions, Fl(X,y) and F2(x,y), are continuously differentiable
on D. Let S(x,y) be a continuously differentiable function on D
and suppose F(x,y) and S(x,y) are related by the partial differential
equation
v • F = S (C.I)
Integrating (C.I) over an area RCD,.we obtain
f [(FI) + (F2) ]dxdy = f SdxdyR x y R
Application of Green's Theorem to the integral on the left yields the
conservation relation
(FldY - F2dx) = f Sdxdy (C.2)C R
where C is the boundary curve of Region R taken in the proper
direction. In a physical sense the line integral represents a
flux through the boundary of R, and the integral over R a source
within R.
Now, in the transformed plane (C.I) becomes
C-I
C-2
!j {[yn(Fl)_ _ y_(Fl)n ] + [x_(F2)n - x (F2)_] } = S (C.3)
But also note
(YnFl - x F2) _ + (x_F 2 - Y_FI) n = [yn(Fl)_ - y_(Fl)n]
+ [x_(F2) _ - xn(F2)_] + [Y_nFI - x_nF 2]
+ [x_nF 2 - y_nFl]
which, since the last two terms cancel, implies that (C.3) becomes:
(YnFl - xnF2) _ + (x_F 2 - Y_Fl) n = SJ
Integrating over the area R* in the transformed plane and applying
Green's Theorem as before yields the relation
[(y F1 - xnF2)dn - (x_F 2 - Y_Fl)d_] = f SJd_dnC* R*
(c.4)
where C* is the boundary curve of R* (R* and C* are the images
of R and C respectively). To see that (C.4) expresses the
conservation relation in the transformed plane parallel to that
expressed by (C.2) in the physical plane, consider calculating
the components of F normal to lines of constant _ and n. Utilizing
equations (A.26) and (A.2_) we have
n (n) _ Y_Fl)/C_y• ~ = (x_F 2
F • n(_) = (y F1 - xnF2)//_-_
C-3
Let r be the position vector describing the points along the curve
C*. Then, utilizing conventional techniques,
Idrl = /y(d_)2 + Bd_dq + _d_)2
Along a line of constant q we have
Idrlr _ = _ de
and along a llne of constant
Idrl_ = _ dn
Thus, the flux across a llne of constant q is
F • n(q)~ Idrlq~ = (x_F 2 - Y_Fl)d _
and across a llne of constant _ becomes
• _(C) Id_l_ = (YqFl - x F2)dq
These are the relations appearing in the flux terms of (C.4).
The flux terms thus have an exact analog to those appearing in
(C.2). Hence, the conservative relation (C.4) in the transformed
plane expresses conservation in the physical plane over the non-
square area formed by intersection of the curvlllnear _ and n
coordinate lines in a manner which is precisely equivalent to
the conservation expressed by (C.2) over the square area formed
by the intersection of the x and y coordinate lines.
APPENDIXD
FINITE DIFFERENCEAPPROXIMATIONSIN THETRANSFORMEDPLANE
This appendix contains a compilation of the second order finite
difference expressions used to approximate partial derivatives in the
transformed plane. Computational molecules for the derivative approximations
appear to the right of each expression given. Combineddifference forms
for the transformation parameters _, B, _, J, o, and T are also included
here. Since the field step size is immaterial in the (_,q) plane, it
is taken as unity for all approximations and does not appear explicitly.
The following definitions are used throughout this appendix:
f=f(_,q) - a twice continuously differentiable function of _ and
n.
x---x(_,q) - coordinate transformation function defined by Equation
(13).
y=y(_,q) - coordinate transformation function defined by Equation
(13).
a, 8, Y, J, c, and _ - transformation parameters defined
in Appendix A.
In addition the following notational convention is utilized to indicate
the position at which functions and derivatives are evaluated in the
discrete ($,q) plane:
fi,J m f(_i'nJ ) '(f_)i,J _ f_(_i'nJ )' (f_)i,J m fn(_i'nJ )
D-2
(fnn)i,j fnn(¢i,nj)
Derivative Approximations
First derivative, central differences:
(f_)t,J = (f_)t,J/2
where
l i i
i
(D.la)
t(f_)i,j fi+l,j - fi-l,J
(fn)i,J = (f'n)l,j 12
where
i
(D. Ib)
(D.2a)
(f'n)i,J - fl,J+l - fl,J-i(D.2b)
First derlvatlve, forward differences:
(f_)i,j = (-fi+2,j + 4fi+l,j - 3fi,j)/2
(fn)i,J -".(-fi,j+2 + 4fi,j+l - 3fi,j)/2
Second derivative, central differences:
(f_$)i,J = fi+l,J - 2fi,J + fi-l,j -
(fna)i,j -_ fi,J+l - 2fi,J + fi,J-i - (f'na)i,J
(f_n)i,j = (f_n)i,j/4
where
! I |
i
i
i
(D.3)
(D.4)
(D.5)
(D.6)
(D.7a)
D-3
fi+l,J+l - fi+l,j-I + fi-l, J-i - fi-l,j+l (D. 7b)
Transformation Parameters
ui, j = a' j/4i, (D. 8a)
where
=_,j - (x')2 _ + (y_)2n 1,3 i,J(D. 8b)
!
Bi, j = Bi,j/4 (D. 9a)
where
(D.9b)
Yi,J = YI,j/4 (D.10a)
where
_' = (x_)'_ + (y_)2i,J ,J i,J(D.10b)
= ' /4Ji,J Ji,J (D.lla)
where
Ji,j- (x_)i,j(Y_)i,j- (x_)i,j(y_)i, j (D. llb)
!
ai, j --" oi,j/2 (D.12a)
where
O' - [(Y_)i,j(Dx' - (x_) i (Dy') ]/J_,ji,J )i,.J ,J i,j (D.12b)
D-4
= ' /2 (D.iBa)
where
T_,j E [(x')i _(Dy') - ') (Dx') ] 'n ,J i,j (Yn i,j i,j /Ji,j (D.13b)
and where
(Dx')i, j - 2B_,j (x_11)i,j!
+ Y_,j (xnn)i,j (D.14a)
m' (y_) ' - 2B_ (y_q) + ' ' (D.14b)(DY')i,j i,J i,J ,j ' i,j Yi,j(Yqq)i,j
APPENDIXE
CONTOURPLOTS
This appendix presents a detailed discussion of the methods used
to determine contour plots of a function defined in the ¢,n plane.
The numerical procedures which are used to transform the contour to a
physical plane representation are also covered.
Determination of Contours in the _n Plane
Let _ = _(_,n) be a function defined in the region D* (Figure i)
possessing continuous second derivatives. Since D* is closed and
bounded, let m(_) _ min {_(_,_) i [_,n]e D*} and M(_) E max {_(_,n) I
[_,n]E D*}. If _ is a number such that m(_) ! _ !M(_), then we define
the _-contour of #(_,n), CT(_), as the set
CT(_) = {[_,n] [ [_,n]E D--*and _($,n) = _}
Graphically, CT(_) is the curve created by the intersection of the graph
of _(_,_) and the plane _(_,n) = _. For plotting convenience the curve
is usually projected onto the (_,n) plane. These ideas are illustrated
in Figure E.I. The contour, CT(¢), is also often referred to as the
level set of _(_,n) through _ .
Now suppose that _(_,n) is known only in a discrete fashion. That
is, let the net function _i,J E _(_i'nJ ) be known on the discrete set
D--**- {[_i,_j] I _i = i-i for I ! i !IMAX and nj = J-i for i ! J !JMAX}
E-I
E-2
¢
c(¢)
Figure E.l. Contour - _ of _(_,n)
(The fact that _i,j may only be an approximation to #($ ,n ) is im-
material to the current discussion). If similar defintions for m(_)
and M(_) are made for _i,j on the set D**, then the _-contour of #i,j'
denoted CT(_) again, can be defined as
CT(_) = {[_k, nkl I 05_ _--k< IMAX-I; O< _--k<--JMAX-1;
#(_k,nk) = _; k=l,2,...,N; N _ 2}
The bars over 6k,nk, and _ are to indicate that [_k,nk] may not be an
element of D** and that _ is not necessarily one of the values of the
net function _i,J" This is readily apparent from the discrete analog
[o Figure E.I given in Figure E.2.
E-3
3
Figure E.2. Contour of _(_i,_j)
Numerically, the import of the above discussion is that interpo-
lation between the points of the discrete set D** is required to deter-
mine CT(#). Consider a portion of grid D--**as shownin Figure E.3a
where i < Ii < 12 < IMAXand i < Jl < J2 < JMAX. Each grid block is
labeled by the (i,J) coordinates of the lower left hand corner of the
the block. Block (m,n) is shownon a larger scale in Figure E.3b.
In order to improve the plotted resolution consider subdividing
each block into four triangles, as shown. The value of _ at (m+ ½,
n + ½) is taken as the four point average
_m+ ½,n + ½ = (_m,n + _m+l,n + _m,n+l + _m+l,n+l )/4
A local (_,_) coordinate is affixed to each grid block as demonstrated
E-4
J=J2
(m,n+l) (m+l n+l)
m,n
Iqm,n2
a)
Figure E.3.
j=Jl i 1
l=Ii i=12 (m,n) _m,n (m+l,n)
b)
Sample Grid in Set D
in Figure E.3b. In order to standardize the interpolation procedures,
a local (_,p) coordinate system is also placed on each of the sub-
triangles as illustrated in the series of drawings given in Figure E.4.
r 7II!
m,
_2
r
m,n
ii
P3,
I i
rl tm,n i m,n
!
_m,n _ P_'m,n
_4
a) b) c) d)
Figure E.4. Local Coordinate Systems for Triangles
Interpolation is carried out on each of the four triangles for each grid
block in the segment (Ii ! i ! 12, Jl ! J ! J2) of the set D--** specified.
In particular interpolation is performed along each of the three sides
of each of the triangles if the contour value, _, lies between the values
at the ends of the sides. Let d' be the directed distance from a given
E-5
triangle vertex to the point on the triangle side where the contour
intersects that side (denoted bye). This distance is illustrated in
Figure E.3b for triangle Oand is defined in an analagous manner for
the other triangles. If _i is the value of $i,J at a particular triangle
vertex and _2 the value of $i,j at the other end of the side, then d'
may be expressed as
d' = (side length)(# - _i)/(_ 2 - _i )
For example, along side 1-2 of triangle_), d' is given by
di_ 2 = (I.0)(_- Cm+l,n)/(¢m,n- ¢m+l,n )
Noting that the sides of the triangle have lengths 1.0, 1.0//2, and
1.0//2, the contour intersections can be expressed in the local triangle
coordinates as
Side _£ _A
1-2 l-d 0
2-3 d/2 d/2
3-1 (l+d)/2 (l-d)/2
where d _ (# - _i)/(# 2 - _i ) and where £=1,2,3, or 4 denotes the triangle
number.
Once the contour intersections have been determined in the local
triangle coordinates, (_£,p£), they must be transformed to the grid
block coordinates (_m,n,nm,n). This is done in the conventional fashion
using orthogonal rotation matrices. If [_£,p,p£,p] are the coordinates
of an intersection in triangle £, then
E-6
"(_m,n)_,
(nm,n)A,p
= A£
where
[I r1A1 _- 1 0 A2 0 10 1 -- l-I 0
L
[-'o:I o-'IA3 -- ' =1
-- - -- L1 o
Note that up to three contour intersections can occur for each triangle
(i.e., one on each side). Finally, the point [(_m,n)£,p, (Dm,n)£,p]
is transformed to (_,n) coordinates by a simple linear transformation
producing an element [_'k' q--k] of the set CT(_ ).
Transformation to the Physical Plane
Since contours in the (_,n) plane are of little interest, CT(@ )
must be transformed to the physical plane. This is made possible
through the use of the coordinate transformation functions x(_i,qj )
and y(_i,qj). Again interpolation is required since almost all elements
of CT(@ ) are not elements of D** on which the discrete functions x(_i,qj )
and y(_i,nj) are defined. As illustrated in Figure E.5 this implies
a double linear interpolation must be performed. If [_k' n'-k]denotes
an element of CT(@), the first step is to locate the _ and n values
bracketing Ck and nk. Denoting these by _i' Ci+l and qj, _j+l as shown
in the figure, the values of xk and Yk are calculated as follows
'='(q'k - nj)(Xj+ 1 - Xj)/0]j+ 1 _ nj) + Xj
E-7
where
Xj = (_k - _i)(Xf+l,J - xl,j)/(_i+l - _i ) + xl,J
Xj+l = (_k - _i)(Xi+l,j+l - xi,j+l)/(_i+l - _i ) + xi,j+l
for all k=l,2,...,N. Similar expressions are used to calculate Yk"
Figure E.5. Interpolation for x and y
NASA-Lan_I ey, 1977
