•» •«'«»"•»liwminpfpwumBiw" Ml AD-A016 9;4 SMITE - A SECOND ORDER EULERIAN CODE FOR HYDRODYNAMIC AND ELASTIC-PLASTIC PROBLEMS Samuel Z. Burstein, et al Mathematical Applications Group, Incorporated Prepared for: Ballistic Research Laboratories August 1975 DISTRIBI ,T ED BY: mi] National Technical Information Service U. S. DEPARTMENT OF COMMERCE
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AD-A016 9;4
SMITE - A SECOND ORDER EULERIAN CODE FOR HYDRODYNAMIC AND ELASTIC-PLASTIC PROBLEMS
Samuel Z. Burstein, et al
Mathematical Applications Group, Incorporated
Prepared for:
Ballistic Research Laboratories
August 1975
DISTRIBI,TED BY:
mi] National Technical Information Service U. S. DEPARTMENT OF COMMERCE
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CONTRACT REPCRi NO. 255
SMITE - A SECOND ORDER EULERIAN CODE FOR
HYDRODYNAMIC AND ELASTIC-PLASTIC PROBLEMS
Prepared by
Mathematical Applications Group, Inc. 3 Westchester Plaza Elmsford, New York 10523
D D C ^Uil^
August 1975
Approved for public rcleise; distribution unlimited.
Copy ovc;;;--. .. . uoc-s noi gensit fully logibb reproduction
USA BALLISTIC RESEARCH LABORATORIES ABERDEEN PROVING GROUND, MARYLAND
B«p(jdu<«d by
NATIONAL TECHNICAL INFORMATION SERVICE
US D«pirtment ol CommMcm Spnnotnld, VA 23IS1
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Destroy this report when it is no longer needed. Do not return it to the originator.
Secondary distribution of this report by originating or sponsoring activity is prohibited.
Additional copies of this report may be obtained from the National Technical Information Service, II.S, Department of Commerce, Springfield, Virginia <;21S1.
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The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents.
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UNCLASSIFIED SECURITY CLMtlFICATION Of THIt PAOI Clim Dmm taMn>4
1 REPORT DOCUMENTATION PAGE READ mSTRUCTIONS BEPORE COMPLETINO FORM
11. RCPOAT NUMBCR
BRL CONTRACT REPORT NO. 255 i. OOVT ACCCUMM NO >. RECIPIENT'S CATALOG NUMBER
1«. TITLE f«n«Ju»l(II.)
ISMITE A Second Order Eulerian Code for lllydrodynaraic and Elastic-Plastic Problems
S TYPE dt REPORT t PERIOD COVERED
Final ». PERFORMING ORQ. REPORT NUMBER
h. AUTHOR^
jSamuel Z. Burstein lllarold S. Schechter IF.. L. Türkei
t. CONTRACT OR GRANT NUMBERCaj
DAAD05-:j-C-0512
1» PERFORMINO ORGANIZATION NAME AND AODRcis jMathematical Applications Group, Inc. 13 Westchester Plaza IFlmsford, New York 10523
10. PROGRAM ELEMENT, PROJECT, TASK AREA t WORK UNIT NUMBERS
| Approved for public release: distribution unlimited.
1 17. DISTRIBUTION STATEMENT (ol Ihm •»•Iracl mltrmd In Block 10, II dllltrmnl Iron Ktporl)
I II. SUPPLEMENTARY NOTES 1
1 19. KEY WORDS (Contlnum on rmvmtao old* 11 nacaaaary and Idmnttly by block numbarj
iBallistic Impact Hydrodynamic | IPenetration Elastic-Plastic Code 1 iFinite Difference 1 Second Order Numerical Methods 1
JHulerian Code j 1 20. ABSTRACT (Contlnum an rmvmmo «Ida If nacaaaarr and Idontllf br block number)
A computer code for hydrodynamic and elastic-perfectly plastic problems is 1 described. The model uses a second order finite difference method based upon 1 a Eulerian formulation of the basic differential equations. The algorithm j utilizes an individual grid for each material domain, determined by j Lagrangian type boundary points which are subject to free surface and inter- 1 face conditions. The only communication between domains is through their j common interfaces. |
MFORM I JAN 71 1473 EtMTIOM OF I NOV »t IS OBSOLETE UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PACE f»t>an Data EmaradJ
This manual describes the SMITE (Second order Moving Interface Two dimensional Eulerlan) code for hydrodynamic and elastlc-perfectly plastic problems. The code is based on an I ilerlan formulation for the numerical model. Eulerlan methods are characterized by a mesh which la fixed in space for all times. The materials are allowed to move freely through this grid. In the SMITE code, each material has its own Independent grid. Thus, the mesh spacing and number of mesh grids in one material is in no way affected by that in another material. The equations are also solved in a trans- formed plane. This allows for transformations to be used which concentrate the mesh points in regions of greatest Interest.
The extent of each domain is determined by particles or material points which define the domain boundary. These points are moved in a Lagrangian sense by integrating the ordinary differential equations relating their positions and velocities. The values at the boundary points are subject to free surface and interface conditions. The interface conditions provide the only communication butweon the various material domains.
The model upon which the SHITE code is based is fully described in the report "A Second Order Numerical Model for High Velocity Impact Phenomena" by the same authors. All references in this manual are to sections of the above report.
Preceding page blank
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II. SUBRDOTiy; LOGIC
2.1 Main Program
The main program controls th« overall logical flow of the code. INITAL is called to read input and initialise the run. All tine increments are added to the starting time to determine when the processes dependent on these Increments will next occur. The cycle count is incremented to start the main loop.. The time step is set to the minimum of the CFL time step determined for all materials. If this time step is lass than the minimum allowed, a message is printed and the run aborted. The time is incremented and MVBND called to move all material boundaries. ADJINT is then called to determine the new inter- face locations and adjust the boundary positions accordingly. Initial boundary values for each material are extrapolated fron the interior by calling BOtJDRY DENSB Is then called to satisfy boundary conditions- The routine now loops through the material. GENVAL sets the proper material dependent mesh and array bound values. FDIFF Is called next to solve the finite difference equations. FINISH is then called to determine if any shifting of the domain or rezoning of its boundary is necessary. The positions where the boundaries cross the mesh lines are determined by calling BPOSN and BVALU is called to calculate the crossing coordinates and the variable values at these crossing points. Values are then interpolated at all interior mesh points for which the finite difference equations could not be used by calling INTRPL. This ends the material loop. The error fla<j is checked and If it is set the run is aborted SECOND is called to obtain the elapsed running time. If the running time or problem time are greater than their maximum values, the run is ended. The problem time is compared to the time at which the next printer plot is desired. If this time has been exceeded PRNPLT is called and the printer plot increment is added to the printer plot time. The same is done with PLTOÜT for plotter output, OUTPUT for printed output and SAVE for restart output. The next time cycle Is then started. At the end of a run all desired output is generated regardless of the respective time increments,
2.2 ADJINT
Subroutine ADJINT determines the locations of material interfaces and adjusts boundary positions accordingly. In its present state, it is a rather ad hoc routine which makes strong assumptions regarding material orientation and relative position. If a particular problem does not meat these conditions, ADJINT will have to be modified or rewritten for that problem. It is assumed that all interfaces exist at the start of the problem. If the boundaries of the two materials penetrate each other, it is assumed that tho first material (material A) that appears in the interface specification is the predominant material. The interface points of the second material (material B) will be re- placed by those of material A. The task of ADJINT is to determine the exact extent of the interface. All boundaries consist of three segments which are divided by the boundary points into linear sub-segments. It is assumed that the interface starts with the first point of material A and the last point of material B and hence that material A is on the right of material B. In order to find the end of the interface, the logic is to find a point on each boundary not on the interface and then trace the boundaries toward the interface until they first intersect. It is further assumed that the interface lies entirely within the third segment of material B. The maximum z coordinate of material B
Is found and then the last point of material A whose z .-oordlnate Is less than this maximum value. The next point on material A Is the endpolnt of the last material A sub-segment that can Intersect the aoundary of material B so that no additional points on the material A boundary are considered. The maximum r coordinate of the valid material A points in determined and then the first material B point less than this value. This point defines the endpolnt of the first material B sub-segment than can Intersect the boundary of material A. A box is drawn around the material B sub-segment and the first and last material A sub-segments that enter this box are de- termined. Only those sub-segments of material A between these end segments can possibly intersect the sub-segment of material B. Starting with the last material A sub-segment, each sub-segment is tested for an intersection. If an intersection is found then the lower sub-segment endpolnts define the end of the respective Interface segments. If an intersection is not found, a box is drawn around the next material B sub-segment and the search continues. Once an intersection is found, the material B interface segment is replaced by the material A interface segment. Since this replaceaient may add to the total number of material B boundary points a test is made to see if the number of points on the material B boundary exceeds the maximum allowable. If so, a message is printed and a flag is set to Abort the run at the end of the cycle.
If material B represents material domain 3 special logic is provided., In this case, the interface segments as defined upon entering ADJINT are used. The points of the material B segment are simply replaced by the points of the material A segment. No assumptions are made about the segment location or boundary orientations,
2.3 BONDRY
This subroutine obtains values for all variables on the boundaries of the material domains. The boundary point coordinates, which upon entering BONDRY are in z-r space, are first transformed to a-H space. The arc length along the boundary at each point is then computed. Boundary values are extrapolated from the interior by choosing the closest interior mesh point to a boundary point and assigning the dependent variable values at this mesh point to the boundary point (5.a). The four mesh points surrounding a boundary point are examined» If the closest mesh point is interior to the domain it is chosen; if it is not interior, the next closest is selected, etc. If none of the four points are interior to the domain, the boundary point retains the values associated with it from the previous time step. These extrapolated values are then smoothed to prevent a step function appearance. The smoothing formula used is
b. =a. + -r r (a.^, -2a. +a. ,) i i 4 i+l i i-l
where r is the ratio of the time step used to the time step calculated for the material. This ratio prevents the smoothing operation from propagating signals through a material at a rate greater than the CPL stability condition would allow.
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2.4 E?OSN
The positions where the boundaries cross the mesh lines are determined by subroutine BPOSN. Two similar arrays are dsflned, one for the mesh lines in each coordinate direction. Thase arrays are computed by systematically following the boundary. The logic for both sets Is identical, so that only the computation of the crossings of the lines 2=constant will be described . Let Int(a) be the largest integer less than or equal to a and Az be the mesh spacing. Then i=lnt(2 /&z) implies that the boundary point p with axial
m m coordinate z lies between mesh lines 1 and 1+1. Starting with n=Int(2./Az)
values of i are determined for succeeding boundary points until i is not equal to n. If 1 is less than n, mesh line n has been crossed; if 1 is greater than n, mesh line n+1 has been crossed. Let p ^ be the boundary point where i
m+1 changed value. Then the boundary crossed a mesh line between point m and m+1 The integer m is entered into the array for the mesh line that was crossed and all entries for that mesh line are ordered according to increasing radial coordinate values. A total of |n-l| mesh lines may have benn crossed and an entry is mada for each of them. The Integer n is then reset to n=Int(z .,/Az) and the procedure continues until the entire boundary has been traced
2.5 BVALU
BVALU determines the coordinates of the points where the boundaries cross mesh lines and the values of the dependent variables at these points. The co- ordinates of a crossing are defined as the intersection of a mesh line with the straight line segment between the boundary points on either side of it The dependent variable values are then obtained by interpolating along this line segment.
2.6 DEHSB
Boundary conditions are satisfied by subroutine DENSE INFACE is called to satisfy conditions on all interfaces. All non-interface, i.e free surface, points are then determined. The normal stress at these points is then set to zero (5.b) .
All interfaces are examined and an array is created which specifies, for each material, the first and last points of all boundary segments that are interfaces. These segment endpoints are arranged in ascending order with a zero after the last endpoint to signal the end of the array. Each material is then processed. The boundary coordinates are transformed from a-ß to z-r space and the transformed values are stored in a temporary array. Each free surface segment, if any, will be all those boundary points lying between the high end- point of one interface segment and the low endpoint of the next interface seg- ment The local slope at a point (z.,r.) is obtained from tan ^ = (e -z. )/ (r. ,-r, .), For the normal stress to vanish, we must have 1~
i+l i-l 2 2 p = S sin 'ii + 2 Z sin ij* cos ty + S cos ty
This condition on pressure is satisfied by finding the internal energy and using Newton-Raphson iteration to solve the equation of state for the proper density,. The d:nsit'/ at the free surface boundary point is then set to this value, If the iterition does not converge, a message is printed and a flag is set to abort the run at the end of the cycle
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2.7 FDIFF
The finite difference equations at interior mesh points are evaluated in subroutine FDIFF (4.a). The entire mesh is shifted in memory two radial lines to the right (toward higher index of the axial mesh). It should b« noted that the solution at any mesh point depends only on the mesh lines through that point and on either side of it. Hence, if the solution proceeds along radial mesh lines and the answers are stored back two lines to the left, only room for two additional mesh lines need be added to the memory arrays to allow the old solution to be overwritten by the new solution. The eight nearest neighbors of a mesh point are examined to determine if any lie outside the domain. The neighbors are ordered and if the nth neighbor is missing 2n is added to a code word. The neighbors are ordered counterclockwise with the four nearest neighbors first starting with the right and then the four outer neighbors start- ing with the upper-right. If the code used is non zero, all neighbors are not interior to the domain and ONESTP is called. If all neighbors are present, the two
3 L
7 ' 1 B
step solution may be used.
At each mesh point in the nine point solution lattice, the transformed seven component vector w and its vector functions f, g and h are computed and stored in 3x3 matrices. In both steps of the two step method, the difference equations for the first four components of w are evaluated first and then the difference equations for the three stress components are evaluated. Predicted values at the four midpoints oi the boxes shown in the above figure are obtained. That portion of the second step which depends on values at time t is also ob- tained. The artificial viscosity is included in the partial second step eval- uation.
The predicted values are used to define the f, g and h vectors at the lattice midpoint. The remainder of the second step of the solution that de- pends on the predicted values is then obtained. This solution is transfoimed back to its non-conservative form and stored in the solution array. A test is then made to determine if the stress components satisfy the yield condition, If they do not satisfy the yield criteria the stresses are modified to force them back on to the yield surface (3„b) . Finally, the CFL stability parameter at the mesh point i« evaluated. If the solution appears to be going unstable, ONESTP is called in an attempt to obtain a more stable value. After all mesh points have bet-n evaluated, the L array is shifted back to it original locations The CFL stable stepsize for the material is then computed.
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2.8 FIMISH
This subroutine checks the position rf each domain relative to storage and of the boundary points of a domain relative to each other. As a domain moves through its mesh it may cover sauie mesh points and uncover others. The algorithm requires at leaut one line of exterior mesh points in all dir- ections at the start of a tine step. This ensures the existence of mesh points to allow the domain to expand in any direction. The boundary is checked to see if any point on it has exceeded the mesh. If the mesh 13 ex- ceeded in one direction, the opposite direction is checked. If the opposite direction has more than two exterior mesh lines, the domain is shifted as far as possible in that direction. This is done by shifting nil interior arrays in storage and performing a linear translation on the proper a-6 coordinate., If a shift is not possible, a flag is set to abort the run at the end of the cycle. A message is printed together with the domain limits and the boundary coordinates in a-6 space.
The arc length of each boundary segment is compared with the length of that segment when it first contained its present number of points. If the arc length more than doubled, the number of points is Increased by fifty percent. All boundary points above the segment are shifted up to make room for the additional points. Tracer point and interface endpoint information that refers to the shifted boundary points is also adjusted. If the arcsize has not doubled the segment is also checked for relative distance between adjacent points. If the deviation from the average is too large the points in the segment are redistri- buted. The actual redistribution of points, whether the total number remain the same or has been Increased, is performed by subroutine RELABL,
2.9 GENVM,
Values of various variables that depend on material properties and mesh geometry and which vary from one material domain to the next are sot by sub- routine GENVAL.
2.10 INFACE
The boundary conditions on interfaces are satisfied by subroutine INFACE, These conditions are the continuity of normal velocity and stress (5,0,) „ The local slope at ä point (z.,r.) is obtained from tan ij; = (z. -z. )/(r. -r. ,) ,
The boundary values obtained from the interior on each side of the interface are used to calculate a normal and tangential velocity and a normal stress, A weighted average of the normal velocities and another weighted average of the normal stresses are used to define a common normal velocity and a common normal stress. The common normal velocity together with the previously calcu- lated tangential velocities on both sides of the- interface are used to obtain the adjusted axial and radial velocities. The common normal stress provides a condition for pressure on both sides of the interface. Fixing the internal energy, a Newton-Raphson iteration is used to solve the equation of state for the proper density. If the iteration does not converge a message is printed and a flag is set to abort the run at the end of the cycle.
10
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2.11 1NITAL
All input data Is read In subroutin« INITAL. General run parameters are read first and the interfaces are defined. If the run is a continuation of a previous run, the interface parmeters are overwritten by those on the restart tape. Input is then read for each material domain in turn. The domains are numbered according to the order in which they are input. The bounds of the array segments associated with the material are calculated and these segments are then zeroed out. Non-dinensionalization factor.» are then generated. Length is non-dimensionalized by mesh size so that the mesh size is unity. The scaling factor used for time is set to that for length so that velocity is non-dimensionalized by a cpeed of unity; hence non-dimensional velocity numerically remains the sane as dimensional velocity. The initial material density is used to non-dimensionalizc density« The constants needed in the coordinate transformations are calculated and the input boundary coordin- ates are transformed to a-ß space. The transformed mesh size is unity for both coordinates so that the arc length of a boundary segment is an approximate measure of the number of mesh segments it traverses. Thus setting the number of points in each boundary segment to twice its arc length will result in approximately two boundary points per mesh segment. Values of r and its deri- vatives are calculated for each radial mesh point and mid-point,, RELABT, is called to generate the proper number of equally spaced boundary points from the input values. These points are then set to their initial values» The points where the boundary crossed the mesh lines and values at these points are de- termined by calling BPOSN and BVALU. The interior points are set to their initial values and IPOSN is called to determine which points are interior and which exterior to the domain. A line of reflected points below the axis of symmetry is then set. The CFL stepsize for the material is calculated as is the initial arc length of each boundary segment. If this run is a continuation of a previous run, the initial values are read from the rescart tape. The restart domain is then shifted to the positions required by the first axial and radial mesh point interior to the domain as specified by the input. For a restarted problem thore portions of INITAL that initialize domain values are skipped. After all materials have been input and initialized the time incre- ments for the various types of output are examined. If the increment is nega- tive it is set to a large number so that it will never be reached and its corresponding type of output will never occur. Otherwise the associated output subroutine is called to output the initial data.
2.12 INTRPL
The solution at mesh points that were too near to the domain boundary, that have just become interior to the domain or where the finite difference solution appeared to be going unstable is evaluated by interpolation from neighboring interior and boundary crossing points in subroutine INTRPL. First subroutine IPOSN is called to determine which points are interior to the domain. An axial mesh line below the axis of symmetry is reflected from the mesh line above the axis of symmetry. The mesh is then swept to determine which points remain to be computed by interpolation. Linear interpolation from adjacent interior or boundary crossing points is used. The four nearest neighbors to the mesh point are examined. Each may be a boundary crossing point, an in- terior point that was obtained from the finite difference equations or an in-
11
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terlor point that must be Int-jrpolatad. These three posalbllltles at the four neighbors represent 81 possible configurations. Each configuration Is accounted for and a point on either side '. the mesh point Is selected, If possible, for the Interpolation process. Pol ts that themselves must be Interpolated are not selected since they may not as yet have bean evaluated. After the entire mesh has been swept the points that ware Interpolated are redefined a* normal Interior points. A line of reflected values la again generated below the axis of synnecry to .account for any recently Interpolated values. The 81 possible configurations are represented by the codes In the following table. A 0 represents a boundary crossing point« a 1 represents an Interior point that was obtained from the finite difference equations and a 2 represents an Interior point that must be Interpolated. Reading the numbers from right to left, the digits represent the point below, above, to the left and to the right of the mesh point. The points that were selected ioz use in the interpolation are underlined. In scune cases only one point wa •• used. Here the interpolated values were set directly to the values at that pcin .
1. 0000 22. 0210 42. 1112 62. 2021_
2. 0001 23. 0211 43. 1120 63. 2022
3. 0002 24. 0212 44. 1121 64. 2100
4. 0010 25. 0^20 45. 1122 65, 2101
5. 0011 26. 0221 46. 1200 66. 2102
6. 0012 27. 0222 47. 1201 67. 2110
7. 0020 28. 1000 48. 1202 68. 2111
8. 0021 29. 1001 49. 1210 69. 211 2
9. 0022 30. 1002 50. 1211 70. 2120
10. 0100 31. 1010 51. 1212 71. 2121
11. 0101 32, 1011 52. 1220 72. 2122
12,. 0102 33. 1012 53. 1221 73. 2200
13. 0110 34. 1020 54. 1222 74. 2201
14. 0111 35. 1021 55, 2000 75. 2202
15. 0112 36. 1022 56. 2001 76. 2210
16. 0120 37. 1100 57. 2002 77. 2211
17. 0121 38. 1101 58. 2010 78. 2212
18. 0122 39. 1101 59. 2011 79. ?220
191 0200 ■lO. 111C bL , J012 Pf'. ?221_
20. 0201 41. 1111 61. 2020 81. 2222
21. 0202
12
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2.13 IPOSN
The L array, which spcclfle* whether a mesh point is In a domain or not, is set in subroutine IPOSN. The crossing points of the boundary with the mesh lines are used to determine what is exterior and what is interior to the domain. The mesh is first swept along the lines r - constant. All mesh points along such a line up to the first boundary crossing are outside the domain. All points fron the first crossing to the second crossing are inside. The next set of points are outside, then inside, etc. If a point is outside the domain, L is set to 0. If a mesh point is inside and L has been set to 2 because the point could not be calculated from the finite difference equations, it remains set at 2. Otherwise L is set to 2-L. Thus for all exterior points L Is 0, for all interior points that were calculated fron the finite difference equations L is 1 and for all interior points that have not yet been calculated L is 2. Interpolation errors in obtaining the boundary crossings may cause a mesh point to appear interior when one coordinate direction is considered, yet exterior when the other coordinate direction is considered. The algorithms require a mesh point to be interior no matter which coordinate direction is considered. To allow for this the boundary crossings of the mesh lines z ■ constant are also considered. The crossings are followed as before except that only exterior points need be considered. L is set to 0 for these points so that L will be non zero only for those mesh points which appear interior no matter which way the mesh is swept. For the z « constant mesh crossings the assumption that all mesh points until the first crossing are exterior is valid only if the lines are swept from above the domain toward the axis of symmetry.
2.14 MVBND
In this subroutine the domain boundaries are moved in time by using a simple Euler integration (5.a). The coordinates of the boundary points are trans- formed from o-B space to z-r space and the velocities are integrated with res- pect to time to find the new boundary positions. These new positions are then in the z-r coordinate system.
2.15 ONESTP
The one step finite difference equations for mesh points whose nine point solution stencil does not lie entirely in the domain are evaluated in subroutine ONESTP (4.b). As described in FDIFF, a code word is generated to specify which neighbors of the mesh point are missing. If this word is divided by 16 the result v/ill refer to the four outer neighbors while the remainder refers to the four nearest neighbors. First derivatives and non mixed second derivatives may ba approximated to second order by using the two points on either side of the mesh point in the required direction. These two points will be either mesh points or boundary crossing points. The four outer neighbors are required only for approximating the second mixed partial derivatives. Each of there outer points may be represented by a Taylor's series about the central mes i po'nt. Combinations of these series will yield formulas for the mixed derivative in terms of the outer points and the first and non mixed second derivatives De- pending upon the combinations used, these formulas may be first oz second order accurate. However, first order accuracy in the second space derivatives is all that is required for the overall difference scheme to remain second order accurate. By thus splitting the four outer neighbors from the four nearest neighbors the 256 possible combinations of missing points may be dealt with by first considering the 16 possibilities relating to the four nearest neighbors and then the 16 possibilities relating to the four outer neighbors.
13
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The derivatives are evaluated firat in the axial and then In the radial direction. In either case there are four posaibilitiea to consider. There may bo neah points on both aides, a mesh point on either side and a boundary crossing on the opposite side or boundary crossings on both sides. When there Is a boundary croaeing, the distance to the boundary is calculated. If this distance is less than the allowable diatance the point is skipped to be later evaluated by interpolation in subroutine INTRPL. After the derivatives in each direction are obtained the nixed derivatives are r-valuated. These space derivatives are combined into the expressions appear 1.1 in the one step equations and these equations are computed. The artificial viscosity is then added. A test is made to determine if the stress components satisfy the yield condition. If they do not satisfy the yield criteria the stresses are modified to force them back onto the yield surface (3.b). Finally, the CFL stability parameter at the mesh point is evaluated. If the solution appears -to be going unstable it is discarded to be later evaluated by interpolation in subroutine INTRPL.
The following table will illustrate the code used for missing points.
Code
1. 0000
2. 0001
3. 0010
4, 0011
5. 0100
6. 0101
7. 0110
8. 0111
9. 1000
10. 1001
11, 1010
12. 1011
13. 1100
14. 1101
15. 1110
16. 1111
Date Used for Derivative Approximation
use all points
use lower right and lower left
use lower right and upper right
use lower right and lower left
use uppe~ left and upper right
use lower right and upper left
use lower right and upper right
use lower right
use lower left and upper left
use lower left and upper left
use lower left and upper right
use lower left
use upper left and upper right
use upper left
use upper right
skip and interpolate
A 0 represents the presence of a point while a 1 means it is outside of the do- main. The codes may represent the ordering of the 16 possibilities for both the four nearest neighbors and the four outer neighbors. Reading from right to left the digits represent, for the nearest neighbors, the points to the right, above, left and below. For the four outer neighbors the order is upper right, upper left, lower left and lower right. In representing the outer neighbors the codes are followed by an indication of which of the points are used in eval- uating the mixed derivatives.
14
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2.16 OUTPUT
This routine prepares all printed output. All output is of dimensioned variables and in z-r space. The output proceeds according to material domains. The boundary points appear first. If both 1PRINT and JPRINT are not zero, the Interior mesh points appear next. They are numbered along axial lines with the line nearest to the axis of synmetry first. Only those interior points which are also interior to a box of size IPRINT tines JPRINT are printed. This box starts at the left boundary if IFRINT is positive or ends at the right boundary if IPRINT is negative. It starts at the bottom boundary if JPRINT is positive or ends at the top boundary is JPRINT is negative. The tracer particles are output next. Their values are obtained by interpolation from the proper boundary points. After all domains have been printed the boundary points that are interfaces are identified.
2.17 PLTOUT
Information is prepared in PLTOUT for processing by separate plot programs. The z-r space boundary coordinates, the tracer particle coordinates and the tracer particle velocities are calculated and then written out. All mesh points which are in a plastic region are then obtained. The coordinates of these points are then written out in blocks of 100 points.
2.18 PRNPLT
This subroutine prepares printer plots of the domain boundaries. The boundary points are transformed to z-r space and then scaled to inches of plot. The printer plot routine PRNT is then called for each point to plot the arrays.
2.19 PRS
PRS is a function subroutine to evaluate the equation of state. If the logical variable PDER is true it must also evaluate the partial derivatives of pressure with respect to density and internal energy. The routine supplied uses the Tillotson equation of state but may be replaced by any user written routine.
2.20 REXABL
The points defining a boundary segment eure redistributed over that segment in subroutine RE1ABL. The positions of tracer particles and interface end points are also redefined in relation to the new boundary poir .s. The point coordinates and function values are considered to be functions of arc length along the boundary. The total arc length is divided into evenly spaced intervals by the number of points in the new net. At the end points of the evenly spaced intervals, new point coordinates and function values are interpolated from the old coordinates and values. The arc length value at a tracer particle is also used to find the exact new interval in which that particle lies and its relative position in that interval is then calculated. The new boundary points which define interface endpolnts will be those points bounding the largest segment wholly contained in the old interface segment.
15
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2.21 SAVE
Information needed to continue the calculation in a aubeequent run is output by subroutine SAVE. Before being output, all dooains are shifted as far to the left and bottom as possible. Only those aesh points interior to the smallest mesh rectangle containing the domain are than output. This allows for the possibility of the restarted problem being run using smaller arrays and less computer memory.
16
r wmmmmmmmmmm* iii.ii.iiiiiin.iiiin^wiiwi>iHiii mmm mmmmm m i wm IPI" ' i m»"
III. GENEHM. PROGWM DESCRIPTION
3.1 Flow Chart
Read Input and
Initialize
Clean Up and Stop
Yes
Determine Time Step
Start Material Loop
Solve for Interior Points
A Ves
No
Move all Boundaries
Produce Printer
Plots
Adjust Values for Boundary Conditions
Shift and Rezone if Required
Interpolate Interior Points Not Yet Found
Write Restart Data on Tape
Determine Interfaces
Obtain Boundary Values
Determine Positions of Boundary Crossings
Interpolate [Values at Boundary
[rrossinos
.Yes
Write Plot Data on
Tape
No 'Printer ^ {>.
atput?^
Produce Printed
Output
17
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3.4 Storag» Arrangement
The number of neah point» In each coordinate direction and the number of boundary points may vary from run to run and from material to material. In order to efficiently utilise coaputer memory the storage arrays have no fixed dimensions for these quantities. All materials are stored consecutively. The area reserved for each material is ccnputad fron input parameters. The index of any point in a material is computed relative to a reference value for that material. If KS is the reference value for a material then the point in that material with index K would be referred to in the array with index KS+K. The axial arrays are those in ooonon BNDCRS whost name contains T. and their reference is IS. The radial arrays are those in oomncn BNDCRS whose name contains J and all arrays in coimnon MSHFCN. The radial reference is JS. The first dimension of these arrays controls the maximum number of points for all materials and the second dimension controls the maximum nvanber of boundary crossings in these directions. Thus, if IMAX and JMAX are the axial and radial dimensions, respect- ively, and for each material the maximum number of axial points is NI and the maximum number of radial points is NJ, then the requirement is
lm < IMAX MNJ+1) < JMAX
where the summation is over all materials. The addition of 1 to the NJ is to take account of the reflected line of lata points below the axis of »ynmetry. The arrays for interior mesh points are in blank conoon and have reference US. A mesh point of axial index I and radial index J is referred to in the arrays with the single index US + (J-l) (NI+2) + I. If the array dimension is UMAX then the requirement on the materials la
S(NI+2) (NJ+1) <_ UMAX
The 2 that is added to NI allows for shifting as stated in the description of subroutine FDIFF. The boundary arrays are in common BUDVAL and have reference IBS. For dimensions BMAX and maximum per material of NB the storage require- ment is again
INE < BMAX
Temporary arrays for boundary points that are used separately for each material are in common SCRTCH. The storage requirement here is just that the maximum number of boundary points for any material be less than the array dimension. The maximum number of materials is the first dimension of the arrays in common ZONES, the second dimension in MATARR, the first dimension in TRCPRT and the second dimensions of the array NONIN in comnon INTFC. The maximum number of tracer particles per material is the second dimension of the arrays in common TRCPRT. The maximum number of interfaces is controlled by the array INFC in common INTFC which has six entries per Interface while the maximum number of interfaces one material may have with all other materials is controlled by the first dimension of NONIN. NONIN must have storage for two entries per Interface plus one additional entry to signal the last interface for the material.
20
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There are three types of input data» Integer (I), real (R) and alpha-
numeric (A). An integer is a number without a decimal point which must be
right justified in its field. A real number has a decimal point which may be
followed by an exponent of the form E+N. The +N represents the power of 10 by
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and the E need not appear if either the + or - is present. If an exponent is
present it must be right justified in the field. Alphanumeric input consists of
exactly the punched characters and blanks appearing in the field. A description
of the necessary input data for SMITE follows. The card number refers to the
type of input. If more than one card is necessary for this input, there may be
several cards of the same type number. A number in parenthesis indicates an
optional card.
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Complete listings of the input decks for the two Silver Bullet compu- tations (without and with a sheath) (6.0) appear on the following pages. In addition, listings of sample input decks for the following five geometric and material combinations are given. These five cases have not actually been run. They are presented only as Illustrations of various possible input con- figurations.
Case 1. A blunt truncated mild steel rod (1020) impacting a 0.635 cm (1/4") aluminum target at a striking velocity of .371 km/sec (1220 ft/sec).
Case 2. A Bearcat rod with a hemispherical nose impacting a 1.27 cm (1/2") aluminum target at a striking velocity of .343 km/sec (1125 ft/sec).
Case 3. A hollow Bearcat cylinder impacting a 1.905 cm (3/4" aluminum target at a striking velocity of 1.22 km/sec (4000 ft/sec).
Case 4. An ogival projectile with an aluminum core and a tungsten carbide sheath impacting a 2.54 cm (1") target of rolled homogeneous armor at a striking velocity of 1.372 km/sec (4500 ft/sec).
Case 5. A hemispherical cap projectile with a tungsten alloy core and a maraging-300 steel sheath impacting a .9525 cm (3/6") target of rolled homogeneous armor at a striking velocity of 1.524 km/sec (5000 ft/sec).
Target-projectile configurations are shown for each rase. Sketches are not to scale and all dimensions are in centimeters.
35
1 ■ ■"" "in M««niTCmT^<mi«pnimmni«H*imin<nnpp«mivRipn«v<m«iF '
TOO MANY POINTS ON INTERFACE OF MATERIAL XX WITH MATERIAL XX - In subroutine ADJINT when attampting to replace the interface boundary points of material 2 with those of material 1, the material 2 boundary would exceed it's maximum allowable size,, Rerun allowing for more points on material two boundary,
AT POINT (XX,XX) IN FDIFF AT T-XX.XXXXX STAB-XX.XXXXX MAT-XX - Solution may be going unstable at this point. If it does not correct itself try larger artificial viscosity coefficient.
AT POINT (XX,XX) IMAGINARY SOUND SPEED IN FDIFF MAT=XX ESY=XX.XXXXX T=XX.XXXXX - Solution unstable at this point and will be recomputed in ONESTP. If it does not. correct itself try larger artificial viscosity coefficient.
AT POINT (XX,XX) IN ONESTP AT T-XX.XXXXX IPOINT=XX STAB-XX.XXXX RHO-XX.XXXX U=XX.XXXX V=XXoXXXX DELA-XX.XXXX DELB-XX.XXXX B SUB R-XX.XXXX MAT-XX - Solution may be going unstable at this point. If it does not correct itself try larger artificial viscosity coefficient.
AT POINT (XX,XX) IMAGINARY SOUND SPEED IN ONESTP RHO^XX.XXXXX U=XX.XXXXX V=XX.XXXXX T=XX.XXXXX MAT-XX - Solution unstable at this point and will be re- computed by interpolation. If it does not correct itself try larger artificial viscosity coefficient,
AT T-XX.XXXXX DENSITY DOES NOT CONVERGE ON MATERIAL X BOUNDARY AT POINT XX - In subroutine DENSB or subroutine INFACE iteration for density to obtain pressure that satisfies proper normal stress boundary condition does not converge. This is usually a sign of an instability in the solution.
MATERIAL X BOUNDARY EXCEEDS DOMAIN - RUN ABORTED (ZMAX,RMAX)/(Z (I) ,R(I)) - A shift was attempted in subroutine FINISH but there was no room. The boundary coordinates are printed in a-3 space. The minimum is zero and the maximum is the printed value. Rerun allowing more points in the proper coordinate direction.
AT X=XX MATERIAL X BOUNDARY CROSSED MORE THAN X TIMES T-XX.XXXXX - The boundary crossed a mesh line more than the maximum number of times allowed in subroutine BPOSN. The boundary points should be examined to see if this indi- cates an instability or if more crossings should be allowed.
48
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4.4 Sample Output
A sample of the output for the two Silver Bullet runs (6.0) follows. The Input data for all materials appears first. Printer plot and printed output at each selected time interval will appear next. The printed output is described in the writeup for subroutine OUTPUT in Section 2. A listing
" the complete output for the two problems at time t-0 is given.
49
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10 KT ■ KT*1 DT « l.OEö DO 20 MAT « l.cJMAT IF (FMAT(?2,MAT),Rfc,lT) ßi) TO 20 ÜT • FhAT(2?,MAT) MT ■ HAT
2ü CONTlNLiE IF (UT.GT.DTMIN) (!C TO 30 22 » FLOAT(IMAT(ö.fT>-l)«M''*T(l.MT) RR « FLOAT(!MAT(V,MT)-2)«^HAT(2iMT) -Rl'fc (6,120) AKT,fT,;ZiBW,PT GO TO 100
30 AKT « AKT»DT CALL MVSMC CALL ADJINT CALL B0NDRY CALL DENSE) DO 40 MAT i l.flHAT CALL GENVAL CALL FDIFF CALL FINISH CALL RPOSN CALL BVALU CALL INTHPL
40 tüwTINUE IF <ERH) RO TP IOC CALL SECOND (T) IF (T.GF.TCOMP.OH.AKT.GF.Tf.AX) GO TO 90 IF (AKT.LT.TPRT) OC Tn bQ CALL PRNPLT TPRT ■ TPBT*TPRPL
10 DC 030 MAT ■ 1,NH»T l\'ITAL41 REA[i(5,l»0) M.MJ.ir.IL.jF.JL.NLF.KTr.KBT.NFVAX.NTR.HATMC.IPMNT, OF A/
IJPHINT
READ (5.410) ZLES.YNAX.THETAiCAR.D.VIS.vnT.CfL REAl; (5,410) klu.^C,RHn,iJl IF (MATNJ.GT.C) GO TO 20 MAi»'0 ■ -MAT\n HEAD (5,410) APLCATNnj.BPLC'ATNOj.ArpCATMOl.BPP'MATKOl.ALPl- (H»T^
140 CONTINUE }M}Tl?n3 YStrAUS*l> ■ ySEA(JS»2; INJTAZDJ feSY*(jb*i) . ESYA(JS*2) i'J!!l?n* AST*(JS*X» « -ASyA<JS*2) I on! DyA(jS*l) ■ -DYA(JS*2) !!»,„, IF (ISTAHT.NE.Q) RO TO 260 !.!2u DC 240 M • i.30 J TA2
GC Tu (WO,180,190). M {.,!.,,,
BC TO ^0U IMTA21Z ien LF « LU*NTP.1 I^ITIPM
Ü0 TU 200 M TAPit 190 LF » La*NRT-l M TA916 200 LAUD . NfcND(MAT.»1)-LF M I*,:,
If ■ Ht-ir IM1TA217 IF (LM.EU.O) Gü TO 240 IIÜT^I« DC 210 I • l.NTR M rl^O IF UTH(M*T,n.LT.LF) GO TO 210 l« !!!,« ITK(MAT.1> ■ lTR<MAT,I)*LADO L, I*,,,
210 CONTINUE WirlUl NIN2 » 2»NINFC }. ,i,oI DC 2Un I ■ 1,NIN2 XJTAI« IF <IN>C(J«I-2),ME.MAT) SO TO 220 l% HI IF (INFCOM-D.GE.LF) INFC(3.I-1J ■ INFC (3« 1-l)*LArn Mi?!«? IF (INFC(3«1).GE.LF) INFC<3*I) ■ INFC(3*I)*LADD IMITA227
240 CALL RfeLAÖU (M) ^ TA,tp DC 2b0 I • l.NF WlAllo weilHS^I.l) . 1.0 M TA3«n WBUHS*I,4) . 0.0 WIVAAI
250 WB»IÖS»I,Z> « UI M TA54? 260 CALL BPOSN M TAS43
CALL BVALU WlAlA IF (ISTART.NE.0) GO TO 320 M.TAO!^ DC 2Ö0 J - 2.NJ \l\ltlll JJ=IJS*<J.1)»NIP2 \n\Al47 DO 280 1 - l.NI J rlllh
W(1J,1) ■ 1.0
99
IMITA249
M<iJ,2) ■ UI DC 27(1 KK ■ SiKMAx
270 M(IJfKK) '0,0 KlJ) ■ 1
280 COTINUB C*LL IHOSN DC 2V0 I ■ 1,N1 WUJSM.ll ■ W(IJS*?«NIP2*I,1) WtlJ8*l»2> • W(IJso«NlP2«I,j:) «(US^l.J) ■ -W(IJS*2»M1P?»I,?)
290 LIUSM) • L(ljS*2«MIPZ*l) SB'1 a u.U PttH • .TRUF. DC 300 J ■ 2,NJ jj=IjS*(J-l)»NlF2 DC 300 I • l.Nl IJ=JJ*1 IF(L(IJ),tQ,0) 00 Tf) 300
6 ' W(IJ,4) P = PR!i(HH0,E) C? » PW*P*PE/RH0**2 C = !>QHT(C2) UC « (AB!»(W<IJ,2)-*<5Y(JS*J)*,.'(lJ,3))»Eir1r«jS*J)*r)«r,AX VC « (ABS«W(IJ,3>)*r)«ESV(JS* ))«D*V SR » AMAXKUCiVf) IP (SB.LbiSBM) GO TO 300 S6^ ■ SB ISd « I JSU a j CCNTJNUE UC 3X0 M ■ 1,3 NKd ■ NBfcG(MAT,M) NNf ■ NEN0(HAT,M) *RCS1Z(MAT#M) ■ FRS( IBS*MNF)-rRS(IPS»NNP) DTM ■ CFVSBM DTMN « AMlNKDTM.nTMN) fMATt22,rt*T) « DTM
300
310
320
330
340
350
360 370
380
1HAT(8,MAT> ■ IS3 |MAT(9,MAT) ■ JS§
CONTINUE If (ISTAHT.EQ.O) DTMIN « 0.0F*DTMN IF (TSAVb.LT.O.O) TSAVt ■ 1.CF8 IF <TPHPUiGE.0.O) GO Tu 340 TPHPL • i,0F8 GC TO 350 CALL PHNTS (999IT1TLE) CALL PHNPLT IF (TPLOr.GF.0,0) GO TO 360 TPLOT » 1,0E8 QC TO J70 CALL PLT0UT IF (TPHIN.GF.0.0) GO TO 380 TPHIN • liOEB REIÜKN CALL OUTPUT
M2»IbS2*NENt)(MAT2,3) *D'J' 'I*! ZKX2.0.0 APJ!KT20 DC 2 J.Mi,M2 AnJP:I"
2 Z''X2iAM*XX<rRX(J),ZMX2) An-INT2Z K«^! AnJlM2J DC A J.N1.N2 AnjlfTZ^ IFlFKXtJ) ,GT,ZKX2) r,0 TO 4 AflJlM?!) K,J AnJlNT26
4 CONTINUE AnJINT27
N2 = MtN0(K*l,N2) AEJ!M^ HMXI.0,0 AnjIf.T29 DO 5 J.N1,N2 APJir.T3
5 RfXl»AMAXl(rRY(J),RMXl) ^Ü^ÜI.l DC 6 KiMl,M2 APJINTi? IFirRY(K),UE.RKXl) r,0 TP ft Anjuns
6 CCNTINUF APJIUT3« 7 K.R*1 AnJINT35
a RHX2»AMAXl(FRY(K.i)lFRY(K)) AHJ!I!'l,!? Rh1>*2»AMINl(FRY(K»l>.FRY(K)) ADJIM37 Zh'X2«AMAXl(FRX(Krl),rPX(K)) {JHIUT« ZHN2iAMINl(FRX(K.l)1FRX(K)) AnJIMJV DC 10 J«N1,N2 ADJINT40 IFtFPY(j),GETRMN2.AND.FRX(J).r,E.ZMNJ) GP TO 11 AnJlM41
10 CONTINUE APJINT42 6C TO 7 AnJlNT43
11 J1«MAX0(J.IBS1*2> AOJIf.T44
DC 12 J«Jl,N2 ^,m5 lF(FHY(J),LT.RMN2,0R,FRX(J).f!T.ZMX2J 110 TO 12 AnJI.. ! j2sj ADJINT4n
lYSt*(7«),E3VEI78),El0r(7JI),ASV(7B),Ayy<7l'>,ASVA(7«) MSHFCN i COMMON /M4TARR/ NMAT,IM*T(12,S),FHAT(23,5).HAT M4T*HR ? COMMON /COMV*L/ KT.AKT.DT.DTMIN.lTER.NEIBR.IPOINT'.EBR.EPS.EPSl.nDICOHVAL ?
ISi lOP.UAUiKMAX.PORQ.PSCL.LB.I.r COMVAL >) COMMON /MATVAL/ AK, »Y.CFL.VIS.VIST.Mt,VC.RHO,XQ.PO,TO, XZ,»,ö,ABL.OHiTVAL 2
lABL,CHAN,BMAX,FACT.ANQ,AF,SBH,ALiAS,AK1HAL,HAS,HAl'1CAt.OASlOAK,DAXMATVAL 3 2.CAV,HDAX,HDAV.0Ax2.DAy2.HDAXY,0ßAXV#DTR,HALrHU.TWtCMU,rRT6MU.60VMMATVAL * 3U. ISEOTM.NI.Nj.NFMAX.IS.JS.IJS.IBS.rSB.JSn.NIMi.MJMl.N'D'.Nrj.NF.NElMATVAL i 4.^^.N1P2|NTR MATVAL 6
COMMON /PHESS/ POER'.PR.PE.MATNO PRtSS 2 REAL MU FTIFf 1 LOülCAL PDER.AXIS FTIFF 1.) ÜIMENSION WW(3,3.7). F(3,3.7). 0<3,3,7>l M^.S^)', T(3.3.3) F01FF H DIMENSION WP(2,2,7)', UN<7) FniFF 1? SBM ■ O.ü FPIFF U NIH1 ■ Nl*l FnlF> 14 DO 10 I ■ l,Nl F01FF lb M = MPl-I F^IFF 1A DO 10 N ■ l.NJ FniFf 17 MN ■ IJS»(N-1)«NIP2*M F3IFF 1b L(MN*2) ■ L(MN) FniFF 1<3 DO 10 K ■ l.KHAX FniFF 20 W(MN*2|K) ■ H(HN.K) FniFF 21
10 CONTINUE FPIFF 22 DO 260 M ■ A.NIPl FDIFF 2J DO 260 N • 2#NJM1 FniFF 2* MK » |JS»<N-1)«NIP2*M FniFF 25 IF (L<MN),EQ.O) GO TO 260 FDIFF 26 NEIHR • 0 FHIFF 27 IF <L(MN*1)1EO,0) NPIBR ■ NEIFIR»! FP1FF 2H IF l\.tm»NlP2),EQ,a\ NEIBR • NEIBR*2 FPIFF 2V IF (UMN-D.EO.O) NEI«R ■ NEIRR»4 FDIFF jn IF <U<MN-NI^2>.EO,0> NEIHR ■ NEIBR^B FDIFF 31 IF (L<MN*NIP2*1),EOIO) NEIBR ■ NEIB«*i6 FDIFF 32 IF (l(HN*NIP2.1)lFO10) NEIBR ■ NEIBR*32 F01FF 35 IF (L(MN-NIP2-1),E0.0) NEIBR ■ NEI8R*64 FI21FF 34 IF IU(MN-NIP2*1»,E(M) NFIBR ■ NEIBB*12« FDIFF 35 IF (NEIBR,NE.O) SO TO 210 FniFF 36 MMü ■ M-2 FOJFF 37 NM<: ■ N-.J FDIFF 3P AXIS ■ .FALSE, FDIFF 39 IF (N,feO,2) AXIS ■ .TRUE. FDIFF 40 ESYN ■ FSY(JS*N) FDIFF 41 Djl ■ ÜV(JS»N)«OAK FDIFF 42 DJ<i ■ ,5«ÜJ1 FDIFF 43 PCbR ■ .FALSE, FDIFF 44 DO 20 J ■ 1,3 FDIFF 45 MJ ■ NM2*J FDIFF 46 YSbJ ■ YSE(JS*F«J> FDIFF 47 AR ■ -ASY<JS*Hj) FDIFF 48 JJsIJS*(MJ-l>«NlP? FDIFF 49 DO 20 I • 1,3 FDIFF 50 MI ■ MM?*I FDIFF 51
107
■a. liitüi
FDIFF &"! IJ»JJ»MI FOIFF S3 RMU • HdJ.l) FDIFF 5* U » h(lJi2) FOIFF 6? V » H(IJ|3) F0irF so E « N(IJ.«) FDIFF 07 SH • W(JJ.5) FOIFF So Si* ■ W(IJ,6) FDIFF y S2* ■ W(!J.7) FDIFF 60 P = PRSCHI-CE) FPIFF 6: WV.lI,Jil) • flHO*YSEJ FPIFF 6? Hhll.J.Z) ■ WW(I,J,1)»U FDIFF 63 WW(t,J,S) ■ HH(I.J.1)*V .„.i^iMi FDIFF 6«
l,5.5X4MMAT»l2/aH W»7El5.5» .„,,«., TM^riNiPY SOUND SPEfC 1^ FD1FDIFF?93 300 FOKH*T (UH AT POINT «p'l«'^'1"»3^' i^Ji^g 5? FD1FF294 irFJ){4HHAT»12,3x4Hesv»F-9,».3X2HTir9,5/3H W«7ElJ.5> FPIFF29'.
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112
»■ v«ty
SlbROUTINE ONESTP <!I.JJ) ONtSTP d COMMON// U(2200.7),L(2200) 1NIV*c I COMMON/BNDCRS/ HB! (109,8,7),PBI (10«,a»# INDa09.9)i ?N„Huc T 1WBJ(78I8|7>,PBJ(7«.H>IJND<78,9) BNUCPS 3 COMMON/MSMrCN/ Y<78).DV<78).DYA(7«),E8Y(7«).YSE<7B),ESY*(78), MSMFCN d lYSb»<7»)#ESYE(78),ElQF(7«J»#»8Y(7e»l«YY(7e),ASYA(78> "
ST^O I
COMMON /MATARR/ NMAT. IM»,(ia,§).rMAT(23,5».«AT "llf"" ^ COMMON /COMVAL/ KT,»KT,nT.OTHIN,ITER,NEIPR,IPniNT,E«R.EPS.EPSl,RDlCnMVAL 2 lS|IOP,DAl.,KMAX,P0RC,P8CU.LP,Lr «E0?V^,L I COMMON /MATVAL/ AX,AY.CrL.Vl8.VIST,MU,YC,RH0,XO,PO,T0.XZ.A,B.ABL,DMATvAL 2 lABL,CHAN,BMAX,rACT,ANO,AF,SRH,AL.AS,AKlMAL,HA3,HA«,OAL,OA«!,'3AK,OAXMATVAL i 2,D*V#H0AX.HDAY,DAX2,DAY2.HDAXY,0DAXY,DTR»HALFMU.TWTrMU,rRTCMU,00VMMATVAU * SU.IREOTM.Nl.NJ.NFMAX.IS.JS.IJS.IBS.ISB.JSB.NIMi.NJMl.VDlNDl.NE.NElMATvAL b 4,N>,NIP2,NTR "*JVAL o COMMON /PRESS/ POER.PRiPE.MATNO „ ^!;D,n COMMON /i)ERl/ RT,UT,VT#ET.S11T,812T,S22T 2
AXIS ■ FALSE ONtSTP25 IF (J.60.2) AXIS - .TRUE. D^TP?7 AI - FL0AT(I.3).AX mtlVpll AJ - FL0AT(J.2).AY ntl74l DJ ■ ESY(JS*J)»OY(jq*J) n^cTP^O DO 10 KK ■ l.KMAX „ ^CTD.I Hta(KK) » WUJ.KK) S^^TO,'
10 CONTINUE O^STP3< 1 PCbR » .TRUE. 0NfcSTP33 PP • PHS(WW(l),W^(4)) SN^3o^ PPH i PR 0NfcSTP35 JJ^ J ll 0NfcSTP36 JPUIN^ » NE1BR/16 n^cTP^H IPOINT ' NEIBfl-16«JP0INT^l ONL?TP3« iPilThiT a JPnTNT*1 ONtS""^" GO to «20.40,20.«0,120,80.120.60,20,40,20.40,120,80.ISOieo). IPO INONtSTP40
lT 0Nt:STP41 C USX IS U SUB X UXX IS U SUB XX UXY IS U SUB XY ON»:STP4Z C USY IS U SUB Y UYY IS U SUB YY 2^!!-!? C UT IS U SUB T UXT IS U SUB XT UYT IS U SUB YT UTT IS U SUB TT 0^tSTP44 C SECTION 100 IS FOR WHEN NEIT"FR (IP,J) NCR <IH,J) IS MlSSlKG S^H^J! _ 0NLSTP4O
20 DO 30 KK ■ l,KMAX S^IIPI« HSA(KK) ■ <W(lj*l,KK)-W(IJ-l,KK))«HDAX SI^ETOIQ HSAA(KK) ■ (W<IJ»1,KK)-2.0*WW(KK>*W(IJ-1,KK))*DAX2 0NfcSTP49
230 N ' 1ND(|S*IM2.1) IIIIAIA DC 240 K • 1,N ZclAVb IF UJ.LT.PBKISMM?,«)) CO Tn 250 Zllrill
2.0 CONTINUE g-ST19.
Jf ". ^-pBI{Is*,k1^•,', Ztlnl* IF <K2,LT.RDIS.AY) GO TO ABO TulVrlll ^ ' 1./«K1.K2.(K1*K2)) ITtlViV?
DC260KK11KMAX u«Nca i eu f WSU(KK) ■ SK2MW(lJ«NIP?,KK)-WM{KK))*SKl»(HH(KK>.WBI(IS*lM2.K.KK))ONfcST20fl WSdBCKK! - K2MW(ij^IP2.KK>.HH(KK)>.Kl*{WH(KK)-H«I(!S*|H?,K,KK>) OVfcST209
260 CONTINUE ZlllAl Pl ■ PRS(W(IJ*NIP2,1).W(1J^NIP2,4)>-PP n^cT9i5 PRl • PR-PPR ^tVrlll PEl ■ F*E«PPE ONfcSTZlJ P2 • PP-PRS(WPI(IS»!M2,K,l>.MflI<IS*IKZ,K,4)) S^CT^Ü
PEbB ■ S|<2*PE1*SK1*PE2 PBH ■ K2«Pl.Kl.f»2 GC TO HO
270 N » IN0(1S»IM2,}> 00 260 K ■ 1,N IF <AJ,Lt,PBmS»lM>.K)J 00 TO 290
280 CONTINUE 290 Kl ■ PB1(IS*IM2,K)-*J
If (K1,LT,RDIS**Y) nO TO 680 If U.t0.2) GO TO 310 K2 ■ *J-PB1(IS»IH2,K-1) IF (K2,LT,RDIS«*Y> 10 TO 680 UELY » *MIN1(K1,K?> DK ■ 1,/(K1«K2«(K1»K2)) SKi ■ 0K*Kx**2
UT « .UU»USA*DR*<T11A*2,*B«S12B» VT ■ n, E*U ■ UR«tTll*USA*2.«B«T22*VSR> ET ■ -IJU«ESA*EAD PT ■ RT*PH*ET*PE SUT » -uU«SllA*FRTnMU»(USA-B«V8B> S1<!T ■ 0, S2iT i -00«S22A-TWTnMU«(USA-n«y9B)
700 FOHMAT (ilH AT POINT <I3.1H,12,17«) IN OfiESTP AT f.F9.5^8H IP0INT»0NbST593 lI2.3X5HSTAB«F7,4,3X4HRH0.F7.4,3X2HU"r7,4,3X2HV-F7;4.3)'5HDEU*»F5.3.0NbST594
23X!>H0El.B«rs,3.3X8HB SUB R.F7t4,3X4HHAT»I2> e,SN^Ico! 710 FORMAT (ilH AT POINT <I2,1H, 12,33H) IMAQINARY SOUND SPEED IM PNEST0NbST596
SUB^UUTINE BONDRV COMMON// W(2200.75,L<2200) COHMON/BNDCnS/ HB1(109,8,7>,PBI(10«,8>,tNO(109.«), lHHJ(7a,a,7).PBJ(7a,8)iJND(7«|9) COMMON /BNDVAL/ WB(600l7)lrRX(*00)irRV(«00).n(S(600) COMMON /MATARR/ NMAT.lHAT(l2«9)irN«T(23,9).HAT COMMON/SCRTCH/ THB(300t7)«TrR)((300)«TrRy(300)
BONDRY 1NTVAL BNDCRS 8NDCRS BNDVAL MATARR SCRTCH
COMMON /ZONES/ NBEG(9>3).NEND(9<3)«ARC8IZ(9.3)«OlSMAX(9l3).ni9HIN(ZONES 19.3) ZONES COMMON /COMVAL/ KT,AKT.OT.DTMIN,ITER#NEIBR.IROlNT.ERR.E^S.EPSl,RDICOMVAL IS.TOP.DAL.KMAX.RORC.PSCL.LB.LF COMVAL COMMON /MATVAU AX,AYiCrL.VII,VI8T,MÜ,YC.RM0,X0.P0,T0.XZ,A,B,ABUiDMATVAL lABL.CHAN.BMAX.rACT^NR.AF.SBMiAL.AI.AK.HAL.HAS.HAK.OALiO'j.OAK.OAXMATVAL 3 2,nAY.HDAX,HDAY,DAX2,DAY2,HDAXY#00AXV#DTR,HALrMJ
CcffiT'bNDVAL/B-fl(*00,7).FRX(600).rpy(600),fRS(600) BNÜVAL Z
COMMON /M4TARR/ NMAT, 1MAT<12,9)iTHif(23,5)#MAT MATAPR 2 CCMMQf' /MATVAU/ »X.AV.Cru.yiS.VIST.MU.YCjRHOjXO.PO.TO.XIjA.B.ABL.DMATVAL 2
lABL.CHAN.BMAX,FACT,*NO,AF,8»H.AL#Ai»AKi«AL,HAS,HAK, OAL.aAS.OAK.DAXMATVAL i 2,C*v<HDAX,HDAY,DAx2.DAY2.HDAXV,0DAXY#OTR,HALFriU,TWTDMU,RRTCMü,00VMMA7VAL < SU.IRtOTM.NI.NJ.NFMAX^S.JS.IJfi.lBS.HB.jSB^IMl.NJMl.NO.NOl^E.NElMATVAL 5 4 IU NIP2 MR MATVAU o
70 HB(J,1J ■ -WB(J,l) SINIB 48 DO 220 MAT . l.NMAT Jf^J ^ CALL GENVAL ll^ll H DO 80 J ■ l.NF DENSB 50
126
I » IHS*J DEN^B bl TfHYCJ) • R(rBr{I)) DEN3B ^^
ao TFKX(J> ■ z(rRx<l).TFHr(j)) DENSB »J U ■ 1 DENSB 54 Nl • 1 DENSB Sb N2 ■ NONIN(l,MAT) DENSB bt If (N2,Nfc,0) GO TO 100 DENSB *>7 Nl « IdS«l DENSB 5« N? • IbS^NF DENSP bV 00 TO 11(3 DPNSB 60
90 U ■ IJ*2 DENSB 61 NJ ■ NONlNdj.j.MAT)*! DENSB 62 N2 » NONJNdJ.MAT) DENSB 65 IF (N2,NtiO> GO TO 100 DENSB 64 IF (NliGT.NF) CO TO 1V0 DENSB bb Nl » IdS^Nl DENSB 6e N2 » IHS*NF DENSH 67 GC TO HO DENSH 6«
100 IF <N1,E0(N2) GO TO 90 DENSB 69 Nl » IHS-Ni DENSB 70 N2 « iaS«N2-i DENSB 71
110 DO 100 I ■ N1,N2 DENSfi 7? j z i.IBS dtHSB ti IF (J.NE.l.AND.J.NE.NF) GO TO 140 DENSB 74 IF (TFWY(J>,GT.l,0E-4) GO TO 120 DENSB 75 CCbPSl ■ 0.0 DENSB 76 SINPSI « 1.0 DENSB 77 GC TO 16U DENSB 7«
120 IF (J.hQ.NF) GO TO 130 DENSB 79 jp • ? DENSB eo jH=NF-l DENSBC21 60 TO 15Ü DENSB 02
X30 JP»2 DENSBC22 jf- a NF-j DENSB 84 GO TO 150 DENSB «5
Nl > IbS*lNrc(6*K«4) N2 * IUS*lNrC(6*K-3) NF ■ NbNü<MATl-3) MAt ■ hATl J * iMAU7,MAT2)*INrC(6«Kl -1)*1 DO 60 I ■ N1,N2 J » J-l IJ ■ i-ias IF (IJ.Nfc.l.AND.IJ.NE.NF) GO TO 30 IF (FRY(l).GT,1.0E-4) QO TO 10 COSPSI « 0.0 SINP5I ■ 1.0 GO TO 50
52 M*1N0»IMAT<10,M*T) IF(MATNO,QT,103) GO TO 96 EIMB(H«4) PPsHÜ(M,5)*SINS0»2,n«HB(M,6)«SINC0S*WB(H,7)«C0SSQ»TK,N DC 54 N«1»MAXIT RhU«-WB(M»l) p*HRS(HHOtE) COHRi(P-PP)/PR HBCM.DiWaCM.D-CORP ir(AHS(C0PR),LEiARS(W6(H,i))*FPSl) OC TO 96
lWbJ</8,e,7),pHj(78,M.jMn(7«,9> Lmwii5 i COMMON /BNDVAL/ *Ruoo,7).rRX(600),rpY(600),rRS(6P(n , .^„.„JSr^c 9 COMMON /ZONES/ NBFG(5,3).tlENr(5.3).ARCSIZ(5i3».niSM*X(5.3).niSMIN(Z0NFS 2
lAfiU.CHAN, dMAX,FACT, ANn.AF.SRM.AL.*S.AK, MAL.HAS.HAK,rAL,IAS,OAK, PAXMATVAL S ?,CAV,HOAX,HDAV.DAX2.DAY?.HDAXV,0DAXY,üTR,HALrHU,TWTrHII,FRTrMl,0OVMMATVAL < 3U,IHfc0TM,NI,NJ,NrMAX.lS.JS.IJS,I0S,IfR.JSn.NPl.NJM1,^D'.Nin.Kl:,NElMArvAL b
4 ^^ NIR? "TR MATVAL " COMMON /INTFC/ NIMFC. INFC(60).NONIN(20,5) i^'J^cu,^ LOülC*L fcHR F 1HI3 ZbMAX I AX.FL0AKNIM1) r cu,!« RhMAX • AY«FLnAT(NJ.2) c K cuK
I«: '-ill' n?!!^ WX ■ ZMX Rf'NiZMN
DO 10 I ■ l.NF r,N.,cu,Q Z^M ■ AM1N1(ZMN,FKX(IBS*I)) r N CW90 ZMX ■ AMAXl(ZMX,rRX(IBS*I)) nrAv TA RMNtAMlNltRMN,FRY(IRS*I)) r,ficu91
10 RMX I AMAX1(RMX,FRY(IBS*I)) „i.v li IFCRMX.Gfc.RPMAX.OR.PMN.LT.-l.OE-O (?C TO 60 r.^cu^
IFCZMN.Lt.O.O) GO TO 15 r N SH7« IF(ZMX,LT.ZRMAX) GO TO 70 r ^ cu« ISH»1NT(ZMN»DAX) K<U
ÜC TO 18 15 ISM«INT( (ZMX-7BMAX)«DAX)
t<«Nl*i ISC»-1
20
00 30 KK ' l.KMAX 3D
FINISM17 OFAX 3b fIMISHl?
FINISH26 .._. FIMSH27 ,r T^ 1« FINISH2H
FINISH29 FINISH30 FINISH31 FINISC21
16 ISH.lSM/i jSM»lAbS(ISH) F1NISC2Ü IFUSH.LE^) GO TO 60 FINlSN3i ZD^ ■ FLOAT(ISH)«AX FIMSH34 XZsXZ-ZDF FINISC23 FfAT(l2,MAT) ■ XZ FINISH36
FIMSH37
FRX(I0S*I)«FRX(I9S»I)-ZDF c ^ CUAO
ISH.JSH.l F N |S3S
IMI sS FINISH42
f IMISH46
W(JJ*K,KK) ■ W(JJ*I,KK) FlNi&M'
131
40
50
60
70
DC 40 j « 1,9 1NU(IS*K,J) a piD(IS«IiJ) DC 50 {»liJSH K"H*1NC DO 50 J ■ 1,NJ JJ ■ IJS«(J-l)«NiP2 L( Jj«K)iO HRlTb (6,170) UAT.ISH GO TQ 70 fcRN ■ .TRUE,
WRiTfc uIzoO) ZBMAX,RPH*X.(rPX<lBS*n#FPy(IBS*I>,I ■ 1»NF» DC 150 H « 1,3
eo
90
100 110
120
» NfcMO(M*T.M ■ NUEQlMATiM)
130
LE DELS » FHS(lBS*LF)-rRS(irs*Lr> IF (Z.O'AHCSIZjMAT.Mj.GT.PF.Lr» 00 TO 130 LAuD « (LF-LB)/2 IF <NF»LAÜD,GT,NFMAX) Gn TO 130 Lf • NF-tf IF (LM.EU.O) OCJ TO 110 DO 60 1 ■ 1,NTR IF tlTH(MAT,n,LT,LF) GO TO 60 ITH<MAT,1) • ITRCMAT,I)*LAPn CONTINUE NIN2 « 2«NINFC DC 90 I « 1IMN2 IF (INFC(3«I-2),NE,MAT) GO TO 90 IF (INFCJJ«1.1),GE.I F) INrC(3«l.l) ■ INFC(3*1-1>*LA0D IF «INFC(3*1),GE,LF) INFC(3«I) • INFC (3* I^LAOD
CCNTINOE NF1 ■ NF*1 DC 100 UC ■ 1»LM LI • NF1-LL L2 » L1*U*DD FRX(IBb*i.2) « FRX(IPS*L1) FRY(IBS*12) ■ FRV(IRS*L1) DC 100 KK ■ l.KMAX wBllÖS*L2iKK) ■ ^R(IBS*Ll.KK) CONTINUE DO ViO LI ■ M,3 IF (LliEQ.l) GC TO 120 NBtG(MAT,Ll) ■ NEND<«ATiLl-l) NENDJMAT.LI) ■ NEND(MAT,L1)«.LADD NF ■ NF*L*D0 NE ■ NENU(MAT(2> NEl ■ NE*1 NC • NbND(MATil) NDl • ND*1 WRITE (6,190) KAT.M,DELS,ARCSIZtHAT.p),AKT ARCSIZ(MAT,M) E DELS GO TQ 14Ü DELS « DELS/FLOAT(LF-LB) IF (DISMlN(MAT,M)lGE,(1.0-DAL)*OELS.AND.riSMAX(MAT,H)iLB.(1.0*DAl.)
l«DtLS) GO TO 150 WRITE (6,160) MAT.M.DELS,DISHIN(MAT,M,D!SMAX(MAT,M>,AKT
» NtMÜ<MAT,r) » NdEu(MAT,H) IS THE ÜEGINMINC UF THF OLH IS THfe BEGINNING OP THE NEW IS tHfe END OF THF OLP SET IS THE END OF THP NEW SET
SET SET
NBi ■ N8*l NFM1 s MF-1 DELS « (FNS(IPS*LF).FRSnF;S*LB))/P|.OAT(MF-NB) S = FRS(töS*LP) K » LB*1 DC 20 J ■ 1,NTR IF (ITH(MAT.J).GE,La) C1 TO An
20 CCNTJNUE 30 STN > l.üEin
IJ ■ l>*i GC TO 00
40 IJ « ITR(MAT,J) IF (IJ.GtiLP) GP TO 30 STH ■ FRSUPSMJ>*TnS(HAT.J)«(FRS<lPS*lj*l)-rns(IRS*lJ) )
50 CCNTINUF DC 110 I ■ NBl,Nf>i S = S»DELS
60 IP <FRS( lHS*K).r,E.S) GO TO 7r K * K*l GC TO 60
70 AO ■ (S-FHS(IHS*<))/(PPS(IRS*K-l)-FRE(IRS*Kn Al ■ 1,0-AO TFHXtM • A0«FRX<I8S*K-1)*A1*FRX(IES*K) TFKV(I) « A0«FRY( IB<!*K-l)*Al«rRY( IBS*K) DO 80 KK « l.KHAX
80 TUd(I.KK) » AO*wa(lBS*K-l,KK)*Al*WB(lPS*K,KK) <»0 IF (STH.üT.S) GO TO 110
ITH(MAT.J) = 1-1 TH!>(MAT.J) ■ (STR-S*neuS)/nELS J = J*l IF (J.GT.NTR) GO TO 100 IJ » ITR(MAT,J! IF (IJ.Gt.LP) (.0 TO 100
STrt i FR!»(IBS*IJ>»T0S(H*T,J)«(rRS(lBS*lJ*l)-FnS(19S»IJ)> UC TO VO
100 ST« ■ X.UklO U » LF*1
110 CCNTINUE IF (U^t.LF) GO TO 13Ü UC 120 1 ■ J.NTR ly » rTR(MAT,I) If (UiGE.LF) GO TO ISO STH « rRS(IBS*lJ)*TBS<M»T,I»*<FRS(lF,S*lJ*l>-'nS<lHS*lJ)) ITH(M*T,p ■ NFM1
1BSM)-FRX(1BS*I-1)>*«2) N1N2 « 2«NINFC Nl ■ IbS*NBl N2 ■ IliS*NF DC 220 I • l.NIMZ IF «INFC»3*I-2),VF.M*TJ GO TP 22" U « MüD(I,2) Kl »rj'I-U 11 « INFC(Kl) K2 ■ 3«I-1*IJ 12 » I>(FC«K2) If Ul.Lfc.Ln.OP.U.nT.LF) GO TO 190 S = TFHXUl)
170
180 190
200
DC 170 J « Nl,N2 IF <S.LE.FRS(J)) CCNTtNUE J s N2 IN^(Kl) ■ J-IbS IF (12.Lfc.LB.0p. S - TFHX(I2) DC 2C0 J ■ NliN2 IF tS,LT,FRS(J)) CONTINUE J = N2*l
20 jNiJ<jS«J,l) x 0 1A s FHX(IBS*l)*DAX JA = FHy(IBS*l)*DAY lFiAbS(rHY(lBS*l>),LE.1.0F-«) JA»-1 DC 170 L " 2,NF It » FRX(IBS*L)»DAX Jb » FHY(IBS*U*DAY IFlABS<rRY(lBS*L)).(,>.T,1.0F-4) GO TO 30 lAslb JF--1
30 IC » Ib-iA IF (ICJ 40#100,50
4n \P « IA*1 1* « IA-1 ÜC TO 60
50 l« » IA*i '.P » IA*.
6U K = INU(li IP.1)*1 IF (K.uT.MXSI) GO Tf) 90 INJ(IS*IP»1) - K
70 IF (K.fcO.l) GO VO 80 LA ■ INOdSMP.K) IF (rRY;iHS*LA).LF.FRY(IPS*L-l)> GO T0 PC IKJtlS*lPiK*H « !Nn(IS*lP.K) K = K-l GC TO 70
JNU(JS*JPil) ■ K IF (K.kO.l) GO TO 110 L* » JNDUS*JP,K) IF {FPX(IHS*L*).LF.rRX(IBS*L-l)) QO Tu 150 JNÜ( JS*JP»K*1) ■ JNnUS*JP»K> K « K-t GO TO 14U JNU(JS*JP»K*1) • L-l GC TO 10U WBlTfe (61I8O) JH, JP.MAf.MXSJ.AKT EfiH i .THUE, COiNTJNUE HFlURN
JS*J.l) • W(IJS* JS*1.2) ■ W(IJS* JSM.3> > -udJS JS*1.4) ■ W<JJS* JS*l.b) ■ W( US* JS*1.6) • -V,(lJS JS*I.7) ■ W(1JS* JS*l) • L(IJS*2* 580 J ■ 2,MJMl ■ IJS*(J-1)«NIP2 S = .FALSE, (j,tQ.2) AXIS 580 I • ?,NIM1 « JJ*l (L(lJ)iNF.2) 50 WY ■ U - FLOAT(I-1)*AX « FLOAT(j-2)*Ay INT = KIJ-NI??) INT ■ IP0INT/9 INT x IP0INT-9*J (AXJS) IPOINT s INT « JP0INT*1 TO (20.230.30,10 TO (220.480,220. TO (470,480,210. TO (300,480,300, TO (100,480,150, TO (470,480,90,3 TO (280,480,28(1, TO (470,480,460, RY « .1 INOMSM,!)
DC 140 KK ■ 1,KM*X 140 WtlJ»KK)«*0*WPI<IS*l«K,KKJ*Al»HtlJ*NIP2,KK)
GC TO S7U 150 DC 160 KK ■ IfKMAX 160 H(IJ#KK) ■ W(IJ*NIP?IKK)
QC TO &70 170 DC 180 KK ■ l.KMAX 180 MdJ.KK) » .25«<W(IJ*1,KK»*W(|J.1,KK)*H(IJ*NIP2.KK>»W(IJ
Gf TO &70 190 PC 200 KK « l.KMAX 200 W(UiKK) ■ UBl(IS*I,K,KK)
(iC TO 1»7Ü 210 bKuRY • -I
GC TO J3U 220 BNURY » 1 230 N = IND(IS*1,1)
DC 240 K • 1,N IF (AJ,Lfc,PRI(IS*I,K)) GO TO ?50 CCMTINUE K = N D s AJ-PdI(IS*IiK> IF (BNURV 190,260,320 AO » AY/{AY-D' Al • 1,0-AO DC 270 KK ■ l.KMAX WUJ.KK) ■ Afl*WBI(I^*I.K,KK)*Al«W<lJ-NIP2,KK) GC TO 57Ü DC 290 KK ■ l.KMAX W(1J,KK) = W(IJ-MtP?.KK) QC TO 570 DC 310 KK « l.KMAX WUJ.KK) » 0.5*(^(IJ*NIP2,KK)*W(1J-NIP2.>'K)) GC TO 57U
240
250
260
270
280 290
300 310
320
330
340 350
360
370
380
390
400 410
DZ « AJ-PHI(IS*I,K-1 ) AO « D2/<D2-D) Al » 1,0-AO DC 330 KK « l.KMAX W(IJ.KK) GC TO &70 BKURY ■ -1 N « JND(JS*J,1) DC 360 K ■ 2,N IF (AI,LT,PEJ(JS*J.K)) CCNTINUE K = N*l K = K«l IF (UNURY) 440.380,440 D = PBJ(JS*J,K).AI AO > AX/(AX-D) Al » 1,D-A0 DC 390 KK ■ 1,KMAX W(IJ.KK) GC TO 570 00 410 KK ■ l.KMAx H(1J|KK) » W(IJ*1.KK) GC TO »70
« AO«''lni(IS»I,K.KK)*Al*WPni?*!.K-l.''K>
130 TO 370
« AO^'JfiJtJS^J.K.KKWAl'WC IJ*l,KKt
I*.T I'll INT 1'IT INT INT U:T
•MP2.KK)) INT INT INT INT 1MT I'-lT INT INT 1MT I'.'T [MI
I'MT I'T INT I N T INT I'T I'MT INT I'T l"T irT If T INT 1MT INT
INT INT INT INT INT INT INT P'T INT IM INT INT INT INT I"T I"T INT I'-'T IvT INT INT h'T
BFCll
Rrit»-» 0PLi>') RHLbo WF'Lb^
RKLfc'' DFL6'
ot-Lfc.< kFLfc^ PKL6'' prL6" 3PLft7 '• F 1.6 »■ i FL6V
600 L<1JS*1> ■ L<lJS*2«NlP2*n INTRP154 REIURN ISjTRPlSb END
142
10 20
DC 11 12 If UC U
DC U IF
SLbRUUTIhE IPOSN CCtlHÜN// W(?200,7),1.(2200) COMMUN/BNOCPS/ wai(t09,«,7),nPl(109.p)ilND(109.9), lHBJt78,8,7>,pBj(7S,«),JNn(7fl,9) COMMON /MATVAL/ *X,»Y,crL,Vir.VlST,MUYC,PHO»XO.PO, lABL.CHAN.BMAX.FACT.ANQ.AF.Sfl'I.AL.AS.AX.WAL.HA^.HAK, 2.DAY,H0A)(,HDAY,PA)(2.DAY?,HDAXV,0DAXY,DTB.HALFMU.T^T 3UilHfcOTM,NI,NJ,NFMAX.IS,JS.US,IBS,TSB,JSfi.NI Ml, NJM 4,N^NIP2,NTR UC 60 J ■ 2.NJ J4 s tJS*lJ-l)«NlP2 12 » n N = JND(jS*J,l) IF (N.tO.D) Qfi TO 5"
<0 K ■ 2,N,2 « U*X » INT(PBJ<JS*J,K.l)«nAX)*l (u.RT.m no TO 2n 10 I " 11,12 « JJ*l
LUJ) ■ U ii « ia*i 12 • INT(PBJ(JS*J,K)*PAX)*1 IF ( ll.r.T,I2) no TO 40
30 I « 11,12 « JJ*l (KIJl.GT.U GO TO 60
LUJ) ■ Z-LCIJ) CONTINUE CONTINUE II * 12*1 DO 60 I • 11,Nl I- ' JJ*I LUJ) » Q DC 90 I » 1,M Jl » 2 N = INÜ(IS*I,1) IF (N.NE.O) Jl DC 71) J ■ J1,NJ IJ « IJS»(J.1)«NIP2»1
70 L(IJ) « Ü IF (N.LE.l) GO TO 90 K s M DC 80 KK ■ 2,N,2 K = K-2 Jl » 2 IF (K.NE.O) Jl » INT(P3I(IS*I,K)*DAY)*.3 J2 ' lNT(PBI(13*I,K*l)«nAY>*2 IF (J1,GT,J2> GO TO 90 DC 8g J i J1,J2 U = IJS*(J-1)«NIP2*I
YCM ■ (),99*YC N = 0 00 50 j ■ 2,NJ JJ » IJS*(J-1)«NIP2 DO 50 I « 1,M IF (L(JJ*I),EO.O) GO TO 50 YltLD » W?JJ*I,5)**?*W(JJ*I,6)»«2*W(JJ*I,7)** IF (YIfcLD.LT.YCM) GH TO 50 N = N*l TR^ i UUAT(I-1)«AX TRH s FL0AT(J-2)»AY Ch=CHAN«»(BMAX-TRR*YZ) TFHY(N) ■ A-B*<CH-1.0)/CCH*1,0) TFHX(N) ■ X0«(TRZ-X7)*ANG*TFr,Y{N)*TnF*ATAN(AF IF (N,LT.100) GO TO 50 WRITE (2) N,(TFRX(K),TFRY(K),K ■ 1,N)
CChMON/HNUCRS/ WBI<lO9.«,7).Pni(lO9,e>,IND(109,9), BNUCHS Z
COMMON /BNDVAL/ WR(AOO,7).rRX(600),rnY(600).rRS<600) 8MUV*L 2 COMMON /MiURP/ NMAT,IM*T(1?,5),FMAT<23,5)»MAT „«ITWAI ? COMMON /M*mU *X,4V,CrL.VIS.VlST,MU.VC,RH0#X0.P0,T0.XZ,*,P.*BL,DMATVAL Z
lAPL.CHAN.riMAX.FACT.ANG.Ar.SBH.AL.AS.AK.HAL.HAS.HAK.RAL.BAS.OAK.OAXMATVAL J 2.CAy,HDAX.HnAV.DAX2.DAY?.HDAXY.QDAXy.DTR,HAI.rMy,TWTrMU.fRTCHU.00VM^ 4 3UiIRfcOTM,Nl,NJ,NFMAX.IS.JS,!JS,IBS,lSB,JSP.NIMf..NJMl,ND.Nnl,NE,MElMATVAL i
4CCMASiP/^i[Js/ ISTART.TMAX.TPRIN,TPRPL,TPLOT,TSAVE,TCnMP.TZ TVALS ? COMMON /THCPRT/ NTR«;(5),lTR(r.29).TR«(5,25) „„.I'MW., N COMMON /COMVAL/ KT.AKT.DT.DTMIN.lTER.NEIBP.IFOlNT.EPP.EPS.EPSl.RDICnMVAL ,
iS.IOP.UAU.KMAX.PORG'.PSCL.LP.Lf "^*L t COMMON /i'Urc/ MNFr.INFC(60>.NONlN(20.9) ^JJ' ': COMMON /PRESS/ PDER.PR.PE.HATNO ^" ^ DIMENSION WCUT(8) "^I L fcCUlVALENCE(YZ,TO) ^T^ ^
2TEKIAL 1> 0TPTC1 7 ISO FCNMAT (»H0M*TEBULt2illH FROM P01NT|4,9H TO P0INTI4,31H HAS *N INOTKT 103
lTEHr*rfe WITH MATERI*LI2»11H FROH P0INTI4,9H TO POtNTJ«) OTPT 104 190 FOHMAmH059Xl3) OTPTC1 H
EKU OTPT IDb
149
SUflRUUTINF SAVE COMMÜfJ/V W(2200.7t,LJ22Un) COMMON /dNDVAL/ WU(600,7),rRX(600).rPY(600),FHS(60 COMMUN/SCRTCH/ Thn<Jon,7)lTrhXM0O),TrRY(3O0> COMMON /ZONES/ NBECt5i3),NEN0(5.3) . A8CSIZ(5.3>.MS
SAVt 11 SAVF 1? SAVfc n SAVE 14 SAVH 15 SAVf: If SAVE 17 SAVE 11 SAVE 19 SAVE 20 SAVE 21 SAVE 22 SAVE ^^ OFAX 46 OFAX <7 orAX 48 SAVE 25 SAVE 26 SAVE 27 SAVE 25 SAVE 29 SAVE 3n SAVE 31 SAVE 32 SAVE 33 SAVE 34 SAVECl 1 SAVEC1 2 OFAX 49 OFAX 50 SAVECl A SAVE SAVE SAVE SAVE OFAX OFAX SAVE OFAX SAVE
I • 1,3),X7.M.NJSAVE OFAX 54 SAVECl 6 SAVECl 7
(TFRX(1),TFRV( I),SAVr:Cl 8
150
f
k-RlTfc (i) NTR.(jTR<M*T,n,TMS<MAT.I),I ■ l.NTR) DO 2* J«JB,JE JJ»tJS*<J-1)«MIP2
IT2»JJ*IE 25 NRlTkU> ((b(I>KK),KK-l,KHAX).l.(I)>IalTi«IT2) JO CONTINUE
iND FILE l RRTURM fcND
S*vKCl 3 S*Vf *" SAVEClin SAVfeClll S*V6Cll2 SAvecm S4W^C114 S*VE 62 S»VE 5S SAVE t>4 SAVE fes
151
&uäi
SLbHüUTINfc FRAME COMMON /COMPLT/ IXO.IYO.IXN.IYN.IPEN.IMAX.JM
■ inxtxs*x)*jxo ■ IFIX(YS«V)*1V0 « JP <UdS{IP).E0,2) IPPN » .rnuE, (I»ÜS(IP).E0,3) IPEN ■ .FALSE, «,NOT,IPEN) GO TO 40 » MJNÜ(IY,IYN) ■ M*X0«IY,IYN) » MINOdX.IXN) « MAXU(IX,IXN) df^T, IMAX,Ci?,lL.LT,1.0H,JF.OT,Jf'*X.CP.JLiLT,l) GO
iUbRQUTINt PRNT COMMON /COMPLT/ iGiXSiYb LOUICAL IPEN,Fu*0 IX P IP IF IF IF IF IU JF JL {F » 1 r , u i e s i'' Cx a JX-IXN ICY • lY-IYN IF (lAdS(IDX),GT, lABSUDY)) 00 TO 20 IF UDY.tQ.O) GO TO 40 IF « M*XO(IF,1) IL » MINOCIL.IMAX) DC 10 I ■ IF.IL J = IXN*(IDX*(I-lYN))/inY IF «J.LT.l.OR.j.GT.JMAX) GO TO 10 C*LU StTPT <I,J,0) IF ( ,NOT,FLAG) GO Jr 40
10 CONTINUE GO TO 40
20 CONTINUE JF ■ MAXÜ<JF,1) JL « MlNO<JL,JM*X, DC 30 J ■ JF,JL I = IYN*(IDY«fJ-IXN))/inx IF (l.LT.l.OR.I.GT.IMAX) GO TO 30 CALL SbTRT (I.J.O) IF (.NOT.FLAG) GO TO 40
30 CONTINUE 40 IXN ■ IX
IYN ■ IY IF (IP.GT.O) RETURN IXU a IXN pu a JYN REIUHN ENU
SlöROUTINfc SEEK U,wP.PP,K) OlnENSlON A(l) IMEÜEH AiP.PP IT ■ NP IF (JT.FO.O) GO TO ^0 18 ■ 1 p s pp IF <<P-*(l>)/64) 10,20,30 IF ((P-A(lT))/64) 40,50,60 I * 1 GC TO 10Ü I * -1 GC TO 10Ü I * -IT-l ÜC TO 10« 1 - IT GC TO 10U IF ( IT-Id,E0.1) GO TO 90 I = Ud*lT)/2 IF {(P-A(I))/64) 70,100,80 IP ■ I GC TO 60 IT » I GC TO on I = «IT K = I RE HIHN
fUNCTION PRS (RHOO.FE) P»S ? COMMON /PHESS/ PDFR.PR.PE.MiTNO PRfcSS i COMMON /EOSTP/ *PL(lO)iHPL(lO).*PB<lO)i0PR(lO>.*LPH(in);BFT*(lO).EEQSTP 2
lS<10),bZfcHO(lO),ESPRM(10),RHn2RO(lO),P*IM10) EOSTP i LCUICAL MlN PBS b LOGICAL PDER PRS h REAL MU PRS 7 D* I A APl..aPL,APB,pPP,A|.PH,EE"pA#RHO2RC.EZEROiE5,f;SPRK,PM|N/0,5,0.5,PBb e
70, i 02, ü. 056, 0,10 2, 0.180, 0.15, U, 0 »0.0, 0,0, 0.0, 0.0,. 0.0 35',-0,0 0371,-PRb l"5
80,035.-0,0033.-0.035,0.0,0.0.0.0,0.0,0,0/ PRS 1^ C HATUJ'TAROET, MAT(2)«PROJ , MAT(3)«PRnj SHEATH PPS> 17 C PRb li:
MAI ■ MATNO PpS 1« ETA ■ HHOO PPS ZU IF (ETA.üT.0.0) GO TO 10 PRS 21 fcTA ■ -ETA PRb 22 MtN ■ .FALSE. PfS 23 GC TO 20 P^S ?4
10 MIiN » ,THJE. P^S ^^ 20 CCNTINUE PBS üh
E = EE PPS 27 R = fc-5A«RHOZRn(MAT) PPS 2H ML s ETA.1.0 PRS 2<5 irsÄc(S(MU'.LT,l,OF-(i) iinn.o PRS 3n E- ■ (-.•iL(MAT)/(E/(E7ERÜ(MAT)«FTA**2)*l,0) PRS 31 IF (fcTA.Ufc.l.0.OR,E.Lt,FS(H*Tn GO TT 30 PRS M IF <E,GE.eSPRM(MAT)» r,0 Tfl 50 pi'S 33
C PPS 34 C CUMPRfcSSlON PRS 35 C PRb 3ft
30 PRSC « (APL«MAT)»FX»*E*R»(APn(MAT)*BrB(KAT)*^l)*M(J PRS 37 IF (MlN) PRSC « AMA)(l(PMSr,PHIN(MATn PRS 30 IF (,NUT,PDER) GC To 40 PpS 39 PRL ■ (APL{MAT>«E*)«E*(APP(MAT).2,0*RPB(|'AT)«MU)/PHrZRO<MAT)*2l0*(PRb 40 :E*tX/ETA)»«2/(EZER0(MAT)«PPL(MAT)) PRS 41 PEL ■ (APL(HAT)*E)(-F«(EX/'ETA)««2/(EZER0(MAT)*BPL(MAT)))*R PRS 4?
40 IF (ETA.LM.0,ANn.F,OT.FS(MAT)) GO TO !C PPS 43 PRb ■ PRSC/RHC2R0(MAT) PRS 4* IF (,NUT,PDER) RETUPN PRS 45 PR ■ PHC PPS 46 PE ■ PfeC/HHOZRO<*AT) PRS 47 REIUHN PRS 48
50 EV » EXP(-ALPH(MAT)«(MO/ETA)««2) PPS 4V EZ ■ EXP(HETA(HAT)*MU/ETA) PRS 50 pfibE » APL(MAT)»E»R*(E*R»FX*APb(MAT)»ML»F7)»EV PWS 51 IF (MIN) PRSE « AMAX1(PRSF,P>,IN(MAT)) P^S 52 IF (.NQT.PDFR) r,3 TO 60 PRS 53 PRt ■ APL(M*T)*E«((F»R«F)(*APn(MAT)«ML*EZ)«2,D»ALPw<^AT)«Mu/(P*ETA*PRb 54 l*2)*E*6X*i.Q»(E»Ev/PTA)*.2/(r7ER0(M»T)*PPL(MAT))*(APB(MAT)/Ph0ZR0(PRS bb 2MA1 )*ApB(MAT)*MU»RfcTA(MAT)/(rTA«R>)«"Z)»FV PRb 66
160
Pfet ■ UPI.(MAT)*(EX-EMEX/FTA)««2/<EZER0(MAT)«BPL<MAT)n*FY)*R 60 IF (E,LT.ESPRM(«*T)) r,0 TO 7C
Pfiä ■ PRSfc/RHOZR0(M»T) IF (,N0T,PDER) RETURN PR s PHE Pfe « PEE/HM02RO(M»T> REIURN
70 A = (E-Eb(MAT))/(ESPRM(MAT)-E9(HAT)) B s (FSPRN(MAT)«E)/(ESP»»M(MAT)-BS<MAT)) PHs ■ (*«PRSE*B*PPSC)/''^02R0<MAT) IF (.NUT.PDER) RETURN MR i A*PRfe*B«PRC PF. ■ (A*P6E*B«PEC)/RH0ZH0(MAT) RE I URN