AUTODYN™oss.jishulink.com/caenet/forums/upload/2008/01/27/... · · 2015-09-12Chapter 2. ALE...

AUTODYN™ Explicit Software for Nonlinear Dynamics

ALE Tutorial Revision 4.3

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Table of Contents

PREFACE........................................................................................................................................... 1

Chapter 1. Introduction................................................................................................................. 3

Chapter 2. ALE Tutorial Problem No. 1 (ALEXP1)...................................................................... 5

Chapter 3. ALE Tutorial Problem No. 2 (ALEXP2).................................................................... 25

Chapter 4. ALE Tutorial Problem No. 3 (ALEXP3).................................................................... 31

Chapter 5. ALE Tutorial Problem No. 4 (HCDA) ....................................................................... 35

Appendix A. Subroutine EXALE(2D) for Tutorial Problem #3 ................................................... 41

Preface

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Preface AUTODYN Tutorial Manuals AUTODYN tutorial manuals provide detailed tuition on particular features available in the program. The manuals assume that you are proficient in setting up, reviewing, executing, and post processing data for simple problems such as those presented in the AUTODYN-2D or AUTODYN-3D demonstration manuals. If you have worked through the problems in the demonstration manual, you should have no difficulty following the tutorials. Most tutorials are interactive and you are expected to have access to AUTODYN while working through the tutorial. Some tutorials have associated files which contain sample calculations used in the tutorial. Existing manuals are continuously updated and new manuals are created as the need arises, so you should contact Century Dynamics if you cannot find the information that you need.

Chapter 1. Introduction

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Chapter 1.

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Introduction The ALE (Arbitrary Lagrange Euler) processor in AUTODYN-2D and AUTODYN-3D allows you to use a combination of the best features of both the Lagrangian and Eulerian methods. The multiple processor feature of AUTODYN includes Lagrange, Euler, and Shell capabilities, as well as ALE. In a Lagrange numerical mesh, the gridpoints move with the particle motion of the material. Typically, Lagrange methods are very good for solids behavior wherein the material motions do not create large mesh distortions. Material interfaces and history dependent material behavior are well described using Lagrange. The major disadvantage of Lagrange is that for severe material distortions or movement, the numerical mesh may become tangled causing inaccuracies in the solution or even complete stoppage of the calculation. In Euler, the numerical mesh stays fixed in space and the material flows through it. Material is transported from cell to cell and, therefore by definition, there is no possibility of mesh distortion. Euler is well suited for fluid and gas behavior and can be applied to solids where very large distortions may occur. The primary disadvantage of Euler is that material interfaces, free surfaces, and history dependent behavior are more difficult to model than in Lagrange. The ALE processor in AUTODYN-2D and AUTODYN-3D allows the numerical mesh to either:

Move with the material as in a Lagrange mesh

Stay fixed in space as in an Euler mesh

Or move in an arbitrarily specified manner to provide continuous, automatic rezoning

ALE, while encompassing both Lagrange and Euler methods, has some important limitations which can restrict its applicability to certain problems. The primary limitation of ALE is that material interfaces and free surfaces must still be treated as Lagrange. Regions within a single material may be rezoned according to an ALE specification but not at material boundaries. Since many numerical difficulties tend to occur at material interfaces, the ALE technique does not necessarily avoid such difficulties. Nevertheless, ALE provides a substantial capability with certain advantages over purely Lagrange and Euler formulations for many problems of interest. This tutorial shows you how to apply the ALE technique to some simple problems. These problems are included in your AUTODYN distribution files. The following discussion pertains to both AUTODYN-2D and AUTODYN-3D. Short note on terminology Various terms are used interchangeably throughout the AUTODYN manuals in referring to portions of the numerical mesh. For clarification please note the following: A numerical mesh is composed of a rectangular I-J (2D) or I-J-K (3D) index space. I-lines (lines of constant I) are sometimes referred to as “columns” while J-lines (lines of constant J) are called “rows”. In 3D, reference is made to “I-planes” (constant I), “J-planes” (constant J) and “K-planes” (constant K).

Chapter 1. Introduction

A column or row is made up of a series of “nodes”. These nodes are sometimes equivalently referred to as “vertices” or “gridpoints”. A set of four (2D) or eight (3D) nodes defines a “zone”. A zone is generalized quadrilateral (2D) or brick (3D). Such “zones” are also interchangeably referred to as “cells” and “elements”. Basic Theory In AUTODYN, the calculation of an ALE subgrid (a subgrid is defined as a single independent I-J (2D) or I-J-K (3D) index space) is identical to that of a Lagrange subgrid except for the addition of a possible rezoning phase at the end of each computational cycle. During this rezone phase, motion constraints are applied to grid vertices and then relaxed, as specified by the user, to new positions. For example, you could choose to keep vertices (grid points) at the average position of their surrounding neighbors. Once these new vertex positions are defined, the corresponding advective flux calculations (as for an Eulerian calculation) are performed to allow material to move from cell to cell. The net effect is a type of automatic rezoning. With the judicious selection of the “right” algorithm for the specification of the grid motion, you can obtain a more efficient as well as more accurate solution to your problem. Because of the additional rezoning phase, a computational cycle (timestep) will take longer to compute than for an equivalent purely Lagrange cycle. However, the savings from the elimination of time consuming discrete Lagrange rezoning (see the AUTODYN Rezone Tutorial) and a possibly larger timestep provided by the improved mesh can more than compensate for the additional computation required.

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Chapter 2. ALE Tutorial Problem No. 1 (ALEXP1)

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Chapter 2. ALE Tutorial Problem No. 1 (ALEXP1) Your AUTODYN distribution contains the sample problems used in this tutorial. This first ALE problem is similar for both AUTODYN-2D and AUTODYN-3D. A highly compressible foam material is impacted against a rigid wall. Using a purely Lagrange technique it is shown that the mesh compacts to very long and narrow zone sizes. Such zones are not very accurate. Moreover, they can lead to small time steps or even termination of the calculation due to mesh distortion and tangling. Employing an ALE technique to automatically rezone the mesh allows a much more efficient and accurate solution. The ALEXP1 problem in AUTODYN-2D is a cylinder (axial symmetry) while the AUTODYN-3D problem is a 3D rectangular block. Appropriate velocity boundary conditions have been specified in both problems to obtain a one-dimensional problem for purposes of illustrating the ALE technique. Begin by activating AUTODYN on your computer in the normal fashion: type AUTORUN2 or AUTORUN3. Then load Ident: ALEXP1. The screen images shown below are from AUTODYN-2D but will be virtually identical for AUTODYN-3D.

Choose Modify from the main menu and then View the problem that has already been set up by selecting Matplot, Vecplot, and Boundplot:


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In both the 2D and 3D simulations only one-dimensional (horizontal) motion has been allowed. This is accomplished by constraining the y-velocity in 2D and both the y- and z-velocities in 3D. (For the 3D case, you may view these boundary conditions by selecting the displayed boundary condition names as they appear in the pop-up window.) Constraining the x-velocity to 0 simulates a rigid wall on the right side of the subgrid. The problem has been initially defined using a Lagrange processor. If you run this problem you will obtain the following grid at ~1 microsecond:

(Note: the 3D solution requires a slightly different number of cycles to reach the 1 microsecond wrapup time) Note that the first several columns on the right have been compressed to a very small width. This can be better seen using the Zoom function as shown below:

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Such highly distorted zones are generally undesirable for reasons of accuracy and efficiency. One solution would be to use the interactive rezoner to respace vertices in the impact region and make the zones larger. An alternative solution is to run the calculation using the ALE processor to maintain an optimum grid. We will employ the ALE processor by modifying the above sample problem. The foam subgrid will be changed from Lagrange to ALE and then suitable motion constraints will be applied to the vertices of the numerical grid. The Lagrange and ALE processors are fully compatible in that the user may switch between the two formulations at any time before or during a calculation in any given subgrid. We first reload cycle 0 of ALEXP1. To change the processor type for the foam subgrid (CYLINDER), first select Subgrid from the GLOBAL/SUBGRID menu:


(Note that a Material Location plot has been displayed above, previously obtained by View-Material-Location) We have only one subgrid named CYLINDER (in 3D the subgrid name is ONE) which is displayed in the popup window. We select CYLINDER (or ONE for 3D) by pressing return.

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We then select Options.


We select Processor and then switch from Lagrange to ALE by pressing the space bar once. Note that AUTODYN only allows switching between Lagrange and ALE. Other processors (e.g. Shell, Euler) are not compatible and thus are not offered as options. The Active option is not used here. Active is only used in situations where you want a subgrid to be activated or deactivated in a calculation starting at a specified time. This might be useful in saving calculational time, when you know the subgrid will have no activity for a period of time. We press <return> to close the popup window.

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We return to the Subgrid menu by pressing <escape> once. A new Motions option, applicable to ALE subgrids only, now appears. The Motions option is used to specify the motion constraints for the ALE vertices. We select Motions.


The following Motion options are then available:

Block: Assign a motion constraint to a block of vertices in index space. Note that within a subgrid, different blocks may use different motion constraints.

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I-line: Assign a motion constraint to vertices along an I-line

J-line: Assign a motion constraint to vertices along a J-line

Node: Assign a motion constraint to a single vertex

Spacing: Set geometric ratios used in constraints that utilize geometric spacing

Frequency: Define the frequency for ALE rezoning. By default the rezoning will occur every cycle. It is recommended that this default be taken for most cases.

Relax: Define the relaxation parameter for ALE rezoning. Relax has a value 0.0 ≤ Relax < 1.0. By default, Relax is set to 1.0, which denotes that the node will be moved completely to its newly defined ALE position. Numbers less than 1.0 specify that the node will be moved that fraction from its current position to the new ALE position. It is recommended that the default be taken for most cases.

View: View the current motion constraints as applied to the subgrid

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If you select View, you will see that initially, by default, all vertices are free of any motion constraints and are treated as Lagrange. An ALE subgrid without motion constraints will be equivalent to Lagrange, except that it will take computationally longer.


Press <escape> once to return to the Motions menu. The first four options of the Motions menu allow you to assign motion constraints to various portions of the subgrid. We select Block and enter the above data. By default, the entire subgrid range is selected. We wish to apply the same motion constraint to the entire subgrid so we simply press enter four times. We then have various options for the motion constraints:

Equipotential: This option is perhaps the most general-purpose choice when the expected flow is unknown. An equipotential algorithm is used to position the vertex relative to its nearest neighbors.

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Free (Lagrange): This is the default. Each vertex moves in a free Lagrangian manner

Fixed (Euler): Each vertex is fixed in (x,y) space.

Equal spacing in X: The x-coordinate of each vertex (i,j) is moved to the average position of its four neighbors. The y-coordinate is unchanged.

Equal spacing in Y: Analogous to equal spacing in X

Equally spaced I: Each vertex is moved to a point equidistant between adjacent I-lines (columns)

Equally spaced J: Analogous to equally spaced I

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Equally spaced (I, J): Each vertex is moved to the midpoint between adjacent I-lines and J-lines

Geom. spaced I: Analogous to Equally spaced I except that the specified geometric ratio will be used to space I-lines in the direction of increasing I.

Geom. spaced J: Analogous to Geom. spaced I

Geom. spaced (I,J): Analogous to Equally spaced (I,J) except that geometric ratios can be applied to the spacing

Flow averaged: This option is useful in situations where the flow is diverse and none of the previously described options are obviously applicable. Flow averaging is designed to maintain good spacing of grid lines in the direction of flow. An octagon is constructed around each (i,j) vertex by joining the eight surrounding vertices. The velocity vector at (i,j) intersects this polygon at two points. The vertex (i,j) is moved to the midpoint of these two intersections.

User defined: This option allows you to specify your own motion constraints using subroutine EXALE. By including your own custom subroutine you may specify an arbitrary positioning of vertices.

For this example problem, which is one-dimensional, we choose Equally spaced I. Note that in AUTODYN-3D the same motion constraints as above are available with the addition of the analogous features for the z-direction and k-index.


Selecting View, you are shown the above picture(without I,J notations) which indicates that I=1 and I=IMAX will be treated as Free (Lagrange). Remember that free surfaces and material interfaces in an ALE subgrid must be treated as Lagrange, except for certain special conditions. Thus, even though we have specified Eq.Sp.I for the entire grid, AUTODYN automatically resets any inconsistent motion constraints. After redefinition to Lagrange, any other specified boundary conditions (stress, velocity, etc.) will then be applied. In our case, the previously defined fixed velocity condition at I=IMAX, representing a fixed wall, will be utilized. Any nodes that are co-linear (2D) or co-planar (3D), even though they may be on a grid boundary, may have an ALE constraint so long as the nodes remain co-linear or co-planar. In our example case, J = 1 and J = JMAX which are initially co-linear and will remain so because of imposed velocity conditions. Thus, they will remain as “Equally spaced I” even though they are on a grid boundaries. We can verify this by selecting Next.

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We now see that the vertices at J=1 and J=JMAX will use the Eq.Sp.I motion constraint.


We return to the Motions menu by pressing <escape>. Additional Motions options are available. The Spacing option allows you to set the geometric ratios used with motion constraints which utilize ratios (e.g. Geom. spaced I, etc.) . The default ratios are 1.0 as shown above. You can define one set of ratios for each subgrid. The equally spaced I-line specification does not use geometric ratios and so we can skip this option by pressing <escape>.

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The Frequency option allows you to specify the frequency of cycles where ALE rezoning will be performed. The default as shown is every cycle (1). This is the recommended value for most cases. For our problem we will keep the recommended default of 1 by pressing return.


The Relax option specifies a relaxation coefficient for the ALE rezoning. The relaxation parameter defines the rate at which vertices are relaxed to the positions specified by their motion constraints. The coefficient value must be 0.0 ≤ Relax < 1.0 . If the value is 1.0, vertices are moved directly to their constrained positions immediately. If the value is 0.5, they would be moved halfway from their current position to the constrained position. The default, as shown, is 1.0 . If there are significant differences between the initial coordinates and the constrained positions, it is recommended to specify the relaxation coefficient less than 1.0 to allow the vertices to move over several rezones. For example, a coefficient of 0.2 will allow the gradual movement of the vertices to their constrained positions over several ALE rezone cycles. In our example, the initial coordinates and the constrained positions are both equally spaced I-lines, so there will not be an immediate large ALE rezoning required. The default of 1.0 can be therefore be taken. ALE rezoning will be most accurate when the remapping phase does not involve large advective fluxes (i.e. large differences between current vertex and constrained positions). Generally, it is recommended to always utilize the default frequency of 1 and the default relaxation coefficient of 1.0. This has been proved to provide a more efficient and more accurate solution as the advective fluxes in a given rezone cycle are minimized. The input for this example is now complete. We return to the main menu by pressing <escape> several times. We can Save the database and then Execute.

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Upon reaching the Wrapup time the gridplot below appears. Note that now that the I-lines are equally spaced and that reasonable zone sizes are maintained as compared with the purely Lagrange calculation (shown on page 9).

There are very different internal mesh sizes between the two calculations. The ALE simulation provides a very regular mesh while the Lagrange approach has extremely small zones. Moreover, in this example it only takes the ALE calculation 29 cycles to achieve the 1 microsecond wrapup time vs 103 cycles for the Lagrange calculation. This is a direct result of the larger timestep allowed by the larger zonal dimensions of the ALE simulation. (Note: if you are comparing the 2D and 3D solutions there is a difference in the number of cycles required between 2D and 3D. The timestep is calculated differently in the two programs because of the additional dimension in 3D). Final note: As in most situations, there is always a trade off in using different numerical techniques. It has been shown above that the ALE technique can provide a much more regular mesh. On the other hand, it is also true that the spatial resolution at the impact front will be much finer for the Lagrange calculation. In the Lagrange case, many cells define the gradient that will exist in the impact region versus ALE where a single “averaged” cell must resolve this the spatial variation. This can be readily demonstrated by producing a profile plot of pressure along the foam subgrid. Utilizing the Post Processing profile plot feature produces the following ALE and Lagrange results, respectively:


ALE Calculation

Lagrange Calculation

The large pressure peak is averaged out in the ALE calculation. However, the impulse at the wall (an integral quantity) will be comparable for the two calculations.

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Chapter 3. ALE Tutorial Problem No. 2 (ALEXP2) A second ALE tutorial problem is provided in which a solid unstressed steel cylinder is translated with a fixed velocity. As the cylinder translates in space, the internal vertices undergo major rezoning through use of the ALE processor. This example, while artificial, does show the power of the ALE processor. Load cycle 0 of this problem (Ident: ALEXP2) and select Modify, View, Gridplot:


Selecting Vecplot we can see the constant translational velocity.

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The ALE motion constraints may be viewed by selecting Subgrid, Motions, View. Equally spaced (I,J) has been specified (free Lagrange overwrites the option of equally-spaced (I,J) ). Note that the outer free surface will be automatically treated as Lagrange with no motion constraints. Choosing Next we see that the interior nodes will be constrained according to Eq.Sp.(I,J):


The Frequency has been specified as 1 (every cycle) and the Relaxation coefficient as 0.2 . Use of a Relaxation coefficient less than 1.0 is important as the initial coordinates initially differ greatly from the specified ALE positions ( Eq. Sp (I, J) ). This will allow the coordinates to be adjusted gradually. Note that once the coordinates have “found” their natural position, the user may reset the relaxation coefficient to 1.0 .

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Execute the problem until wrapup (10 cycles) and you will see the grid has been gradually rezoned to the much smoother zoning shown above. Advective fluxes are accurately computed during the ALE rezoning so that the cylinder remains unstressed and maintains its initial constant velocity throughout the calculation.

The equivalent AUTODYN-3D problem also produces a similar mesh. Note that the velocity in 3D has been specified at an angle as a variation on the 2D problem.

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Chapter 4. ALE Tutorial Problem No. 3 (ALEXP3) If the standard motion constraints offered by AUTODYN do not meet your needs, you may provide your own constraints by including the user subroutine EXALE. A detailed discussion of how to include user subroutines in AUTODYN is given in the User Subroutines Tutorial. If you are not familiar with the procedure, please refer to that tutorial. The EXALE capability exists for both AUTODYN-2D and AUTODYN-3D. However, this third tutorial problem is only provided for AUTODYN-2D. A subroutine EXALE.TUT which is a small Fortran program is included with your AUTODYN-2D distribution. This EXALE.TUT should replace the regular (dummy) routine included in USRSUB2.F . Start AUTODYN-2D and Load ALEXP3. If you quickly view this problem you will see that it consists of a tantalum cylinder impacting a rigid wall. The grid is similar to the one used in the first sample problem, but the outer surface of the cylinder can deform radially in this case.


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The subroutine EXALE shown in Appendix A and included with your distribution will provide a constraint that keeps all I-lines vertical by setting the horizontal (x) position of each vertex on a given I-line to the same x-coordinate as the vertex at the outer (JMAX) row. In addition, the subroutine equally spaces the vertices along each I-line. The subroutine logic is straightforward and is documented within the routine. Executing the problem for 100 cycles provides the following grid:


The user defined constraint works properly with all of the I-lines remaining vertical and the nodes along each I-line spaced equally.

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Chapter 5. ALE Tutorial Problem No. 4 (HCDA)

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Chapter 5. ALE Tutorial Problem No. 4 (HCDA) The above examples were relatively simple in formulation. The following example illustrates some of the real power of the ALE technique used in a three-dimensional simulation. The problem is a TNT explosion inside a water filled tank. The tank walls in this example are taken as rigid. When calculated using a Lagrange formulation, the TNT rapidly expands, in the process compressing the surrounding water and causing large distortions and “crushing” of the water cells. This can lead to inaccuracies and the premature termination of the calculation. An alternative is to employ ALE which will maintain the water region with a much more regular mesh. The HCDA (Hypothetical Core Disruptive Accident) problem is only included with the AUTODYN-3D distribution. Start AUTODYN-3D and Load Ident: HCDA. The problem is already completely defined and ready to Execute. View the problem setup by producing various plots:


You see two identical 1/4 cylinders. The left cylinder uses a Lagrange processor while the right cylinder uses an ALE processor. The ALE motion constraint is specified as equipotential for the entire subgrid. AUTODYN will automatically redefine material interfaces and free surfaces as Lagrange as required. Running the Lagrange and ALE formulations side by side allows you to observe the differences between the two approaches. Planes of symmetry are specified for X and Y.

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A small charge of TNT is located along the axis of each cylinder. This is detonated at time=0.0 . The expansion of the explosion and compression of the surrounding water is apparent. The water free surface is accelerated upward eventually impacting on a steel cover plate. Note that the ALE solution provides a much smoother mesh in the water region. This allows a more accurate and potentially much longer problem time to be attained. The Lagrange solution will eventually terminate because of tangled and distorted water cells.


It is instructive to view the Motion constraints for this problem. At cycle 0 the entire ALE subgrid is specified with an Equipotential constraint. AUTODYN then checks the ALE motion constraints and automatically redefines any material interfaces and free (non-planar) surfaces as Lagrange. Note that nodes that lie along symmetry planes, even though they are on the surface of the mesh are not interpreted as free surfaces and will still be constrained to be rezoned according to the equipotential specification. Moreover, nodes on the top surface of the mesh which initially lie on a plane will be treated as Equipotential until the free surface becomes non-planar due to the dynamic upward movement of the water.

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You will also note that the internally joined nodes necessitated by the Type 2 cylinder predef are treated as Lagrange.


Viewing the ALE motion constraints at cycle 150 after the upper surface of the water starts to move and is no longer planar, the nodes are now treated as Free (Lagrange) and will not be subject to rezoning. At a later stage, should these nodes again lie on a plane due to impact on the upper plate, they could again be subject to equipotential ALE rezoning. Thus, AUTODYN will automatically apply the ALE motion constraints whenever it is possible to do so and will dynamically redefine these constraints, between Free Lagrange and the specified ALE motion constraints. You are referred to the publication “Calculation of Blast Loading in the High Performance Magazine with AUTODYN-3D”, Department of Defense Explosive Safety Seminar, Miami, 1994 for further illustration of the use of the ALE processor of AUTODYN-3D for the calculation of explosive-structure interaction problems.

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Appendix A. Subroutine EXALE(2D) for Tutorial Problem #3

Appendix A. Subroutine EXALE(2D) for Tutorial Problem #3 SUBROUTINE EXALE (IREZ,JREZ,NREZ,XREZ,YREZ)

USE mdgrid

USE kindef

USE locsub

USE wrapup

IMPLICIT NONE

INTEGER (INT4) :: IREZ, JREZ, NREZ

REAL (REAL8) :: XREZ, YREZ

INTEGER (INT4) :: IJMAX, IJ

REAL (REAL8) :: DYB

! ******************************************************************

!

! THIS SUBROUTINE COMPUTES THE CONSTRAINED GRID VELOCITIES FOR

! VERTICES ASSIGNED THE "USER" MOTION CONSTRAINT

!

! INPUT PARAMETERS:

!

! IREZ - I INDEX FOR VERTEX TO BE CONSTRAINED

! JREZ - J INDEX FOR VERTEX TO BE CONSTRAINED

! NREZ - SUBGRID # FOR VERTEX TO BE CONSTRAINED

! XREZ - CURRENT X-COORDINATE OF VERTEX

! YREZ - CURRENT Y-COORDINATE OF VERTEX

!

! OUTPUT PARAMETERS:

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! XREZ - CONSTRAINED X-COORDINATE OF VERTEX

! YREZ - CONSTRAINED Y-COORDINATE OF VERTEX

!

! ******************************************************************

!

! THIS LOGIC SETS THE X-COORDINATE OF EACH VERTEX (I,J) TO THE

! X-VALUE AT VERTEX (I,JMAX) AND EQUALLY SPACES THE Y-COORDINATES

! BETWEEN (I,1) AND (I,JMAX)

!

IJMAX = IJSET(IREZ,JMAX)

IJ = IJSET(IREZ,JREZ)

XREZ = XN(IJMAX)

DYB = YN(IJMAX)/FLOAT(JMAX-1)

YREZ = DYB*FLOAT(JREZ-1)

!

! TERMINATION OF SUBROUTINE EXALE

RETURN

END SUBROUTINE EXALE

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