AUTODYN. Explicit Software for Nonlinear Dynamics. Jetting Tutorial. Revision

Transcription

1 AUTODYN Explicit Software for Nonlinear Dynamics Jetting Tutorial Revision AUTODYN is a trademark of Century Dynamics, Inc. Copyright 2005 Century Dynamics Inc. All Rights Reserved Century Dynamics is a subsidiary of ANSYS Inc,

2 Century Dynamics Incorporated 1001 Galaxy Way Suite 325 Concord CA U.S.A. Tel: Fax: Century Dynamics Limited Dynamics House Hurst Road Horsham West Sussex, RH12 2DT England Tel: +44 (0) Fax: +44 (0) Century Dynamics Park Ten Place Houston TX USA Tel: Fax: Century Dynamics Röntgenlaan DX Zoetermeer The Netherlands Tel: Fax:

3 Table of Contents CHAPTER 1. INTRODUCTION... 1 CHAPTER 2. JETTING THEORY Standard Jetting Analysis Optional Jetting Analysis...6 CHAPTER 3. SETTING UP A JETTING CALCULATION Review of problem setup Review of material models Jetting definition Setting the detonation point...22 CHAPTER 4. POST PROCESSING DATA FROM A JETTING CALCULATION CHAPTER 5. COMPARISONS WITH EXPERIMENT AND OTHER METHODS CHAPTER 6. REFERENCES... 36

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5 Chapter 1. Introduction Chapter 1. Introduction Shaped charges are used in a variety of commercial and defense related applications. The phenomena associated with the high explosive detonation and extreme deformations of the shaped charge casing and liner poses a challenging task for numerical analysis. AUTODYN provides the analyst a number of different options for shaped charge design. These options range from using a highly efficient jetting option to the use of the full Euler/Lagrange/Shell processor capabilities within the program. The jetting option, discussed in this tutorial, uses a combined numerical and analytical approach. Two-dimensional finite difference grids (Euler and/or Lagrange) represent components such as the explosive and casings, while shell grids model the liner and other thin components. For the liner, modeled as a thin shell, an additional analytic jetting option is invoked which allows the liner to behave as a shaped charge jet and slug. Although thickness is considered in the shell formulation, the shell thickness is not included in the geometric representation of the shell. Thus, a shell grid is simply a series of nodes joined by linear segments. This representation has two distinct advantages over a regular 2D finite difference grid when modeling shaped charge designs: If a full two-dimensional explicit finite difference grid is used to model a thin component, several zones have to be used through the thickness in order to model bending effects accurately. This results in a small time step for the calculation. In such cases bending effects can be modeled more efficiently by a shell formulation which has a stability timestep dependent only on the segment lengths, not on the shell thickness. If a two dimensional Lagrange grid is used to model the liner, the large cell deformations that occur in the stagnation region during jet formation will cause the run to be terminated early unless substantial rezoning of the grid is performed. AUTODYN has one of the most powerful interactive rezoning capabilities currently available in a finite difference code, so it is possible to carry out a complete analysis in this way. Another approach is to use an Eulerian grid for the definition of the liner. By definition, this avoids any mesh distortion. To accurately perform such an analysis using Euler requires a large, finely zoned mesh. Full Lagrange or Euler analyses are certainly within the capabilities of AUTODYN but another engineering solution is also available, the jetting option, which allows very quick turnaround design calculations. In the jetting option the liner is modeled as a shell grid, with the explosive modeled with full Lagrange or Euler detail. In this way, the full hydrodynamic equations of motion are used to compute the collapse velocity and angle for each shell mass point. These values are then used in conjunction with an analytic jet formation algorithm to obtain the jet and slug masses and associated velocities. 1

6 Chapter 1. Introduction Section 2 of this tutorial outlines the jetting theory used in AUTODYN. Section 3 shows you how to set up a jetting calculation. The post processing of jetting data is described in section 4. The jetting example used in this tutorial is based on a tested design for a 90 mm charge with an 18 degree conical liner and results of the calculation are compared with experimental data in section 5. 2

7 Chapter 2. Jetting Theory Chapter 2. Jetting Theory 1. Standard Jetting Analysis During an AUTODYN computation cycle, the coordinates of each shell mass point are updated from time t n-1 to t n using: u u n 12 x n 12 y = = n n ( ) n u + f + f dt x n 1 xs xe m n n ( ys ye ) n u + f + f dt y n 1 m n n 1 n 1 x = x + u 2. dt n 1 2 x n n 1 n 1 y = y + u 2. dt n 1 2 y where (f xs, f ys ) ( ux, uy) (f xs, f ys ) m dt internal shell forces external forces from the explosive velocity components mass of the shell point time step With the new coordinates known the jetting analysis proceeds as follows: When a string of jetting points is created, AUTODYN automatically arranges them in the order in which they are expected to jet (the point closest to the axis is expected to jet first). Assume that the point j-1 has jetted on a previous cycle and we are testing for jet j+ 12 / formation at point j. Also assume that point j+1 exists. We define the angle β 0 to be the slope the segment from j to j+1 at the instant that point j jets. 3

8 Chapter 2. Jetting Theory j+ / dy tan( ) = dx j+ 12 / 12 0 β 0 j+ 12 / 0 j+ 12 j+ 1 dy = y y 0 / j 0 0 j+ 12 / j+ 1 dx = x x 0 0 j 0 where the zero subscript indicates values at the instant of jet formation. The angle β at point j is then computed as an average of the two adjacent segments. dy j tan( ) = dx + dy j 12 / j+ 12 / 0 0 β 0 j 12 / j+ 12 / 0 + dx0 Note that this equation has no geometric interpretation since the jet formation times of the two adjacent segments are different. The jetting algorithm assumes that, on jetting, the mass point j splits into a slug and a jet according to the theory given by Pugh, Eichelberger and Rostoker in reference [1]. Using this theory, the jet and slug masses and velocities are: m m j jet j slug = m j = m 1. j 1+. j cos( β 0 ) 2 j cos( β 0 ) 2 j j j j j u = u.(sin( a ) + cos( a ) b ) jet 0 u u a a B j j j j j slug = 0 (sin( ) cos( ) ) where j 2 2 u = ( u + u ) 0 x0 y0 4

9 Chapter 2. Jetting Theory u j tan( a ) = u b j j x0 j y0 j sin( β0 = )j 1 cos( β ) 0 The mass point j is assumed to split into a slug and a jet at the time that its radius j equals the anticipated slug radius,. This radius is related to the slug volume by: y slug where V j slug 2 12 π ( yslug ).( d0 + d = 2 j j / j+ 12 / 0 j 12 / j 12 / 2 j / d = ( dx ) + ( dy ) ) From the jetting equations, the slug volume is related to the pre-jetting volume (computed for each mass point at the start of the calculation) by V j slug V = j j ( 1 + cos( 0 ) 2 β ) We can solve these last two equations to give us the slug radius Y j slug = j j 2V ( 1+ cos( β0 )) j 12 / j+ 12 / π ( d + d ) 0 0 If mass point j-1 or j+1 does not exist, the above equations are still valid if the non-existent point is assigned the same coordinates as point j. These equations are used to determine if jet formation occurs at mass point j during the current cycle. If the radius of point j is less than the computed slug radius at the end of the cycle then a binary search is performed to determine the exact time at which jetting occurs during the cycle. Once point j has jetted, focus switches to point j+1. The logic allows more than one point to jet in a single time step. After mass points have been tested for jetting, the cumulative slug mass and momentum is calculated by summing over all jetted points. From these sums, a mean slug velocity is calculated and assigned to the x-velocity component of each jetted point (the y-velocity component is set to zero). This averaging keeps a stable profile for the slug but has no effect on points that have still to jet. 5

10 Chapter 2. Jetting Theory 2. Optional Jetting Analysis An optional jetting analysis has been included in AUTODYN which uses post calculation values to provide an improved estimate of the collapse angle, β, and the dependent jetting parameters. The following diagram shows the vector triangle formed by the collapse velocity,, the velocity of the stagnation point, u, and the velocity of the jet relative to the stagnation point, u r. c u 0 u r u o a ß u c u c and u r are related to u jet and u slug by: u u c r = = u u jet jet + u 2 u 2 slug slug From this diagram the following relationship can be obtained: tan( β ) = u0 cosa u u sina c 0 The optional jetting analysis uses the fact that once a calculation is complete, the locus of the stagnation point is defined by the values of ( t, x ) obtained for each jetted point and thus u c can be obtained by differentiating this curve. The differentiation is performed by constructing a quadratic through the points j-1, j and j+1 and taking the derivative at point j (except for end points where the derivative is taken at point j-1 or j+1 as required). With u c known, the above equation can then be used to obtain new estimates for the collapse angle, β. In general these values will not be the same as those computed during the calculation jet jet 6

11 Chapter 2. Jetting Theory and because they yield a more consistent set of data, tend to give better estimates of the jetting parameters. 7

12 Chapter 3. Setting Up A Jetting Calculation Chapter 3. Setting Up A Jetting Calculation 1. Review of problem setup Cycle zero of a sample jetting calculation has been included on the tutorial diskette. Activate AUTODYN on your workstation and load this file with has ident JETTNG. (This model is similar to the INTER3 Interaction Tutorial Model. See the Interaction Tutorial for details on setting up this problem.) The calculation is based on a tested design for a 90 mm charge with an 18 degree conical liner (results are compared with the experimental data in section 5). A majority of the input for the calculation has already been generated. This allows us to skip over the standard input required for most calculations (geometry, material assignments, etc.) and focus on the special input required for a jetting calculation (jetting points and explosive burn). The input units used are (cm, gm, µsecs). Before we proceed to input the remaining data, take some time to review the problem as it has previously been set up. Choose the Modify option from the main menu followed by View, Gridplot & Matplot to view the four subgrids that have been generated and the associated material assignments. 8

13 Chapter 3. Setting Up A Jetting Calculation Gridplot of shaped charge model Material Location Plot 9

14 Chapter 3. Setting Up A Jetting Calculation Enter the subgrid menu and view the zoning, material allocations and boundary conditions for each subgrid. The following figures illustrate the four subgrids. Liner (Shell subgrid, copper, constant thickness of 2mm) 10

15 Chapter 3. Setting Up A Jetting Calculation Explosive (Euler subgrid, Octol and initial void regions) 11

16 Chapter 3. Setting Up A Jetting Calculation Case (Lagrange subgrid, steel) 12

17 Chapter 3. Setting Up A Jetting Calculation Detonator (Lagrange subgrid, aluminum) 13

18 Chapter 3. Setting Up A Jetting Calculation Viewing all of the subgrids together using View Gridplot (available from the Global/Subgrid menu) and turning on the Mirror option (F5 function key) produces the following plot of the entire system: Once you are familiar with the subgrid data, proceed to the Global menu and Review the Material data for the problem. 2. Review of material models 14

19 Chapter 3. Setting Up A Jetting Calculation AUTODYN uses only the reference density for copper. A shell subgrid containing jetting points is automatically treated as a string of mass points with no material strength. 15

20 Chapter 3. Setting Up A Jetting Calculation 16

21 Chapter 3. Setting Up A Jetting Calculation A JWL equation of state for the explosive is defined. The data was extracted from the material library (EXPLOS) supplied with AUTODYN which contains JWL parameters for most conventional explosives. Note: You must specify a JWL equation of state if you wish to use the burn logic included in AUTODYN. If you do not have JWL parameters for a particular explosive, you can model the explosive as a gamma law gas by setting the first two terms in the JWL equation of state to zero. You should now be familiar with the problem setup, so we will proceed to complete the definition of the problem. 3. Jetting definition 17

22 Chapter 3. Setting Up A Jetting Calculation Our first task is to define which points are to be jetting points. We do this by moving to the Global, Subgrid (choosing LINER), Options, Jetting menu. To set our jetting points we enter the data as above. AUTODYN allows one string of jetting points to be defined at the start of a calculation. These points must be a consecutive string of nodes within a shell subgrid. It is not necessary (or usually desirable) to define all nodes of the shell subgrid to be jetting nodes. However, all nodes of a subgrid containing jetting points will automatically be treated as mass points with no strength. When a string of jetting points is defined, AUTODYN automatically numbers the points in the order in which they are expected to jet (the point closest to the axis is expected to jet first etc.). If we wish to redefine a string of jetting points we must first Clear any existing points. You can only Define and Clear jetting points at the start of a calculation. However, you can use the Reduce option at any time to remove unjetted points from the end of the string. This might be useful if an ill behaved late jetting point were to adversely influence the jetting of an adjacent point. 18

23 Chapter 3. Setting Up A Jetting Calculation The Jetting menu is displayed with the following options: Reduce Reduce the number of points in the current string of jetting points Clear Clear all jetting points Setwrap Set wrapup to occur when jetting completed View View the current jetting points We set Setwrap to Yes in order to wrapup (stop) the calculation when all the specified jetting points have jetted. 19

24 Chapter 3. Setting Up A Jetting Calculation We now select View to check that the points have been assigned correctly. Note that we have not included the first and last two nodes of the shell subgrid in the string of jetting points. The first node is on the axis of symmetry and is therefore not able to jet. Instead we have already assigned a boundary condition which fixes this node in its starting position: 20

25 Chapter 3. Setting Up A Jetting Calculation Fixed node on axis The last two nodes have been omitted because from past experience of similar geometries, these nodes will probably receive a smaller impulse from the explosive than other nodes and consequently are likely to lag behind during the liner collapse. The nodes will probably not jet, but more importantly the averaging of segment quantities (length, angle) used to determine jet parameters would cause the node (if included) to adversely effect the jetting of the adjacent nodes. Note: If you are unsure whether or not to include such nodes, include them initially and then use the Reduce option to remove them later if necessary. You cannot add nodes later. Since we have elected to have the calculation terminate automatically after the last point jets, the wrapup time and cycle limit have already been set to large values so that they will not stop the calculation. 21

26 Chapter 3. Setting Up A Jetting Calculation 4. Setting the detonation point Our next task is to set up the detonation of the explosive. We do this by moving to the Global, Options, Explode menu which has the following options: Node Define an detonation point Plane Delete Review View Define a detonation plane Delete detonation points/planes Review the detonation points/planes specifications View all detonation points/planes Choose the Node option to define a single point detonation at the coordinates shown above. After entering this data we are asked if we wish to limit the range of influence of the detonation point we have defined. This is useful if, for example, we want to detonate around a wave shaper which we may define by multiple detonation points with different detonation times. However, this is not required for the present analysis. Enter no. Choose the View option to check that the detonation point has been defined correctly: 22

27 Chapter 3. Setting Up A Jetting Calculation Detonation point The rest of the problem data has already been setup, so that we are now ready to proceed with the analysis. For further details on specifications of the interaction of the various subgrids (Euler/Lagrange coupling) please refer to the Interaction Tutorial. We return to the main menu by pressing <Tab> a number of times. We Save the data and then Execute the calculation. On most machines the calculation will take less than one hour to complete. The problem wraps up when the last specified jetting point has jetted around 35 microseconds. 23

28 Chapter 3. Setting Up A Jetting Calculation Final cycle, t= 35.5 microseconds We save this last cycle and exit to the main menu by selecting Wrapup. 24

29 Chapter 4. Post processing Data From A Jetting Calculation Chapter 4. Post processing Data From A Jetting Calculation Once a calculation is complete we can post process jetting data to produce graphical and printed output of the jetting variables. In our sample calculation, jetting is complete by cycle 373. The save file for this cycle is included on the diskette for this tutorial. Return to the main menu and load this file (Ident JETTNG, cycle 373). Then select Post proc., Jetting. A jetting summary plot appears showing the jet velocity as a function of the cumulative jet mass and the following menu is presented. Variables Define new variables for the horizontal and vertical axes Analysis Numpts Zoom Reset Define the type of analysis to be performed on the jetting data Limit the number of jetting points to be included in the plot Zoom in to a portion of the plot Reset the window to show the entire plot 25

30 Chapter 4. Post processing Data From A Jetting Calculation Examine Output Determine (X, Y) coordinates of any position Output a jetting summary to a disk file The default variables of jet velocity (vertical axis) and cumulative jet mass (horizontal axis) can be changed by variables from the following list: Variable X-ZERO Y-ZERO LIN.MASS LIN.THICK T-JET X-JET Y-JET UX-JET UY-JET DX-JET DY-JET BETA ANGLEA U-ZERO U-JET U-C U-REL SLUG MASS JET MASS CUM J MASS CUM J KE Definition Initial X coordinate Initial Y coordinate Initial liner mass Initial liner thickness Time of jet formation X coordinate of jet formation Radius Y of jet formation (i.e. slug radius) X component of collapse velocity at jet formation Y component of collapse velocity at jet formation DX of segment at jet formation DY of segment at jet formation Liner angle at jet formation Angle A in the jetting equations Collapse speed at jet formation Jet velocity Velocity of the stagnation point Velocity of jet relative to stagnation point Slug mass Jet mass Cumulative jet mass Cumulative jet kinetic energy Jetting Variables For example, in the following plot the horizontal variable has been changed to X-ZERO. 26

31 Chapter 4. Post processing Data From A Jetting Calculation The jetting theory given in section 2 describes two types of jetting analyses - the standard analysis and an optional analysis. We can select which of these we wish to use with the Analysis option. The default is to use the standard analysis. We now select the optional analysis. Notice that the analysis type is shown in the top right corner of the plot window: 27

32 Chapter 4. Post processing Data From A Jetting Calculation The Numpts option allows us to exclude some of the later jetting points from the plot if we wish. There are a total of 21 jetting points for our calculation. You have the option of plotting data from a lesser number if you choose. 28

33 Chapter 4. Post processing Data From A Jetting Calculation If Output is selected a summary of the jetting data for the current analysis type will be output to a disk file. If the standard jetting analysis is currently being used the data will be written to the file JETOUT.STD. If the optional analysis is being used the data will be written to the file JETOUT.OPT. Data output to these two files for our current problem is reproduced on the following pages. 29

34 Chapter 4. Post processing Data From A Jetting Calculation Standard Jetting Analysis (JETOUT.STD) SHAPED CHARGE JETTING ANALYSIS LINER MASS = E+02 LINER MOMENTUM = E+01 LINER KINETIC ENERGY = E+00 STANDARD JETTING ANALYSIS JET MASS = E+01 JET MOMENTUM = E+01 JET KINETIC ENERGY = E+00 J X-ZERO Y-ZERO LIN.MASS LIN.THICK E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 J X-JET Y-JET T-JET ANGLEA U-ZERO E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 30

35 Chapter 4. Post processing Data From A Jetting Calculation J BETA U-C U-REL U-JET JET MASS CUM J MASS E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+01 31

36 Chapter 4. Post processing Data From A Jetting Calculation Optional Jetting Analysis (JETOUT.OPT) SHAPED CHARGE JETTING ANALYSIS LINER MASS = E+02 LINER MOMENTUM = E+01 LINER KINETIC ENERGY = E+00 OPTIONAL JETTING ANALYSIS JET MASS = E+01 JET MOMENTUM = E+01 JET KINETIC ENERGY = E+00 J X-ZERO Y-ZERO LIN.MASS LIN.THICK E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 J X-JET Y-JET T-JET ANGLEA U-ZERO E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 32

37 Chapter 4. Post processing Data From A Jetting Calculation E E E E E-01 J BETA U-C U-REL U-JET JET MASS CUM J MASS E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+01 33

38 Chapter 5. Comparisons with Experiment and other Methods Chapter 5. Comparisons with Experiment and other Methods As mentioned earlier, the calculation used in this tutorial is based on a tested design for a 90mm shaped charge with an 18 degree conical liner. The following two figures compare the experimental data with the calculated jet velocity versus cumulative jet mass using the standard and the optional analyses respectively. AUTODYN vs Experimental Data 9.00E E E-01 U-JET 6.00E E E-01 U-JET - Standard U-JET - Optional U-JET - Experimental 3.00E E E E E E E E E E E E E+01 Cumulative Jet Mass AUTODYN Comparison with Experiment This test was one of a number that were used in a comparative investigation of various analytic and numerical shaped charge liner collapse models. The results of this investigation were published in reference[2]. 34

39 Chapter 5. Comparisons with Experiment and other Methods There are considerable discrepancies between the results of individual models and the lack of experimental data on parameters such as the collapse velocity makes it impossible to say which model best predicts such parameters. However, in all cases the AUTODYN results fall within the spread of the other results and compare favorably with the experimental data for jet velocity as a function of cumulative jet mass. 35

40 Chapter 6. References Chapter 6. References [1] Pugh, Eichelberger, Rostoker. Theory of Jet Formation by Charges with Lined Conical Cavities. J. Appl. Physics, Vol 23, No [2] Dullum, Haugstad, Gustavsson, Nordell & Arvidsson. A comparison Investigation Of Various Analytical And Numerical Shaped Charge Liner Collapse Models Proc. 9th International Symposium On Ballistics, R.A.R.D.E., Shrivenham, England, April,