Boundary-Fitted Curvilinear Coordinate Systems for Solution of Partial Differential Equations on Fields Containing Any Number of Arbitrary Two-Dimensional Bodies

NASA CONTRACTOR NASA CR-2 7 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

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY _1

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. 1. Z r 1. Report No. 2. Government Acc_sion No. 3. R_ipient's Catal_ No. NASA CR-2729

• 4. Title and Subtitle 5, Repo_ Date Boundary-Fitted Curvilinear Coordinate Systems For Solution of July 1977 Partial Differential Equations on Fields Containing Any Number of 6. Performing Organization Code Arbitrary Two-Dimensional Bodies

7. Author(s) 8. Performing Organization Report No. Joe F. Thompson, Frank C. Thames, and C. Wayne Mastin

10. Work Unit No.

9. Performing Organization Name and Addre_

11. Contract or Grant No. Mississippi State University Mississippi State, Mississippi 39762 Grant NGR 25-001-055

13, Type of Report and Period Covered

12. Sponsoring Agency Name and Address Contractor Report

National Aeronautics & Space Administration 14. Sponsoring Agency Code

Washington, DC 2n546 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 at Langley 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 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 regardless 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.

17. Key Words (Sugg_ted by Authoris)) 18. Distribution Statement

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TABLE OF CONTENTS

Page

Ie Introduction ...... 1

II. Mathematical Development ......

Preliminaries ...... 7 Doubly-Connected Region ...... i0 Multiply-Connected Region ...... 13 Simply-Connected Region ...... 15 Coordinate System Control ...... 15 Time-Dependent Coordinate Systems ...... 19

III. Numerical Solution ...... 21

Difference Equations ...... 21 Multiple-Body Fields ...... 23 Multiple-Body Segment Arrangements ...... 24 Initial Guess ...... 26 Convergence Acceleration ...... 30 Optimum Acceleration Parameters ...... 33

IV. Examples of Coordinate Systems ...... 36

Coordinate System Control ...... 36 Various Body Shapes ...... 37

V. Examples of Application to Partial Differential Equations ..... 38

Potential Flow ...... 38 Viscous Flow ...... 42 Velocity-Pressure Formulation ...... 44 Pressure Distribution and Force Coefficients ...... 45 Loaded Plate ...... 46

_I. Instructions for Use - Coordinate Program ...... 47

Coordinate System ...... 47 Dimensions ...... 48 Input Parameters ...... 48 Field Size ...... 48 Plotting ...... 48 Initial Guess ...... 49 Body Contours ...... 49 Re-entrant Boundaries ...... 50 Acceleration Parameters ...... 51 Coordinate System Control ...... 51 Convergence of Very Large Fields ...... 54 Listing of Program TOMCAT ...... 56

iii 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

VII. Instructions for Use - Factors Program ...... 123

Scale Factors ...... 123 Listing of Program FATCAT ...... 124 Sample Case: Single-Body Field ...... 140 Input ...... 140 Output ...... 141

Vlll. References ...... 147

IX. Tables ...... 150

X. Figures ...... 170

XI. Appendices ...... A-I

iv BOUNDARY-FITTED CURVILINEAR COORDINATE SYSTEMS FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS ON 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 regardless 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 representation 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 difference 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: LTV Aerospace Corporation, Dallas, Texas.

§Associate Professor of Mathematics, Ph.D. elliptical boundaries, etc. Finite difference expressions at, and adjacent to, the boundary may then be applied using only grid points on the intersections 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 generation of a curvilinear coordinate system with coordinate lines coincident with all boundaries (herein called a "boundary-fitted coordinate system" for purposes of identification) is thus an essential part of a general numerical solution 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 differential system in the physical plane, with Dirichlet boundary conditions on all boundaries. One coordinate is specified to be constant on each of the boundaries, 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 coordinates 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 boundary-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 orthogonal 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 regardless of the spacing of the finite grid in the physical field, all computations, 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 coordinate system is independent of the set of partial differential equations to be solved on this system. For example, numerical solutions for inviscid 7 and 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 same computer 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 transformed plane [_,n] is given by the vector-valued function 8 (1) (x,y)J

The Jacobian matrix for this transformation is

(2) J1 -- [ _x qy

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) 9 Partial derivatives are transformed as follows:

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 o about (xo' yo ) such that the inverse function (3)

exists and (6) holds for all [x,y] in N . It is readily apparent that the o

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 superharmonic 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, respectively, to the constant H-lines forming the bottom and top sides of the transformed region. The more general case of segmentedbody contours transforming 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

Sxx + _yy = 0 (10a)

qxx + qyy = 0 (10b)

with the Dirichlet boundary conditions

[x,y] e r I (10e) r! B 1

_2(x,Y) , [x,y] e r 2 (lOd) n 2

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 F 1 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 F 3

to F 4. Therefore, boundary conditions are neither required nor allowed on

P3 and P4" (A graphic analog to the above ideas can be found in most complex 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 uniform 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

(llc) a - x n 2 + yn2

(lid) B -= xEx n + YEYn

(lle) _, - x 2 + y 2

with the transformed boundary conditions

fl(_,_l) , [_,nl] e FI* (llf) f2(_,n I)

n2 )] = , [_,n2]_ (llg) [g2(E,n2)j r2*

The functions fl(_,nl), f2(_,nl ), gl(_,n2 ), and g2(_,n2) are specified by

the known shape of the contours F 1 and F 2 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 illustrated 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 _

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 transformed 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 transformation. 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

nI)

[$,_i ] c F7* (llh) hl(_' I h2(_,n I)

= , [$,nI] c F8* (lli)

2(_,Ul)]

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 transforms to the entire rectangular boundary of the transformed field. The manner in which the physical boundary is split into four segments for placement 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

- 2_x_ + yxn_ + J2(pxE + Qxn) = 0 (13a)

_ - 2BY_n+ YYnn+ j2(py$ + Qyn) = 0 (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' = _max is a branch cut in D, this branch 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 17

$xx + _yy = - Z a.i sgn(_ - _i)exp(-cil_ - _iJ) i=l

m - Z b.3 sgn(_ - _j)exp(-dj/(_ - _j )2 + (n - njYr-) j=l

- P(_, n) (14a)

nxx + nyy i=l a.1 sgn(q - qi)exp(-ciJq - qil)

m Z b.3 sgn(rl - qj)exp (-dj,/(_ -- _j)2 + (19 - rl.)2j ) j=l

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 problems arise in later applications, this function can be replaced by 2 Arctan(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. Many successful 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 much under investigation, and other control functions are being evaluated. It is anticipated that new coordinate control packages will 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. 19 Time-DependentCoordinate Systems

If the coordinate system changes with 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 transformation 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

(15) + yt(f_x n - f x_)/J

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 transform, 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 approximation 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),

I + Xi+l, j) - 8'i,j (x_n)i,j/2 xl,j " [_i,j(Xi_l,j

! + T' + -1)]/[2(ai + Ti, j) ] (16a) i,j(xi,J+l xi,j ,J

+ Yi+l,j) - $'i,j (Y_n)i,3' ./2 Yi,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:

(17a) (x_)i, j = (x2, j - XIMAX_I,j)/2

(17b) (x _ ) i,J = x 2 ,j - 2x I ,j + XlMAX-I,J

(17c) (xEn)i, j -- (x2,j+ 1 - xz,j_ I + XlMAX_I,j- 1 - XIMAX_I,j+I)/4 23

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.

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

(18a) (Xn)ll+l,l -_ (Xll÷l,2 - xi4 - 1,2)/2

(18b) (xnn)ii+l,l = Xll+l,2 - 2Xll+l,l + x14_l, 2

(x{n) II+i, i -_(x14,2 _ x14- 2,2 + Xll+2, 2 _ Xli,2)/4 (18c)

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 many segmentsas 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 segments, 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 examples of 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. 26 Initial 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 numberin 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 -i 2 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 particularly 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 SegmentBoundary Points 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 therefore 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

(Z dijk)(Z X_k) - Z dijkX k k k k

xij N E dij k (20) k

with dij k = /(xij - Xk)Z + (Yij - Yk )2

Here x k 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).

(e) 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 28 to 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, the o 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 exponential 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, 29 6, 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 difficult, 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 discussion 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)

(22) + b2(fi ,j+l - f"1,j-i ) + cf..13 + d..13 = 0

(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:

Case #i: a12 >_ b12 and a22>_ b22

2 = (over-relaxation, i < _ < 2) (23) i + _--_- _

Case #2: a12 _ b12 and a22 _ b22

2 w = (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

a I = _ a 2 = y

bl = j2p2 b2 = j2Q2

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 32 2 2 Jij JPij J Jij JQij 1 Case #2: _i-'j< 2 and Yij < 2

(under-relaxation) (27)

lJ i + /i + Pij z

where

2 J .P • = I _ij - _ cos MAX - i PiJ _ij _ Yij

2 + Yij - 4214 (28)

j2JpJ j2jQJ In the remaining case where a _ --and2 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

(29a) Pl = _ +-----_ 4 cos --_ _ 1

(29b) 02= +y y2_ 4 - i

If over-relaxation is _pecified then m is calculated from Eq. (26) using

P = Pl + P2" The same expression for p is used in Eq. (27) if under-relax-

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)

if _ _ J__P_i2 " Similarly, using p 2 in place of p, m 2 is calculated from

Eq. (26) if y > _ else by Eq. (27) The average is then formed by 2 '

Plml + p 2m2 = (30) Pl + P2 OptimumAcceleration Parameters

In order to provide someguide to the selection of acceleration parameters 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 segment 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 variable 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 example of 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 much sharper 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 moves outward. 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 absence of 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 coordinate 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 manneris 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 mesh spacing 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 contraction 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 38 and 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 examples of 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

(32a) _(x,y) = _k on the surface of the kth body.

_(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- 39 foil chord and the free stream velocity. Whentransformed to the curvilinear coordinate system this equation becomes

_$ - 2B_n + Y_nn + °_n + _ = 0 (33)

The transformed boundary conditions are (see Figure I).

qJ(_'"11) = )o on n = n I (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 nondimensional variables is

Cp = 1- IvI 2 (35)

On the body surface this becomes

Cp = 1 __I_Y (36) j2 _D 2 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

(38a) CL = _ Cp (-x¢cos0 - yEsin0)d¢

(38b) CD = _ Cp (y¢cos0 + xi,sin0)d_j

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 airfoil, 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 streamlines 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, mesh spacing, 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 differential system. 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)

a_ - 2B_n + Y_nn + °_ n + _ = _j2_ (40)

with boundary conditions:

= constant , Aj _g = 0 on body surface (41a)

= 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, -_j _n was below

some tolerance: 43

(k+l) (k) mil - mil Fyy (k) (k-l) I _I (k)

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 _). dr r ~ ~ ~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:

Fx = + _ py_ d_ - _2 _ mx$ d_ (44a)

Fy = - _ px_ d_ - _2 _ _y_ d_ (44b)

Finally, the lift and drag coefficients are given by

C L = Fy cos0 - Fx sin0 (45a)

C D = F y sin0 + F x cos0 (45b) where O is the angle of attack.

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 !vl

0 = _ V p • dr =- _ [_t + V(---_2-)

i - v x m +_ V x a] dr

= _ 1 _ (V x a) • dr R (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

(49) + gv n + Tv[)/RJ 2 45

aP_ - 28P_n + YPnn + oph + Tp_ = - (YnU_ - y_un) 2

- 2(x_u - xnu_)(ynv _ - y_vn)

(x_v n - xDv_) 2 - J2D t (50) where

D _ (ynu_ - ygu + x_v - xnv$)/J (51)

Equation (50) is the transformed Poisson equation for the pressure,

obtained by taking the divergence of the Navier-Stokes equations.

The boundary conditions are

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 numberi000 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 make minor 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 iteration 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 49 plotted 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 transformed 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 segments 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 physical 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. Then by (8)

Ynf_ f_ f - x x_y n x_

The difference approximation then would be

fi+l- fi-1 f = + T. x xi÷ 1 - xi_ 1 i

where T. is the local truncation error. l

about f. then yield, after Taylor series expansions of fi+l and fi-i 1

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 u t = -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 £f in 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 toward 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_ITwTAPEblOUTPljTe ITAPEIO,TAPEI|) t E E ******** MT$$T$$rPPT STATE _-0 80DY-_ITTED CnOR_TNAT_ _YSTE_ ******** 3 C * ($I_PLY nR MULTIPLy CON_ECTE_ REGIONS) E * U S e * {DEPARTMENT OF AEROPMY$IC5 a_ AEROSPACE E_INEERI_G ) 6 E * ( _I$$1$STPP! 8TATE !J_TVERBITY tg?5 } ? C * {DEVELOPmEnT @PO_@nREO BY _ABA,L*_GLEV RESEA_C_ CENTER) o C * DIRECT IN_UTR_F@ TO _R, JOE F, THUMP@ON c • _A_ER A 1o 11 * _I_BISBIPPI C * _TATEw _S ]976_ t2 13 C *, PHnNEI bnl_3_S-]_5 15 C ********************************************************************* ;Y DI,EN$10_ X(7np60), Y(TO,60], RETA(?O,_), RxT(70,bO), _ACC(70,bO) I, T*CC(?O,bO), XPLOT67_), YPLOT(7_) EO l), LBI(6)_ LR_(6), LBDY(6], LR8_D(6)_ LRI_6), LRE(6) e L?$ID(b)_ LI El _1(6], tIE(_)_ LTYPE(6), LSE_(6), IwER(_} INTEGER TACC E2 2x DATA NbIM,NDIV! /70,60/ 2, OATA RA_/S?,EOST?OS1308/ E5 _ATA _NBBEG,_N_BEG /6,61 Eb _ATA ZERO /I,OE-08/ E7 C iB 29 _0 31 C * 32 C *** CARDS(S) , CIICE/BnY - FOR_AT{BAt_) 33 C i (NAy RE BLANK) 3. C * 3S _6 C * C_ AND C_ • BO CHARACTER A/h ARRAYS ._ICH ARE PRINTED AT ]T C *w TH_ TOP OF EACH OUTPUT P'_GE AhD DN ANY PLDT$_ $8

C * BOY • NA_E OF BODY BEING TRAN$FOR_ED (BO CNARACTER$ NAX), taO C _*l CARD I IMAX,JMAXeNBDYeITER_IGEB,IDISK,t_R_I_INTL_I_FIN,IGED C C * I_AX = NUVBER OF XI•LI_EB, .5 C * J_AX . NUMBER OF ETa-LINES, C * .7

C * NBDY • NUMBER OF BODIES _N THE FIELD. .e C * (ZERO FOR B_NPLYeCONNECTEO REGION) C * 50 5t C ** _TER = _AXI_U_ NUMBER OF ITERATIONB ALLOWED * 5_ 53 C * IGE$ • INITIAL GUE8B TYPE I (BEE IGEO AL80] S, C * (IOME BEGHENT ARRANGEMENT8 _ILL NOT.CONVERGE 5S C * WITH THE |TANDARD _NIT_AL _UEBB, THEREPORE _6 C * SEVERAL ALTERNATIVES ARC PROVI_ED,] C * (IGE$mO I8 TYPICAL _OR BINGLE•BOOY FIELOI OR E8 C * MULTiPLE=BODY FIELOB HAVIN_ ALL BODIF8 ON THE 5, C * SAME 81DE OF THE TRAN|FOR_ED FIELD. _GE$sE DR -_0 bO 58

61 I$ IYPICAL FOR OTHER MULTIPLF-_O_Y FIELDSe EXCEPT c _Z F_R THOSE HAVING TWO @_DIE@ IN SINGLE SEGMENTS c 'if e_ 0_' (}PP_$ITE SIDE@ OF T_E TRAN@ FORME_ FIELD, IN THE C @g LATTFR CASE TRY IGE§II O_ A,) 6S (IGED.NE,O T@ MOR_ FFFECTI_F FOR CASE@ ,ITH TWE f. b_ riOTER BOUNDARY O_ THREE @IOE$,) f, el c. t bB mt - wEIGWTED AVERAGE OF FDL!R PRPJECTE_ C ()@ BOUNDARY VALUE@, c t 70 mO - SAME kS I EXCEPT ZERO I $EO I N PLACE OF VALL_E@ f- t T1 O_ CUT@, c ?Z I_ - _AM_ i_ % EXCE pT _ OUNDARY VALUES ON CUTS c OMITTE_ IN AVERAGE, c 7a I_ - MOMENT PROJECTIONI C t TS WI(@U_D,@UMXmSUMXD)/@UMD, DIVIDED BY IOTAL C t NUMBER OF BOI_k$OARY POINTSe c 7'7 w_ERE $UMWmSUV OF ROU_ARY VALUES, t 7_ c @UMDISUP OF DISTANCE@ TO HOUNDARY POINTS, c t 7'; @U_XDmSUw OF RRO_UCT@ OF ABOVE, t SO m_ - SAME k@ ] EXCEPT NO _IVISlON BY TOTAL c t @1 NUMBER OF BOLINDARY ROIMT@, c @Z )a - _AME A_ _ EXCEPT EXPONFNTIAL _EIGMT RATHER TMAN c @]5 LINEAR= CONCF_TRAT_ON I@ T_wARD LUwER VALUES c t S_ OF XT AND/OR ETA WITH DECAY FACTOR OF C t 8'3 K O,I_(IGESea) , 8b ¢0 - EXPONENTI AL PROJECTIONI C t ST XISUMXE/@UME; wiTH EXPONENTIAL DECAY C @B FACTOR EOUAL 10 IAMS(IGES), C @9 *WERE StJMEm@UM OF EXP{._ECAY*OISTANCE), c t _0 SUMWEISUM OF PRODUCT OF BOUNDARY VALUE C @ 91 AND ABOVE EXP, c 9E (DISTANCE IS NDNDIMENSIONALIZEO RELATIVE c t 93 TO DIAGONAL OF RECTANGULAR TRAN$F _RMED FIELD) c ea C 95 C t IUISK - DISK READ/WRITE CDNTROLI 96 C mO START ITERATION pROM INITTAL GUESS, DON_T STORE C_ORDINATE $VSTEM ON DISK, c t _8 m_ START ITERATION FROM INITIAL GUESS, C 99 @TORE COORDINATE SYSTEM ON DISK, C tO0 mE CONTINUE ITERATION OF k PARTIALLY c 101 CONVERGED _OLUTION READ FROM RESTARTFILE C ON DISK, STOKE COORDINATE SYSTEM ON DISK, C t tO$ m_ A@ #E EXCEPT _ONeT @TORE COORDINATE SYSTE M, C tOU C 1.OS C t lob NOTEI IDI$K OF 1 _R Z CAUSES THF COOROINATE SYSTE _ TO BE _RITTEN c t tot C , TO DISK IN TWE FOLLOWING FORMkTI 108 c t t 10@ C t t ,RITE(tOet) C1 110 c ,RITE(IO,1] C2 11t *RITE(tOet) IMA.X,jMAX,NBSEG,NRSEG,LISEGeNBDY tti_ *RITE(tO,I} (LBSID(L]_LBL(L)_L@_(L)_L@OY(L)*LSEN[L)_ C I * IL$ C * * I Lmt,NBSEG) _RITE(10_I) (LRSIO(L)eLRt(L),LR_(L).LISIO(L}eLIt(L}eLIZ{L]_ C t 1LTYPE(L)_LmieNRSEG) It@ ,RITE(IO_I) ((X(I_J), I=I_IMAW)_JII'JMAX}_ C * * tL? 1 {(y(ImJ)_III_IMAX)_JI_JMAX) 118 119 MERE LISEG IS THE TOTAL NUMBER OF BODY @EGMENTS(EXCLUSIVE tEO OF OUTER BOUNDARY SEGMENTS) PLUB 11 LBEN(L) IS et IF 59

c LP2CL_>LR1(I } ANn IS -I {_T_E_ISE(LRI a_D LR2 ARE C t 121 I_TE_CHA_GEO I_TERNALLY AFTF_ LSE_J TS _ET IF NECFSSA_Y C t ¢t C t J_ S_ T.AT LB2>LR1}I LTYPE IR Im2,],_,_,OR _, _ESPFCTZVELY, 125 IF THE TWO SEGMENT8 Or A :E-E_T_a_T PaIR k_F tl) n_F ON t" t I2(J C mOTTO_ AND O_E _ T_P, (_) RPT_ _ ROTTO_. _) ROTH O_ TOP. {_) ONE ON LEFT AND O_E 0_' RIGHT, (_) B_TH nN LEFT_ OR c t C ¢1 fb) _OT_ 0_ RIGHT! X _D Y IRF THF _ARTEST_N COP_DIN_TF$. 127 C ¢t c ¢t IF CONVERGENCE IS NOT aTTAINED TN TWE ALLOWED N(_BE_ OF C ITERATInNS T_E PARTIALLY CONVERGE_ S_LLJTIO_ I$ _Tf_E_ ON 150 c. * DISK FOR RESTART, T_E ITE_ATInN _AY THEN HE C_NTT_UE_ Ry 131 ***** SETTING IOISK TO 20_ 3 AN_ INC_EASTN G ITEm, C t t' 135 c T*I_ - _0 nO_T PRINT EAC_ ITERATION ERROR NOR_, c _I PRINT EACH ITERATION ER_n_ NO_, C 13b c t T*TNTL - m_ OON_T PRINT _NTTIaL _UF$$, 157 11 INITIAL c PRINT GtJE_S, I_ C t I.FI_ * *O PRINT C_ORDIkaTE SYSTEM, C C II _ONeT PRINT CDORDINATF SYSTEM, I_I C t C ¢1 IGED - CONTROLS OIRECT]O_J OF wF]_TED AVERAGES FO_ INITIAL GUESS TYPES 0,1,2, AND ,GT,U I C t I_ C • 0 . AVERAGE IN ROTH Xl AND ETA _IRECTTONS (FOU_ POINT AVEragE) C dr I_? • ! * AVERAGE IN nNLY ETA DIRECTION c # (T._ ROINT AVERAGE) ¢ I - AVERAGE _N _NLY Xy nI_ECTTON C • E 150 (T*O _UI_T AVE_AGF} C t C t IS$ **t CARD I IPLOT_IPLTR_NCOPY,LIN*T_,LIN_TE.NUHBR_NU_RR|_ C *** 18KIPI,ISKIP2, - FORHAT(qI_) C 15S C I IPLOT - PLOT OPTIONS1 c I 157 C t mO NO PLOTS (INPUT OF REST OF CARD NOT REQUIRED), 158 xl PLOT COORDINATE SYSTEM, C li c I l_ PLOT INITIAL GUE88 aND COORDINATE SYSTEm, It)O C li I_ IPLTR - I1 PLOT _ITH OOULO _800, C i B2 PLOT kITH GERRER C I OTHER ( CALL P_EUO0 - DEVICE INDEPENDENT . C l m_ VARIAN AND CALCOMP DO NOT HAVE LINE*T C t CAPABILITY] C I 165 C Ir NCOPY - NUMBER OF COPIES O_ PLOT DE8IRED, C I C il C t LINwT1 . PLOT LINE wEIGHT DESIRED POR PLOT TITLES, 17o C Q (=_ "I_ O_ I_ _ RESPECTIVELY FOR TRIPLE_ DOUBLE_ 17I NORMAL_ HALF_ THIRD WEIGHT LINES) C il C l) (THIS APPLIE| ONLY TO THE GOULD _800) C l C 1) LIN_T2 * PLOT LINE WEIGHT DESIRED FOR COORDINATE 8YSTEH, ii {|[E NOTE BELOW LINWTI] C i (TH_8 APPL_F8 ONLY TO THE GOULD 4800) 177 C l C t NUHBR . NUMBER OF ETAmLINE8 DESIRED FOR PLOT, [ZERO VALUE PLOT$ ALL] 60

C _.LJMmPl .. _ll,_E_ OF XT..LI_ER r_E._Tt_E r_ Ircl_ PLC1T, r (ZF.R{] VALLIE PLUT$ ALL) c t $ba F t 18S c TSKIPt - SKIP PARAMETER FOR XI-LT_E$, (1PL_TS EACH LINE, 2 PLOTS EVEPY SECO_ LT_E,ETC, c 187 c 188 ]SKIP2 - SKIP PkRA_ETE_ FOR ETk-LI_ES, c t (REE NOTE BfLnW ]SKIP1) c t Tel c *** EACW B_OY A_D TMF OUTFR _0 IjN_A_¥ IS nIVI_E_ I_T0 ONE OR c *** M_PF SEGMENTS PLACE_ ON TM_ BI_ES nF X_F RECTAN_ULA_ T_ANIFO _EO c *** 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 * c *** CARD I N_SEG.NRSEG - FORMAT(2IS) C c , NBBEG - T_TAL NUMBER OF _O_Y A_O O_TER BOUNDARY SEGMENTS, c E00 C , NRBEG - TOTAL NUMBER 8F PAIRS OF WE-ENTRA_;T SEGMENTS. FnR $1MPLY-CON_ECTED REGION) r , (ZENO CO2 C _05 r *** C&RD$(N_$EG) I LbSIDeLR%eLR2#LRDY " FOP_AT(_IS) 20_ C * E05 C * L_$1D - BIDE OF RECTANGULAR TRAkSFOHMED FIELD nN wHICH EO_ R_DY _EGwE NT OR OUTER BO(IK, nARY SEGMENT I8 LOCATED, C * _0? C * (BOTTOM IS 1_ LEFT IB _, T_P I8 ], RIG wT IS _,) ZOb C * _09 C * LHI_Lm? - FIRS T AND LAST INDICES OF REGMENT, C * (Lm} MAY EXCEED LB2) C * 2%2 C * LBDY -RQDY NUMBER (BUDIES ARE NIJMRERED CONSECUTIVELY C * FROM | TO NBDY (- , OUTER BOUNDARY IS BODY NUMBER 0 OR Nb_Y_%, C * NUMBER 0 CAUSES OUTER BOU_DIRY TO BE CALCULATED C * INTERNALLY IS A CIRCLE, P_STTIVE nR NFGATIVE NBDYe% CAUSE8 OUTE R BOUNDARY TO RE READ AS OTHER C * ElS C * BODIES,) C * NOTE I EACH BODY IS RFA_ VIA A SINGLE CADD OR SET OF CARD8 wiTH _0 RFPEATED POINTS_ NOT EVEN _EO C * TME FIRST POINT, BODY SEGMENTS MUST BE C * CONSECUTIVE ACCORDIN_ TO LBDY, WITH THE E22 C * OUTER BOUNOARY LAST REGARDLESS OF ITS C * C * LBDY, _OINTS IN FACM SEGMENT ARE READ CONSECUTIVELY AND PLACED ON BOUNDARY 2_5 C * Z2_ C * FROM INDEX LBI T_ INDEX LB_, C 228 C *** CAROS(NR_EG) I LRSID,LRI_LRE,LISIO,LII,LIE " FORMATCbIS} , (OMIT FOR 8IMPLY-CONNFCTEB REGION) C _0 C LI C LR$ID,LISI_ - SIDES OF RECTANGULAR TRANSFORMED FIELD ON WHICH IEOMENTB OF A RE-ENTRANT pAIR ARE C R C LOCATED, BEE NOTE BELOW LBSIB FOR 8IDES, C (LIIIb MUST EQUAL OR EXCEED LRBIO) C C LRteLREeLIIeLIE = FIRST AND LAST INDICES OF EACM SEGMENT OF A REtENTRANT PAIR, (I_ LISID II EQUAL !]? C TO LRBID THEN LII MUST EXCEED LRE, IF C C LIiID T_ NOT EQUAL TO LRBI_ THEN Lit MUST EQUAL LR%. IN ANY CASE LRE MUST C 6i

C * EXCEED L_I, ANn LI_ MU_T EXCEED LII.] C t c *** CARD I _(1)eR(2),R(3)eYINFI_oAIN_IN,XOT_.fjYOINF,_JI_F - C FORMAT(?KIn.0,15) c c R(I) - GAt,$$-$EIDEL ACCELE_ATIn_ PArAMETEr. C ZERn VALUE CAUSES VARTARLF ACCELERATION PARAMETER c FIELD TO RE CALCULATE_ tNTEPNALLY, C (TYPICALLY 1,85) C c R(2) - CQNVERGENCE CRITERION Fn_ X ITE_ATInN EPROk NOR_, C (TYPICALLY 0,0000|} r ¢ R(3] - CONVERGENCE CRITERION FO_ Y ITEWATIO_ ERPn_ NO_M, c (TYPICALLY 0,00001} c C YINFIh = RADIUS Or CIRCULAR OUIER _ULINOARY, C C AINFIn - ANGLE OF FIRST PnINT nN OUTEP _O_NDARv, (nEGQEE$) (ANGLE t_ POSITIVE COIJNTER-CLOCK_ISE FWnM POSITIVE c X-AXIS, PC)INT8 ARE CLOCKWISE FROM T_T$ ANGLE.) C XOINF,YOINF - CENTEN OF CIRCilLAR O_ITER BOUNDARY, C C k,INF= NUMBER _F UNIQUE POI6T$ ON DUTE_ ROUNDARY e C C HOTEl YINFIN,AINFIN,XOINF_ & VnlNF _AV BE RLA_K IF f, OUTER BOUNDARY I_ TO _E READ, C C *** CARD I IffvmIAIT,R(10] - FORMAT(@IS,FIO.01 (BLANK CARD MAY e_ INPUT IF COh_TANT ACCELERATIOk C PARAMETER I_ USE_} C C IEV - CONTROLS COMPLEX JACOBI FIGENVALLJE PROCEnURE. c [OPTIMUM ACCELERATION WITH R_AL _IGENVALUE_ IS c OVER-RELAXATION. OPTIWIJw _ITW IMAGINARY I_ C UNDER-RELAXATION, NO THFORETICAL OPTIMUM 18 KNOWN C FOR COMPLEX EYGENVALUE$,) C -l I UNDERmRELAX= 0 I WEIGWTED AVERAGE, C el I OVEHuRELAX, C C IAIT - NONmZERO VALUE CAUSES VARIARLE ACCELERATION C PARAMETER FIELD TO _E PRINTffD, C C R(|O) - CONVERGENCE CRITERION FOR VARIARLE ACCELERATIO h C PARAMETER FIELDm FIELD I$ FROZEN wWE_ _AXlMUM c ABSOLUTE CHANGE ON FIELD I_ LESS THAN RCtO]. C (TYPICALLY O,OI) C C *** CARD I INFAC,INFACO - FORMAT(EI_] C (CAN RE USED TO AID CONVERGENCE BY CONVERGING A C SMALLER FIELO FIRST AND USING TwI! REiULT TO PRODUCE ¢ &N INITIAL GUE$! FOR k LARGER FIELD, BLANK CARD C MAY RE INPUT If ?HI8 FEATURE I_ NOT TO BE II!ED, C !TANDARD I! TO NOT U!E TWI! FEATURE,| ¢ C INFAC = NUMBER OF !TEP! IN ATTAINMENT OF OUTER BOUNDARY. C POSITIVE DOUBLE! OUTER BOUNDARY AT EACH 8TEP_ C NEGATIVE MOVE! OUTER BOUNDARY LINEARLY, 62

c t {INCREASE MAGNITUDE IF DIVEWGENCE _CCUR$) 30t c t 30_ r * INFACn - INITIAL STEP IN ATTAINMENT OF OiITER BDU_DAWY, C , (LINEAW ATTAIk_E_T ONLY) C _05 r *** CArD I $1ZE,RATIO,_IST,XBIw_B_YRIeYm_ - FORmAT(IF)0,0) 306 C (C)MIT IF NO PLOTS DESIRED= I,Fo= IF IPLOT IS ZFwO) $01 c 30B r SIZE = PLOT SlZE IN Y=DIWECTION, (TNCH_B) 30e c [TYPICALLY 8,0) c C RAIl0 = I0 X AND Y PLOT _CaLES ARE E_UAL, c >0 X A_D Y PLOT SCALES ARE AnJIJSTE_ 80 THAT THE 313 c PLOT I$ SQUARE, c (IYPICALLY 0) c 316 r nlST - GOULD PAGES REOUEST(X=DTRECTIDN)p EITHER 10 OR 20, c (TYPICALLY tO,O) 3IS c 319 c _BI,XB_ - _INIWUM A_D MAXIMUM W-VALUE8 TO BE PLOTTEO_ 3_n C (ZERO8 PLOT ALL) c C YBIeYB2 = MINIMU_ AND uAXI_UM Y=VALUE$ TO BE PLnTIED, 3_3 c (ZERO8 PLOT ALL) c c *** AFTER THIS INPUT, READ IN BODY COORDTNATF$ = FORWATCEFtOmO) c _et c *** TF NO COORDINATE SYSTEM CONTROL 18 TO RE USED, FnLLOW TMESE CARDS C *** *ITH T_REE BLINK CARDS, TF CONTNnL I$ TO BE USED, *lSF THE 329 C *** FOLL0.1NG I_PUT RATwFR THAN THF BLANK CARDSl 330 c C *** INPUT FOR COORDINATE SYSTE M CONTROL= USE TWO 8ETSe ONE FOR C *** XI-LINE CONTROL AND ONE FOR ETA=LINE CnKTROL, 333 C *** (THIS DATA 18 RFA_ I_ SUBROUTINE SOR.) I 35. c # c *** CARD I ITYPwITYP,NLN=NPTeDECwAMPFAC = FDQMAT(Ab=I_I_F|O=0) 336 c 337 C ATYP = TYPE OF ATTRACTION e (Xl FON XI'LINE ATTRICTION_ _38 c * ET_ FOR ETA-LINE ATTRACTION,) LEFT JUSTIFIED, 33q C c * ITYP - ZERO GIVE8 ATTRACTION ON BOTH $1DES, c * NON=ZERO GIVES ATTRACTInN ON CONVEX SIDE AND C , REPULSION ON CONCAVE 81_E, 3.3 C C e NLN = NUMBER OF ATTRACTION LINES, C l 3_6 C • NPT = NUMBER OF ATTRACTION POINTS, C 3_8 c * OEC - NON-ZERO DEC U_E8 DEC FOR BECAY FACTOR, 3wg C 3S0 c • AMPFAC= NON=ZERO AMPFAC WULTIPLIE$ ALL AMPLITUDES C • BY AMPFACI C 3S$ C **• CARO$(NLN) I JLN_ALN_DLN - FDRMAT[§W=IS_F[0,0) C • (OMIT IF NLN I8 ZERn) C 35_ C • JLN = ATTRACTION LINE INhEW, C c * ALN = AMPLITUDE (NEGATIVE REPELS) FOR LINE ATTRACTION, C e 63

* nL_ = DECAY FACT_ FnR LT_E XTT_AC?IO_J, 3hi C * 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 * 367 C * APT - A_PLTTU_F (NFGATIVF REPEL$_ WOQ POINT ATTrACTIOn. 368 C * 36q * f)PT - _ECAY FACTO_ FO_ P_INT ATT_ACTI_. 370 C * 371 C *** 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) ]75 C * (CAN BE USED T_ AID CONVERGENCE HY CO_vERGING ;IELn 37k * _IT_ LESS ATTRACTION FIRST A_O D_TNG TWIS RESULT 377 C * A_ TH_ INITIAL GHE_S F_ ST_N_E_ ATTRACTION, ]T_ * BLANK C_RD -LIST RE INPUT TF TwTS FEATURE I_ _OT USED. 3To C * 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,) 386 C * 38T C * IRIT - NONmZERO VALUE CAUSES INWO_nGENFOUS TERM TO BE $_ C * PRINTED, 38q C * 390 C * EFIC - MULTIPLE OF CONVERGenCE C_ITERIO_ TO RE USE_ FOR ]gt C * INTERMEDIATE CONVERCENCF BETWEEN AOnITI_NS OF ]9_ C * INNOM_ENEOU8 TERM, (TYPICALLY |00,0 _ _ C * $9_ C ********************************************************************* SqS C $qb C READ INPUT PARAMETERS ]97 C 3qO *RITE(_,6_O} Sqq C _00 C A CALL TD PSEUDO INITIALIZES THE GRAPHIC OUTPUT SYSTEM, THIS MUST _Ot C BE THE FIRST CALL FOR GRAPHICS IN ORDER TO GENERATE PLDT DATA Q02 C IN THE FORM DF A DEVICE INDEPENDENT PLQT VECTOR FILE, _03 C k CALL TO LEROY _LOW8 TMF 8PFED FO_ GERBER F_R LTOUIO INK PEN, _0a C IT I$ OK TO LEAVE CALL LEROY IN FOR T_ OTHER DEVICES, _OS C _Ob CALL PSEUDO _OT CALL LEROY _08 RE_O (§_bSO) C|,C_BDY QO_ READ iS,k20) IMAW.JMAXeNBDYeITEReIGE8.1DISK,I*IR,TWINTLeIWFINeIGEO _10 IF (IMAW,GT,NOIM) wRITE (6e700) IMAX_NDIM _11 IF (JWAW,GT,NOIM|) WRITE (6,?_0} JMAX.NDI_[ U12 IF (IMAX,GT,NDIM,OR,JMAW,GT,NDIM[) STOP [ _15 READ (Se6_O) IPLOT_IPLTR,NCOPY_LINWTI,LINWT_NUMBReNUNBRI_I _1_ tIKIPI_IIKIP_ _[_ REAO (§_q60) NBSEG.NRSEG _16 IF (NB|EG,OT,MNBBEG) WRITE (_,qtO) NBSEG,MNBOEG _|7 IF (NR|EG,GT,MNRDEG) WRITE (6#9_0) NRBEGeMNRBEG _18 IF (NB|EG.GT,MNB|EG.OR.NRSE_,GT_MNR8EQ) 8TOP Z _19 READ (_,880} (LROID(L]_LBI(L}.LRI(L).LRDY(L)eLBt_NBB[_) 0_0 64

tL)wLIE(L)wLSloNQ$_G) _EE RFA_ (_eO00) (_(1)_Tule$),YT_FI"_eAIKFT_,XnI4FjYOI_F,_I_'F u23 _An (S,_20) IEV,IAIT,_(In) RFAb (_b20) I_FAC_I_rACn IF (IRLOT.hE.O] _FAP(_M30) !TZFogkTIOenI!T,X_leX_2wvR1wYR2 UE/_ _27 _WITF T_PLIT PARAMETFR$ (_28 uEq *RITE (6,1n60) _rRITE (6,990} I"_W_J"_X,N_DY,ITKR,I_ES,T{)ISw_I*IR,IWINTL,IWFI _ u3! %,IGEO IPLOT,TPLTW,_COPY_I. IN-T_,I I_WTP,NU_BP_U'_R|

_RITE (_,I010) wWI_E (_,Ih205 _RITP (a,tO3h] I_V_I_IT,R(IO}

$1ZE,_ATTC},I)T!T_XBI,XB_,YHI_YB_ (_3q *RITE (6,101o) _W0 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) UWQ III(L),LI)(L)eLmI,NNSEG) IF (INFACO.EG.O) INFACOml IF (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) Q_0 SET {JP P_RAMETERS

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} 18KIP2ut _SB IF (NBDYoLE.O) NRDYm| _sq IF (I_LOT,EQ,O) G_ TO 10 /4b0 IF (kB$(XBI),LT,ZERO) XBIB=t,0EEO IF (AB$(XBE).LT.ZERO) XS_le!lO[_O

I_ (AB$(YRE).LT.ZERO) YB_meI.0[_O /4e/_ IF (DIST,LE,O,O) DTSTmlO,O I0 CONTINU_

RE_D POINT8 ON BOOIE_ AND READ OR CALCULATE _OINT5 ON OUTER ROUNDARY /4bq

DO 20 LmI_NB$E$ E0 If (IAB_[LBDYtL]},EQ.NBOYeI.DR.LBDV(L).EG.O) GO TO $0 LISEGmNBOEG_I GO TO _0 30 LISEGmL _0 DO SO luleZM_X DO _0 Jmt,JMkX X(Z,J)uO.O SO VCI,J)u0,O k_OYOmO _0 LBDYCNB8EG+I)me|O0 _81 nO _EO LmSmNBSEG .82 XImLBI(L) _8_ T_mL82(L] _8_ ISENml _B5 TF (Tt.GT.I_) TBENB-I ,86 LB[N(L)mTBEN _87 KImMTNO(II_T2) _8 K_mMAXO(II.I2] ,89 LBI(L)mKI U90 LBZ(L)mK2 _91 Ie (LBnY(L).EQ.LBnYO) IIIIIeISEN _2 IF (L_DY(L).NE.LBDY(LeI]] I2mT2-TSE_J _93 IF (LBDYtL)mNEmLBOYO) I_mTI ,q, KImMTN0(TI,12} .9_ K2mMAX0(IIwI2} _q6 IGOT0mLBBIO(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$10(L}pYI_FI_eAINFINmXOINF#YOIN_wNINF_ _0_ INDIM} 50b GO TO 130 S07 90 DO _00 XmKl,X_ 50B IIII÷(KmKI)*IBEN S09 100 RE_ (S,850) W(T,J],Y(I,J} 51_ 1_0 I_ (LBOV(L),EQ,LBDYO) GO TO I_0 SII I_mj 51Z x._mXCla,la) 513 Y_6tY(I_,Ia) 51Q _0 TO t50 SIS IQO CONTINUE 51b XCII-18EN.J]mX]5 517 Y(II-IBEN,J)uY$_ 518 150 ISwJ 51q ISmI_ 520 X$SmX(l$_IS) 5_1 Y$SIY{I$115) 5_ IF (LBOY(L),EQ,LROY(Lel]) GO 70 2bO 5_3 IF (NR_EGmEOm0eAND_NBOYmGTmI) GO TO _0 5_ X(II+IBEN_J)mW.6 5_ YfI2+IBEN_J)aYQa 526 GO TO _0 5_? C*** LEFT OR RIGMT S_8 160 IF (LBSIDCL],EQ,Z) Iml 5_9 I_ (LBBIDCL),EQ,#) TmIM_X 530 IF (NROEG,EQ,O) GO 70 190 S]l IF (L,LT,LTSEG) GO TO 190 S$Z IF (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) CALL WAXMIN (X,T_AX,JMaXpNDI_,XB_AX,XRVTN,IDLwM,InlIM,IDUM,IDUMel, it) Cn_TINUE LROYOuLB_Y(L) _Oo cONTINU_ DO _90 LI|aNMSEG )9O L_DYfL)mI_B$(LBnY(L))

CLASSIFY RE-ENTRANT SEG_MENT_

IF (NRSEG.EO.O} GO Tn 3_0 no ),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)mi IF (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 CONTINUE 3_0 CONTINUE

PULL OUTEP BOUNDARY IN FOR INITIIL GUE$_ Eft IN@_C NOT ZERO

Ir (IN_C.GT.0) ClN_Cm!.0/FLO_T(E**(INF_{-!)) IF (INFAC.LT.0) CIN_A_m_LOAT(IN_ACO)/AB_(FLOAT(TNFA_)) IF (LISEG.GT.N68_) GO TO DO QO0 LmLISEG_NB_EG ItmLBt(L) I)mLS)(L) IGOTOmLBSlO(L) _0 TO (_60_380_60,_80)) IGOTO $60 IF (IGOTO,EQ,I) Jml IF (I_OTO,IQ,)) JmJW&X DO 370 Imll,IE 67

370 Y(I,J)mY(I,J)*CINFAC GO T_ _00 380 IW (IGOTO,_Om2) Iml IF (IGOTn,_Q,_) ImI_AX DO ]_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_.NB IEGeLRIpLR2,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_RATIO QT0 CONTINUE IF (IWINTL.EQ.O) GO TO SO0 wRIT( (_?_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_WAC IC,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 STO WRITE (5_6_0) WRITE(6_6_O} CI wRITE(_.b_O) C_ W_ITE (_770)

O0 TO _0 wRIT[ (keT_O) ITER 68

b_1 530 ,_ITE {6,8nnl 662 t665 _QITE (6,_50) TXERCI),TXFR(2)eIYF_(1),IVF_(_) *mITE (_,7_0) b6_ _0 550 JII,JMiX ,mITE (_,75o) J 6e7 550 _ITE (6,71n) Cx(Y,J)oI¢IoIMaX) *_ITE (6e7_0) Dn 560 JsleJ_X 67n _RITE (b_75_) J _71 S_O ,_ITE (6,71e) (Y(I.J).ImI.I_AX) _RITE (6,6Q0) e73 570 TK (ZDISK.EU.O.n_.TDISK.EQ._) _P TO 6_ _RITE (10,q80) C1 675 wRITE (lO,gSO) C2 TF (IDISK.EQ.,) GO TO $80 *RITE(IO_620) I_AX,J_X,_aSEG,_'_$£G,L?SEG,_BDY 678 _RITE(IO.qFI)TLSBT_CL),L_](L),LR2(L),LBnY_L_.LSE_rL),Lmt,N_SEG) 67_ wRITE(IO.ga_)(LRST_(L),L_I(L),L_?(L),LIST_(L),LIIfLI,LI2(1), tLTYPE(L),LBI,N@$EG) GO TO _0 580 _RITE(IO,_I) IMAX,J"_X 5_ *_I?E(10eg83)[(XC_,l),Im1,I_AX),Jm1,J_iX),(CY(led)_lileI_al)_Jlle 68U tj_X) 600 COnTInUE bBb 68T PLOT 668 68_ IF (IPLOT.GT.O) _0 Tn 61n STOP 69t 610 CALL CORPLT (X,Y,NDI_,4tJ_RR1,NU"BR,C1,C_,RATIO,$IZE,NCOPY_LINWT1,L 69_ IIN_T_,ISKIPI,IB_IP_eXPLOTeYPLOT,DTST_TPLT_NPLT$,LBSIDeLBle 6q3 _LB_,LSOYeL_$ID,LISIO,L_t,LH_,LII,LI_eLTYPEe_BBEG,NRSEGeLI$EG,JMAXe 31_AX_X_t_XR_,Y_I_YR_) 69_ IF (_PLOT.GT._) CALL PLOT (O..O..g09) b_6 STOP 010t 6_7 698 6ZO FORMAT (1615) bq9 6]0 FORMAl {8F10.0) 7O0 b_O FORMAT _IH1) 701 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 TO_ _UMBER OR ETA-LINES • *,I_t/ItqX,* ITF_ATION PARAMETEmS! $OR ACCEL ?Oh ]ERATION PARAMETER • *,FS,St_ZX,* MAXI_u_ NUMBER OF ITERATIONS ALLO _OT _wEO u *,I=/3_X,* ALLOWABLE ITERATIO_ ERROR NORMSl Xl *,E10,_/75 ?08 SX_* Yl *,E|O,S) bBO FORMAT {I_IX_*NO PLOTS DESIRED*) 710 650 FORMAT (/EIX,*PLOT PARAMETERS1 COPIES DESIRED • *I3/_?x,* LINE*, T11 1*wEIGHT _ES_RED • *I3t_qxe*PLOT SIZE TN YoDIRECTIDN • *F§,]/3qX_*R Tt| |ATlO I *_F8,$) Tl$ ?00 FORMAT (*0--- ERROR ---- IMAX TO0 LARGE*,IOX,*IMAXs*,TS_SXn*MAXI_U tM 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 _,*) TiT 710 FORMAT (_Xe10F11.S) 710 ?_0 FORMAT (*Omo- ERROR ..o= JMAX TOO LARGEteIOXetJMAXi*e_eSXeeMAXtMU 715 IM IS,,I_/t6X,*INCREASE NDIMt TN DATA STATEMENT AND*,,' SECOND DIMEN 69

2SION OF X,Y,WETA,RXIewACCoTACC**IIbX,,ALSn,** INCREASE _IMENSION O T21 3F WPLOT AND YPLOT TO MAXIMUM OF*,m NDIM A_O NDIM| PLUS 2**) T22 ?]0 FORMAT (/21We*INITIAL GUESSES FDW X AND Ye) 723 7UO FORMAT (/_Xe]3(JM,)wq_p,XmARRAymSw_](|Ht)} 72Q TSO FORMAT (SXw*JmeI_} 725 T60 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$1 Xl *wE|O.Se* YI _eE 7_2 tlO,Sw* AT ITERATE i |*/2IX**FINAL ITERATInN ERRQP,,, NORMS1 Xl 7_ 2*wElO,Sw* YI *eE|O,So* AT ITERATE #*,I_) 73Q 820 FORMAT (215,F10.0) 830 FORMAT (2F|O.O) 7_5 736 850 FORMAT (21Xo*LOCkTION OF MAXIMUM TTE_ATIO_I ERRORI Xl I=*I_p*,Jm*Z5 737 l/SOXe*Yl II*I_we J=*IS) T]_ 860 FORMAT (/21XweNUMBER OF BODIES TN FIELD 1.I5) 739 880870 FORMAT (_15)(*OUNIFORM ACCELERATION PARJMETER USED**) ?gO 890 FORMAT (61S) 7WI 7_2 qO0 FORMAT (TFIO,Op2IS) 7_3 q|o FORMAT (*O_w. ERROR ._._ NBSEG TOO LARGF*,IOX_*NBSEGIe_I_eSXe*MAXI 7_ IMUM IS*,I3/16X,*INCREASE MNRSEG IN _aTA 8TATEMEN T AND,o, DIMENSIO N TQ5 28 OF LB8IO_LBt,L_2,L_OY_L8EN,,) 7_6 q20 FOWMAT (*O*=w ERROR .-.= NRSEG TOO LARGEI,IOX_*NRSEGle_I$_§W_*MAXI ?at IMUM 18*_I3/IbX**INCREASE MNR8EG IN. DATA STATEMENT AND*_* DIMENSION TQ8 2S OF LRBID_LRI_LR2_LISIO_LII_LI2,LTYPE,,) TWq q30 FDRMAT (cO--BODY 8E_MENTS''*//$W,*L*_2W,*LB8IDe_QX_,LBI._X_.LB2,_ TSO I$W_*LBOY*//(Ia_I?)) ?St 9_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?)) 753 qSO 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_ NMENT OF _NFINITY,,IOW_, ?S6 ]INITIAL STEP (LINEAR CASE) m*_I_) ?ST _60 FORMAT (2IS_2F10,0) 7SB qTO FORMAT (*OINITIAL GUESS TVPEl*eI_) 7Yq qO0 FORMAT( OAlO) q81FORMAT(SI_} 760 982 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_) ?b8 1010 tlS)FORMAT (*OR(t)_R(2)_R(])_YINFIN,AINFIN,XOINF,YOINF_NINF I*/?F15.8, 76q ?70 |030 FORMAT (*OIEV,IAIT_R(IO) 1*_2IS.FIS,8) f?! tOaO FORM.AT (*OINFAC,INFACO I*,_IS) T72 10§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,)) 775 1080 FORMAT (*OSIMPLYwCONNECTED REGION*) C IT6 END 77T ?T8 TTe 780 7O

781 ?BE 8UHROUTTNE BNORY {X,y,IJ,TI,12,LBI_,R,A,XO_YO,_:TNF,N] C '_8S C ********************* CIRCULAR OUTER BOUNDARY *********************** 786 C t ?ST t $ THIS ROLITINE CALCULATE8 NINF XeY COORDTNATE8 AROUND A CIRCLE OF ?88 C • RADIUS R AT EQUALLY 8PACED ANGULAR INCREHENT8 STARTING AT ANGLE A 789 C * (POSITIVE COUNTERoCLOCKWISE FRO_ POSITIVE x-AXIS] ANO PROCEEDING ?eO CLOCLOCK_ISE FROM THI8 &NGLEe Tgt C * ?BE C i_,il**ililllli.i.,lllllll.li_i_lilil*_lil*illl'li**'lllli_ll**illll* 793 C _IMENSION X(Nol)w Y(N,1) 79_ DATA PI/_.tU1_9_65_91 T96 ?R? KIsMIN0[II,I_) ?98 KEmMAX0(II,12_ TR9 ISENel 800 IV (II.GT.IE) ISENI=I 801 IXMIININF 80E Dm.E.OtPI/FLOAT(IXMI} 80_ GO TO (I0,30,I0,$0], LSID BO_ Ct*** BOTTOM _R TOP 80S 10 00 EO KIKteKE 80_ ImIle(K-Kt]*IBEN SOT XtI,IJ)mR*COBtA)+XO 80B y(I.IJ)IRI$1N(A}eYO 809 EO Asked 810 GO TO SO 811 _0 O0 _0 KmKIoKE C***t LEFT OR RIGHT Jmlle(KmKl)ilBEN X[IJ,J)mR*COS(A)+XO y(IJeJ]BRIBIN(A]eYO _0 ksA÷O 817 SO CONTINUE 818 RETURN 819 C BEO END BEE 8E$

B_S SUBROUTINE CORPLT (%,YwNDIM,NUNBR|P NUMBR'CI'C_pRATIOpBIZEwNCOPYwLI BEb INWTI,LINWTEwlSKIPI,18KIPE,XPLOT,YPLOT.DISTeIPLTR'NPLT$tLBBI BET EO.LBI,LB2,LBOY_LRSID,LISID_LRt_L R_'LII_LI_'LTYP_'NBSEG_NRSEG_LISEG SEB $.JJMAX,IIMAX.XBIeXB_YBI_YB_] 81$ C 050 C **_ PLOT ROUTINE • PLOT8 COORDINATE 8YBTBM AND SEGMENT DIAGRAM e_mwe C * C *********************************************************************

C DIMENSION LBBIO(1), LBt(1)e LBE(I]e LBOY(t), LRBIDCt]. LZ8IO(i]e L IRI(I), LRE(I), LIICI), LIE(t)e LTYPE(I] DIMENSION X(NDIMel], Y(NDIMeI]_ XPLOT(I), YPLOT(I]* CI(I], C_(l)e 857 lkXISL(_)e XA(6), YA(6] 85S OATA 8CAL /O.II DATA 81 I0,08?S/ DATA 8_ tO,l?S/ 71

DATA HA 11,01 DATA Hg /Oe5 / C***** ISYm2T 8_3 JMAWmNUMBR IMAXINUW_R| 8_5 HlmO.5*81 M2si.S*81 _3m0.S,8_ H_ml.S,S2 HSmO.2S,8! 8_n H6m2.0,81 H?ml.$,8) IMAXMImIMAX.I _53 C C JXl8 M_NIMuM 8 AN_ $CkLE FACTON$ 85_ C

I,I8KIPE)CALL MAXMTN _XeIMAX,JMAX,NOIMoXMAX,X_IN,IXMXe3XMXeIX_NejXMNelSKIp |

IoISKIP_)CALL MAXM)N (y. IMAX,J.AX,NDT_,yMAxeyMTN.IX.X,JXMX,IXHN,JXMN,IBKIPI 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 I0 Y2m(YMAX-YMIN)IAXIBL(2) 869 X2mY) _70 AWISL(I)m(XMAX-XMIN)IX2 GO TO 30 _72 2o AXISLCI)mAXISL(2) 8_3 W2m(xMAx-XMIN)IAWISL(I) Y2I(YMAX-YMIN)IAXISL(2) 87S

SET UP PLOTTER

30 CONTINUE 87q NPLT|I2

LAB_L$ 882 883 88_ IF(IPLTR,E@,I.OR,IPLTR,EG,_) CALL LTNwT@IPLTR,LINWTI) CALL PLOT (,5,,_,-3) 8aS 886 CALL 8YMBOL(O,,.OOIw,08?S_CI,90_,80) CALL 8YWBOL(,3e,OOI,,OS?Sec_,qo,eSO) CALL PLOT (1..0.e-$) 8_8 CALL PLOT (0..9'99.|) 889 CALL PLOT (.Se.Se-)) C C PLOT LINES OF CONSTANT ETA C 8q] eq_ IFCIPLTR,EQ,[,O_,IPLTR,EG,2) CALL LINWT(IPL_R_LINWy2) DO TO Jm)_MAX_ISKIP) 8q_ WmO 8q_ O0 _0 Iml,IMAX,18KIPl 8qT

_qq IF (Y(I_J),OT,YSI,OR,Y(I_J),LT,YSI) GO TO (0 qO0 72

901 KIK÷I 90E XPL_T(K)WX(I,J) 905 YPLOT(K)aY(I,J) _ CONTI_OE 905 xPLOT(Ke|)IX_ _06 XPLOTrK+E),X2 90? yPL_TCKel)mYI 90P, yPLOT(KeE_sY2 q09 50 CaLL LINE (XPLnTwYPLOT_KwI*O_Oe.OT} OtO

PLOT LINE8 OF COhSTANT XI 915 On TO II],IMAXjISKIP] KmO 915 OO 60 JmlwJMkX,lSKl p2 6n I; (X(I_J).GT,XBE,OR,W(IwJ},LT.WBI) 911 IF (y(IeJ).GT,YBE,OR,Y(IpJ),LT,YBI) bO OrB xmKeI 919 WPLDT(K)mX(I,J) 9Z.O yPLOT(K)mY(IpJ) b_ CONTINUE 9E2 XPLOT(K_|}BX[ 9;t$ XPLOT(Ke2}BX@ 9Z_ yPLOT(K+_)mvl 9ZS YPLOT(KeE)mY2 9Zb ?0 CaLL LINE (XPLOTpYPLOTpKRIpOeOp,O?) 0/.? C 9Ze C SEGMENT CONFIGURATION nIAGRAM 9Zq C 9ZO CALL PLOT (AWISL(1)*E,Ow[,O,-3) 9],1 JMAXmJj_AX IMAXBIIMAX 9],$ C C**** BODY SEGMENT8 **** 93S C O$6 00 100 LslwNBBEG OlBLBI(L)*SCAL 9]IB DEBLB2(L)i$CAL IGOTOBLBBID(L) 9_0 GO TO (&OegO_BOigO). IGOTO If (LBBID(L),EQ,%} DSmBCAL 9_Z _OTTOM OR TOP IF (LBSlD(L},EQ,$) D]mJMAX*BCAL LwTm-_ I_ (L,GE,LIIEG) LWTI'I IP(IPLTR.EQ.t.OR.IPLTR.EG.E} CAL.L LINWT(IPLTRwLWT} CALL PLOT (DtwO_,$) 9_8 CALL PLOT (OioD]el) IF(IPLIR.EQ,I,DR,IPLTR,EQ,E) CALL LIN_T{IPLTR_O) IF (L88ID(L},EQ,I) MIwM_ 9St IF (LBBID(L),EQ,$) MuM208! 9'JE MMIBIGN(MgwM)-_2t(I.O_BIGN (t,OeM)}*O.$ 955 HMM|BIGN(M]eM) _FLOAT(LBt(L)},O.O_'t} CALL NUMBER (D[eMt_OSeM_B| R_'J CALL NUMBER (O|-Wt_O|eMell _LOAT[LB_[L})eO,O_=t) CALL LINWT(IPLTRe-I) _Sb IF[IPLTR,EQ'.|.OR.IPLTR.EQ.|} 9S? IF (L,LT.LIIE;] CALL NUMBER ((DteDE),O.S=M3,DSeMM.IEeFLOAT(LBDY( eSe IL)),O,O_e[) _)IH) IF(LeGE,LIBEG)CALL BYMBOL((DteDI)_O_S-M$_DSeHM_B_IBY_OeO_=I] _)bO IF(IPLTR.EQ.i.OR.IPLTR.E_.2) CALL LINWT(IPLTR_O} 73 '_

CALL PLOT (_].D3-HHH_2) 96] CALL PLOT (D2pD$+Hww,]) 962 CALL PLOT (D2_D].Hw_w2 }" 963 GO TO £00 ¢b. C**** LEFT OR RIGHT 9bS 90 IF (LBBID(L).EQ.2) O]w$CAL oh6 IF (LRBTDCL).EQ._) _]II_AX,BCIL gb7 LwTx-2 9_B ¢b9 IF (L.GE.LIBEG) LWTm.[ g70 IF(IPLT_.EQ,I,OR,IPLTR.EQ,2) CJll LIN_T(IPLTRoL_T) g71 CALL PLUT (D].Dle_) gT_ CALL PLOT (_3_npw2) g75 IF(IPLTRoEg. I.OR,IPLT_mEQ,2) CALL LINwT(IPLT_a_ ) Q7Q IF (LBBTD(L),EQ.2) HI-H6 Q75 IF (LB$1D(L)mEQ,_) waN6-8! g7b wMmBIGN(HTww)'B2*(|'O+$TGN(l'_'w))*n'l q77 HMHmBIGN(MS,H) g7_ CALL _IJMBE_ (D_W_DtwBIeFLDATCLBI(L_),O,O,.I) 979 CALL NUMBEP (D)+MeDP#SIjFLDAT(LR_(L)),O,Op.)) q_O IF(IPLTR,EO,lo0_eIPLT_mEQ,@) CALt. LIN*TtTPLT_.I) 081 IF {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) get CALL PLOT (n]+wwH.02,_) _8_ CALL PL_T (D]-wHM,D2_2) 100 CONTINUE 9Bg C ¢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 gg6 D_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_ 1000 C**** BOTTOM OR TOP - FIRST OF PAIR 1001 lln IF (LRBIDLL),EQ,I) n3mBCAL _00_ IF (LRBID(L),E_.]) DBmJM_Xe$CAL 100_ CALL PLOT (01_D]_$) [OOQ CALL PLOT (D2_D3_2) [OOS GO TO 13o C_*_* LEFT OR RIGHT - FIRST OF PAIR 1006 [007 120 IF (LMBIO(L),EQ,_) D3o$CAL 1008 IF (LRBIO(L),EQ,_) D3BIMAXm$C_L 100g CALL PLOT (D$_DI,)) 1010 C_LL PLOT (0],02,2) 150 D_mLII(L),BCAL 1011 101_ D_mLI_(L)_BCAL 1015 IGOTOeLIBIDtL) _0 TO (laOelSO,lOO,l_O)_ IGOTO 101_ C,m** BOTTOM OR TOP - BECOND OF PAIR 1015 laO IF (LIBID(L).EG.I) D6mBC_L lOlb 1017 IF (LI81OtL),EQ,3) D6BJM_X*BCAL 101B CALL PLOT.(D_D_$) 101g CALL PLOT COS_O6_) 10_0 74

lO)t GO T_ lb0 LEFT O_ RTG_T - _ECO_n OF PAI_ I0_$ I0_ IF (LI$ID(L).EQ._) _blIMAX*SCAL IO_S CILL PLQT (Dbm_Uw31 I0_6 CALL PLDT (nb,DS#2) 1027 1028 _PTTED LINES CON_EC TI_'G HE-E_THANT I0_ I0$0 ]hA TF(|PLTg,Eq,],OR,TPLTR,EQ,2) CALL L;N*T(IPLT_,_ 1 lO$t IGOTOILTYPE{L) I0_ GO TO (170_l_OpiQO,200e_lOw22_), IGOTe Io31 C**** ONE 0_ BOTTOm, O_F _ TOP I05_ i?n XA(I)m(_t+_2)*O.S 1015 yA(t)mn3 ln_6 Xi(2)mx*(t) 10_7 yIc_)uVk(1)-H_ 1058 XA(_)ITMAX*SCAL+HB tO)_ IF (2,0*D_,LT, FL_TfTM_x)*$CAL) XA(3)m-H8 I0_0 ¥A(_)NYA(_) I0_2 yA(Q)IJMAX_$CAL÷_8 I0_1 XAC_)uCD_÷D_)*O.5 lo_a

xA(_)mX4(5) 1o_6 Yi(b)lO6 lO_T NAmb 1o_8 GO TO 23_ IOU_ C**** _OT_ ON HOTTO_ loso 180 XA(l)m(nl*O2)*O.5 1051 Yk(I)ID] I05_ IOS1 yA(_)IYAC1)-HQ 105_ 1o_$ y_(3)lYk(_) I056 I0_? 1058 _kmQ lOng GO TO 2_0 1060 C**** BOT_ ON TOP lOet t_0 XA(1)m(nl+O_)*O,5 106_ VA(1).O$ 1061 XA(2)mXA([) lOb_ Y_CZ)|VA(I)_Hg lOb_ Xk(3)m(O_eO_)/O,_ 1066 YI(_)IYA(_) rob? lOb_ YA(_)mDb lOb_ N_N_ I070 GO TO 2)0 I011 C**** O_E ON LEFT_ ONE ON _IGHT 107_ 200 X_(I)103 1073 YA(|)I(D]_D2)*O,_ 107_ XA(E)mXA(t)-H8 107_ Y_(2)mYA(I) 1076 XA(])IXA(2) 107? y_($)mJMAX*BCAL*H8 1018 IF (2.0,D2.LT.FLOAT(jH4X)*BC_L) YA())mm#8 10?_ XI({)mlMAXISCAL+_8 1080 YA(_)mYA()) 75

XA(S)mXA(_) YiCS)mCO,tO_),n.S 1081 XACb)aDb I0_ Y*(6)mvA(S) 1083 _Am6 I08. GO TO E30 In_b _OTH O_ LEFT lOeb Xa(l)mO3 I087 YA(1)I@OI÷_E),n.S 1o88 XA(E)mXA(1).M9 108q Y*C2)mYACI) lOgO Xi(_)mXA(2) I0@I YAC$)m(O,÷DS),n._ IOq_ Xa(_)mO6 YA(_)mYA(3) I09, Inq5 GO TO E$O log6 80Tw ON RIGHT 10qT XA(1)mO$ IOqM YaC1)m(_leO2)eO.5 I099 Xi_2)uXA(l)÷M9 1100 Va(E)uY,(1) II01 XaC3)mXaT2) IIoE YAC$)m(Do+D_).O.5 1103 XA(_)m06 110_ YAC,)mva($) 1105 NAN_ 1106 1107 PLOT DOTTED LINES 1108 11o_ 1110 230 CALL PLOT CXACI)wYA(1).)) DO EaO NIE,NA 1111 111_ E,O CALL PLOT (XA(N),Va(N),E) 250 CONTINUE 111_ Ilia TERMINATION SEQUENCE 1115

E_,n 1117 CALL PLOT ($CALiKL_JT(IMAX)_/.Oj.2.Se.3_ CALL NFRAME 111_ RETUR_ 1119 1120 END 11E1 11EE lIE3

lIES llEb SUBROUTINE LINWT(IPLTR,IPEN) C lIE? C C t******************** SETS PLOT LINE wEIGHT ************************** 11_ C 11]0 C 1131 ;0 TO (IO.EO). IPLTR 1112 10 CaLL LI_EWT(IPEN) 1133 RETURN 11_ EO CILL PENSL (IPEN) R_TURN 1156 END 1137 1138 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*** II_7

115n TIA_S{V_EW-V_Lb) 1151 _o ?0 10 IF (T.LF.ERROL_I I15_ IVER(1)uIFLn 1153 IVE_(_)IJFLO ERRO_IT 1155 RETuR_ 115_ F_WOwIEPWOL_ 10 tlS? RETUW_ 1159 E_c) 1160 11_1 11_2

11_

1166 1167

11bq 1170 1171 117_ 1175 C_ 117_ TGEBuO 1175 IE(IGE_.EQ.2.0N.IGE_.GE.51 1176 J,XwIIJWAX-| 1177 I_XMIII_IX=I

C 117q C PROJECTION FROM BODY SEGMENTS 1180 C 1181 Ie CIGES,NE,O) GO 10 )0 1182 TIIO,I*tIGESS-"] 1185 _RG_ITI*JMWM1 118= T_II,0/(EXP(ARG2)'I,O) ARG)ITI*IMWMI 1186 TSII.O/(EX_(&RG$)-I.0) 11BT DFACII,0 118B neACCIl,O llBq IK(IGED.EQ.$) DFACCIO.O 1190 1191 DO _0 JI_,JMWwt IF(IGES$,GE,5) GO TO 1195 FACIFLO&TCJ-I)/FLOATCjMXMt) GO TO ? 11qS _RGtITt*(J-t) FACI(EXP(ARGI]-I,O)=TZ CONTINU_ DO 10 II_IMXMI 11qq T_(IGES$,GE,S) GO TO 8 1_00 FkCCIFLOATCI-I}/FLOkT(INXMI) 77

GO TO 9 8 ARGImTI*(I=I) 1_01 12o2 FACCI(EXP(kRGI)-I.O].T3 g CONTINUE lZn3 120_ X(ZeJ)m(X(Zel)e(XCZejMAX)=X(Zel))*FAC)*_FACe(CX(ZM&XeJ).X(Iej) l)*FJCCeX(teJ))wOFJCC 1205 1206 Y(ZwJ)m(Y(]et)etY(ZeJMAX)'Y(Ie]))*FACI*DFkCe(/y(]MAXej)=y(|wj) 11*FACC+Y(|wJ])*OFkCC lEO? 1208 CF (XGE$8,EO,R,OR.ZGESS,GE,S) GO TO 1_ 1209 ]F(DFACzEGeO*OmORmDFk_CmEO.O,O) GO TO 1_ X(%.J)mO.S.X(r.j) 121n 1211 Y(IeJ)aO.S*Y(ZwJ) 10 CONT|NUE 1212 ZO COkTZNUE 1213

RE-ENTRkNT SEGMENT8 1215 121_ 1217 30 ZF (hR$EG.EO.O) GO TO t,O 1218 O0 15n LBL._RSEG ZloLRI(L) 1219 tZmLRZ[L) 1220 ZlisZt+i ZZE=Z2-i 1222 1223 ZGOTOaLRSZD(L) 122_ GO TO (_OebOe_OebO)w IGOTO 12Z_ "0 re (LRSID(L).EO.t) Jzl 122a IF (LRSYD(L),EO,3) JzJ_kX OXmX(IE,J]=X(I1,J) 12_7 OVlYtlZeJ]-V(IteJ) OXm_X/CIZ-%[] 1_29 OYNDY/(12*II) 1250 1211 DO SO ]mll1,11Z DDXm(I-II],DX 1232 ODYm(I*II)*DY 1233 123_ X(leJ)mX(llwJ]+ODX SO Y(I,J)IY(II,J),ODY 1215 GO TO 80 1237 60 IF (LRSXD(L).EO.2] Zml 1238 IF (LRSID(L)mEQ._] TmZHAX 123_ DWmX(I,IZ)-W(I,II) 12,0 DYmY(I,Ii)mY(I,II) OXIDX/(II-II) 12=1 DYIDYI(_-X[) 12_E O0 70 JslIt,lt2 DDXm(J-II),OX 12_5 DDYm(J=Zt)*DY lZa_ XtI,J)mX(Z_It)÷O_X 12_? 70 Y(leJ)IV(Iolt)÷OOV 80 IImLII(L) I)mLII(L) lllmll+! 12So IIi=IE-I 12S1 IGOTOeLI|ID(L) 1_§2 IZS$ GO TO (90oltOjgOwllO)e IGOTO 90 IF (L_$IDCL).Eg.I) Jet 12SS IF (LI$_DCL).EG.3] JIJ_X

OX|DX#(_-]I) 1_$8 OYIDY#(]_||) I_So 1260 78

lee! 1262 12b3 IEb_ 126_ 1_66 1267 1268 IR_q 1270 I_71 I_72 1273 IE7_ 1275 127_ 127T 1E7B IE7e 1E80 1281 LINEAN PROJECTTO_ o IGE881 128_ 1283 %F (IGES.NE.I) GO TO 170 IE8U OFiCl|,O 1285 OFACCIi.O 128_ %e[%GEO,Eg.t] OekCCmO.O 128T ]V(TGEO,EQ,2) DFACIO.O 1288 DO lbO JxEeJMXM! 1289 VACxVLO[T(j-i)/FLOkT(JHXM1] 1290 DO 150 %mE,%MXM! FACCIFLOATC]-|)tFLOAT(]MX_I) X(I,j)x(XIT_tlefXIT,JMAX_=Xfl,I)),KkCI*DFAC*(_X(IHAXJJI=X(I,J] 12_2 12g_ I),FACC_X(twJ))*OFkCC 12q_ y([eJIBIY(Iel)e(Y(IeJM&XI=Y¢I_I_] *FAC)*DVAC+((YflHAX°JI.Y(IeJ) 129S t],VACCeY(IpJ]]*DFACC %V(DVACoOT,O,OoOR,DFACC,GT,O.O) GO TO 1_0 l_gT x(%,J)mO,5*X(T,J] Y(%eJ]lO,_*Y(TwJ) 150 CONTTNUE I$00 160 CONTINUE RETURN 15o_ 13o3 NOMENT OR EXPONENTTAL PROJECTTON I]0_ 1305 1?o BNII, CIMAX÷JMAX)-_ 1306 DNm$ORT(FLOAT((%_AX-I]**E*(J_AX-I]**2]5 1_07 OMmtTABB(%GE$])/OM 1308 DO ZIO Jm|,JMXHt 150_ O0 _|0 II2o_XH| 1$10 $XBO_O I$II |YmO,O 1312 |OlO.O I]I_ 8XOlO,O IYDIO,O 1315 IXNIO_O 1316 If_lO,O tE_O_O 1318 DO 180 TIml,IMkX 131_ |XllX+X(%Iel)eX[%_eJ"kX) |YISYeY(Zlel]eY(l_.J"AX) 13_0 79

DmIAB$CTI=I) I)ImTAB$(1-J) 132Z DEmlAB8CJM&X.J) 1323 OmD**Z 132. OInSORTCD+DI**2) D2mS_RT(O_D_**2) 13E6 _OnSDeDI÷D_ 13_7 SXDsSXDeX{TIet).DIeX(II_JMAX)._2 SYOaSYOeYtIIe|)*()|÷Y(TIeJMjX).D_ 1329 EImEXP(-DIeDM] 1330 EEuEXP[-DE*OM) SXMISXM÷X{II,I)*EIeXCIIpj_AW).E2 1332 8YMBSYMeY(IIel}*EI÷Y(IIwjMAW),E? 1533 9EBSEeEtsE_ 180 CONTINUE 1_35 D0 190 JJn2eJMX_! 13_6 8XmSXeX(teJJ)÷X(IMAXeJj) 1337 SYeSYeY(toJJ),YCIMkX_JJ) 1358 DmIABSCJJ=J) 133_ DIuIABS(I-I) 13.0 D2eIIO$(IMAX-I} OLD**2 01mSQRTtOeDI,mE) 13.3 O_mSQRT(Oe_2**2] 13,, SDISDeDIeO_ 13.S $X0mSXDeX[lejJ]*0teX(IMJX,jJ}.D2 13.6 8YOmBYDeY(I.JJ)*OI÷Y(IMAX.JJ),O? 13.7 EIBEXP(eDI,DM) 13_8 E2|EXP(wD2,0M) 13.9 SXMISXMeW(IoJJ)eEIeW(IMAXeJJ)tE) 135O 8YMISYMeY(|oJJ),EIeY(IMAX.JJ),E_ 135! SEB$EeEI÷E2 1362 lqO CONTINUE 1353 IF (IGESoEQ.3) GO TO ZOO 135" 1355 EXPONENTIAL PROJECTION - IGE$cO 1356 135? TImI.O/$E 13§8 w(I.J)mTI*SxM 1359 Y(IoJ)oTI*gYM 13e0 GO TO 210 1361 200 CONTINUE 1363 MOMENT PROJECTION - IGES= ] OR . 136_ 13bS TIIt.O/BD 13b_ IF (IGES.EG.$) TImTI/8N I$b7 X(I.J)u($D*SW-BXO)*TI Y(IeJ]O(BO*SY=BYD]tTI 13b_ ZIO CONTINUE 1370 RETURN 1171 137_ END 1375 137, I]?S

1377 |UBROUTINE MAXMIN (XBIMAX.JMAXeNOIW.XMIWeXMIN.INXoJMXeIMNeJMNwIIKI I$?| teleIlwIPl) l]?e 1300 80

c **** TwI$ SU_ROoTIN_ CALCSJLATES TM_ _XI_;_ A_ MINIMUM VALUFS ****** C **** (IF A T*O-_IWENSTnNAL DATA A_RAY. ****_* l_SE C * C , T_PLI? DATAI C * C * Xm2-D _TA ARPAY _OSF MAXIMUM & _TNIMU_ IS T_ BE O£TERwINEn 1_86 C • I_AXmLARG_SI VALIJF or ;IRST SUBSCRIPT nF X TO RE SCANNED 1387 £ • JMAXsLAR_ES? VALUE _r SECO_JO SUR$CRIPT _F X 70 BF SCANNED 118R C * NDIMmIST DIME_$IOK' OK X I_89 C * ISKIPIaSKIP PARAMETFR FOR |ST INDEX OF X (?ME l INDEX) 1590 C * ISKIP2mSKIP PARAMETER FOR _ND I_OFX OF X (THE J IN_[X) C . • OUTPUT OATAI 15_3 C * I$9_ • _AXmMAXI_UM OF X ARRAY 119S C * I_X,J_XmI,J L(}CATIP_ OF W-AX 1596 C * W_TNmMINI_U, OF X ARRAY C * I_N_J_kmI.J LOCATT_ OF X.IN I$9_ C * 15_9 C ********************************************************************* 1_00

I_02 l_O_ X,AXsX(I.t) l.Oa XMINIX(_I) I_XI| I_06 J_XI| I_O? I_O8 J_NBI ISO9 DO 20 II|_IMAX_IBKIP| I_I0 O_ 20 jm]_JMAX,ISKI_I l_ll IF (W(I_J).LT.XMAW) GO TO I0 X"_XmXCI.J) I_I_ IMXm| I_I_ JMXmJ I.IS 10 IF (X(I.J).GT.XMIN) GO TO 20 X,INmW(I_J) IMNml JMNmJ CONTINUE I_EO RETURN 1_ ENO 1_$ 1_

l_m I_? SUBROUTINE _ARA (WWI_YXI,XETA_YETA,LACC,CFAC_RETA,RWI,CBI_CSJ,_ACC la_8 I.R_TeACCL.ACCLI.NDIM.I.J.I"ER.TACC_IEV) la$o **** C_EFFICIENT| AND VARIABLE ACCELERATION PARA_ETERB ON ********** la$l **** RE-ENTRANT 8.EGMENT8 ********_* l_ 1_33

DIMENSION RETA(NOIM,I)_ RWI(NDIW,I)_ *ACC(NDIM_I}_ R(|)_ T(|) DIWENSION TACC(NDIM_t) DIMEN810N IWER(1) 1_$8 REAL JACBS,JBAG I_ INTEGER TACt I_0 BI

LOGICAL LACC I..1 Ctt**t I..2 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_**E I_ AGI].O/(ALFA÷GAMA) JAC_8=(XXI*YETA-XETA*YXI)*tE*CFAC JSAGmJAC_8*AG*O.O_2S la.9

VARIABLE ACCELERATIO_ PARA,ETER Ia52 YF (LACC] GO TU t_O 1.53 TE_IaJACBS*O.125 81|-TEMI*RXICYwJ) 1.55 _BeTEMI*RETI[Zej) 1.56 81siSB(Bl] 1.57 REmABB(R2] l_Sm TIBALFA*QE-BI**2 T2mGAHA**2-B2**2 l_eO *TIm*B$(Tt) ATZmABS(T2] I.62 IF (TI,_EaOIO.AND.T_.GE,O.O) _0 TO BO I.65 _F (Tt,LT,O,O.AND.T_,LT._,O] GO Tn 90 Ia6. RJtBSGRTCAT1)*CSI*kG IF (TI,GE,O,O] GO TO IO 1.6b TEMPwlmUkCC(RJI) l.b7 TACC(I,J)IIO GO TO 20 1_69 10 TENP_lsOACC(RJ1) TACC(I.J)a20 I_71 EO RJEBSQRT(ATE)*CSJ_AG 1,72 IF (TE.GE.O._) GO TO Sn T_MPwESUACC(RJ2] TACC(I,J)BTACC(I,J),I GO TO _0 _0 T_MCwEBOkCC(RJE) l_T? TACC(IwJ)mTACC(I,J)÷E I_78 O0 IF (IEV) $0,60e?0 SO TEMPwmUACC($QRT(RjI**E_RJE_wE]) I_80 GO TO 100 60 TEH_m(Rj|eTEMPW_e_J_eTEHP_E)I(RJ_eRJ2) 1.82 GO TO 100 1.8] ?0 TEMPWmO_CC($ORT(RJI**EeRjE**E)) GO TO 1o0 1085 80 RJB(8_RT(ATI),CS_,SORT(kT_)_C_J_wk_ 1_86 TEHP_BOACC[Rj) T_CCCI_J)m_ 1.88 GO TO 100 1_89 90 RJ_(8_RT(ATI)*CST,8_RT(AT_)*CSJ]*AG laeO TEMPwmUaCC(RJ) 1.91 T_CCCI_J)_! lace I00 R(I|)mERROR(TEMP**w_CC(I.J].R(I_).I.J.IWER) wACC(_#j]_TEMP_ l._O |10 CONTINUE

COEFFICIENTS l,e? 1098 T(S]wJSkG*RETA(Z,J] 1O99 T(O]mJ8kG*RXI(I,J] 1500 82

T(L)B,5*ALFA*AG 1501 T(2)B,5*_AMA*AG 1502 T_3)I,25*(XXI*XETA+YXT*YETA)*AG 1_u3 A_CLB*ACC(ItJ) 15o_ *CCLImI,0oACCL 15o5 R(12)RR(12)+ACCL 1506 RFTURN 15o? 1508 1509 1510 1511 1512 1513 151_ 1515 1516 1517

1519 I_O 1521 I DPT(IOO)w IPT{|O0], JRT(IO0) IS_ INTEGER XIwETA,BLKoATYP 15_$ DATA ZE_O/I,_E-tn/ DkTA XI,ETAeBLKI6HXI o6HET& 1525 C***** 1526 IF _NLN÷NPT,EQ,O) RETURN 15_7 *RITE (6,200) 1528 IF {ITYP,EQ,O) *RITE (bw2_O) IS_ IF (ITYP,N[,O) wRITE (6e270) 1530 IF (ATYPoEQ,ETA) wRITE (6o280] 15$1 IF (ATYP,EQ,XI) *RITE (6_Z90) 1532 IS_$ SET (_P JMPLITUDE AND OECAY FACTOR 153a 1535 IF (NLN,NE,O) REaD (5,210) (JLN(L}_kLN(.L),OLN(L)_LB%,NL N) 15_6 IF (NPT,NE,O] R_AO (S,220] (I_TtL},JPT(L),_PT{L)e_T(L)_LB|,NPT) IS)? IF (DEC,LT,ZERO) GO TO _0 15}8 IF (NLN,EO,O) GO TO _0 155_ OO |O LBIwNLN 1_0 10 OLN(L)IOEC 15_1 20 IF (NPT,_O,O) GO TO _0 DO 30 LBI,NPT 15_$ ]O D_T(L)BDEC _0 CONTINU_ 15w5 GO TO _o 15_6 IF {NLN,EQ,O) GO TO 60 15_? DO 50 LmI_NLN 50 _LN(L)B_LN(L)*IMPF_C _0 I_ (NPT,[Q,O) ;0 TO _0 1550 DO TO LBI_NPT 1551 70 a_T(L]ma_TCL),aMPFAC 80 CONTINUE IF (NLN,N[,O) _RITE (6,_0) (jLN(L),kLN(L},OLN(L),LBI_NLN) IF (NPT,NE,O) _RITE (6,_0) 155_ 1556 C_LCUL_T[ INHOMOGENEOU_ TEEM 155_ DO 190 I|I_IMAX 155_ DO 190 JmI,JMAX 1560 83

ATT_ACTINN 15o] IF (ATYP,EQ,ETA] IjIJ 1562 IF (ATYP,EQ,xI) IJml 15o3 T2m_mO 15h_ IF (NLNmEOmO _ GO TO 100 150_ O0 90 LBIaNLN 15e6 TmAL_(L.]*EXP(-_LN(L)*IaHSfIJ-JLN(I ]5) I_b7 IF (ITYP.NE._) G_ TO 90 156_ TmT*ISIG_(I,IJ-JL_J(L)) 1569 IF (IJ.EQ.JL_(L)) Tmh,O 1570 9O T_mT_-T 1571 C**** POINT ATTRACTION IS72 IF (NPTmEQ.O) GO TO I_0 157_ DO 170 LsI,_JPT 157" ItmIPT{L) 1575 IF (ATYPaN_oETA) G[} TD t=n 1576 IlaJ-JPT(L) IS77 GO TO 150 IS7_ 1uo IImI-IRT(L) 157_ 150 18NuISIG_'(I,II) 15_0 IF (II.EQ.O) I$_.mO 1581 Te(I-IPT(L))**2÷(J-JPT(L))**2 15_2 TxAPT(L}*EXP(-OPT(L)mSQRT(T)) 15_3 IF (ITYPaNEoh) GO TO 160 158_ TmT*IBN IF (I._QoIPTCL)oANDoJaE_.JPT(L)} TxO.O 1586 160 T2mT2-T 1S8T I70 CONTINUE 15_ l_n RETA(IjJ)eT_ 1589 190 CONTINUE 15RO C 1591 C PRINT INHOUOGENEOUS TERM 15_2 C 1595 IF (IRIT.NE.O) WRIT[ (_,250) {(I,J,RETA{I,j),TBI,IMAX),JmI,jMAX) 159. C***** 15_5 RETURN 1596 C 15RT _00 FORMAT (*OEXPONENTTAL DECAY RMS.] 159_ 210 FORMAT (I10,2FIOofi) 1590 220 FORMAT (2IS,IFIO.O) 1600 2]0 FORMAT (IOATTRACTION LINES*IIQW,*J*oITX,*AMP*olSW,*DECAYmt(ISe2F2O 1601 1.8]] 160_ 2_0 FORMJT (tOATTRACTION POINTS*t/QXe*I*_X,*J*,I?XwtAMP*_ISX,*DE_AY*e 1605 1/(215,_F20,8)) 160_ _SO FORMAT (*O_HS*//b(]X,*I*,3X,*J*.6X,*R[TA -*)/(6(2I_,1E10,_* -*))) 160_ _&O FORMAT (* - ATTRACTION -*) 1606 _?0 FORMAT (* - ATTRACTION TO CONVEY _IOE, RE_UL$10N TO CONCAVE -*) 1607 _80 FORMAT (*0--. ETa EQUATION RH8 e-at) Ib00 290 FORMAT (*0--- Xl E_UATION RH8 -.-i) 160_ 1610 END 1611 1612 1615 161_ 1615 SUBROUTINE 80R (ReX,Y,IMAWeJMAX.ITERelXER,IYEReIENO,NOIMe$wEReNB 1616 IDY_ACC,RETA,RXI,LRI,LR_,LIt,LI_,LTYP[,NRIEG_IWIR,IAIT,INRAC_INF_C 161T _O_NBSEG_LIIEG,LBI,LB2,LBSIDeTACC_IEV_IOISK) I&18 I&19 16_0 84

16E2 * LOGICAL COe, T_(_I_S I LACC - VArIAbLE ACCELFWAIIn_ PaRAUETER rilL n 16E3 . CO_veRG_ (TWRFLEVA_T IF LCONmT_LIE) . IbES * LCrAC - AnDITTO_ eF I_Fn_nGE_r_i!$ TER_ COMPLETED 16E6 w 16_? , L_C_ _ - CO_ST_T ACCELFkATIO_ PARAMETER. 1628 16_9 * ACCEL_WATI_ _ PA_AmETE_ TYPE l I - ROT_ _V T_AGINARY {U_J_R_RFLAX) 16_0 * 2 - @OT_ EV REAL (OVER-RELAX) * 12 - XI EV T_AGINAWYw ETA EV REAL * 21 - Xl Ev REAL, ETA _v I_AGINARY 1633 16_ 1635 1636 IXFR(1), IY_R(11. T(5), LRI( 163?

nTwENSIO_ -ACC(NDT_,I), RETA(_DIM_I), RXIt_DI_,i), TACC(_nIM,I) I_3_ DT_SIDN LRI(%]_ LR_(1)_ LBSID(1) 16_o DOUBLE PRECISION AG 16QI I_TEG_R ATYP,_TA,XI,TICC 1a_2 REAL JSAG,J_CR_ LOGICAL LACC,LCFAC,LCON IM_ _ATA XI,_?A/_Xl ,_wETA .' 16_5 n,TA PII$,I.ISq?_53591 16_6 16.7 _ACK(X,T,I,jeTI,12,JI,JE,A_AI_TE,T_]mAI*W(I,J)÷**(T(1)*(X(12,.I)ex( 16_8 111,J))+T(_)*(X(I,J_)÷X(T,JI))-T{3_*(X{I_,32)-X(12.JI)+X{II,JI)-X(I Ib_q 21,J2))÷T(S)*T2÷T{_)*T1) 1650 M*CKECW,T,I,J,II.I_,JI,J_,A,AI,T_.TI)mA_*X(T,J)÷A*(T(1)*CX(12,J)_X 1651 I(II,J))_T(?)*tW(I,J_)÷X(JI_JE))-Tt$1*tW(I;,JE)-X(JI-I,JE)÷X(JI+I,J 165E 1653 MACW3(X_T_I,J,II,T2,jI,j2_A,AI_T2.TI)mAI*X(I,J)_A*(T(q)*(X(12,J)÷W 165a I{I$,J))*T(E)*(XfJ_,J_)÷X(I,JE)}-T(3)*(WtJI-I,J_)-X(I_,J2)*X(It,J_) 1655 _-X(Jt*I,J2))+TCS)*T_+T(_)*T_) 1656 MACK,(X,T,I,J,II,I_,JI,J2,A,AI_TE_TI)mAI*X(I,J)÷A*CT(1)*(X(12_J)÷X 1657 t(I_,II))÷T(2)*(W(I,J2)+X(T,JI))-T(])*(W(I_,JE)-W(IE,JI)_X(I_,II.I) 1658 I-X(I_,II-I))*T(S]*T2eT(_)*TI) 165. _ACKSCX,T,I,J,II,IE,JI,JE,A,At_TE,TI}mA_*X(I,J)÷A*(T(1)*(X(IE,II)÷ 1660 IX(12,J))÷T(2)*(X(I,J2)÷XtI,JI))-TCI)*(xfII,TI-I)-X(12,11*I)_X(IE_J 1661 EI)-X(12,J2))_TCS)*TE÷T(_)*TI) 166_ UACC(RJ)mE.O/(I.0*SQRT(I.0÷Rj**2)) 1665 OACC(RJ)wE,O/(I,0÷SGRT(1,0-RJ**E)) 166_ C***** 1665 WRITE (6,630) 1666 C 1667 C READ COORDINATE ATTkACTIDN PARAMETER8 AND CALCULATE 1668 C INkOMOGENEOU8 TERM 1669 C 1670 KOBI 1671 REA_ (§,610) ATYP,ITYP_NLNeN_T#DEC,AMPFAC 167E IF (ATY_.EG.ETA) CALL RH8 (R_IMAX_jMAW_NDTM_NLN_N¢T_ATYP_ITYP_D[Ce 1673 IAMPFAC,_ETA] 167_ I_ (ATYPeEQ.XI) CALL RH8 (R_IMAXeJMAX,NOIMeNLNeNPTeATYPeITYPeDECeA 1675 IM_FACeRXI) I676 READ (5_610) ATYP,ITYPeNLN_NPT_EC_AMPFAC 16T? IF (ATYP,EU,ETA] CALL RH8 (R,IMAX_JMAX_N_M_NLW_NPT_ATYP_ITYP_O_C_ 1678 IAMPFAC,RETA) 1679 IF (ATYP.EO.XI) CALL RHa (ReIMAXeJMAX,NOIMeNLN#NPTeATYP_ITYPeDECeA 1680 1MPFICwBXI) 1b81 _EA_ (_p_qO) TFACpI_ITeEFAC 1682 1683 TNITXAL SETUP 168a 168S IF(R(I).GToO,O) GO T_ _0 1666 IF (I_V) lOe_nl30 1687 10 _RITE (6_6_0) 1688 GO TO _0 168Q 20 _RITE (6,650) 1690 GO TO _0 1691 $0 _RITE (6,660) 1_92 _0 CONTINUE 1693 IF tIDIRK,EQ.2,0R,IDISK.EQ,3) GO TO 50 169U ;0 70 6O

Se REA_ (116720) LkEC,LCONwCFAC,LCFAC,IWCC.I!_,INCOONeCINFACp_eIEV,IFkC 1696 IeEeAC,jWXMI,IMXMt,JMXPI,CSImCSJ,NPT,I_FkE,_INF,IEND 1697 READ (11,730) tR(I)el=l,t]) 169_ REkD (It,7]O)((Xtloj),V(I,J),RVI(IpJ),_[Ik(I,J],wACC(Xpj),ymt,iMA 1699 lX),Jlt,JM&X) 1700 RE*IND tl 1701 KOIK_| 1702 IF (IENDeNE,3) G_ TO 100 1705 KOst 170_ GO ?0 80 170S 60 le (IFACoEO.O) IFACII 1706 WRITE (6ekO0) IFACeEFAC 1707 EFACmAMINI(R(2),_($)),EFAC 1708 IF {IFJC.GT.O) C_AC=I.O/_LOAT(2**(IFAC.i}} 17UQ Ie (IKAC.LT.O) CwACmI.O/A_8(WLOAI('IW,C)) 1710 W_ITE (6,670) CF_C 171! IRCOUN_I 171_ IF (IWR.GT.O) *RITE (66570) 1713 JMXMIIJM_X.I 171_ IMXMIIIMAXB_ 171S JMXPIBJM_X÷I 1716 CSJBCOS(PI/FLO_T(JMXM1)) 1717 CSImCO$(PIIFLOkT(IMXMt)) 1718 NPTm(IMAX_)wCJM_W._} 1719 IF (NRSEG.EQ.O) GO ?0 7_ 17_0 O0 70 LIIeNR_Ee 17_1 70 NPTBNPTsLR_(L)-LRt(L).! 17_2 7S IF (INFAC.Ee,O) INFACII 17_ IF (INF&C.GT.O) CINFACBIeO/FLOAT(2mI(yNFAC.|)) 17_ IF (INFAC.LT.O) CINFACmFLOAT(INKACO)IAS_(FLOkT(INF,C)) 17_5 wRITE {6,710} CINeAC 1726 INCOUNIt 17_7 IF (INFIC.L7.O} INCOUNmINFACO 17_8 EINPmEFIC 17_9 C 1710 C SET UP ITERkTION C 17]_ 175_ LCFACI,FALSE_ 173_ I_ (IABS(IF_C).LE.I) LCFACI.TRUE. 175S DO 90 ImI_IMAX 17]_ DO _0 Jlt,JM_X 1737 _0 *_CC(IeJ)=l,0 1738 w_lTE (6,6_0) 1759 17#0 86

80m YTERATZO_ 17_I 17_E 100 IEh_sO 17_$ DO _50 XlXOwIT{R XF (K.LE,2) LACCm.TRUE, 17_5 17_6 IF {KmNEm_} GO TO I_0 LICC1,FALSE, 17_7 7F (LCON) LACCtmTRLJE, 11"8 TF_.R(1) 17_9 DO 11o yul,I_aX 1750 O0 110 jm1,J_aX 1751 I10 *aCC(I,J)=TE_ 17_E 1755 120 R(_)m_,O 175_ R(5)a_,O R(11)m0,0 17_S R(1_)m0,0 17S6 C 1757 1758 C**** VTFLO C I?S_ O0 E60 JJIE.jMXM! 1760 JIJMXHIwjJ+2 1761 J_lIJ=t 1762 _763 JPlmJ+l 176_ DO 260 ITII_,_HXMI III_XHI-III+2 17e5 IPlmI+1 1766 IF [I,G7,1) GO TO 130 1767 %_IzTMXMI 1768 GO TO I_0 1769 1770 130 I_III=! 1771 I_0 XXIIX(IPI,J)=X(T_I,J) YxIIY(IPI,J)-Y(IMIoJ) 177_ XETAmX(IpJPl)=W(I,J"I) 1773 Y[TAmY(IwJPI)°Y(I,JMI) 177_ ALWAmXETA**)+Y[TI**) 177S GJ_AmXXI**E+YII**E 1776 AGIi,Ot(ALFAeGAMA) 1777 JACBSm(XXI,YETA=XETA*YXT)**Z*CFAC 177g jSAGmJACBB*AG*O,O6ZS 1779

C##WW VARIABLE ACCELERATION PARAHETER 1780 IF (LACC] GO ?0 _)0 1781 TEHIIJACB8*O,I_S 178E BIImT_I*RXI(_J) 1783 BEI-TEHI*RETk(I.J) 178_ BI_ABI(BI) 17aS B_mABS(BE) 1786 Tlm_LFA¢*2-BI**2 1787 T)IGAM&i*2-BEiIE 17_8 kTIBABB(Tf) 1789 ATIIAB$(TE) 1790 IF (TI,GE,O,O,_ND,T_,G[,O_O] GO Tq EZO 1791 IF (TI.LT.O.OmAND.T_..LT.O,O) GO T_ 179_ RJIII_RT(AT1)*C$%*AG 1793 IF (TI.GE.O.O) GO TO 1SO 179_ TE_P_I_UACC(RJI] 179S TACC(I,J)_IO 1791 GO TO lbO 17_7 150 TE_PWtIOACC(RJi) 1798 7kCC[l,J)_ZO 1799 1_0 RJEIiQRTCAT_)*CSJ*AG 1800 87

IF [TE.GE.n.O) _0 TO i?O l_ot TEMP*Emt.ACC(R,YE) 18o_ TaCC(XoJ)uTACC(T,J).! 1803 GO TO 180 180_ 270 TEMPw2uOACCCRJ2) 1805 TACC(IwJ)ITACCCItJ)+E 180b 180 XF (X[V) 190.EOOpElO 1807 lqO TF_PwmUACC($ORTCRJI**2+RJ2**_I) 1_08 GO TO E_O tSOg EO0 TE_Pwm(RJL*TEMPw1+RJ_ITE_P_)I(_jI+Rj_) 1810 GO TO E_O 1811 EIO TEMPwmOACC(8_RT(RJ|**_eRj_**_)) GO TO 2_0 1813 RJm(SORT(ATI)*CSX$$ORT(ATP),CSJI,AG TEMP_IOACCCRJ) 181S TACC(I,J)u_ 181b GO TO E_O 1817 E3O RJB(SORTCAT|)tESZeSQNT(AT_),C_J),AG 1_18 TE_P_uUACC(RJ] 181q TACCCX,J)ml 1_dO _ao R(I|)aERROR(TE_Pwe_ACCCIeJ)eRCI|)._.J,XwER) WACC[XeJ)mTE_P_ 18E2 _so CONTINUF 18E3 ITERATE T(S)mJBAGtRETACTpj) 18_S T(_)wJ8AG*RXI(_,J) 18E6 T(I]m,S*_LFA*AG teE? T(_)mtStG_k.AG 18E8 Tf_)te_(XXZwXETAeYXTwYETA)wAG 182q ACCLmXACC(I.J) 1830 _CCLImI,O-ACCL 1831 R(I))mR(IE)+ACCL 183E TEMPXNHACK(X_T.I_J._MI.IPi,J_I.JPt,ACCL.ACCLI_XFTA.XXI) 18_$ TEHPYIHACK(Y_T.I_J.IHt. IPieJ_i,JPI.ACCL.ACCL_.VETA.YXI) 183_ R(_)IERROR(TEMPX_X(Z_J)_Rf_)_I_J_TXER) t8$S R(S)mEHROR(T[MPY_Y(ZeJ),R())_T#J,)Y(R) 1836 X(I,J)mTEM_X 1837 Y(Z.J)ITEMPY 1838 IF (XtGT.I) GO TO 2_0 183g X(_MAXej]mTEMPX 18_0 Y(IMAX,J)mTEMPY Z60 CONTINUE 18_E ¢ 18_3 C***, REENTRANT SEGMENT8 **** 184_ C 18_S IF (NRSEGs[O.O) GO TO _00 18_b DO SqO LNIeNR|EG 18_T IIoLRI(L)÷t 18_8 X_|LRZ[L)-[ 18_q ISmLXt(L}et 1850 XamLX=(L}=t IGOTO_LTYPE(L) 185_ GO TO (_?O.EqO,$tO.$30,3$O.$TO)_ XGOTO 1853 ONE ON BOTTOM. ONE ON TOP 185_ 00 180 JJ_Xt_IJ IBS5 ZaX_-JJ+X! XXIIX(IeI.I)*X(I-I_|) 18S? YXZtY(I#|et)eY(]e|,|) 18S8 XETk_X(_e|]*X(Z.JMXMI] 18Sg YETAEY(_)oY(|,JMXM|) 1860 88

CALL PAPA (XXT,YXI,XETA,YETA,LACC,CFACp_FTA,_XIoCSI,CSjo_ACC 10b1 t_.T.ACCL.ACCLt._IM.T.I._FReTaCCeIEV) 1092 tE_PXm_ACK(XaT,I_I,YoI,I÷I,J_XMlmPpiCCL,i_CLImXFTA,XXI) 1063 TE_PYm_ACK(Y,ToTptwT-t;T_Iwj_X_|_wACCL,ACCLIJYETapVX_) 186_

_(S}xE_RO_tTEMPVeY(_,I)e_f_)eT;I,TYF_) 1866 xtI_I)mTE_PX 10_T Y(I_I)zTE_PY IRbO X(_,J_AX)mIF_Px I_b9 Y(I.J_AX)BTE_PV 1070 28o CO_TINuE 1071 GO ?_ }90 1072 ROT_ _ _nTTO_ 1_7_ 290 _ 300 JJmIlo_2 107_ _mI_-JJ÷I! 107S ITmT_-(_-II) 187b XXIaX(?+I.t)-X(I-t.t] 1077 YXIIY{_÷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 1081 1._.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 1086 X(IeI)wTE_PX 1887 Y(Z.I)xTE_PY 1000 X(II_I)mT[_PX 188q Y(II,I)NTE_PY 1090 300 CO_TI_U[ 1891 GO TO _90 189_ C**** _OT_ ON TOP 1093 3t0 oO $20 JJsZt,12 189_ IsI2=JJ÷It 1095 IIBI_-(Z-II] 1896 XXIIX(_.I,JMAX)-X(I-I,JMAX) 1897 YXIxY(I*I,J_AX)-Y(I-I,J_AX) 1896 XET/IX(II_J_XNI)-X(_.J_X_t) 1099 YETAIY(II.jHXMt)=Y(I,JMX_I) 1900 CALL _ARA (XXI,YXIeXETAeYETA,LACC.CFAC,qETA_RXIeCBI,CSJewACC 1901 1,_,7,1CCL,ACCLI,NOIN,I,J_AX.IWE_eTACC,IEV} 190Z TEMPXIHACK$(X,T,|,IMAX_I_IeY_t,_,JqXMt_ACCL,ACCLI_XETA,XXI) 1905 TE_PYI_ACK$tY,T,I,I_AX,I*{,I*I,T_,J_XMI,ACCL,ACCLt_YETA_YXI) 190_ _(_)IERROR(TEMPX,X(IeJ_AX).R(_).I.JMAX,ZX_R) 190S R(S)IERROR(TEHPY,Y(I,jNAX_,RtS),I,J_AX,_YER] 1906 XCI_JMAX]ITEHPX 1907 Y(I,J_AX)mTE_PY 1900 X(I_J_AX]mTEHPX 1909 ¥(%%,jMAX]xTEMPY 1910 SEo CONTZNUE 1911 GO TO ]gO IgIZ c**** ONE ON LEFT, ONE ON RI_T 191$ 330 O0 I_O JJmIl,I_ 191_ JzIE-JJ¢Ii 1915 xXIwX(_,J)-X(I_X_t,J) 191_ YXIIY(_,J)-Y{I.XM1,J) 1917 XETA_X(l_J÷t]-X(l,J-l) 1918 YETAIV(t_Jel)-Y(I,J-I) 1919 CALL PARA (XXIeYXI,XETA_YETAeLACC.CFAC,RETA,RXI,C$I,CSJ_WACC 19|0 _9

I'RwT'ACCL'ACCLI'_IMel'JP_*E_wTACC'IEV) 19_! TEMPXBHICK(XwTeI°JP_XMle_JeIP'T_IeACCLwACCLIwXETAeXXI) 1921 TE_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) 192S X(I,J]sTEMpx 1926 Y{]ej)mTEMpY 19_? X(IMAX,J)sTEMPx 1928 3_0 YfIMAXwJ}ITEMPy 1929 CONTINUE |930 GO TO $90 ROT_ ON LEFT |9$1 $S0 I912 DO 360 JJxIl,I_ 1935 JmI2-JJ*I! 193_ I_sr_'(J=It) 195S XXI=X(_.J)-X(2,I%) 1956 YXIIY(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_3 R(_]sERROR(TE_PX_X(|_J)_{_],t,j,_XER] IRUa

X(I,J)ITE_PX 1R_6 Y(I,J)ITEHPY 19_7 X([_II)sTE_PX 19_8 360 Y_I_ZI]mTE_PY 19_ C_NTINUE 1_0 GO TO 390 _95! 370 ROTH ON RIGHT [9S_ DO $80 JJsli,I2 tRS$ JmTZ-JJ+I! 195_ Ilit_'[J'II) 19§S XXImX(I_XMI,II)-X(I_XM!_j] 1_§6 YWIIY(IMXM|eII)eYCIMXMteJ) |_S? XETAxX(I_AX_JeI].X[_AX,J.!] leSS YETABY(ZMAX_Je!)=Y(IHAX_J.I] 1_9 CALL _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) [qiS X(I_AX,J)BTEMPX 1966 Y(ZMAX,J)mTE_PV 1e6? XCI_AX,II)mTE_PX 1q66 ];80 Y(I_AX_ZZ)sTE_PY l_b9 39O CONTINUE 1970 CONTINUE 1_?| CONTINUE 1_11 1975 STORE INITIAL ITERATION ERROR HORN| le?_ IST5 R(IZ]PR(II)/FLOAT(NPT) 1976 ZV (K,GTs|) GO TO _10 R(6)eR(_) 1917 R(?)IR(§) 1978 [97S 1$B0 9O

C _AITE ITERATION ERROR NORM8 1981 1982 C UIO IF (IWIR.GT O) WRITE (6.§_0) W.R(_)._(S).R(II}.RCI2}.LACE.LCFIC_ 1983 • 19@_ ILCON 198S C 1986 C CHECK TO SEE IF ITERATION I8 COMPLETE 1987 C C,Wt* CHECK VARIABLE ACCELERATION PARAMETE_ FTELD C_VERGEhCE 1988 1980 LACCmR(tl},LT,R(tO) 1990 IF CLEON} LACCm,TRUE, 1991 IF (LACE) R(II)eO,O C*,** CHECK INTERMEOIA'TE FIELD CONVERGENCE REFO_E INEREABIN_ 1992 1993 C**** COORDINATE ATTRACTION 199_ IF (LCF&C} GO T_ _0 IF (R[_],GT,EVAC,OR,R(_),GT,EKAC) GO TO _20 1905 1996 C**** INCREASE COORDINATE ATTRiCTION 10qT R(U)xI,O 1998 LACCI,FAL$[, 1999 IF (LEON} LACCm,TRUE, IV (IFAC.GT.O_ CFICuE,OICFAC 2000 IF (IFACoLT,O) CFACICFACtVLOAT(IRCOUNel)/FLOAT(IWCOU_} _OOi 2002 IRCOUNmlRCOUN÷! E00$ XF (CFACmGT,I,0) CFkCml,0 LCFACIABB(CPAC.1,0),LT.O.O00001 200_ _RITE (6e67_) CFAC 200_ U20 CONTINUE 2006 C**** CHECK INTERMEDIATE FIELO CONVERGENCE REVORE _OVING 200T Cew*m OUTER BOUNDARY 2008 IF (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 _OZ6 WRITE (69680) _OIT DO _0 JSl,JMAW wRITE (6,690) J 2018 wRITE (6_700} (TACC(I_J)eWACC(I,J)* I_I_IMAx) 2019 20_0 _0 CONTINUE 20_1 _0 CONTINUE 2022 C 20_3 C ITERATION 00E8 NOT CONVERGE 202. C C_*_* 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 _027 WRITE (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$1 Z032 IF (IAIT.EG,O} GO TO _70 _033 WRITE (6,680} 203u O0 _60 j_I_JMAW 205S wRITE (6_690) J WRIT[ (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 E039 IENDIt E0_0 91

RETUPN OAO IENDm2 20_! WETU_N 20_ 20_ ITERATION CONVE_GE8 20_ 20_S Qgn IFNDI] _0_6 C**** MOVE OUTER BOUNDARY Zoa? 20.8 OINFACmCINFAC 20_9 IF (INCn(IN.GE,IAR$(INFAC)) GO Tn 550 _050 IF (INFAC.GT.O) CINFACn2.0*CINFAC 2051 IF (INFJC.LT.O) CINFACmCI_FAC*FLOAT(I_COuN+I)/FLDAT(I_COUN} _052 DINFACmCINFAC/OINFAC Eos_ *RITE (6_710) CINFAC 20SU I%COtJNmTNCOUN+_ _05$ 00 S_O LmLISEG,NBSEG IImLBI(L) _05_ 12mLB2(L) _057 IGOTOmLBSID(L} _059 Gn TO (_00o520_S00_5_0)p IGOT_ 2060 50n Ir (IGOTO,[Q,I) Jml 2061 IW (I_OTO.[Q._) J|J_AX 2o_E DO SIO Imlt,12 _o_$ X(I,J)mX(I,j)*DINFAC 20b_ SI0 Y(IoJ)mY(IeJ}*DINFAC GO TO SUO 20bS 20b_ IF (IGOTO.EQ.2) Im! $20 EO_? IF (IGOTO.EQ._) In|MAX EO_8 00 S$0 JsIl,12 EOb9 W(IeJ)IW(IaJ)*DINFAC _n?O S$O Y(IoJ)uY(I_J)*OI_FAC _UO CONTINUE Z071 ZOT_ IF(LCFAC) IFACml KOml _o7_ _o?a IR(K.EO.ITER) GO TO _Y5 GO TO Oo E07S _O?& C**** FINAL CONVERGENCE 550 CONTINUE _077 Ro/# ITERIK _079 C,i** PRINT VARIABLE ACCELERATION PARAMETER FIELD EO#O IF (IAIT.EQ.O) RETURN E081 _RITE (6e6_0) _08Z DO _0 JI|$JMIX 2083 _RIT[ (&_&90) J _08_ wRITE (&,TO0} (TACC(I_J),_ACCfI,J%,IBI,IMAX} $60 CONTINUE ZOeS RETURN Z086 _08? _088 S?O t*|*#|SX,mITERATION NORMS*//} FORWAT ERROR _ose SO0 FORMAl (I_)[IS.S_F_O.S_SL|O) 2090 S_O FORMAT {IIS,IFlO.O,I_) _oel 600 FORMAT {eO*#I_#* STEP8 IN ADDITION OF INMOMOGENEOU8 TERM. INTERM _o_ IEOI&TE CONVERGENCE FACTOR I*_F|_,_) 1o93 _o9_ &20 FORMAT (*0----- MAXIMUM NORM8 OF ITERATE CHANGESI._OX.eLOGICAL CON io_s |TROL_m//eOITER_TEs_$W_,X_NORMm_gX_,Y_NORMm#?X_eACCmNORM_?X,_AVG,A _oeb _CCmPARA**_bX_*LACC*_SX_ILCFACI_X_wLCONI ) _Oq? &]O FOR.IT (*OLOGICAL CONTROLS I LACC - VARIABLE ACCELERATION*_, PARAM _058 IETER FIELD CONVERGED (IRRELEVANT IF LCONITRUE)I//_OWtiLCFA C . AODI _TION OF INMOMOGENEOU_ TERM COMPLETEDe//_OW_,LCO N B UNIFORM ACCELER ilO0 92

ATION PA_A_ETER*//EnX) FORMAT (*OCOMPLEX EIGENVALUE P_CEDIJR¢ ! UN_EW°_ELAX*_ EIUE EIO_ FORMAT (*OC_MPLFX EZGE_VACUE DR_CEDU_F I _ETGWTE_ AVERAGE*} 65O E1o_ FOR.AT (,OCn-PLEx _TG_NVALUE PR_CEnURF I _VE_=_ELAX*) FORWAT {*Oi,FiS,R_* (IF I_HOMOGENEOU$ TEPK*) 6?0 EIO_ FORMAT (,OACCELERATIO_z PARA_ETER$*//} 68O aI07 _qo FORMAT (*Oj m*,I}/) EIO_ FORMAT (IO(IX,*(t,I_,*]t,FT,_,IX)) TO0 EIo_ 710 FORMJT (*0,.FI_.8.* OF OUTER BOUNDARY*} FURMAT(_LIOpEI6,B.LIO,21_,EI6,8/3T_,E_,B,_IS._I_,_/_I_,E1_,8'IS) _11o ?En a111 FDRMaT(_EI_._) 730 EIIE END E_5 E116 E11? EIIB BUBROUTTNE PLOT(X,Y,I} CALL CALPLT(X,Y,I) EI_O RETUNN _IEI END _I_

EI_ EI_S BUBROUTINE _YMBOL(X*Y_M_TFXT, ANGLE_NC_p} CALL NOTATE(X_Y,M_TEXT_ANGLf_NCHAR) EIE8 RETURN EIE9 END EI$0 E131

_133 EI_ _135 EI_6 EI_? E_38 Z13e

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 21 0 100 O L L 0 0 0

IPLOT,IPLTR,NCUPYeLI_*II,LI_T_,NUM_,N_MB_t,I$_I_I,I_KIP_ I 1 ] I 0 0 0 0 1 1

NB_EGeNH$EG I U Q

_(I},_(_},N(_),¥INFIN,AINFI_,XU1NF,YUI_FeNINF S l,BO000OO0 .0_010000 .00O]O0_0 0,00000000 OmOOOOO000 O,O00OOOOO O,O0000000 0

IEVpIAII,_(10) I o 0 0.0DOQ0000

I_FACeINFACQ 8 0 0

SIZEFRArI0,OISTeXBI,XB_,YBL,YB_ B,0OO00000 0,0VO00000 10,00000000 0,00000000 0,OOO00O00 0,00000O0o ODD0000000

$1MPLV-C0_NECTED NEGIUN

INITIAL GoES$ T_PEI 0

--BODY _EGMENT$--

L LB$1U Ldl L_ L_DY

I I 1 _1 -1 _ l _t -I _ _I 1 -1 ol

=-OUTEN BUU_qDIRY= _

R4OIU@ • g=O0000000 INITIAL ANGLE • 0o00g0000O 0_IGl_ AT X • 0,0000000,) , Y • g.000O0000

N_BEH OF POINTS • 0 I _TEPS I_ aTTAINmENT OF INFINITY INITIAL STEP {L|NEAW CA_[) • 1

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=FITTEO CUOWDINATE SYSTEM SIMpLy=CONNECTED REGION KEY SEAT 5HAFT 21 21 0 100 0 1 1 0 0 0 0 0 1 t 0 1 1 21 =1 1 21 o1 3 21 1 "l 2 21 1 °1 0,0 0,0 0 le8 O,O00l 0,0001 0,0 0,0 0 0 0.0 0 0 0,0 8,0 0,0 I0,0 0,0 0,0 0,0 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 =,92718 o_@932 °,77715 ,8290_ =,55919 ,9563 -,29_37 I, ,9563 o29237 o8290_ .55919 ,b@932 ,77715 ,J7_61 ,92718 o125 o99215 ,125 ,98b ,125 .978 .125 .97 .125 .962 ,125 ,95_ ,I_5 o9Q6 ,125 ,91b ,t25 .932 ,125 .9_e o125 ,92 ,125 ,91_ ,t25 .908 o125 .903 ,1_5 .898 95

.125 ,8q3 ,125 ,888 ,125 .88U ,122 .88 ,877 ,125 ,875 ,12 ,872 ,115 ,d7S 011 ,872 ,1 .875 ,09 ,875 cO0 ,875 ,Oh ,875 0675 ,02 ,875 ,875 -,02 ,075 =,0_ .872 =.Oh ,875 -,08 ,872 =,09 .875 .875 ",11 ,875 =,115 ,875 ",12 ,872 °,125 .875 °.125 m877 °,125 .88 =,125 m88_ =,125 ,888 -,125 ,89_ ",125 .898 ".1_5 .90J ",125 ,908 =.125 =,125 ,92

=,125 t92_ ".125 ,9_2 ".125 ,936 ",125 .9_b ".125 .95_ °,125 ,962 ",I_5 .97 "o125 ,978 ".125 .98b ETA l 0 0,0 OoO 10.0 0,00! XI 0 0 0,0 OoO 1 100,0 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 _I 0 IO0 0 1 ! 0 0 0

IPLOT,IPLTReNCUPVeLIN_IteLI_T_,N_M_,NUMBRt,IS_IPI,ISKIP_ I l ] 1 0 0 0 0 l 1

I,_0000000 ,00010000 ,00010000 OmO000000O 0,00000000 0.00000000 Q,00000000

IEv,laIT,R(IO) | 0 0 0,00000000

I_FAC,INFAC0 1 0 0

$_ZEeRATLO#OIST_XBIf_B_rYBI,Y_2 8,OUO00000 O,OUOOOODO lO,O0000000 OmO0000000 O.OUOOO_O0 OtOOOOOOOO OmO0000000

SIMPLv-CONNECTE0 REGION

IN|TI&L GUESS TYPEt 0

*-BODY S_GMENT$--

L L_SIU LBI u_ L_OY

1 I l _l -I -I -I _ 21 I -I

--OU_ER _OU,WOaRY-°

RaOlU_ • O,O00000OO INITIAL aN_Lk • 0e00gO0000 OR_IN _T X • 0,00000000 _ • Om00000000

I STEPS IN alT&IN_LNT OF INFINITY INITIAL ILINEA_ • NUMBER OF ROINTS • 0 STEP CASE)

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 INTL_MEO|AI[ CONVENGENCE PAGTQR • 100,00000000

1,000U0000 OF INHOM0_ENLOU$ TE_

I,0000U000 UF 0UTb_ _OUNDJ_Y

----- MAXIMUM NOHM$ OF ITERkTE CHANGES LOGIck_ CONTW06I

ITEHATE Y-NORM _VG,_CC PA_A. _CC L_FAC _CON I .77jb7bo01 .187_0E_00 0 OOO00 T T T .755_3E-01 .179_5b_00 0 00000 T T T $ ._ESBbE_00 .52_70E+OU 0 8000O 1 l T :192_2E*00 ._2021E+00 0 8OOOO T T T 5 ml_070E_o0 .3e712L+0_ 0 80000 T T T b _I09J_E÷00 .2177_+00 0 800QO T l T 7 .81_BdE-01 ,_3q7_E+OO 0 80000 T 1 T 75859_-01 .I,885E+00 0 80000 T T T 9 _7_32E-01 .|,OSbE.00 0 80000 T l T 10 W75@0E-01 ,gb_23E_Ol 0 80000 I T T Ll ._8bTbEo01 0 80000 T T T 299_bE-01 .55B_1E°01 0 80000 T T T 13 2700_E-01 .57719_-01 0 80000 T T T .17_5_E-01 .bIZ_Ob-Ol 0 80000 f i T 19b_oE-ol .,52_E-01 0 80000 T T T 10 .29517E-01 0 8O0OO T T T 17 _70_bE-01 m327._E-01 0 80000 T T T 18 ._0b_0Eo01 0 80000 T T T 19 _3gE=o2 BOO00 T T T _0 99U_-U2 ._bT55Eo01 0 80000 T T T 21 10,0dE-Ol ,22_90E-01 0 80O00 T T T .11177b-01 .20057L-01 0 ,80000 T T" T 2_ _15_bqE_Ol 0 80000 T T T .710bbb-Od .1297bE'01 0 80000 I T T eSbbb_E-U2 .10755E'01 0 80000 T T T ._1eSIE-02 ._085bg-0_ 0 BOO00 T T T 27 ._}512E°02 80000 T T T 2_ ,265|_E_02 ,_b27E-02 0 80000 T l T _Q .21gl_E-02 .328b/E-0_ 0 _0000 T T T 3O mlg00W_02 ,255_3L-02 0 80000 T 1 T 31 _l_0bSb-0_ .ITq,SE-02 0 BOO00 T T T 32 ,|021_E-02 .1055_E°0_ 0 BOO00 T T T 35 .Tbb_2E-0$ .89375_-03 0 1.80000 T T T .b_99_E-O_ .B3blgb-03 0 1.80000 I I T .5_767_-0] 0 1_80000 T T T 3b ,.2585E°OJ 0 l.BO000 T T T 37 .2858iE=0J ,361_7E-0_ Om 1,80000 T T T 3_ ,36811_-OJ .3g2O_-0J 0 1,80000 T T T 39 .3072bE-0J 0 lm80000 T T T _U .270_gb-03 ._I)0oE=U_ 0 lmB0000 T T T .18_I,E-03 ._bb87L-O_ 0 1.80000 T T T .IwO_UE-0J ,289dSE-OJ 0 IMP0000 i T T ,gBbBB{-U_ _Ig891E-0_ 0 1,80000 T T T ,II059b-0_ .I_S/E-0_ 0 ImSO000 T T T _5 ,IU_b_E-O_ 0 lmSO000 1 I T _o ,5507_E-0. .78_OOEo0_ 0 lm_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 I FINAL ITk#ATION [RHOR NURNSI Xl ,5507gE-0_ YI 178160[-0g AT _TERAT[ # 4b LuCATIO_ 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+O0 Jm $

-._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_||eOO m1_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_OO J= 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_00 J= 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_1 m_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÷O0 J= 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*O0 J= 11

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*O0 J= I_

.l_bOO_*OO J= l_

.t_bOOE_O0 J= 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*O0 J# t7

.b_a7E-o_ ._JOObE-O! ._67b3b*_ .08010E-O ._eegTE*01 .IO_17E*O0 .tta_?E*O0 .lZ$S6E+O0 .1_9_6E*00 .1300_[+00 .t_5OOb*O0 J= lO _5_E_'1_E*_'_a_E_11_9E*_-_1_93_*_973a9E_1°_8_8_E_1*_59_-_aa_a_E_a6_E_ .a7_79E-05 ,_aeSE*oi ,ua_SqE-g] ,baotaE-0 ,8_egbE=01 ,_7379E*01 ,1093bE*00 ,ttOb_E*O0 ,_gZE+O0 ,IZ6OSE*O0 .1_500E*00 J= 19 _+_Sj7_÷_1_$_*_*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_O0 J= _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÷O0 J= _ -_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[*00 J. ._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*O0 J= 3 .97800E+00 ,95897EegO .85767E*O0 .7_515E_OU .55959[+00 ._Ug/_EeO0 ,5b38=E-Ot-,IgOS?E_OO-,399_OE*OO-.$a_gSEgO0 -.b_b75E+OO*._9_E*OO=.$_qJ_E*OO-.tgO_E+O0 .Sb511E-Ot .30989E+00 ,51970E+00 ,?_5_[$00 ,8S770E900 ,95098E900 .q7800E÷O0

.97000E_00 .q$O_SE*OO .87507E*00 .77539E_00 .b_680E*O0 .a568_E$O0 ,ZS2OIE+O0 .38149EoOle,tS_O_E*OOo._OS?OE_O0 -.33_5_E÷OO*._8565E+OO=.ISt93E*O0 .3833_E-0[ ._5_bE÷O0 ._5901E+00 ,63696E_00 ,?75_?E+O0 .87370E*00 ,_3_OE+O0 ._7000E+O0 J= 5 .ge_oOE+O0 .gJ|8_L900 0_055E_00 .80abgt_OU .70_lbE_O0 .57_60E900 ,a_tO8E+O0 ,2bObOE+O0 ,1;3_7E*00 ,719SJEoO_ -,$t77eE-O! ,7_859E-o_ ,lt_E*u_ ,_608=E÷00 ,_13_E*00 ,57_83E*00 ,70_JIE+O0 ,80=73E+00 ,8_0$9E_00 ,93;_0|_00 .9_OOE+O0

,95_001900 ,q_bb_eO0,08_G[*O0 ei_||_)O0 i?_O/_eOO e/_i4_)O0 ,$_OSek*O0 .4_TeSEeoe ,)&7_4*00 ,&Se_O|+OO ,_$0_5[*00 ,_5l=lE*O0 ,$_7§0E*00 ,_ZTb_E*O0 .5587aE*00 ,bmtSSE_O0 .71114|,0o .011101"00 ell|4&|*ti ,i|(Nk||itl ._5_00[*00 i00

J= 7 .qaeogE,0o .9_10oE*00 .88Z33E,O0 ,830_]E*00 ,7o7@e£*00 .i95|#E*00 .b_lTZ*OO ,S35_S[$00 ,IkiOSEeO0 ,41i30|$it .]gqagE_oo .atoJeE_o0 ._oa_?E*o0 ,etZqTE,o0 ._qJ_E$O0 ,717$0E*00 ,l$OqTleO0 ,ll_$1keO0 ,$||_Olell .gaeOoE,O0 Jx 8 ,g_OOOk÷O0 ,9_5_£*_0 ,_81gqE_gU ,83717L#ou ,7_53E_0O .72515E*O0 .beEebk*o0 .60aSaE_O0 .S$lTtEegO ,StT_aE_O0 ,ogasgL_oo .bb_7tE÷gO .7_blTk*O_ ,7_beL*O0 ,85721E*00 ,881L_E*O0 ,qL_EeO0 .qJbOOE+O0 Jx g ,gJ_oOE_OO ,g1oo_EeO_ ,87q2qE÷O0 .8a136E÷O0 ,797_0E÷00 ,Ta8etE*O0 ,bg8eSL÷UO .6b|bgE*O0 .i|_7_E*O0 ,_868_E900 .bI/77E_O0 ,56087L+00 ,_[E78E_O ,e517_L÷O0 ,69B_OL÷_O .7_6_E_00 ,7eT_E*O0 ,8_1_3[*00 ,87g$OE*O0 ,91005E_00 ,9J_OOEeO0 J= .10 ._2000E_O0 .gOQ85L÷QU .S??ZJE+Un ._aa_EE,O0 ._O?O_E*O0 ,70683E÷O0 ,7@b35E*O0 ,b8885E*O0 ,eb8_aE+o0 ,eSllOEgO0 ,e3107L+O0 .O_llE_O ._S8_OL_OU ,bB687EtOO .7_bSbE_gO ,TbbBeE*O0 .80705L.00 ,8_35E$00 .871@5E_00 ,gO_gOEeO0 .q2bOOE_O0 J= tt =9_00U£$00 ,_ggg_EeOU .e7bt_E_u .8ueS_E*O0 .SLag7E_O0 ,7_tb_E*O0 ,7aB53E+O0 ,?L83EE#OO ,bq]qtE+O0 .b77%gEtO0 .b7@qSk÷O0 ,b7rqgkeOO .bg_W[EeO0 .7183_Ee00 .7=852L_00 .TBlb3E+O0 ,8l_gg£*O0 ,8_bSBE$O0 ,87517E$00 ,89997[_00 .q_OOOEeO0 J= L_ ,g_ooL÷o0 ,_gb25E+OU ,_73|0£.00 ._qBa2E+O0 .SZ175E*O0 ,Tq_lOE*O0 ,76705E*00 .7_@5qE900 ,7@$01E*00 ,71g3|EeOO .705g|E÷O0 ,710_UE_OO .7E3OOL_OU ,7_25bE$OU ,76h_E÷O0 ,79_u8E*O0 ,82175E*00 ,8_8_E_00 ,87316E*00 .sq53oEeO0 ,g|QOCE+O0 Jx I_ .gOSOOk_O0 ,BeO_gL*O_ ,_71ZbE*O0 .8501LE+O0 ,8_778E*00 ,80_99E*00 ,7829_E*00 ,?e3aOEgO0 ,7_7_8£*00 ,757$a[_00 .7_38_£+00 ,7_7_L+00 .l_7abE*O0 .7_tOE#OU .78gqOE+_O ,_0_97E_00 ,8@77qL_0 ,8S015[_00 ,87t_tE$O0 ,8909aE_00 .90_OOE_O0 J= la .qUJOOE_O0 ,_6667£÷OO ._bgoq£_O_ .8btB3E_oO ._3BE*O0 ,81_8LE*O0 ,7970lE÷00 ,78|17E_00 ,7_86]Ee00 ,760SkE#O0 ,7577TE÷00 ,TbOS_E_oU ,7_SqE_uu ,78tlJLeo0 ,79egeE÷O0 ,81_8LE+O0 ,8)_]eE+O0 ,85187E$00 mB_ST_ZeO0 ,IBb_]E_O0 .qO$OOE_gO Jm tb ,8b_TIE÷O0 m83877E$OO ,8_3q_EeO0 ,80q78E÷O0 ,7972_E#00 .7875@E$00 ,78094[_00 .7787_E÷00 ./80gQE*OU .787_BE_UO ,7971qEgO0 .8097%E+O0

J= |b .Sg_OOE*Oo .87995E÷O0 .Sb/7_E_O0 ,8bSggE_O0 .SaalYE÷O_ .83259E900 .821e3[*00 ,81_89E_00 ,80_40E*00 ,79%aTEgO0 .79757E*00 ,/gg_bE+UU ._OalbE_O0 ,8|LBSE÷UU ,8_lbOE+O0 ,_3@58E*OU .8_15E*00 ,sssq_EeO0 ,8_77_[÷00 ,87qqSE#O0 ._gJOUE_O0

.OS_a9E÷O_ ,8_beE*_O .SatOaE,O0 .83ESbL*O0 .8ab53E*O0 .81973E*00 .81bOaEgO0 .O|u7bE+O0 .SIoolE$O0 ,8lg70£÷00 ,825UqE$OO ,8)282E*00 .B_IOLE_O0 .8aqesE*O0 .B_BSOE*O0 .857_ke00 _877_EeO0 ,_8800E+00 Jx _ .8_OOEeO0 .8751IE÷OO .8_TgbE_O0 ,855a_E+O0 ._ga|E+Og ,8q_bSb÷O0 ,838a_EeO0 ,83_qE*O0 .8516bE*00 ._$07bE#O0 .8_tbbE*O0 .83_27E÷00 ,85_£_00 ,8_bSE_O0 ,8aq39E_O0 .855_1E*00 .8616EE900 .8eTq6E*O0 .8751_Ee00 ,O_aOUE+OO Jx |g .86000E+00 .87JTgE_Og .8_q_,EeO0 ,_bSJq_$OO .B_I53E+OO .B5781E_O0 .85aa7E+oo .85087E*00 .8_818L$00 .Sab_gE$o0 _8_Sg_E÷OO ,8_QE_OU ,8QSl_Ee_O ,85085E_00 .85_2bE*00 ,_57_0E_00 ,8bt51E÷O0 ,86538EeO0 .8bq_][$O0 ,87$7qE_00 .88000_*U0 J= 2u .8bb$$E÷O0 ._6a7aE_O0 .86300E*O0 .8b|6?E_O0 ,SbO8_E_O0 ._O050E÷O0 .8e0_5_0_ .SeleeE_gO ,802qgE÷OU ,QbQ7_E_O0 .8_$3E_00 .SbSO_E+O0 .8_986E*00 .STtS_E*O0 .87558E$00 .8770vE#00

.87S00t*00 ,87_00[*00 .87500E*00 ,87S0_k*00 ,11S00|*00 ,17|00|*00 ,ll|OOleOO ,81500E*O0 ,87_oOEeOg ,s7_ooEeo0 ,87500E_00 ,87500E*00 .87S00E900 .87SOO[*O0 .B?SOO[*O0 ,iTSO0|$O0 ,IT|lOWell i01 Sample Case Plot" Simply Connected Region

! 1 1 102

Sample Case Input" Single-Body Field

T[$T CASE - BUOY-FITTED CUO_DINATE SYSTEM 8INGLE _ODY I KA_A_-T_EFFTZ AIRFOILe _b PU_NT8 K-T AZRFOZL #1 27 20 1 200 0 1 1 0 0 1 J 1 0 0 0 0 ! l 2 I 1 _7 "1 1 _7 0 2 I 20 _ 1 20 1,8 O,OU01 0,0001 5tO 0,0 0.0 0.0 26 0 0 0.0 0 0 8tO OtO 10.0 0,0 0,0 0.0 0,0 ,_9952_ -,000031 ,_98625 omO01_02 ,_9606b ",002760 rUSh293 -,00555_. ,4704b_ -,00850q ,3901_8 -.018591 12bq55_ "10299bb eI_2707 -e0_0790 -.03U802 =,0_8_53 =,187_72 -,05052_ -t3211_b "tO_SBq_ .,_a36eb =,033739 ",_SbSOb "tO165_6 -,50_7_b ,003820 "t_7110_ .026435 o,39690_ ,0_8b_2 ot287913 ,O_blOl o.15375b tO75_qg -mOObb_ .07u531 mlq_O_b .ObtUSE

t3_577_ .0_8262 ._71990 ,011299 ._67103 .006809 ,ugb_19 ,0030_0 .qq87?q tOO1_b LTA 0 _ _ 0.0 OtO 1 I00,0 1.0 2 100.0 l,O I00,0 1.0 100.0 1.0 1 1 100.0 ltO 21 I 100.0 1.0 XI 0 0 0 0,0 0.0 I 0 100.0 103

Sample Case Output" Single-Body Field

_0 1 d_)O 0 1 I _ n 0

1 ] 1 r; U I, u L 1

W(I ) _ w (d}," ()), YL',f" I., AI_F I'., XU(_J" , YUI_F, NI',_ • :

I,_OOuJLJOd , U(),J t HJ(_V , UL) O | I)OU_) 5,,IUOULIUOU O.Ol10OO00 n ))._O00OOOU 6.UOUU_nOI, _0

|['Vw|All)_(|O) ! o !_ ,*, o U n(,,(l 0 t2 U

INI- AC, _,_F _C[ I 0 0

INIIIAL STEP &LLNEA. L_bEJ = | 104

LCFAC = A0_[T|UN uF %N_O_UGk_EUU_ 1E wM COMPLETED

LCO_ - U_IFOQM _CCE_ERA_IUN PAR_wETE_

EXPONENTIAL 0ECAY W_S - ATTRACT[0N -

ATTRACT_UN L_NE_

j =Rp OECAY L 100,0UU00_u_ L,O000000O 2 10U,UUO01) 001) l,O_OOOOOu $ L00,0U0(10"0" L,OOOPOOOO LO0,O_O_O000 1,00000000

ATTR_TIO,_ PUINI_

I J AMp DkC_Y LPl 0 1 1 lOOmO000_}O00 l,OOoO_)OOo d7 I lOCi,OOn_O000 1,00o000o0 0

INTERMEDIATE CONVEWGENCE FACTUR • 100,00000000

LUGICA _ CUNTRO_

ITbdArk XmNLJ _ YmN_RM ACC-NUR_ AvG,_CC,PARA, LACE LCFAC _CUN 1 ,IQ0g_E÷()I ,_bbb2F#00 0, l,OUOOO I T t LeOUO00 T T T ,117geE÷Of ,LOb$2E_Ol O, 1,80000 T I T ,5_5_bE_00 ,_19_E*00 0, 1,80000 T T T 5 ,ETO25E*OO ,178RTE÷O0 0, l,BO000 T T T 105+

b 27q_ak*O0 ,15720[_0_ Oi 1,80000 T ; T I _507_L÷00 ,10559E+00 O, 1,80000 T r T 8 ,IbB88E_O0 O, 1,80000 T T T 9 ,taOOtt*Ou O, 1,80000 t T T IU olS_oE+O0 O, 1,80000 T r T II _OOE_UO ,llOl|[*OO U, 1,80000 T t T 12 .L8500_÷00 O, 1,80000 T r T .18o0_+00 O, ,80000 T T T I9O_gE+UU ,Ie85bE+OO O, 80000 T i" T .15_5qE_Oo O, .80000 T I T lb 15_O_k÷O0 .I_50bE_0o O, .80000 t T T .I1877E*00 835JEE-OI O, 80000 T t T .82_bbE=Ol eIZqlF-ol 0.. 80000 T T T 76787E-01 O, 80000 T I' T 20 .lOl_3k*O0 O, 80000 T T T .qStdSk-Ol 7_oloE-ul 0.. 80000 T l T g_O0_E-OI 0.. 80000 T 1 T _J b_SqSE-ul 190_bE'Ul O, 80000 T T T 2_ bO7qS_-01 _800J7_'0_ O, leSO000 r T T Ogu8OE°o1 ,759J2E-O1 O, 1,80000 T r T 575]3E-of ,o777_E°ul O. 1,80000 T T T 58O55E-oI ,7?u_?E-01 O, 1,80000 T T T 51_78E-01 ,o$51qf-Ol 0,. 1.80000 T T T ,Jg 5_ddE-ol O, 1,80000 T T T ,57Jlek-Ol aSu95EoO| O, 1,80000 T T T ,5_Iblt-ul 509_5E_01 O, 1,60000 T T T 3808_E-ul 0,, lmSO000 T r r _J_Sbbk-Ol 50289E-01 0.. I,80000 T T T ,39_O5L-01 o2_oE-ol 0.. 1,80000 T T T J5 O, 1,80000 I" T T Se O. 1,80000 T T T J7 JSP/SE-01 O, 1,80000 T I T 3_ O, ImSO000 T ! T 370J7E-ul 13118E-01 O, 1,60000 r T T _o 88q_lE-O_ O, 80000 T T T ,bb_=lE-O_ O, 80006 T T T 11689Eo01 ,SleSJi-u_ O, ,80000 T I' T aqO_bE-OE O, .80000 T ! T 5518bE-0_ O, ,80000 T T _5 39505E-0_ O, 80000 T T T 25005(-02 0 60000 T I T _r 1585b(-0_ _57q2E=O_ 0 60000 T T T _8 ,1057_E-0_ I0755E_OZ 0 60000 T T u9 qq_12t-o_ I0031_=02 0 60000 T r T 5d 750_Eo0_ 77_geE-o3 0 80000 T T T 51 6q_J=E-U_ bo$1oE-o3 0 1,80000 T T T 5a 5_780L-03 53b_OL-O_ 1,80000 T T T OJ7_E=OJ 0 1,80000 T T T $9697E-0_ .3538bE-03 O, 1,80000 T T T 55 JO_79E-OJ Im80000 T T T 5_ _$OJlk-O_ ,E_SOE-U_ O, 1,80000 T T T 57 17911E-03 ._3555L-_$ 0,. 1,60000 T 1 T 5_ Ib377E-O5 O, Im80000 T t T 5_ ,1750bL-O_ ,15005E=05 0,, 1,60000 T T O0 1_81bE-03 .150)5E-03 1,80000 T "f T 133_k-OJ ,lJO_OF-03 1,80000 T T T 99_75E°0_ .12qqlE°03 0., 1,60000 T r T 03 .B115UE-u_ .I(IoI_E-03 O, 1,80000 T T T ,I0581t°U_ ,93708E-0_ O, 1.80000 T f T e_ ,81793E-Om O, 1,60000 T T T 106

_J_AL VALU£S

_lt_ATI_ CUn_LN_ES.

[_LTLAL ITE_ATIU_ E_NOR NONM$! XI eL_OQOt+01 Yt ,3b6b_E_UU AT _T_NATt • [

_l L= _5 Js t7

._9955£+00 .a_o_k_OU ._eUgL*U_I .a_o_gE_Oo =_70_OE_C .J_UISE÷O0 ._b955L_uO .I_£TtL_UO'eSaRU_E=U1"*I _TqTE+Og

=_8_5E+OO ,_g37_[+00 =_7tgqE÷O0 .O_710_.eOO ,ugbq_L÷O_ .qgSTSE+O0 .ovqbJE*Uo

.bOeO_E_O0 .50,30E+00 ._97_k÷d_ ._S[=$E+OU .uaO_EtUO .JbTaJEfO0 .EbOoBt+O0 01077bE_OO=.aO750E'01".l _Tb_EeO0 _3e_U_*q3_95_+U_5_*(!U_5_7_÷_U-_=_7E_U_*=_ajE+u_9_5_E_=*_559E*__U_ ,13uO_f+O0 ._eSeuE+dO ._7oltt*OO ,_513_L*UO ._8_15£*0_ ._98_E÷_tl .5050_L_00 ._Ob_It+O0

.5_dObttUO .51o_E*CL) .50ubSE+O_ .aOOTlE+O0 mgJ_95_$UO ._5_1E+00 .2JbSOE+O0 .qagllE'OioeS_b7k'Ul'e_OgOSE+OO

,_5_]_E_O0 ,_b_1_E_uO ._E*O0 ._855_E÷00 ,50Tb_E+O0 .5|7b_E÷O0 ,5_008L÷00 J= a .SaOOgE÷O0 .SJq_l_÷.lU .51797E+0_ m_SBUE+UO * gjUo_E÷uO .J_JOSE+O0 m_SJIL+UO .BJ_q_E'UI'eTgBq7EoOI'e£_T_E*O0

m_JgSE÷UO ,$S/_OE+_IU ._a_b_E÷O0 ,aq_7JEeO0 mSd_E_uO ,531UOE÷O0 ,SaOeqE*O0 J= 5 ,51011b*O0 ,5619_E*00 ,55919k*00 ,_gBIaE_O0 ,a$$_ZL*O0 ,$$8qSE*OO ,_leO_[_O0 ,70BIbE-Oi" 858qSEoOI-,ZaOg_E*O0 _*_1E*_b_E+_5_*_*_S_1_7_9E*_S_$_+__*_ 50_55E'01 ,95351E*01 ,=$eaoE+Og ,$S$1aE*00 ,_eSe_E*uO ,S07ZSE+OO ,5_520E*00 ,Si_O(*O0 ,57017_*00

,el_55E+O0 ,eOl?6E+O0 ,57_5E*00 ,$Zi_tE+O0 ,=_78[+00 =$=OgOE*O0 ,_OBSSi+O0 ,Sb_8OE-Ol-,IOkTOk_OO=t_tTqSE*O0 _*_1$_9_°*S_]__*_'_$_E*_*59_97E*_*_5_*_8_z_B_*_e ?qO0_£-Ot ,8191_Ee01 t_OgOE+O0 ._5888_+0U ._5qo7E*O0 .5_b_÷O0 .5799_E+00 .OOSe2E+O0 .b1_55_00 J= 7 .b7078E_O0 .e570qt_O0 .b_OSE+UO .557b_E÷OO _blbqE_gO ._agtOE+O_ ._0_58E+00 ._O_9aL-ot-.l_bTbeOO-.30ag_E+O0 _5g58E+_5__j_+u_=_e_8_*_-__°_5b=_E+_a$53_E_z_7_E÷_ lOSagE+gO .bT_beE-01 ._tBE+O0 .$7u59[*00 ._BSa=E_O_ .57137E_00 .b_9_bE*O0 .ebtelEgO0 .b707Bk+O0

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J= 7

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oo 7

!7 1 ii0

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 K-T AINFO[L-FLA_ #I o9 20 2 2O0 0 1 1 0 0 I 3 I 0 0 10 0 I I 2 I I I 56 b9 -I a8 "2 3 l 09 0 _2 I _8 5b 2 I _0 a 1 20 1.8 0,00oi o,0o01 510 OtO omo 0,0 b8 0 0 0,0 0 0 8.0 0,0 I0,0 0.0 OmO 0,0 0,0 ,8081W7 -,Ib_W_8 .807780 -.1626_1 ,8070bl ".Ib2bO_ ,80_591 ",Ib19_9 ,800795 -,160050 ,78_14_ -,15279_ ,75abl_ -,1_0182 ,721_b_ -,I_170 ,68_09 ",I001_0 .b53090 ",087505 ,bZ_771 -,OOqT_ ,bO_O09 -,05U319 .59_bO_ -,0_765 .59163W -,036352 ,601308 -,035389 ,b201_8 ",OSgSSb ,b_5928 -,0_9100 ,b76123 -,00W_78 ,708095 -,oe315e ,739231 -,lO_OT3 ,7670qb -,12_998 ,789_2L o,I_353 ,803001 ",15655_

,807858 ",IOl3g_ ,8081_/ -,lb_039 ,_995_ =,0U0031 ,U980_3 -,001_02 ,_gbosb -,O0_TbO ,_e293 -,00555Z ,_70WOq -,008509 ,390_8 -,ulSSql

,12_707 -,0_0790 -,03_802 -,O_bW5_ -,187_72 ",05052_ -,32114o -,0_559_ -,_2388o -,0337_ -,_8_bO0 =,OlbS_b -,502720 ,0038_0 -,_/1102 ,02b_5 -,396903 ,O_8ba_ -,281g15 ,UOblOI -,15315e ,01b_99 iii

o,005b,4 ,07_5_!

i282_51 10_14_8 /395770 ,0_8202 ,_71990 ,0112q_ ,_87.10_ oO0_Oq ,_gbU19 .OO_O_O

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Sample Case Output: Double-Body Field

[_AXeJMAX_N_0Y,ITLH,LGt_,IOISa,I_I_,L_INTL_*_I_#IGt_ ; eq 20 d 2ou u I I U U U

I $ I 0 0 I0 O I I |PLOTwIPLT_,NCUPYwLL_nTImL]_T2t_M_wNUMH_X,IG_IPIaL_IP2 I

H(|)$H(d],R(5),Y|_FI',,A[NFI_mXOI_F,Y_]NF,NIwF ; I,_0000000 ,_O01000V ,OOUIUO_O 5,UOO_OUO_ O,OOOOuOuO O,O0000000 O,O000_OVO O0

IEV,i*IT,N{tOJ I o 0 v,OUOuOOOU

INFAC_INFACO I U U

5IZE_aIIO_OIST_X_I,XHd,VHI,YH2 O,uuOOOUOU O_nOOOOvOo lO,O0000000 o,noooouno O,OOO00000 O,O00000OQ OtOOOOOOOO

INITiaL GUEG$ TYPE! 0

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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) Z r *******,** _I$EI$SlPBI _TATE 2=D COORDINATE TRANSFORMATION *******_** 3

C * CDnRDI_aTE SYSTE M "SCALE FACTDRE*I ALPwA,B_TA,RAMAeSIGMA,TAUe C * JAC_HTAN,DXIDWI,[)XlbETA, C * I)YIOXI, AND DY/DETA ? C * C ********************************************************************* 9 C tO

DIMENSIO_ CI(8)wC2(8).LBSID(b),LMI(6),Ln_f6],LEDY(6}, LRS_D I(6), LRI(6), LRZ(6), LISID(6). LI](_), I I2(6), LTYPE(_), LSEN(6) 13 _ATA NDIM,NDIMteNDIMX 170e_0,70/ ta _***** _S C t6 $1 C 18 C * 19 C * THE COORDINATE @VSTEw IS REAO _R_M nTSKCTAPEt] IN THE $_[ _0 C * • DRW*T USEO BY TO_CAT TO *RITE ON _ISK. THE SCALE F_CTC.R8 C * kRF WRITTFN ON DISK(TAPE|O] IN ONE OF THF FOLLOWING FONM_T8, C * SP_CIFIEO BY IFORM, C * C * *** FORMAT _1 (IFORMmt) *** C * _6 C * _RIT[(tO,t} c] =7 C * Z8 _RITE(tO,t) IM_X,J_W.N_SEG,NWSEG,LISE_,NBDY Z9 C * .RITE(IO,_) (LnSID(L],LB_(L].L_2(L)_L_Yt' },LEEN(L} , 30 C * I LmI,NBEEG} w_ITE(tO,I) (L_BID(L),L_I(L).LRE(L),I_I_ID(L).LIt(L),LIZ(L), C * I LTYPE(L),LmI,NR8EG) C * DO 2 jBIIJMAX-2 C * 2 *RITE(tO,I) ((CtN,T,J),Nml.IO).IB_.IM_X-2) ]IS DO _ Jmt_ $ WRITE(IO,I) ((CB(N,I_j)_NII_|O)_m_wM_XO(IMAX,JM_X} C * ]8 THE _RR_Y C CONTAINS THE F_CTOR$ FROM THE _ECOND POINT TO THE ]9 C * PENULTIMATE POINT_ ON SECOND ROW TO THE PENULTIMATE ROW, aO C * THE *RRkY C_ CONTAIN8 THE FACTORS ON THE RECT_NGULkR B_UNDkRY, C * JmI,2,3,_ IN CS CORRE8RONDINS, RESPECTIVELY, TO THE SOTTOM, C * LEFT, TOPe RIGHT SIDES, ON EiCH SIDE THE POINT8 RUN FROM 1 TO C * EITHER IM_X OR jMAXe A_ _PPROPRI_TE, llll =S C * *** FORMkT _E (IFOWMI|) *** C * =7 C * *RITE(tO,l) C1 C * WRITE(IO,I] C2 a9 C * _RITE(tOwt) IM&X,JMAX,NBEEG,NREEG,LIEEG,NBDV SO C * *RITE(tO,l) (LB$ID(L),LBt(L)_LB_(L}eLBOYtL)_LSEN(L]e St C * I LBI,NBOEO) C * .RITE¢tO_t) tLR8ID(L),LRt(L].LRZ(L)_LIEID(L]eLIt(L)eLIE(L), S_ C * | LTYPE(L]_LlleNREEO} SII C * DO _ JmI,jMkW SS C * wRITE(IOn1) ((C(N_I_J)_NBI,IO)_IBI_IMAX) t6 C * C * THE JRRAY C CONTAIN8 THE FACTORS FROM THE FIR|T POINT TO TME C * LAST POINT_ ON THE FIR*T.ROW TO THE L_ET RO_, S_ C * 126

C tit w_ C _2 C t T_E FACTORS CORREBRON_ TO TMF TNnEx _ AS FOLLL)*SI (}3 C C * qll I JACORIAN bS C * NO2 I ALPHA 6b C * Nm3 t B_TA _7 C , Nm_ I GAMMA C , Nm_ t SIGMA C * NB_ | TAU 70 C * NmT I DXIDXI 7! C * Nm8 l DXlDETA 7_ C , Nm9 l DYIDXI 73 C t Nm10 I.DYIDETA 7_ C t 75 C 76 C TT C *** CARD l IwRTIjI_RTE,IFOR_ - FORmAT(SIS] 78 C 79 C IWRTI - I0 DONeT PRINT COORDINATE SYSTE M FRO_ _HICH 8O C , FACTORS ARE CALCULATFD, C t )0 PRINT COnRDINATE SYSTEM, 82 C t C * T_T2 - m0 _DNWT PRINT FACTOW$, C * >0 PRINT FACTORS. B5 C e _S6 C t IFORM - FILE STORAGE FORMAT CONTWOL, {SEE AMOVE) 87 C t 88 C 89 C 90 C READ INPUT DATA C 9_ _RITEC6eISO) 95 N|m| 9_ REWINO NI READ C5,110] I_RTt,T_RTE,IFOR _ g_ _RITE (_ISO) IWRT%,IWRTE,IFDRM 9? IF (IFORM.LT.I,OR.IFORM.GT.2) _RITE (6,%70} IF (IFORM,LT,|,OR,IFORM,GT*_} STOP | 99 IO0 READ COORDINATE SYSTEM 101 _02 REAO (NI,IOI] C| 1U_ READ (NI_IOi) C_ iOQ REAn (Ni,IIO) IMAX,JMAX,NRSEG,NR8_G,LISEG,NRDY IUS ITEST|IMAX-_ SOb JTE$TmjMAX*2 IF (IFORM,EQ,|) ITESTmlMAX lOS IF (IFORM,EU,_) JTE$TBJMAX 109 IF (ITEBT.GT.NDIM) WRITE tbeL]O] _0 IF (JTEST,GT,NDIM1) wRITE (6_IQO) IF (ITEST.GT,NDIM.OR.JTE8T,GT,NDIMt) STOP IF (MAXOCIMAX,JMAX).GT,NDIMX) WRITE (b_160) li) IF (MAXO(IM&X,JMAX).GTtNDIMX) 8TOP ] READ (N|_IO2]fLBSID(L),LRI[L]_LB_(L),LBDYfL}_LSENfL)_t. mt_NB$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) 127

PRINT LAREL l_t

_ITE (b,lO0) Cl tE$ _ITE (B,lnn) C2 _ITE f6,120) IM_XeJ_JX 125 tea CALCt,LATE FACTORS 12? 1E8 jMxMImj_AX.I

M_IHWm-AXn(I-Axpj_AW) 15! I_X_2mT_AX-) I12 j_XM2IjMAX-p CALL ABG (C,_,Yp*D_,I_AW,JUAX,?_X_le.IpY_I,T_,_T2,_$E_rLISEGwCB,LB IST_,LH_.L_,LB_V,LR_T_,LQI_LP2,L.I$1_,LII.LI2,LTYPF,LBF_NDI_X,NR_E _G,_iDIMI) _36 137 PWIK,T X A_r_ V FIELr)$ 139 IF (I_RTI.EQ.O) GO T{} 10 t_o CALL *_T* (X,I_aw,J_AX_I,2,_DI_} I_I I_2

*WITE F_CTOW$ T_ nISK

I_? *** FORMal It *** t/4(; ITE(IO,IOt) C! 150 _R ITE(IO_IOi) C_ 151 ITE[IO.110] I_AX,J_AX,NBBEG,NRSEG,LISEG.NRDY 152 WRITEttO_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}, ILT YPE(L)_LIt.NRSEG} 155 _0 Jmt_JMXM_ 3n _R ITE(IO_IOQ} ((C(N,I,J)_NII_IO)_Imt_IwWM_} 15T OO _0 Jm1._ _0 _ITE(IO_IOQ)((CB(N,I.J)_mI_IO)_III;MDIMX] IS9 STO_ 01o1 leo t61 *** FORMAT _E ***

So _RITE(tO,t01) C1 *RITE(IO,tOI] CE leS _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] le? _RITE(IOetO_](LR$1_(L}eLR_(L)eLR_(L)eLI$I_(L]eLII(L)eLIE(L), ILTYPE(L)_LmI#NRSEG) 169 DO 60 JJ|I,JMXM_ 170 DO 60 IIat,I_XM2 17! DO 60 NIt,tO I72 JJJIJ_XMI-JJ IIISlMWMI-II JIJMAX-JJ t?S IBIMAX-II bO C(N_I_J)mC(N,III,JJJ] I?? O0 70 IBI,IM_X t?O DO 70 NII_IO 17e C(N,I,I)ICB(N,I_I} 180 128

70 C(N.I.jMAX)uCB(_.I.$) 00 80 Jz2.JMXMI DO FO _uI.10 C(N,I,J)mC_(_',J,2) BO C(N,IMAX,J)mCB(N,Jp_) DO gO JmlpjMA_ ]86 90 *RITE(IO,IO_) ((CCNeT,J)w_II_IOIeIBIeT _AXI STOP OlO_

100 F_RMAT(lXeBAIO) IgO lOl FORMAT(BAIO) 102 FORMAT(SIS) 10] FORMAT(TIS) IV3 10_ FnRMAT(_[16,8) 110 FORMAT (1hiS) 195 120 FORMAT (*OFTELD I IMAX meuIUeSXe*J MAX meet_) 130 FORMAT (*O=-e.- ERROR m'--'*o|OW_*IwAX Ton LARGE. I_CREASE*o* _DI" ]Q? I AND FIRST OIMENBION nF XeY A_JD SECOhn DI_E_JSTON OFew* C .*} 198 |QO FORMAT (*O.--o- ERROR --.--*,IOX,*JMAX TO_ LARGE. INCREASE*ee N_I M 19g 11 AND SECOND DI_[_BION OF XeY A_O THIRD*w* OIMENSTO_ OF C. ,) 200 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* 202 I NOT BE GREATER T_AN SECOND DIM_N$10N OF _B. TNC_FASE THIS*,* DIME _03 2NBION*/ 20, _ ANO NDIMX IN DATA BTAT_M_NT**) 2O5 170 FORMAT (*Om_w-- _RRO_ -----*._OX_*IFOR_ _U_T BE 1 _R 2.*) 206 I_0 FORMAT(tM|/I) ;_07 2O8 END 2O9 _I_ 211

213 _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_ 21S 2LBEN_NOIMX_NB_EG_NDIM|)

218 21q

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 _23 1(I), iIBID(t}, LIt(1)_ LIZ(I}_ LTYPE(1), L§EN(1) DIMENSION FAC(IO}_ 81D[(_) INTEGER FAC_SID[ _ZT DATA FAC /6HJACOBN_bNALPNk ,&MB_TA ,6MGAMMA ,6NSIGMA ,bMTAU 2Z_ ItNX,Xl ebHX,ETA _6NY.XI _bHY,ETA / _Zq DATA 8|DE/bNBOTTOM_bHLEFT ,6HTOP #6MRIGHT t _$0 C***** Z31 O0 TO _60 C C**_* BODY 8EGMENTB **_* 23_ C Z$5 tO DO 100 L_i_NBBEG IIzH,.BI(L) Z37 I|BLB_(L) I$lIiti l_el|-l 129

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,I II,I.It.2,II,3,1,),B[TA,GAMA,IGOTC,JCb,JLP_AI Jml Imll JCx! TeaT1 C(I,IC,JC)mJCB C(2,IC,JC)xALP_ C(_,ICe,|C)BBETA C(_,IC,JC)mGAMA C(5,IC,JC]xSIG C(_,IC,JC)BTAU C(7,1CIJC)BWXI C(_,ICpJC)mX_TA C(9,1C,JC}mYXI C(IO,IC,JC)mYETA CALL 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)

ICmIE C(l,IC,JC]wJCS C(2,1C,JC]mALPMA C(),IC,JC)mBETA C(_,IC,JC)mGAMA Ct5,TC,JC)mSIG C(6,IC,JC)ITAU C(7,IC,JC]IWXI C(8,IC,JC)mXFTA C(9,IC ,JC)BVxI C(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,I I,Y,2,1,),O,I,BETA,GAM_,IGOTO,JCR,_LPH_)

C(t,IC,JC)=JCS C(),IC,JC)m_L)M* C($,IC,JC)mBET_ C(_IC_JC)mG_MA C(),IC_JC)m$1G C(b,IC,JC)mYAU C(?,IC,JC)mXwI C(i,IC,JC)mXETA C(I,IC,JC)mYwI C(IO,IC,JC)mYETA )0 CONTINUE ;0 TO 1oo C**** 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)

Jmll JCm_ ICml! C(I,IC,JC)mJCB C(E,IC,JC)mAL_MA C($,IC,JC)mBEYk 130

C(5,1CeJC)mBIG 3ol C(6,1CjJC)mTAL! )02 C(T,ICeJC)mWXl 30) C(8,ICeJC)mXETA c(geIc,JC)IYXT $ OS C(iO,IC,JC)mYEYA CALL PARA) (XeY,I,12,_OTM,XWI,v_I,I_T?,)wT2,),IP,WETA,YET*,I,12, )&o7 II,I_-I,IaI_-_,Io-I,_ETI,GA_A,IGOTO,JC_,ALPHA) )08 Jml2 ];0(; ICml2 _10 C(I,IC,JC)mJCB 311 CCZ,IC_JC)mAL?_A )12 C($_ICeJC)mRETA C(_,IC,JC)mGA"A C(),ICeJC)mSIG C(6,1C,JC)mTAU $16 C(?_ICwJC)mXXI _&l'r C(i,IC,JC)mXETI C(IelCeJC)mYXl C(iOolC_JC)mYETA DO 50 Iml],l_ $21 CALL ;ARA3 CX,YwI.Iw_DI"_XXT,YXI,I,T,),Y,3_I,XETA,YFTA,I,I-I,n I,O,I,_I,L,O,flETI,GA_A,TGOTOeJCB./LP_A) ICm! $Z(4 C(I,ICwJC)mJCR 3Z5 C(),IC,JC)IALP.A C(),ICfJC)mBETA C(_,ICeJC)mGAMA C(l,IC,JC)m$1G C(b,ICeJC)mTAU _0 C(7,1CoJC)mXXl C(8,1C,JC)mXiTA C(I,IC,JC)mYXl C(IO,IC,JC)mYEYA 50 COnTInUE GO ?o lOO C**** TOP bO CALL PARk) fX,Y,II,JHAX_OIHeXXT,YX),TI,JMAX,ZI+IoJ_AW,IIe2,JMAX IeX[TA,YETA,ZlwjMAX,TIeJMAX-I,II,jMAX-_eI,-IpBETA,GA_A,IGO?OejCBeAL _PHA) JmdMAX Imll JCm$ ICmll )&_(¢ C(I_IC,JC)IJCB )(;5 C(),IC,JC)mALP_ C(3,1C_JC)mBETA C(_,IC_JC)mGA_A C(),IC_JC)mSlG C(b,IC_JC)mTAU _50 C(?_IC_JC)mXXI _St C(8_IC,JC)nXETA C(_IC_JC)mYXI C(iO,IC,JC)mYETA CALL PARA) (X_Y_I2_j"AXeNDI_XXY_YXI.y2_jMAX,I_ml,J.AX_I_.2eJ_AX I_XETA_YETA,I2,JMAX,I_,JMAXeI,T2wJMAX-_,-I,-I_RETA,GAHA_TGOTO,JCB_A 2LPHA) )ST l_Z2 SSe ICml) C(I,IC,JC)mJCO 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! 366 CC_,IC,JC)mWETA 367 CCl. IC.JC)mYxl ]&A C(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 371 I'YFTA.I.JHAX.I..Y"AW=I.I.J_AX').O.-I._FTA.CA,A.IGO_OmJC_.ALpHA) 37) ICm! 373 C(I,IC,JC)mJC_ 37. C(2,!C,JC)xALPHA ]75 C(],ICwJC)mRETA )76 C(,.IC.JC)mGAMA _77 C(5'IC'JClm$1G 378 C(6.1C.JC)miAiJ 37Q [(7.1C.JC)mXWI 3_0 CCA,IC,JC)mX_T¢ ]81 CCQ,IC,JC)mYwI 3_2 CCiAelCwJC)mYEIA 38) ?0 CONTINUE 38. GO TO 100 3_5 C**** WIGwT _86 80 CALL PA_A$ (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 )88 2_A_ 38_ ImlWAX )gO Jml! Sgl jCm. 39_ ICmll ]93 C(I.IC.JC)mJC_ 3_ C().IC.JC)m_LPWA Sg) C(3.1C.JC)mBETA Sg_ C(QmlCmJC)IGAM* 311 C(S.IC.JC)m$1G ]9_ C(6,1C,JC)mT_U 399 C(?,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_,_ _OS 2L_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! _17 C(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) u21 ICmY C(I,ICIJC)mJCR C(_IC_JC)mALPHA C($,ICeJC)mMFTA C(_,ICeJC)mGA"A C(5wlCpJC)m$1G C(aw|CpJC)mTAU C(?elC_JC)mXXl C(_elCoJC)mX_TA C(qwlC_JC)mYWI C(IOwlCwJC)mYFTA Q0 CONTINU_ I00 CONTINUE "35 C C**** REENTRANT SEGMENTS **** _3b C IF (NR$_G,EQ,0) GO TO 2"5 DO 2_0 LmI,N_SEG I_mLRI(L)÷I 16mL_ZCL)-I I)mLII(L)_I l_mLl)(t)-I IGUTOmLTYRE(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,5 _5(1 XETAm(X(I,))-X(I,jwX"I))*O,) YETAm(Y(I,2)-Y(IeJMX'I))*O,5 _SE _53 IlmI-I I_sI÷l JImjMXM! _55 J292 _Sb CALL PaRAI (XXI$,VXISelE,J,TIeJ,XEIA),YETIB,I,J2,T,JI*XXlETI,Y _57 IXIETA,I_IJ2,12,JI,II,JI,II,J2,X,Y,I,j°NnlM} _5_q CALL PaRI2 (wxIwYWI_WETa,YETa,WXIS,YxI$,xETaS,Y_TIS_XXI_Ta,Yx! _sq IETa,ALPHA_@AMA,BFTA,IIG_TAU,jCB,I,J_N_I_) JCgl ICm! _b2 C(I,IC,JC}mJCB C(2,IC_JC)IAL)HA C(3,%C_JC)mBET* _e I] _bb C(§,IC_JC)I$IG C(tmlC,JC)mTAU C(?,I{,JC)mWWI _bq C(i_lC_JC)mXET_ _70 C(q_IC_JC)mYX!

120 CONTINUE GO TO _10 C**** BOTH 0N BOTTOM 130 00 1_0 Iml$,l_ jol _7'_ II_li-(I-I_) _Te XXle(l(I*l,l)-l(I-l,l})*O,5 _7q Yllm{V(l$1,l}-V(I-l,l))_O,_ 133

XETAm(x(T,E)-X(TI,E))*O,5 Y_TAu(v{I,E)=Y(II,2))*O,5 IlmI-I 12ml*i JIull _S5 J2=E CJLL 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 /_B¢; IETA,ALPWJ,GAMA,BETA,_IG,TAU,JC_,I,JeN_I_) JCB1 IC,I Q92 C(I,IC,JC)IJC8 _'{;3 CC_eIC,JC}sALRMA C(3,1C,JC}xBETA C(_,IC_JC)BGAMA C(5,1C,JC)mSIG C(_jIC,JC)BTAU C(7,IC,JC)wXxI _gq C(8,IC,JC)mWETA 500 C(9pICtJC)BYXI 501 C(IOwICeJC)mY[TA 502 CONTINUE 50_ GO TO 2_0 SOU C**** BOTw ON TOP 505 150 DO I_0 Tml_,I6 50_ JBJMAX 507 IImI_-(I-15) XXIm(X(I÷!wJMAX)-X(I-IeJMAX))*O,5 50q YXIm(Y(I÷ItJMAX)-Y(I-IpJMAX)}*O,5 $10 W_TAm(X(IIwJMXMI)-X(I.jMXMI))*O,5 511 YETAI(Y(ITeJWXM%}-Y(IwjMXW|))*O,5 512 IlmI-I 515 12mI*! JIwll 515 J2IJMXMI CALL PARAI (XWI_eYXI_fI_wJwII,JwX[TASeYETA$1JteJ2,1pJ2,XXIETA, 511

CaLL PARA_ (XXI_YXI,XFTA,YETa_XXI8,YXT$_XETA$,YETI$,XWIETA,YXI IET_ALPW_,GAMA,BET_w_IG_TAU_JCB,I_J,NDIM) S_O JC=] ICuI C(1,1C_JC)_JC_ C(E,ICeJC}mALP_A C(3,IC_JC)mBET* S_5 C(_,IC_JC}wG_MI C(5,ICwJC]_SIG 5_? C(6_IC,JC}uTAU 5ZS C(?,IC_JC)mXXI S_q C(8_IC_JC}BXETA 5}0 C(q,IC,JC)wYX! 531 C(IO,IC,JC)mVETA 160 CONTINU[ GO TO E30 Ct*e_ ONE ON LE_T_ ONE ON RIGMT 170 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 SWO 134

YETAm(Y(t,J*I)-Y(IjJ-i})*O.5 I1mIMXW1 5u2 IEmE 5_3 JtmJ-% J2mJ*! 5_5 CALL PARAI (XXISeYXIS,IE,J,I1,j,XFTAS,YETAS,I,j2,I,JIeXXIFTA,V 5_6 lXIETA,I2eJ2_I2eJl,ItwJ1PIlwJ21X,YeI,J,NnI') CALL PA_A2 (XXIsYXIeXETAwYETAoXXI_eYXTS,XETASjYFTASeXXIETAeYXI IETAeALP_A,GA_A,BETAo$1G,TAU,jCB,I.J,_nI") 5_c_ JCm_ 550 ICmJ 551 C(IeICpJC)mJCB 55_ C(2eICwJC)eALP_A 553 C($oIC,jC)m_ETA 55_ C(_elCwJC)mGA_A 555 C(S_ICpJC)m$1G 556 C(6eICeJC)mTAU 557 C(?eICfJC)sXXI C(8,IC,JC)mWET* 559 C(,.IC.JC)mYXI SbO C(IO_IC.JC)mYETA 180 CONTINUE GO TO 2]0 563 BOTM ON LEFT Sbta 1oo DO _00 JoIS,16 cS6S In! 1_6b IIoIW-(J-15) 567 wXIu(X(_,J)-x(2,1T))*O.S 568 YXID(Y(2pJ)-Y(_,II))*O.S 56c_ XETkmfX(lwJ+1)-X(leJ-l))*O.q 57O YETAmCY(I.J+I)-Y(I_J-I))*OeS 571 IImll 572 12m2 JlmJ-I JZtJ*i 575 CALL PARA1 (WXI8,YXlS,I2,J,12,11_WET_9_VETAS,I,J2,I,JI,XXIETA, 576 tYXIETkeI2eJ2,1_,JIel)_II+leI2elIoleX,V_I,J,NDIM) S7T CALL PIRA_ (XXI,YXI_XETA,YETA_XXI_,YWIS,X_TA$.YETA$_XWIETA,YXI $78 1ETA,AL_MA_GAMA_BETA.$IG_TAU_JCB_I_J,NDIM) 57O JCm_ '580 ICmJ C(I.IC.JC)mJCB C().IC_JC)mAL)HA 503 C(),IC,JC)mBETA 58_ C(_.IC.JC)BGAMa 58S C(S.IC_JC)mSIG 586 C(_.IC_JC)ITAU 567 C(?.IC.JC)mXWI 5_8 C(O,IC,JC)mXETA S_9 c(g_Ic,JC)oYXI 59O C(IO_IC_JC)BYETA $91 2OO CONTINUE Sgi_ 60 TO _$0 $93 C**t* BOTW ON RIGHT $9_ ZlO DO 210 JmIS,%6 595 ImIMAX $9_ IImIQ-(J-IS) XXII(X(IMXMI,II}-X(IMXMI_J)),O.S 59_ YXII(Y(IMXMI_II)-Y(IMXMt_j)),O._ $9O XET_m(X(IMAX,J.I)-X(I_AX_J.t)),OmS b00 135

vFTAs_Y(I_AX'J÷I)=Y(I_AX*J'I))eA,5 601 It=IT _OE TEsT_X_I _03 JlsJ=1 bU_ j2sj+! b05 CALL PA_AI (WXIS,VXI$,I_,II.12,j,yETAA,YETAS,I,JE,I,JI,WXIETA. 606 IYXIFTA'IE'II'I*I2'II+I'I2'JI'I_'J2'W'Y'T"TI_I_) bO? CALL P_Rj_ (XXI,YWI,XETJ,YE?A,XXIS_YWT$,WFTA$_yETA_oXXIETA,YXI b08 IETAeALPWAeBAMAw_TAwSTGeTAUwJC_'I'JeN_I_) 60q JCmQ blO ICmJ ell C(IelC.JC)mJCB 61E CCE,IC,JC)mALPMA 61) CC3,1C,JC)mBETA elQ C(_,IC,JK)mGa,a 615 C(5,1C,JC)mBIG bib C(b,lC,JC)mTAU bit C(TeICeJC)sXXI bl_ C(R,TC,jC)mX_TA blg C(Q,IC,JC)mYXT 620 C(IOpICpJC)mY_TA 621 2_0 C_TINLJE 6E2 E]n CONTINUE 6E3 2_0 C_TINUE _ _a5 DO _6 _ml,lO 6_ TF_ • C(_,I_I) * _(_,I,_) 626 C(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) 63E C(N,IWIW_$) • T_M b33 C(_JM_X,_) • TE_ 63_

C(_.l,_) • TE_ b}6

_A CONTINUE 6_ I_ (I_RT_,EQ,O) GO Th 2QO 63R wRITE (b,330) 6WO nn 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_e C 6S0 C**** FIELD **** 6§1 C 65_ _bO CONTINUE 6§) O0 _TO Jm_IJMXM| 6S_ JCmJ-I 655 JMlmJ-t 6§6 JPlmJ+[ 6)? DO E?O ImE,IWXMI &S8 ICml-l b§e IPlml+I bbO 136

IPlmI-1 WETAICXCIpJPl}-XCT,JMI))*Oo_ VfTAmCYCIojPl)mYCIpJ_I))t_),5 XXlmCX(I_1,J)=X(IM1_J))*O,5 _b5 YXIm(Y(IPIeJ)-Y(IMIeJ))*O,5 _e,h ALP_AmXETA**2eYETA**2 ¢",b ? HETAmXXI*XETA+YXI*YFTA e_bP_ GA_&mXXI*m2+vWI**2 hoq JCHmX_ImYETA-_ETA*YXI b70 XXI_mX(IP1,J)-2,0*X(l,J)÷X(i_l,J) yXISmY(IPI_j)-2,0*Y(IwJ)+Y(TMI,J) 672 WETA$mw(I,JPl}-2,0*X(TwJ)÷XfI,J_I_ _73 YETASmY(I,JPl)-_,O*Y(TpJ)+YtI,JMII _7_ XXIETAmO,ES,CXCIPIPjpl)-X(IWI,JPII-WCTPI,J_I)+x(I_oJMI)) e,75 yWIETAmO,_,(Y(IPI,JPl)-Y(IMIwJPl)-Y(TP|pJMI)+Y_I"I*JU%)) _7_ DXmALPHA,XXI$.E,O,FFTA*XWIETA+GA_a*_ETA$ hT? DYmALPWA,yxI$.2,0,@ETA*YxIETA+G_*YFTAB e,78 8IGm(DX*VXI=DY*WXI)/JCB _7q TAUm(DY*X_TA-DW*y_TA)IJ_R /_8t) ABSJmAHS(JC8) CF(I,IC_JC)mjCB e_BE CFCE_IC,JC)BAL.PHA (',B_ CF($_IC_JC]mBET_ _8L& CF(_IC_JC)mGAMA CF(I_IC,JC)m81C CF(_IC_JC)mTAu e_8? CF(7elC_JC)mXWI CF(M,IC,JC)mXETA _.SR CF(I_IC,JC)mYXl CF(IO_IC_JC)mYETA _70 CONTINUE bee C ag_ *_ITE KACTOR$ TO PRINTER C I_ (IWRT_,EQ,O) GO TO I0 wRITE [b,)O0) bgT JWxwEmJMAX-2 6(]B IWXM2mIWkX-2 00 280 jmlmJMXM2 ?On JJmJ÷l ?0% wRITE (b,$10) JJ DO 280 Nml_lO 7O2 ?03 _BO wRITE (_$20) FAC(N),(CW(N_I,J),ImI,I"XMEI ?0/4 GO TO tO EgO CONTINUE RETURN 707 C $00 FORMkT (*0---- SCALE FACTORS ---=*) 708 _10 FORMAT (// * J m*_I)/) TlO $20 FORMAT (*O*mA_t/(SEt_,8)) $]0 FORMAT (// * BOUNOkRYm) ?11 ]aO FORMAT (//2H *_A6) C ?I$ Tl(& END

?I?

Tie fUNCTION _MWMN (NO_Te_,AWeI,J,IXeJX) 'V_O 137

?E! 12E 723

?E6

G_ T0 {|0_20)e NOPT 7Eq 10 IF _A,LT,AX) GO TO _n 730 GO TO 3h 731 EO IF (A.GT.AX) GO Tn ._ 73E 30 AMX_N=A TXu! jXBj _ETUR_ 7_6 .0 A_X_NIAX RETIJRN ?38 "/)9 E_'O

?#3

SUBROUTTNE PARAI (XI,Yt,lt,J1,IP,j2,X_,Y_,T3,J3,1",J",X12jYt2,15,J IS*I_,J6,I?,J7,I8,J_,X,Y,Iwj,NDI_ 7_6

*4*44,44********444 _ECONP OEWlV_T_V_S ,4,44,4*****4***4,44**44,4****

********************************************************************* 751 _IMEN$_ON X(NDIM_I)_ Y(ND_M_I) ?$3 XIBX{It,JI}-2.0*Xt_,j)÷X(_E,jE) vluY{II.JI)=E.O*Y(I.j)÷Y(TZ_J2) ?55 X_BX(T3_J3)-2.0*X{I.J)÷X(T,.J_) 7_6 Y2BY(I3,J3)-_.O*Y{T,j)eY(;_,J_] XIEmO.2S*(X(TS,J_5-X(T_j6)+X(I?,JT)-X(TS,j8]] ?58 Y_2IO.ES4(YCIS_J_)-Y(T_J_)eY(I?_JT_-Y(TS_J8)) 759 RETURN 76t ENO ?eE T63

?iS

BUBROUTTNE PARA_ (XXI_YXI_XETA_YETA_XXIS,VXZS,XETAS_YETAS_XX_ETA_Y IX_ETA_ALPWA_GAMA_BETA_SIG_TAU_jCB_j,NDZM} T68 C ?70 C * 771 C ********************************************************************* ??l C RE_L JCB

ALPHAIXETA*I_eYETA**E BETA_XX_*XETA_YXI*¥ETA GAMAmXXI**_eYX%**_ ??e JCBIXXZ*YETAeXETA*YXZ DXIALPHA*XX_8_,O*RETA*XX_ETAeG_MA_XETA$ 780 138

DYmALP_A,YXI$=2,0*R[?A*YXIETA$GA_A*YETA_ T8] $1Gi(DX*YWI-DY*WWT)/JCB 7B2 TAUm(_YeX_TA-DW*YETA)IJC_ 783 HETU_N ?B5 END 78_ '787 788 78¢_

T91 7_2 793

7W5 7c_ 7g?

79_ Ct_t_ __Or) IF (KI.EQ.O) GO TO 10 8ol xXIm.O,5*(X(I],J_)-_,O*X(I2,J_)*],O*X(II,JI_)*KI Bo2 YXII-O,S*(Y(I$_J])-_.O*Y(I2,J2)÷_,O*Y(II,,II))*KI 803 GO TO _o SOc_ to XXlmO.S*(x(13,J])-x(It,Jt)} 8uS vxImO,S*(v(1),J$)-Y(If,Jt)) Sub _0 I_ (_.EQ,O) _0 TO _0 80T WETAm-O,S*(W(I_,j_)-_.O*X(IS,JS}÷3,0*x(I_,J_))*K_ 80B YETAm.O,5*(Y(16,J_)-_,O*Y(IS,JS)*$,O*v(T,,J_))*K2 80_ GO TO _0 _10 $0 XETAmO,S*(X(Ib,Jb)-W(I_,J_)) 811 YETAmOeS_(Y(16,Jb)-Y(IQ,JQ)) _41_ QO CONTINUE 813 ALPMAmXETA**2$YETA**_ BETAmWXI*XETA_YXI*Y[T_ 815 G_MAmXXI**_$YXI**_ JCBmXWI*YETA-XETA*YXI 817 RETURN 819 END 820 821 82_ 82.3 B2._. 8Z_ 82e 8ZT 8i_8

830 OIMENSION CODE(_] 831 OIMENSION X(NDI_,I) 83Z DATA COOE/bH*_ XAR,&HRk v *_t6H** YAR#_MRAY **/ al,] C_ 85a WRITE (b,ZO} (COOZ(1),Imlt,I_) 00 10 Jst,JMAW 836 wRITE (6,]0) J e$? I0 _RIT[ (6,aO} (w(I,J),ImIeIMAX) 838 _ETU_N ¢ 8wO 139

_u FO_AT ft_I,2OXp2A6//)

_ _0 _^T (_X,1_11.5)

8_S

8_7

8S3 140

SAMPLE CASE INPUT: SINGLE-BODY FIELD

Input for Program FATCAT requires only one card and is described beginning at 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. 141

Sample Case Output: Single-Body Field

INPUT I ]M_TIm 1 ZxHT2m ] 1FO_Ma 2 lEST CaSE ° 80DY-_ITfE_ CO0_OI_ATE SYSTEM @XN_LE UOOY I KA_Mk_-T_EFFTZ AIRFOILa _6 PUINT$

F|EL_ I X_kA • _7 JMAX • _0

Po'- @CALL F4CTOHS *-*-

J • 2.

JkCO_N

.I055021JL-OJ .2_[_|33TE-0_ .5797632_t-03 .lb_2259@E-02 .36@17J_1[-0_ 1557qS_87EsO_ 1@787_787[-02 .@q7OSbOqEeO_ .bI_OS$b7E-02 .uSllu_BeL-02 .25_0_57E-02 .g_BJb)SIE-O$ .@$_7$qOE-O} .13q337_2[-02 .307121@0[_02 ._TqlO_$1EmO_ .b0_7520E-U2 .b57_b@B,E-02 .@[email protected]_ .5|_qTJ22E-0_ .3_O_9_@OE-O_ .157$5155E_0_ .5_@|SqSOE-03 e_|qSBT_|eO] .10507133E=0]

ALPMA

.t3_esl0eE-03 .270_Tq_oE-03 .59_J_O79E.05 .taaEi0aaE-02 .lqea557?E-g_ .R_BeEs_OE=O2 ._]@_o0oat=02 ,_b3975[_0_

.lqJ73959E*02 .2159_bB2E-O2 .2_269@bE-02 .217_qBbOE-02 .1908_055E-0_ .l_lgB2_2E.O_ .Sa77_735E.03 ._bq_q?87_-O$

BETk

.l_6qlq0_E*0_ ,bS]qOJlOE-O$ .2207_bqqE*03 ._86bT_0|E_0_ **_7830160[-05 -_805q_0_E-0@ _._5_q_OqL_O3 _m@_17_75_0$

GAMMA

.201_26_E-01 .1_12212_E*0! .75_qOl_lL°02 .2718530_E-0_ ._3_780[-02 ._ooqit_?E-O_ .9_$_880E.0_ .|51170q$[_0_ e_qBbq22@E'01 ._@_I_5_E-OI .203_6530E-01 .155@Ol_tE-Oi 0918_56@2[-0_ e33150531EeO_ .T@O@_qSeEeO$ ._17_5_$_Eu0$ ,83_15@6bE-0_

$]G_a

-.[2965._OE-O_ ml_O73@?_E_O@ ._125@177E-05 .3_078_5bE-0_ .17|,ql_bE-O3 e@_,_$qq]E_O$ .@|q7B@bZ[*O$ .bbblbq_@[_O$ ._Oq75@7@E-03 .27551002E-03 .@7$_llqSE-O_ .1_17002_E-0_ ._$[2qO_@E-05 ._b_b@_27E-O_ .127_bS08[_05 .31OS$@_@EeO$ ._q$@_q0BE-03 mS@3@3IqTE-03 _52_q_q_E-O_ .352@_)1_E-0) .[_qT@TbOE-O3 ._95_35qSE-0@ ._97155_E-05 .l]5_lb_Tt_O@ o.lloa_bE*Oe

TAU

-.Jbq57JO@[-O@ -.92_V_7_E-O@ .75115305L-08 .@077qqqSE_07 -.9_585592E-07 ._@_@bJE-O@ .3@aBJETOE-Ob *m_OqS_41E-O@ -.@@JO_@qTE-OT -._OI_9_O_E-00 ._5@O_bbT[-OT .Sl$O_eqE-07 -.IO§2bbB@E-@@ -,9qoo7I_SE_O@ mbl$|q22_E-O@ -_3@6_17|0E_07 *m1275_@_2Eo07 -.75@7_0@E-0_ -._2_9_118E-07 .I@lOl9_qE_0b -.3707077@E-07 **958@_06@E-08 mlbl_k?@_[-07 ".|@|5_O@OEeO? .$_$b_lSOE-O@

X.X!

X.ETk

.87qb_@5OLo02 ._@lJ_OE-02 -.27_20250E-0_ -.l/7587qOE-01 -.1@@@7$1OE-O! *.lb_b2qO[-O| -.|$@@790q[-01 _ml|B_@J?qEm@[

-.12JS?_qSE*O! -.t2_OO[E-O| -.tJO$7IlOE-OI *.1_5_0_5E-01 -.15@23705E-0! -.I_@@7_OOE-O| -,TqJg|OOOE*O$ ,5_qq@gOOE-O_ mq@_qqbooE-o2 142 143

MOU _OANY

JACOB_

.Qgg_7oWlE-OS ,_g_w_gk-q_ ,_l_W£w_wE-O$ ,119eO705E-O2 ._55JI61?E-O_ o5707?OSbE-O_ .7Q_7OWOE-O_ ,w_5og]gBE-O_ ,2_71bOgt-O_ ,bU7OBT31E-03 .175W760_E-O_ ,ll_|@qllE*O_ ,_§91591_E_0_ ,obSelQOEE-02 ,o$721ea$E-OE .51700971E-O_ ,_BSIbOE=O_ ,II_7505E_OZ ,EUa_2IOIE=_3 ,w_9_lg_E-O5

ALPHA

,2_701e?L-d_ ,19_2117_L-02 .77q_Og_E-O_ ,_19bSq72L-O_ .577_175_E-0_ ._815_178E-0} .9_O_?_IE-O_ ,22011717E-0_ ,2J120bObE-O_ ._bSOJ52E-U2 ._0512_5_E=0_ .lbSQ138qE-O_ o}8705]7_[_05 ,l_l|_e27L=O_

_LT_

,lqlg3OO7E=O_ ,IeO75598L-uZ ._0977115E-0_ ,35_L370BE-O_ ,ZE7a2_ITE-O_ .209_57_E-02 ,l_81b_8_E-O@ ,oO_b1oouE-o_ ,I/_JgZ_E-UJ .gOSb_778L-_5 .3W=OSo_SE-O0 -,5011_OOEoOa "._970q58bE-O_ -,27795U_2E-O_ -,_15_SqOeE-O_ *._1159E=02 =,_7979_E*0_ -,l_8bTe_OE-O$ ,II_QO25bE-O_

,1g05_72_*05 ,QO_57_|_k-05 ,I/2_BQW/E=03 ,E$53WeglE-02 ,IQ2QbsO@E-01 ,1800_371E-01 ,_qI_8_i-O! ,_O/bWqLE-OI ,2050_2b_k*01 ,IQ_._32gh-ul ,/o_llgObE-n_ ,lqObbObOE=O2 .5_O?}_eSE-O] e$$O1_TqSE=O_ _878_qOO_E=O_ ,|QWblTobE-O| ,Lgg_bTL_E.(ll ._2¢lwbTL-Ol ,_UgU_5_gE=O_ ,[b_blb_OL-O! .q_78iSqE-O_ ,_O0@I$OE-O_ elbb_b78EeO_ ,I_O_.b_bE-Ob

S[G_A

-._57_o8_bE-UW -,_$7_be_bE-OW -.5_750820E-0_ =.33730B_bE-Og -,3_7_08_0E-0_ -._37308@bE-OW -,357508_bE=0_ -.JJ7$O_OE-O_ -,337JOaEbE-Oa =,3$rJOsEbE-Oa -,337308_eE-Oa -,]$7$08@bE=Oe

l_d

,_2V$12e_EoO_ ,J_oS|_JE-O_ ,J_OJ|_bJE-O_ ,J_O_l_b3E.OW .3_0312b$i-O_ e3_O3t_b]E=O_ ,J_OJt_bJ_-O4 ,_O_LLOJE-Qa .$_0312b$E-OW ,_O_12e_E-Oa ,_O$1_b_E-O_

-.6gOOOOOOh-OW -.II210OOOEeO_ -.blbiOOOUE-O_ -.12_iIOOOE-O[ -,_807_500E=01 -,lO0abeOOE_O0 -,15_7_050E_00 -,_5@17700E_00 -.IbbuOgbOE_O0 -.I_I_{)OE,O0 -.tI_7OUE÷O0 -,_2_OO00E-_I -,_gw_oouoE-o! tTTO_OoOOE-O_ ,5_QLISOOE-01 .9159_500E_01 ,I_Ibl_5OE*JO .1_11_50L_00 ,I_gOlOOE÷O9 ,l_Jg_750E_O0 ,t_SBbSOOE÷O0 ,q_BeqsOOE-Ot ,_50b3500E-Ot ,t@_l_500E-Ol ,583_000L-)_ .15b_SVCO_-O_ ",_hSOVOOOh_O_

X,ETA

-,IO_0_050E-0£ -,ogJelJtOE-O_ -,_qS_ia|OE-01 -,_b7750_OE-01 -,_1_$570E=01 -,IO0_=]t_E=01 -,TI_e_QOOE.Q2 -,_UOb$5Ol-O_ -,bSel_BOOE-OE -,71508qOOE-O_ -,8_150_50E-0_ -,qe_?q_g[-O_ -,lO_e_950L-01 -,It$1_I_bE-OI ",LbO_OOSOE-Ol -oIB77_BSE-01 ",_517955E=Ol -._b_lbS_OE-O_ _eS_tOq_OOE_O_

Y,X!

-,l$llbOJV_-o_ -,I}OWSUUOL-_ -,_()IbUOOOL-U2 -._BTwSOOOE-O_ -,bs[gsoooE=O_ -,IU/_500E-01 -,1109q500_*01 -,9_$_OOOE=O_ "eQ_bTOOOOE'd2 ,Iw_OSOOOE_OZ ,_2bOOOE-o? ,taS_J_UOE=Ot ,[877_50UL*01 ,_laBO500E-01 ,_ZWIIOOOL-OI ,t_B_$OOOEe01 ".1551bbOOE-_i -,|ega_OOOE-Oi -,IBOg@5OOE-O| =,lOT_e_OO[*Ol *.el_q$OO_keO_ -.269;SOOOE-02 -o15_55000E-02 -,I_7HSOoOE-O_ 144

YoL_A

-,_b_i_oqOh=O1 -,_5b_O2OrE-O1 -,_7_7_18_E-01 -,d700_o$gE-ol ",lJ_Bq_eE*Ol ,_BOO2_OE-O_ ,L7b_BI79E.OL ,@_iOi7t_EaO|

_LLFT

ALPHA

,_b|Sml++_*+O

-.5OSJ_bO_-O5 -.L_OgblE-O, -.JO7_OOIJE-O_ -.57_7179_L-0_ -.tOO_tOb6t-O$

-,t+;oOJJ+e-Ol

,I+O_++7_E-O+

,U/IO_O_L÷OO

5IG_A

-,13_l_b?uL-oe -._1_330o1L-05 -,_OblSOOg_-uq -,105_5@60_-03 -,1088_189L-03 ",|_OW_o_EoO_ -,55_O_bl_L-O_ -,eSTS_JOSE-O_

TI, L#

,520J;_C-0_ ,_mIWTmj_E-,!+ -,lJq+ooeS_-tJ7 -.7UI_OoIE-Ob

X,X/

-._7_bCOL-02 o."0097000E-0_ -._6_5000E-0_ -e_B|5500E-O_

X,EIA

,J_IJ_iSOL-O_ ,lO_l?_tSL-Ol ,171n7005[-01 ,_50_15L-01 ,J_gJJO_NL'GI ,bOJg_8_E-Ol ,_7_7090_E-01 ,8771_15E'0! ,I|U_JO_E+OC ,L$707_L¢OI1 .IObO_7OoE+O0 ,_O_OdgOE+O0 ,_7_7_0E÷00 ,_?_bo_OL¢O0 o$507_850L¢00 ,_7_OE_O0 .511=0030E¢0_ .OIIOJ_E++)C .7_lloO00E+O0 ._blLbO2_k¢O0

V,XI

-,a750eilOE-oL -,ai_£7_lO_L-Ol -,5_57027uF-03 -,790q_bO_L-Ol -,1051671bL¢00 -)iJSolT]_h+O0 -.17l_g_Oh+oO -._31b_3E¢OO -.20_b_5_L¢O0 -.j_OBJb_UL*O0 -.J_9_O518E¢O0 -._710III_L÷O0 -,5=gb_3_LE_O0

¥,ETA

-,l_O_OOqI£-02 -,i)qe_Ok-O_ -,|_L_5_E-_ -,llT_bBTOk-O_ o,e_38_00_-03 o,55JOBoOOE-OJ -,_e_7_OOk-OI o70116_00l-0$ ,[7@UOeB_L-02 ,JlJOLJ$OE-O£ ,_q78/TTJ_-_ ,lO8%UJTOEoO_ 145

*TOP

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AlPha

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8ETA

-.12_b0_5_E-0! .e_o_Jqr_E-0_ ._SJS317E-01 .J_$173_gE-01 .Jq3q_88LE-01 .q08|7a0aEe0l .37_0507E-0t .$188e$07E=0| .1_739UE-_1 ,7078JIT=E*U_ *._08U_77bE=03 -,e_Tq_SbE-OZ -.119_0_Z_E*01 -,17Jb_SeBE-OL -,_$30_17bE-01 -._00e_851E-01 -._75L057_L-01 -._53_5_28E-01 -.5_J98Jg_Eo01 -.57572_8Lk-01 ".5qTI0z0BE-0t -.57kbb0_8E-01 -.S08790qSi-01

GamMA

,15Lb_$P_E+01 .1_JL7997E+_L .L_3179_TE+01 .lq31799OE$01 .1_317997Ee01 .1_$1799TE+01 .laJt7_qbE$0t .t_]|TqgtE+01 .lUJtT_qTE+0L .la3179qbE÷01 ,IaSJI_TE+Ol .Aa317_97E+01 .tq$179qbE+01 .la_t7q97E*01 .t=]i?qq7E+01 ,t_JlTgqb_+Ol ,1"317_qTE+oL ,t_JJ7997E+V1 ,lq$lTqgok+Ot ,la3l?_9OE+Ol ,la$17997E+01 ,I=JI?99?E*oI ,1_JlT99_E*O| ,L_JII997E+01 .1_J17997E+0t .L51e_373E+uI

SIGma

-.3575002eE-0_ *,JJTJOB_bE-O. -,_7$US_bE-Ua *,JJTJO_eEeO_ -s]JT3OB_bE-O_ -.$37308=eEo0_ -,_]7308_bLeOa -,337308_eE-oe .._jlJU82bEe0_ ".JJ?_08_bE*0U "._7$00_bE'_ -.J37JQE_bE*UU *.JJ730B_OE=0a -.J$730B_bE=0a ".33730B_OE-0a ee_$?$0B_bE-Oa -¢3JTJ0BZeEe0e =.3J/JOS_bL'0a -._$730_OE-D= *ej_TJOb_bL-0;

,3_031_6/L-ga

._20JI_JE=0U .3_031_b3E-0, .J?0Jt_b3E-ou .3_031_b_E*0U .3_U_I2b_E-0= ._031Zb3E=0a .$_0312e3E-0= ,3_OSt_JEeOa ,J_U3L_O3E-OU ._O$l_OJLe0a ,J_O_bJk-_U

X,Xl

*,_L8500EoO_ OmES_$5995E+UO -.55b01770L+0O ".79_"TB_0E+00 -.gBq7b_bSE+00 o,II|BS_O_E+01 e. IIB7BSJqE$0t ".L|8785]gE+0| -._Su7eabSE*00 -.7q_a78_0E+0u -.Sbb0777UE÷00 -._Sb3599bL+00 O, ._8e3bgqbE+00 ._5b07770[900 .7¥_uT_0b+00 ,IL87_53gE+OI ,118785JgE+01 ,11188EO_Eg01 ,gOaTb_eSk+O0 ,7_]aTeZoE$o0 .55_07_70E+00 ,_ObJ5995E+O0 ._|8_00E-0_ x,Era

.8611o0_0E*00 ._$eI_J_0E+O0 .75619U_bE*00 .e3J_bba0E*00 .ql005oo0E+00 ._7_O8810E*00 .TaO_BT_OE=0! *.1$_50_6gE_00 -,]_90eO70E+O0 -.50b05765E*00 =.bb_gO3?0E_00 -.Tb3L0970E÷00 -.SJlg05eSE÷00 -.BbbJl0a0E+00 e._3b$0770E+00 ..7?O|IISOE_OO -,51B_85JOE+O0 -,_eO355L+O0 -,t_885570E+O0 ,5I?O5_50E-Ot ,Zb311qLSE*O0 ,aS_O7565E+O0 ,e_OOZ20_E+O0 ,7_1795E+00 .8_1505i%E_00 ._bllbO_OE+o0

-.11_180_0L+01 -,10595175E+01 o.e95_S170E*00 -.oTq735_SE+00 *.U2U3Lz_SE+00 =,ta_3100b+O0 .l_a_31b0k+00 ,_31_5bE+00 ,bT97J_gSE+OO ,89%b517OE_O0 ,I05gS_7_E+O_ ,11elSOeOE*Ot ,119e578JE+Ot ,11618079E*Ot ,10595[7S[+0| ._9_bSi70L_00 ,_?qTJ_q5[+O0 ,U_JI_55E+O0 .laU2_k_0E+0O -.t_]tb0E_00 *,4_"_|_55E_00 _,b79?$]Q_E_O0 emSg_b_|?O[900 -.1059517_E+0t -,ilblduS_E+01 -,1_$1_87E+01 v.Era

,70_5_J70E-O_ -,_I_U3u_E÷OO -,_Lg_OL2OE÷O0 -,598_10E+00 *,738939_5E+00 -,B3$eT]_OE+O0 -,877200_0E+00 -,87017975Ee00 -,SI_8OE_O0 -,710_U_SE÷OO -,%TbbgO_OE_00 *._u0707bOE+00 *._I0bLB57L+00 "._gb_50Eg[-0_ ,1gOgB369EeO0 .38E|BbB_E_00 ,55_17bqOE+O0 ,b_b3_U95E+OU mbO1775JSE÷OO .8b_bq_BSE+00 .87a_tt75E+00 .83aS10JS[_00 .7_383ZBSE*O0 .bg600715[*00 146

*eIGHT

JkCOBN

O_ Oo gl Oo Oo Oe Oe Oo Oo O, O, O, Oa, O, Oo O, O, O, zlgbOWteqE÷O!

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Oo Oo Oo O| O_ O| O| O, O, O, O, O, O_ O, Oo Op Oo oo ,7_IoWIOgE_O0

eETA

,ItJgO_55boOe Oe go Oo Om Oe Oe O. Oi O, Oi O, Oi O, O, Oo O, -,50_oqSQOE-O_

GAMMA

190J_5_5E-05 O| Oo Oo Oo Oo Oe Oe 0, O, O_ O, O, O, Oo O, O, o, ,Ibt6_375E_Ol

SIGMa

Oi OD Oo Oo Oe Oi Oe O, O= OD O, O, Oo O, OI O, O, O, -.J_TJOQ£bE-Oa

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Om O= O| Ot Oe Oe Oe O, O, Oo 0,, O, O_ O, O, O, o,

-,SbSOOOOOEoO_ Oo Oo _m Oi O_ Oe Oe O, Oo ue O, O, 0,, O, O, O, Oo O, ,w_18_006-0_

X,ETA

Oo C, Oo Oo Oe Oe Oe om O, O, O, O, O, O, Oo. O, O, ,SbllOO_OE*O0

Y,XJ

om go O, Oe Oe O, uj O, D= 0,, Oo Oo 0,, Oo Oo U= -,I_3L}wBTE*OI

Y,ETA

-,_}OIWgqIE-03 Ot Oi O_ Oo Oe Oo O, vo (I, O, O. Ot O, O_ O, oo O0 147 REFERENCES

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). 14N

18. Varga, R. L., Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood

Cliffs, New Jersey, (1962).

19. 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 & & g

0 e,3

! I ! 0 0 o I I I 0 0 o p..

0 r_ o _

0 0 =0 t_ 0 0

o r_

,-q _o,-q _-4_ u_

,-4 ,--I _ ocq

_ _o _o _oo _.--t o4 e-i o

_3 _q

,-4

_g ,-,4 151

! I O O _. _,_ "_

O _ _ I oh t_ 0 _ N 0

I ! o O O f-. O ¢',1 _OM o O I--. O _.._

O o O

¢',1 0,.,_

• lJ ,_ 0

• N _ _ m 0

_ ° _

t-,I

•_ O -N "O

L3O

,-4 _NOO _OO

t.,I OC0 _D

o'J

•,'4 O0 0 l_00000000 0 0 _ ...... 0 .00 t_,l ,--I > ,--I O O _'_ O O O O

,-I o O 0

t,h I-i ,4 ,.-I 152

TABLE 2

Effect of Initial Guess on Convergence

IGES :

Configuration 0 i 2 3 -i -4 -i0 -20 -40 -i00

NC NC -- 69 71 (Fig. 4a) 68 NC 80 NC NC

i00+ 70 63 67 68 (Fig. 5) 75 NC 64 NC i00+

NC NC NC NC NC (Fig. 6) NC NC 1 NC NC NC

NC NC NC NC NC (Fig. 7) NC NC NC NC 2 NC

i00+ 87 84 86 (Fig. 8) i00+ i00+ i00+ NC NC i00+

(Fig. 9) i00+ I00+ i00+ NC NC i00+ NC NC NC NC

(Fig. i0) 99 89 99 NC NC i00+ 91 84 82 86

i00+ i00 93 91 94 (Fig. ii) NC i00+ i00+ NC NC

(Fig. 12) NC NC NC NC NC i00+ i00+ i00+ I00+ i00+

(Fig. 13) NC NC I00+ NC NC i00+ i00+ i00+ i00+ 99

Legend: 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.

Notes: 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 153 Input Parameters for Basic Single-Body Field Used in OptimumAcceleration Parameter Studies Case 1

IMAX 67 JMAX 40 NBDY 1 IGES 0 NBSEG 2 NRSEG 1

LBSID 1 3

LBI 1 1

LB2 67 67

LBDY 1 0 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 66 154

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 TABLE4 155

Input Parameters Varied for Single-Body Field Used in OptimumAcceleration Parameter Studies

Case IMAX JMAX LB2 LR2 LI2 YINFIN NINF ALN IFAC

2 I00.0

3 i0000.0 4

37 4 37 37 36

97 5 97 97 96

6 20 20 20

7 60 60 60

8 5.0

9 20.0

37 i0 37 37 36 I00.0

37 ii 37 37 36 i0000.0

97 12 97 97 96 i00.0

97 13 97 97 96 i0000.0

14 20 20 20 i00.0

15 20 20 20 i0000.0

16 60 60 60 i00.0

17 60 60 60 i0000.0 4

18 5.0 i00.0

19 5.0 i0000.0 4

20 20.0 i00.0

21 20.0 i0000.0 4 156 TABLE4 Continued

Case IMAX JMAX LB2 LR2 LI2 YINFIN NINF ALN IFAC 37 22 37 20 37 20 20 36 97 23 97 20 97 20 20 96 37 24 37 60 37 60 6O 36 97 25 97 60 97 60 60 96 37 26 37 37 5.0 36 97 27 97 97 5.0 96 37 28 37 37 20.0 36 97 29 97 97 20.0 96 3O 20 20 20 5.0 31 60 60 60 5.0 32 20 20 20 20.0 33 60 60 60 20.0

Case NLN ALN DLN IFAC EFAC

34 i0000.0 i 35 i0000.0 2 36 i0000.0 3 37 i0000.0 5 38 i0000.0 4 i000.0 39 i0000.0 4 i0.0 157

TABLE4 Continued

Case NLN ALN DLN IFAC EFAC

40 2 i000.0, i000.0 1.0, 1.0 41 3 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 Field Used 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 0 R(i0) 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 1.0 ATYP 0 ITYP 0 NLN 0 NPT 0

DEC 0.0 AMPFAC 0.0 I FAC 3 EFAC i00.0 160

TABLE6

Input Parameters Varied for Double-Body Field Used in Optimum Acceleration Parameter Studies

Case IMAX JMAX LBI LB2 LRI LR2 LII LI2 YINFIN NINF ALN IFAC 19 29 65 75 45 60 1 60 i 60 19 29 65 75 44 20 1 20 1 20 47 i0000.0 6

46 i00.0 1 49 20.0 48 5.0 i 33 33 43 107 117 43 149 117 149 i 40 i 40 148 43 107 i 149

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

r/l

O O_ OO r-- oool_ r-- oo ._r l> oo CO _ OO O_ o_ O', O_ _-..I i-i o'_ C_

"_ Z ¢1 _ 4-1

u_ r-. O o'1 r-- O oo _.. ,._ ,-I r-.. oo oo _O o-, oo oo O'_ oo oo oo o_ oo oo O" ,-I ,-_ ,-I ,-I ,-4 ,-I

,-4 _0

O ffl

O O ,H _._ oo-..i-I'--. O -._ '.O •..T _,D ,.o r-. oh O_ I'-. cxl oo i-.. ,.._ O r--. oo .H ,-I ,..-I ,-_ •,_ Z

> 4; 0

.,--I _J z 0

•el 4- r_ O r-- -._- oo Lth ..-r _ O -.T -._" P-. oo oo oo oo oo oo oo oo oo oo oo ,4,4,4

.r..I "H O

O •_ m m I_ "_ o o :n 4-;

_ _ .-_._ •r..I _ _ _) _-_ O_t:D

O O O _1 _ O _ _ .;.-I

r/)

(D 1-1 4-1 _ v O z v_ II II II O O O Lth ,_ ¢',,I U OOO u_ k_ O c'q _T ',.O O nnl "_ nn+ O II U II O ,-W ._ II II II .U II II II _.J _.J 4J tY 0 _ _ _ o o_ ,-_ ,-_ ,_ OJ •,_-r_ ._ _ _._ _J

_.1 O

O0 _-_ 0"_ +.--I 0 z 162

TABLE 8

Optimum Acceleration Parameters for Single-Body Field : Variation of Pairs of Quantities

(12) IMAX Up (5) IMAX Up (13) IMAX Up Amplitude Down Amplitude Up 1.89 : 122 1.88 : 137 1.86 : 183 (1.90 : >122) (1.89 : >183)

(2) Amplitude Down (i) Basic (3) Amplitude Up 1.85 : 81 1.84 : 74 1.86 : 114

(i0) IMAX Down (4) IMAX Down (ii) IMAX Down Amplitude Down Amplitude Up 1.79 : 63 1.77 : 68 1.83 : 86 (1.85 : 92) (1.84 : 83)

(16) JMAX Up (7) JMAX Up (17) JMAX Up Amplitude Down Amplitude Up 1.86 : 129 1.85 : 126 1.87 : 147 (1.90 : DIV) (1.90 : DIV)

(2) Amplitude Down (I) Basic (3) Amplitude Up 1.85 : 81 1.84 : 74 1.86 : 114

(14) JMAX Down (6) JMAX Down (15) JMAX Down Amplitude Down Amplitude Up 1.84 : 89 i .81 : 90 1.81 : 139 (1.84 : 94) (1.81 : >139)

(20) Radius Up (9) Radius Up (21) Radius Up Amplitude Down Amplitude Up 1.89 : 131 1.84 : 205 1.81 : 243 (1.88 : >131) (1.88 : DIV)

(2) Amplitude Down (i) Basic (3) Amplitude Up 1.85 : 81 1.84 : 74 1.86 : 114

(18) Radius Down (8) Radius Down (19) Radius Down Amplitude Down Amplitude Up 1.80 : 185 1.80 : 184 1.81 : 176 (1.91 : DIV) (1.91 : DIV) 163

TABLE8 Continued

(24) JMAXUp (7) JMAXUp (25) JMAXUp IMAXDown I_IAXUp 1.83 : 77 1.85 : 126 1.89 : 128 (1.89 : >i00) (1.91 : DIV)

(4) IMAXDown (i) Basic (5) IMAXUp 1.77 : 68 i. 84 : 74 1.88 : 137

(22) JMAXDown (6) JMAXDown (23) JMAXDown IMAXDown IMAXUp 1.71 : 48 1.81 : 90 1.84 : 142 (1.76 : 62) (1.87 : >142)

(28) Radius Up (9) Radius Up (29) Radius Up IMAXDown IMAXUp 1.81 : 77 1.84 : 205 1.87 : 219 (1.84 : 92) (1.90 : >219)

(4) IMAXDown (i) Basic (5) IMAXUp 1.77 : 68 1.84 : 74 1.88 : 137

(26) Radius Down (8) Radius Down (27) Radius Down IMAXDown IMAXUp 1.78 : 102 1.80 : 184 1.82 : 285 (1.84 : >102) (1.93 : DIV)

(32) Radius Up (9) Radius Up (33) Radius Up JMAXDown JMAXUp 1.82 : 117 1.84 : 205 1.89 : 103 (1.84 : >117) (1.90 : >103)

(6) JMAXDown (i) Basic (7) JMAXUp 1.81 : 90 1.84 : 74 1.85 : 126

(30) Radius Down (8) Radius Down (31) Radius Down JMAXDown JMAXUp 1.80 : 176 1.80 : 184 1.58 : 218 (1.91 : DIV) (1.91 : DIV) 164

TABLE8 CoNtinued

Format: (Case Number)Case OptimumAcceleration Parameter: Numberof Iterations (Average Variable Acceleration Parameter: Numberof Iterations) See Tables 3-4 for values of the quantities varied. 165

TABLE9

OptimumAcceleration Parameter for Single-Body Field : Miscellaneous Variations

Experimental Optimum Optimum Number Case Acceleration of Number Case Parameter Iterations

Effect of Numberof Steps in Addition of Inhomogeneous Term (Attraction Up) 34 One Step 1.35 561 35 Two Step 1.61 317 36 Three Step 1.85 119 3 Four Step 1.86 114 37 Five Step 1.85 125

Effect of Intermediate ConvergenceCriterion (Attraction Up) 38 Convergence : -01 1.86 i01 3 Convergence : -02 1.86 114 39 Convergence : -03 1.85 167

Effect of Numberof Attraction Lines (Basic) i i One Line 1.84 74 4O Two Lines 1.83 76 41 Three Lines 1.83 79

I Effect of Convergence Criterion (Basic)

Convergence : -03 1.84 49 Convergence : -04 1.84 74 Convergence : -05 1.84 117 Convergence : -06 1.89 175 166

0 O_ .-T c'_ O', oO u_ Ch c_ oO oO _D ,.Q u_ .I-I oo O O oo c,'_ c-q ,--Ieq I_ 0 _ ,.4 c_ eq ,-.I c..I

.-4 Z ,LI H 0

(U 0 -r-I

-,1" _O o0 -.T u% ,-4 -.I"_D i oO oo r-... 00 O0 _ oo u'3 _._ cO oo

,--t r-t ,--t O a ,.-I ,-4 I o_ O <

0 .r_ O u_ m < tO m .,-4 O

(n 4-J

.r4 O 0 c_ _:_, 0 _U .N -r_ _D In O ° O •r-I -_ .,-4 c.D _ O0 ,-.-t Q) 4-1 ,--_ ,--t ,--t 4-J 4-1 < Z "_ II II II O _ c-4 E m m ,4-i O __1 4.1 0 O "_D "_D '_ O II U II J.J .el •r't ._ II II II 4J .Ira 4-_ 4-1 "O 0 0 (0 -r4 -,q .,-4 O O (D _-I ,--4,-4 0 q_ "_ "tJ "_ r_ u_ u,-4 q_

_.1 o'3 -

Z 167

O CD ._

_ O _ _. u'_ _- _

Z 4.J 4.J

O O

O 0'_ o'_ O_ o_

4.J

(D O J_J

0 0000 I I I I

0 r_

.r..I r/l

_.) r,..) 0 0 g-

4_1 0

> > 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

0 _ 0 _"-t 0 .I-I 0 ,-I _._

-,-i _o-_ _o

"O,-4 _-H _ "_ 0 -H O O_-_ 0 _ _ O _

•N _ _ O •,_ _ _:_ O ,-4-,-4 _

0 _ .H ,-4 U.4 _-_,_

. O _ oo_

0 ,-4 0 O ,-40 °

0 ",-4 _ O',_ _ 0 O O m O 0 •,'4 I_ 0 O _ O 4_ O O _ _ ¢J O 4-_ O O _O _ O 0 0 _0 _ _,--I •_ 0 •I-_ 0 '4-1

O "_ ¢xl _ 0 .,-I

•,-I "_ I"-,. I--I _ r--

4..1 _ o •_ O

_ _._ _ _ ,-_ _ _

._ 0 og 0 _ 0 -I_ ._ _ u_ _ O O,_U_ t_ 0

,-4

,-4 o4

(D (D

m 170

Physical Plane

n i"2

q=q2

I ' I

Region D* F 3

I I il _=N 1 rl

Transformed Plane

Figure I. Field Transformation - Single Body 171

Physical Plane

N=D2

I ' I I

F Region D* 4 F3

J _=n I F7 F 5 FI F 6 F 8

Transformed Plane

Figure 2. Field Transformation - Multiple Bodies 172

j=JMAX ] ..... i _ ..... T

] (

,"""_I I ,," I I I I I ,,[III _,_'

j=l L

i=l i=IMAX I

a) Single Body

j =JMAX " " 7 ...... _ "" ¶ ...... T

.... , , ," , ...... , ,• J

.... r III,,II I J, j=l i I I i=l I1 I2 I3 I

b) Two Body

Figure 3. Computational Grids - Single and Two Body Regions 173

1 37

Figure 4a. Single-Body Configuration 174

1 g I

o -1-t .,_ i

_0 .1-1

= o

o I _J

4 ,I

4-1

o o .,-I o 175

2 16 _ ! s_ 86 2 81

Figure 5. Double-Body Segment Configuration #i 176

_L 31 2L 31

Figure 6. Double-Body Segment Configuration #2 177

11 2 _I

,oo

11 i _I

Figure 7. Double-Body Segment Configuration #3 178

16 16

...... i

! !

\ / \

Figure 8. Double-Body Segment Configuration #4 179

_0 3!

11l 11 I

L 1 1 31

/ /

Figure 9. Double-Body Segment Configuration #5 180

2_ 51 26

2 2

i 71

2 l',il

k \ 281

01 i

Figure ii. Double-Body Segment Configuration #7 182

Figure 12. Double-Body Segment Configuration #8 183

16 76 2 _6 86 !Ol

Figure 13. Double-Body Segment Configuration #9 184

, ++ _" - I,

. _ _----,.. -- z _

-4O

" 7-_- I-

Figure 14. Initial Guess Types - Single Body 185

_'liI//II//M, I ! ! i ! _t//, ili_,/iI,,'//_/(,A it!,i,O7,'//, _////." ;& <./.K7 _',/ /// 4.'/.."_

Figure 15. Initial Guess Types - Double-Body Segment Configuration #i 186

\ / / /

'\_/ \ \ ....,-/" // " X ,\ \, / <,: '" 1 \ X

-4

Figure 16. Initial Guess Types - Double-Body Segment Configuration #2 187

\ ,,, / /

' // i/

, ', ,, ,, \¢i ¢'" . ,...... ', "_" 4

x - /

"" i' / ,, t , \ -4 " \d / ,_/ i \ j,/

Figure 17. Initial Guess Types - Double-Body Segment Configuration #3 188

7, /

/ , 2

1 i t-..

.J i!.-///'// . -2

./ 3

Figure 18. Initial Guess Types - Double-Body Segment Configuration #4 189

i _*t_ / '

-4O

Figure 19. Initial Guess Types - Double-Body Segment Configuration #5 190

-40 i"" _"

\. A,

.%/]

Figure 20. Initial Guess Types - Double-Body Segment Configuration #6 191

J .f

\_///Y'

Figure 21. Initial Guess Types - Double-Body Segment Configuration #7 192

/ , .,/x\

.-/

• / /

j_'TC-

," // /

,, / /' _x

_k J

_. y ,t

\\,_ \

/-"

Figure 22. Initial Guess Types - Double-Body Segment Configuration #8 193 o< i

/ 196

500 50_ 0 BASIC, ONE LINE GTTRRCTION _SD'_ _50 t° _,00 _.-

:_50 L

___5o_ = 2soL. .oL ._ ZOO F

150 I 100_- fOOL I

50._-

0. io"TI_uM" _'" l 50" _OPTI_UM . 1.8t 1 I i 1 I I O. I I I I t I I 1 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1,90 1.65 2.00 1.50 1.55 1.60 1.65 1.70 1.75 1,80 1.85 I.gD 1.95 2.00 _CCELEflRT|ON PRRRMETER ACCELERATION PARAMETER

(a) (b) I00 IMRI DOWN, JM_I DOWN 90. l° / O0o m ! 160 r 70.," iool_ok I

_ 5o.__ £ c I ioo_ i i _ _o. }-. = eo.b i 30. _ 80. m i t 2O.!_ 1 :o.__ i 20. _ 3PTIMUM - 1.71 O. [ I ' ' Io-I_O,- _.- 1.50 t.155 1.160 1.165 1.70 1.75 _ .'80 1,185 1.190 1,19E ;c. O0 O. I I I I I l I RCCELERRT ION Pq_RMETER 1.50 i.55 1.60 1.65 1,70 1.75 1.80 1.85 1.90 1.95 2.00 ACCELERATION PARAMETER

_50 t

_O0_- I s5oL :I°"°!.o?\ I ;4_sooL i::i = 2ooL = 2oo_ I50L

50.:GO t ,o.L i I OPTIMUM - 1.80 O. I 1 I I I I 1 I I O, _ I 1 I t 1 I I !.50 1.55 1.60 1.65 1.70 1.75 1.80 1,85 1.90 1._5 2,]0 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 ?.00 ACCELERATION PARAMETER _CCELERRT ION PARAMETER (e) (f)

Figure 26. Effect of Acceleration Parameter on Convergence Rate 197

(e) (f)

Figure 27. Examples of Coordinate System Control

CSee Table ii for control parameters.) 198

, , P?_/_./,:-" . ..,, /..J_/_.X3J_ L !,'_, ,_v' _ ..... ' -_,

"-'_'I., <"X.

(a)

__L_._., _,.,_. , _,._ , , ¢ - ,,? ,,:',,,_ ,_ ,,

• _,_, ,_ ,I, ,

(b)

Figure 28. Example of Attraction into Concave Region 199

Figure 29. Coordinate Systems- Circular Cylinder And Joukowski Airfoil 200

(a)

(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

o u_ .H _f J

o) 4J

, .÷f_ o o l; 4-1 4-1 1- o _) 4 7-j-/ 1 7.7 _) l °H / / / ,/ 203

c,I

,--I / ,.Q "\ 0 0

0 u,-i M °N /

"'4 /

4..i

g.._ I

4.1

°_..t t-"" :7::7 0 /--i 0 1.1

1 o r_ 204

i /

,.Q

[._

0 J

0 0 i <

0 0

4J 0

._J

\ 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 207 ! ! ! i i i i i i I d

i 21

Figure 37. Coordinate System - Key-seat Shaft, Simply-Connected Region (See Table iZ) 2O8

0 o

4J m m

4.1

.N I-i o o

4_1

4-)

t-¢3

.1-4

\ 2O9

LIFT COEFFICIENT I 2. 535Ii9t_

ORRG C3EFFIC!ENT - 0.0039¼6

L:=T C_;EFFICIENT EBR_ I --0.00S_0_

I. 00 qa

-°-°° r

b r_ \ :0

-2.00 L I

I _ Numert¢il Solu¢lo

-3.00 II -- Analytic Solution I -I,O0 -0.00 I.OO X

Figure 39. 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 ! -- NUMERICAL I 0 : : ; ; d re TEST DATA [2L] ' 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.....i i...... 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.

-0.0_

-$._7 b -t.O_

e. "|$'_ -II.I -:S._ -$._ -ll.I, -LT.q -10. ill

-IO.I -q. O(

-ZI.I -H.I -il.q "M.I °$U. -K.I "N.I

-N.I I ; ._1. Od I I g.M O.fHD I.N -I.M -LID O.N ql,. IlI II me

Figure 41. Potential Flow Streamlines and Pressure Distributions - Double Airfoil 212

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

(b) I-! NUMERICAL - ANALYTIC

-5.00

-0.00 X Figure 43. 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_.

-16.9-15.9_

,,,L-0,50 _oo 0.50 %

Figure 44. Potential Flow Pressure Distribution for Double Cylinder without Coordinate Attraction 215

Figure 45. Alternate Segment Arrangement - Double Cylinders No coordinate attraction. 216

-o,ooI

-1.oo I

-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 83 219

t = 0.118 C, =0._15916

CD=O.c-_a),u CDp=O- CO ='-'l.l]7Ei31S O

I ] i

! ...... :

Z ? (-) i _ ! _ i .i C. iiV______------C_

cr_

i 0 l

r I

"i _N'j. O0 0.25 O. 50 O, 75 1.00

X/C - DIMENSIONLESS CHORD

t = 3.33 CL=0"83051i

CD:0.258096 CDp=0.090_55 CDF:O.1677u, I (::)

"T'l _ !

W I • t _ t t_ !If- ! - , ,...... E

O_ I

L_ - i I

i I '

_ioo o125 olso o'• ?5 .00

X/C - DIMENSIONLESS CHORD

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 i 0.50 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

B_OY z 2,00 1.00 F L I

) i

;.00 _-

_. -o.ooL \ ,._-0.00_ \

-,.ooL i

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 1.00

-O,OC t

-0"00 1

-2.00 i

-3.00 I -I.O¢ L 1 1 A I 0.60 0.80 1.00 -0.60 -0,_0 -0.20 -0._ 0.20 0._0 X x

Figure 52c. Pressure Distribution and Velocity Vectors - Double Airfoil, R = i000, t = 3.02 225

Fig. 53. Comparisonof Numerical and Analytic Solution for Deflection Contours 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 transformed [_,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. i and j are the conventional cartesian

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

A-I A-2

Derivative Transformations

(A.1) fx _ (_f)_x y,t = (Yqf_ - Y_fq)/J

(A. 2) fy _ (_f)_y x,t = (x_fq - xqf_)/J

_f = _f 1 _x ft -- (_-)x,y (_t)_,q - 7 (Yqf_ - y_fq) (-_-)_, q

(A.3) - _1 (x_fq - xqf_) (3_-_t)_, q

fxx E (_x2)Y,t._2f. = (yq2f_ - 2ysyqfsq + y_2fqn) /j2

+ [(yq2y_ _ 2ysynysq + y 2yqq)(xqf_ _ x_fq)

+ (yq2x_ - 2y_yqx_q + y_2Xnq)(y_f n - yqf_)]/j3 (A.4)

(_2f) = - 2x_xqf_q + Ij2 fYY _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

(A.18) V2f = (af_ - 28f_n + Yfnn + °fn + Tf_ )/J2

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 - x_(F I) n + xn(Fl)_]/J (A.21) A-5

Unit Tangent and Unlt Normal Vectors in the _,q Plane

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.

_=K2 _=K3

_=K_ __ n=C3

n(n) n c 2 t(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

F n (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)

F t ($) = t (_) • F = - (xnF 1 + YuF2)/_a (A.29)

Directional Derivatives

_n(N)_--f = n~ (n) • Vf- = .(Tfa - 8f_)/J_ __ (A.30)

af (n> _--_(n) = .t • Vf = f¢l_Y (A.31)

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 (A.34) D D*

Vector Function:

Consider a vector integral in the physical plane of the form

I = / f(x,y) n(x,y) as (A.35)

- S ~ where S is the closed cylindrical surface of unit depth whose perimeter is specified by the contour F 1 in the physical plane

(see Figure A.2) and whose outward unit normal at any point is given by n(x,y). A-8

n(x,y)

r I

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 (A.36)

r I

We now wish to transform (A.36) to the (_,n) plane. 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 (A.38) -/ f([,n I) (x_J - y_i)d_ _mln 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 = n I. 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 B 2 - 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

A*f_ + B*fsn + C*fnn + E*f_ + F*f n + G* = 0 (B.2)

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

= (B 2 - 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 Sdxdy R 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 F 1 - xnF2)dn - (x_F 2 - Y_Fl)d_] = f SJd_dn (c.4) C* R*

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) = (x_F 2 _ Y_Fl)/C_y

F • n(_) = (y F 1 - 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:

l i i

(D.la) (f_)t,J = (f_)t,J/2 i

where

t¸ (D. Ib) (f_)i,j fi+l,j - fi-l,J

(D.2a) (fn)i,J = (f'n)l,j 12

i where

(D.2b) (f'n)i,J - fl,J+l - fl,J-i

First derlvatlve, forward differences:

(D.3) - 3fi,j)/2 (f_)i,j = (-fi+2,j + 4fi+l,j ! I | i

(D.4) (fn)i,J -".(-fi,j+2 + 4fi,j+l - 3fi,j)/2

i Second derivative, central differences:

(D.5) (f_$)i,J = fi+l,J - 2fi,J + fi-l,j -

(D.6) (fna)i,j -_ fi,J+l - 2fi,J + fi,J-i - (f'na)i,J

(D.7a) (f_n)i,j = (f_n)i,j/4

where i 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' i, j/4 (D. 8a)

where

(D. 8b) =_,j - (x')2n 1,3_ + (y_)2i,J

! Bi, j = Bi,j/4 (D. 9a)

where

(D.9b)

Yi,J = YI,j/4 (D.10a)

where

(D.10b) _'i,J = (x_)'_ ,J + (y_)2i,J

= ' /4 Ji,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'i,J - [(Y_)i,j(Dx' )i,.J - (x_) i ,J (Dy') i,j ]/J_,j (D.12b) D-4

= ' /2 (D.iBa)

where

T_,j E [(x')n i,J_(Dy')i,j - (Yn') i,j(Dx')i,j]/Ji,j' (D.13b)

and where

! (D.14a) (Dx')i, j - 2B_,j (x_11)i,j + Y_,j (xnn)i,j

(DY')i,j m'i,J (y_) i,J' - 2B_ ,j (y_q)' i,j + Yi,j(Yqq)i,j' ' (D.14b) 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

Figure E.2. Contour of _(_i,_j)

Numerically, the import of the above discussion is that interpolation between the points of the discrete set D** is required to determine 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

(m,n+l) (m+l n+l) J=J2

m,n qm,n2 I j=Jl i 1

l=Ii i=12 (m,n) _m,n (m+l,n) a) b)

Figure E.3. 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 7 I i I i ! P3, I i

m, rl t _4 m,n i m,n !

_m,n _ P_' m,n m,n

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)_,

= A£

(nm,n)A,p

where

A1 _- 1 0 A2 0 1 [0 1 I-- rl-I 0 1 L

A3 -- [-'o:I' =1 o-'I -- - -- 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 n k. Denoting these by _i' Ci+l and qj, _j+l as shown

in the figure, the values of x k 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