A 2-D Axisymmetric Model of a Stretching Jet in ALEGRA 

by Müge Fermen-Coker 

ARL-TR-3309 September 2004

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Army Research Laboratory 

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ARL-TR-3309 September 2004

A 2-D Axisymmetric Model of a Stretching Jet in ALEGRA 

Müge Fermen-Coker 

Weapons and Materials Research Directorate 

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A 2-D Axisymmetric Model of a Stretching Jet in ALEGRA 

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Müge Fermen-Coker 

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U.S. Army Research Laboratory 

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14. ABSTRACT 

This report presents a numerical model of a stretching copper jet, representative of a Viper shaped charge jet at 35 μs following detonation. A 

liner collapse and jet formation model is used to extract data for the jet geometry and the axial velocity distribution. Numerical simulations are 

carried out using ALEGRA (Arbitrary Lagrangian Eulerian General Research Application), and the results are compared to previously 

published experimental data and numerical results. ALEGRA results at 35 μs are then corrected to more accurately reflect experimental

observations in the new model representing the stretching jet at the point of impact. Sensitivity studies are conducted using the stretching jet 

model to determine the effects of computational mesh at later times. 

15. SUBJECT TERMS

SCJ, shaped charge, ALEGRA, jet formation, particulation 

16. SECURITY CLASSIFICATION OF: 

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Müge Fermen-Coker 

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ii 
Contents 

List of Figures v

List of Tables v

Acknowledgments vii

Summary 1

1. Introduction 3

2. SCJ Formation Model 4

3. Numerical Construction of the Geometry and the Velocity Profile of a Stretching Jet 9

4. Stretching Jet Model Baseline Results and Sensitivity Studies 12

5. Conclusions 19

References 21

Acronyms 23

Appendix A-1. ALEGRA (Version 4.2) Input File Used for Viper SCJ Formation 25

Appendix A-2. Part 1 of Mesh Used to Simulate Shaped Charge Detonation, Liner 

Collapse, and Jet Formation 29

Appendix A-3. Part 2 of Mesh Used to Simulate Shaped Charge Detonation, Liner 

Collapse, and Jet Formation 31

Appendix B-1. MATLAB Program to Determine the Jet Shape at 35 μs, Using Extracted 

Data from ALEGRA Calculations 33

Appendix B-2. The Viper Jet Boundary at 35 μs 35

Appendix C-1. ALEGRA Problem Specification Deck for the Stretching Jet Model 37

iii 
Appendix C-2. APREPRO File to Create a Mesh with the FASTQ Tool 41

Appendix C-3. Parameters File Used in Conjunction with the Files Listed in Appendices 

C-1 and C-2 43

Distribution List 45

iv 
List of Figures 

Figure 1. Viper shaped charge radiographs at 90 μs, 100 μs, and 110 μs.......................................6 

Figure 2. Mesh and initial configuration of Viper shaped charge representation in 

ALEGRA. ..................................................................................................................................8 

Figure 3. Viper jet formation simulation results at 35 μs: (a) radial velocity distribution, (b) 

axial velocity distribution, (c) void volume fraction, and (d) copper volume fraction.............8 

Figure 4. Use of EnSight line tool to extract data along the length of the jet.................................9 

Figure 5. Sample query entity generated using EnSight...............................................................10 

Figure 6. Approximate representation of Viper shaped charge jet at 35 μs following 

detonation.................................................................................................................................11 

Figure 7. Computationally obtained velocity distribution, modified based on experimental 

results, to represent the axial velocity distribution along the length of the Viper SCJ at 35 

μs following detonation. ..........................................................................................................12 

Figure 8. Stretching jet model initial configuration, representing Viper SCJ description at 35 

μs..............................................................................................................................................13 

Figure 9. Effect of refinement of a square mesh on (a) jet break-up time and (b) number of 

particles....................................................................................................................................16 

Figure 10. Effect of cell aspect ratio (∆y/∆x) on jet break-up time..............................................17 

Figure 11. Effect of cell aspect ratio (∆y/∆x ) on number of particles formed (a) at 85 μs and 

(b) at 120 μs. ............................................................................................................................17 

Figure 12. Effect of cell aspect ratio (∆y/∆x) on preserving the jet tip speed..............................18 

Figure 13. Axial velocity distribution for the particulated jet at 120 μs.......................................19 

List of Tables 

Table 1. Summary of experimental and computational results for Viper SCJ characteristics. ......5 

Table 2. Summary of mesh sensitivity investigations using the stretching jet model in 

ALEGRA. ................................................................................................................................14 

v 
INTENTIONALLY LEFT BLANK. 
Acknowledgments 

The author acknowledges the support of the U.S. Army Research Laboratory, Weapons and 

Materials Research Directorate, Armor Mechanics Branch for this work; and wishes to thank 

Mr. Daniel R. Scheffler and Mr. Kent Kimsey for their technical review and valuable 

suggestions, and Mr. Stephen Schraml and Mr. Richard Summers for engaging in helpful 

discussions and sharing their relevant experience during the course of this work. 

vii 
INTENTIONALLY LEFT BLANK. 
Summary 

In computational studies associated with armor applications, it is desirable to have a numerical 

representation of a stretching, necking, and particulating shaped charge jet (SCJ) at the time of 

impact, without having to model the warhead geometry and simulate detonation and jet 

formation each time a shaped charge jet is fired against armor. The objective of the current study 

is to address this requirement by developing a numerical representation of a stretching jet that 

possesses experimentally confirmed jet characteristics at the time of impact, thus allowing its 

performance against various types of armor to be studied. 

In this report, an analytical representation of the geometric shape and axial velocity distribution 

of a fully formed, stretching Viper SCJ is obtained and studied using ALEGRA (Arbitrary 

Lagrangian Eulerian General Research Application), a multi-material arbitrary-LagrangianEulerian (MMALE) solid dynamics code being developed by Sandia National Laboratories 

(SNL), by modeling liner collapse and jet formation to extract the data, and correcting it to 

reflect the experimental observations more accurately. Sensitivity studies were conducted using 

this new stretching jet model that represents a Viper SCJ at 35 μs, to determine the effects of 

computational mesh at later times. 

Of the various jet characteristics, the jet tip speed and length were found to be the least mesh 

dependent, whereas, the breakup time and number of particles were found to be highly mesh 

sensitive. However, computational jet particulation is largely, if not completely, due to 

numerical error propagation. Jet breakup may or may not be observed, depending on the mesh 

selected. The sensitivity study conducted during this research and summarized in this report 

aimed at developing a better understanding of mesh effects on this numerically observed 

phenomenon, so that analysts can pick the properties of their mesh based on the desired accuracy 

of various jet characteristics in the end. In this study, the difference between the experimentally 

observed and computationally obtained breakup time varied between 6% and 44%, depending on 

the cell geometry. For cells with an aspect ratio of 5 and higher in the axial direction, jet 

particulation did not occur. 

Using the stretching jet model introduced in this report, the error in ALEGRA jet tip speed 

prediction (when the jet is fully formed at 35 μs) was reduced from 7% to nearly 0%, as 

compared to the experimental measurements that included a certain percentage of error itself. 

Shortly before the jet particulates at 85 μs, the error in tip speed was reduced from 7% to 

3-4%, depending on the mesh used. By reducing the jet length predicted by ALEGRA to match 

the experimentally observed length at 35 μs (thus eliminating the 11% difference between the 

ALEGRA prediction and the experimental data), the error in jet length at 85 μs was increased 

from approximately 6% to 11%, with the ALEGRA predictions both being shorter than the 

length measured at that time. 

1 
INTENTIONALLY LEFT BLANK. 
1. Introduction 

In computational studies associated with armor applications, it is desirable to have a numerical 

representation of a stretching, necking, and particulating shaped charge jet (SCJ) at the time of 

impact, without having to model the warhead geometry and simulate detonation and jet 

formation each time a shaped charge jet is fired against armor. The objective of the current study 

is to address this requirement by developing a numerical representation of a stretching jet that 

possesses experimentally confirmed jet characteristics at the time of impact, thus allowing its 

performance against various types of armor to be studied. 

A stretching copper jet model, representative of a Viper SCJ at 35 μs following detonation, was 

generated and studied using ALEGRA (Arbitrary Lagrangian Eulerian General Research 

Application) (Boucheron et. al., 2002). ALEGRA is a multi-material arbitrary-LagrangianEulerian (MMALE) solid dynamics code, featuring strong shock physics and large deformations, 

being developed by the Sandia National Laboratories (SNL). The ALE (Arbitrary-LagrangianEulerian) algorithm allows the mesh to be Eulerian, Lagrangian, or Arbitrary. Considering the 

velocities involved, the ALEGRA simulations presented in this report were performed using 

Eulerian meshes, although this may not be the optimal technique for ALEGRA. ALEGRA 

supports Eulerian calculations through its rezoning algorithms, while it conducts calculations on 

a Lagrangian mesh by default. Typically, the computations take several times longer as 

compared to CTH, an Eulerian code that is widely used for SCJ simulations., and is being 

developed by SNL (McGlaun et. al., 1990). ALEGRA was the numerical tool of choice for this 

study, because it allows a velocity gradient along the jet axis, thereby representing a stretching 

jet. ALEGRA’s Magneto-Hydrodynamics (MHD) capability was also a factor in this decision. 

A summary of tasks performed to develop and assess the stretching jet model reported herein is 

as follows: 

1. Detonation, liner collapse, and jet formation of a Viper shaped charge were simulated using 

ALEGRA. 

2. The results of the simulation were imported into EnSight71 (2003), a multi-purpose postprocessing tool by Computational Engineering International. Using the EnSight7 line tool 

feature, output files were obtained by placing the line tool along the length of the jet at each 

radial location that contained jet material. Data extracted included material volume 

fraction and velocity along the length of the jet at 35 μs following detonation, when the jet 

was fully formed. 

 1

EnSight is a registered trademark of Computational Engineering International, Inc.

3 
3. Using MATLAB2 software (The MathWorks Inc., 1997), the extracted data was processed 

to obtain a geometric description of the jet. The resulting processed data was further 

manipulated using Microsoft Excel3. 

4. The tip velocity obtained numerically was adjusted using experimental results; and an 

analytical formula for the jet shape, as well as the axial velocity gradient, were obtained. 

5. A new model was created using formulas developed for the geometry and the adjusted 

velocity, representing the stretching viper jet when it was fully formed at 35 μs. 

6. The baseline results are compared to experimental results. 

7. Mesh dependency of jet tip speed, breakup time, and the number of particles were 

investigated.

The procedure and results are detailed in the following sections. 

2. SCJ Formation Model 

As a first step in developing a stretching jet model, a Viper liner collapse and jet formation 

model was studied using ALEGRA by comparing the results to previously published 

experimental and numerical data. The Los Alamos National Laboratory (LANL) and SNL 

experimental data, and SNL computational results obtained using CTH, are reported in Kmetyk 

et. al., 1991. Their results are summarized in table 1, along with the results obtained during the 

course of this study, using an identical mesh for ALEGRA. The characteristics of the mesh used 

are detailed in Kmetyk et. al. (1991) and summarized below for readers’ convenience: 

• A total of 82 cells in radial direction and 512 cells in axial direction. 

• In the radial direction, from the axis of symmetry to 3.5 cm, 70 cells of 0.05 cm width, 

followed by 12 cells uniformly increasing in width at a16% rate, up to 5cm. 

• In the axial direction, starting from the lower end, 32 cells decreasing in height at 5% rate 

from 0.25 cm to 0.05 cm up to 4 cm; 70 by 200 square cells of 0.05 by 0.05 cm in size, 

used in the jet formation region between 4 cm and 14 cm; 80 cells increasing in height at a 

2% rate from 0.05 to 0.25 cm up to 24 cm; and 70 by 200 cells with an aspect ratio of 5 

(0.05 cm radially, 0.25 cm axially) in the jet elongation region between 24 and 74 cm. 

 2

MATLAB is a trademark of The MathWorks, Inc.

3

Microsoft is a registered trademark of Microsoft Corp. in the U.S. and other countries. Microsoft Excel is a product name of

Microsoft Corp. 

4 
The user-specified options for CTH and ALEGRA runs were almost identical, with the exception 

that the High-Resolution Interface Tracker (HRIT) was used in the CTH run, as opposed to the 

Sandia Modified Youngs’ Reconstruction Algorithm interface tracker that was used in 

ALEGRA. The ALEGRA input file used for Viper SCJ formation simulation was based on an 

SNL-developed model and is listed in appendix A-1. The input file specifies physics, geometry, 

material insertion, boundary re-mesh control, boundary conditions, as well as the material 

models. A two-dimensional (2-D) axisymmetric cylindrical geometry was used. The equation 

of state (EOS) for the copper liner, was defined by table number 3331 in the ANEOS (Analytic 

EOS) file (Boucheron et. al., 2002; Thompson, 1972, 1973). An elastic plastic constitutive 

model that uses a generalized Hooke’s Law for elastic stress-strain response, von Mises yield 

criteria, fully isotropic hardening, and simple radial return was used for copper, with the 

following material properties: Young’s modulus = 1.2e+12 dynes/cm2 , Poisson’s ratio = 0.33, 

Yield stress = 3.5e+9 dynes/cm2

, and Density = 8.94 g/cm3. The Jones-Wilkins-Lee (JWL) 

(Lee et. al., 1968) EOS for high explosives was used for LX-14. The density for LX-14 was set 

to 1.835 g/cm3. The Poisson’s ratio for copper was set to 0.33 for ALEGRA, which is different 

from the 0.345 ratio for CTH as reported by Kmetyk et. al. (1991). This is due to element 

inversion errors observed in ALEGRA when Poisson’s ratio was set to a value higher than 0.33. 

Table 1. Summary of experimental and computational results for Viper SCJ characteristics. 

Experimental Data Computational Results 

LANL * SNL * CTH (SNL computed) * ALEGRA

Time

(μs) 

Length 

(cm) 

Min.

Jet Dia.

(cm) 

Max.

Tip 

Dia. 

(cm) 

Tip 

Vel. 

(km/s)

Length 

(cm) 

Min.

Jet Dia

(cm) 

Max.

Tip 

Dia. 

(cm) 

Tip 

Vel. 

(km/s)

Length 

(cm) 

Jet Dia.

(cm) 

Tip Vel. 

(km/s)

Length 

(cm) 

Jet Dia.

(cm) 

Tip 

Vel. 

(km/s)

35 20.3 total 

13.2 jet 0.3 0.65 9.17 n/a n/a n/a n/a 20.7 total 13.7 jet 

0.3 min 

(0.55 tip) 9.0 21 total 14.6 jet 

0.2 min 

(0.5 tip) 8.6 

65 38.1 jet 0.25 0.7 9.18 n/a n/a n/a n/a 38.75 0.25 min 

0.51 tip 9.0 37 jet

0.2 (min 

cell size)

min 

(0.5 tip)

8.6 

67.5 n/a n/a n/a n/a 41.6 0.25 0.64 9.26

75 49.6 0.2 0.55 9.19 n/a n/a n/a n/a 46.5 0.2 min 

0.48 tip 9.0 44 0.2 (0.5 tip) 8.6 

77.5 n/a n/a n/a n/a 49.4 0.23 0.56 9.2

85 54.1 0.15 0.5 9.19 n/a n/a n/a n/a 54.5 0.2 

0.45 tip 9.0 51.2 0.2 

(0.5 tip) 8.6 

87.5 n/a n/a n/a n/a 58.1 0.25 0.53 9.16 58.0 0.2min 

0.45tip 9.0 52.7 0.2 

(0.5 tip) 8.6 

NOTE: Dia = diameter; Vel = velocity; and min = minimum.

* From (Kmetyk et. al., 1991)

Note that the LANL and SNL experimental results (Kmetyk et. al., 1991) summarized in table 1 

correspond to times prior to jet particulation. Results of several tests conducted by the U.S. 

Army Research Laboratory (ARL) that correspond to times immediately prior to jet particulation 

and afterwards are available. A sample set of Viper SCJ radiographs obtained at ARL is shown 

5 
in figure 1. At 90 μs, severe necking was observed for the two-thirds of the length of the jet 

towards the tip, and particulation was imminent. At 100 μs and 110 μs, particulation was 

progressively more visible, especially towards the tip. At 110 μs, the jet length was 60.8 cm and 

the tip speed was 9.3 km/s. The average diameter of the jet was 0.18 cm, and 0.6 cm was the 

largest diameter at the tip. The errors in tip speed and jet length measurements were estimated to 

be less than 2%, whereas errors in diameter measurements were typically higher— 

5% to 40%. Kmetyk et. al. (1991) provides the description and depiction of jet length 

measurements listed in table 1. 

Figure 1. Viper shaped charge radiographs at 90 μs, 100 μs, and 110 μs. 

Examining the data presented in table 1, it is observed that both CTH and ALEGRA underpredict the jet tip speed and over-predict the jet length at 35 μs, as compared to experimental 

results. The error in jet tip speed for ALEGRA was approximately 6% and was larger than the 

2% error for CTH. Since the jet tip had reached its asymptotic velocity at 30 μs, this error 

remained relatively constant at later times. At 35 μs, the error in jet length for ALEGRA was 

10.6%, which once again was larger than the error for CTH (3.8%), as compared to the LANL 

experimental observations. At later times, the over-prediction in jet length gradually dissolved. 

Note that the CTH results consistently matched the experimental results more closely, compared 

to the ALEGRA results. The model developed in this report attempts to correct the jet 

description in terms of geometry and velocity, especially for ALEGRA, based on experimental 

observations. 

Several numerical tests were performed using the computational grid described above in 

ALEGRA to assess the effect of user-specified options on jet characteristics, although this was 

not one of the prime objectives of this study. When the detonation point was changed to 

detonation line, the tip speed dropped to 8.4 km/s, resulting in a shorter jet. At outer radius of 

the jet however, the velocities were higher. Perhaps consequently, necking was observed to be 

reduced towards the tip. The tail radius was slightly wider with lower material fraction regions 

6 
toward the centerline of the jet. When the constitutive model for copper was changed from

elastic plastic to Johnson-Cook (Johnson and Cook, 1983), warnings for negative temperatures 

and zero or negative energies were significantly increased. Necking was significantly reduced 

along the length of the jet, and velocities were lower at the jet tail and around the outer diameter 

of the jet. In addition to changing the constitutive model, when Poisson’s ratio was increased 

from 0.33 to 0.34, element inversion errors were received. Changing the EOS from programmed 

burn JWL (Lee et. al., 1968) to KEOS JWL (Kerley, 1998) did not cause any significant changes 

in the results. The tip speed obtained was approximately 8.6 km/s for all the simulations 

described above. Tail velocities measured by LANL and SNL were approximately 1.81 km/s 

and 2.2 km/s, respectively (Kmetyk et. al., 1991), whereas the tail velocity obtained using 

ALEGRA was approximately 1.5 km/s 

Using the computational grid described above, the number of cells across the radius dropped to a 

single cell at times, as the jet stretched. In order to get a better description of the shape of the jet, 

a more refined mesh was used. To reduce computational time, a thin, long mesh placed on top of 

a wider block was used so that the detonation could occur in the lower, wider portion of the 

mesh, allowing the stretching jet to be captured in the long, thin portion. For preprocessing, 

FASTQ, a 2-D meshing tool, was used (Blacker, 1988). Two FASTQ files were used for these 

two blocks of mesh, that were then stitched using GJOIN, a program in the Sandia National 

Laboratories Engineering Analysis Code Access System that combines two or more meshes 

written in the GENESIS mesh database format into a single mesh (Sjaardema, 1992a). 

Appendix A.2 lists the FASTQ input file for the lower part of the mesh that contained the 

warhead. The mesh extended 10 cm horizontally in x-direction and 30 cm vertically in 

y-direction. The corners of the rectangular mesh were located at (0,-10), (10,-10), (10,20), and 

(0,20). The mesh was square between the axis of symmetry (x = 0) and 1 cm outward radially 

(x = 1), with a cell size of 0.4 mm in each direction. While the cell size was kept constant along 

the length of the jet at 0.4 mm, it was expanded radially between x = 1 and x =10, from 

approximately 0.4 mm to 0.6 mm. The FASTQ input file for the thinner part of the mesh is 

listed in appendix A-3. It extended radially from x = 0 to x = 1 and axially from y = 20 to 

y = 120 to capture the stretching and particulating jet at later times. The cells were uniform: 

0.04 by 0.033 cm. 

After converting the two FASTQ files listed in appendices A-2 and A-3 into the GENESIS file 

format, they were combined using GJOIN to create the aggregate mesh (shown in fig. 2) that 

contains a total of 108,750 cells. Note that, displacement in x-direction is not allowed along the 

axis of symmetry (x = 0), and void is assumed to exist outside all other mesh boundaries. 

Figure 3 depicts the ALEGRA predictions of radial and longitudinal velocities, as well as the 

void and copper volume fractions at 35 μs. At 90 μs, the jet tip velocity was 8.7 km/s, 

approximately the same as the velocity obtained using the coarser mesh discussed above. The jet 

length was approximately 54 cm at 87.5 μs, similar to the previously obtained value using 

ALEGRA, but shorter than the reported CTH and experimental results, i.e. 58 cm (Kmetyk et. 

al., 1991). 

7 
Figure 2. Mesh and initial configuration of Viper shaped charge 

representation in ALEGRA. 

(a) (b)

(c) (d)

Figure 3. Viper jet formation simulation results at 35 μs: (a) radial velocity distribution, (b) axial velocity distribution, 

(c) void volume fraction, and (d) copper volume fraction. 

8 
3. Numerical Construction of the Geometry and the Velocity Profile of a 

Stretching Jet 

Once the simulation results depicting liner collapse and jet formation were obtained, as described 

in section 2, the output file was imported into EnSight (EnSight User Manual, 2003). Then, a 

data query was performed over distance using the line tool. The line tool was first placed along 

the axis of symmetry, with its origin located at mesh point labeled “1” in figure 4 and its tip was 

located at the tip of the jet (not shown in fig. 4). The component of velocity in vertical direction 

along the jet length and the copper volume fraction were extracted and saved into a text file. The 

line tool was then moved to each successively labeled mesh point (see fig. 4) parallel to the jet 

axis, and the data query procedure was repeated. Hence, in this particular example, 18 text files 

were created. A portion of a sample file that contains the extracted data is shown in figure 5. 

Figure 4. Use of EnSight line tool to extract data along the length of the jet.

9 
Figure 5. Sample query entity generated using EnSight. 

A MATLAB program was written to extract the jet shape from the series of text files generated

using EnSight. The program, listed in appendix B-1, assumes that the text files are named in a 

certain format (see the comments in app. B-1 for details). Upon execution of the program, the 

user is asked to enter the filename and the total number of files to be read. The output is an array 

“B” that contains the coordinates of the mesh points around the boundary of the jet, defining the 

jet shape. The shape extracted for the Viper SCJ is listed in appendix B-2. The curvature of the 

jet boundary may be approximated, as shown in figure 6. A fitted polynomial representation of 

the jet shape at 35 μs after detonation, when it is fully formed, is 

z = -4644.2 * r5

 + 10582.0 * r4 - 9473.5 * r3 + 4176.8 * r2 - 915.87 * r + 81.852 (1) 

where, z is the distance from the tail in centimeters and r is the radial distance from the axis of 

symmetry. The tail and the tip of the jet correspond to z = 0 and z = 13.2 cm, respectively. 

10 
-1

1

3

5

7

9

11

13

15

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

r (cm)

z 

(cm)

z = -4644.2 * r 5 + 10582 * r 4 - 9473.5 * r 

3

 + 4176.8 * r 2 - 915.87 * r + 

81.852

Figure 6. Approximate representation of Viper shaped charge jet at 35 μs following detonation.

As shown in figure 5, the extracted computational data also includes the velocities computed 

along the jet length. As previously discussed in section 2, the axial velocities ALEGRA 

predicted were lower than the observed experimental values and the CTH predictions. To 

minimize the error in initial conditions for the stretching jet model, the computed velocities were 

adjusted based on experimental results to provide a better representation of the jet at 35 μs. The 

polynomial representation of the modified velocity distribution is 

 Vz = 0.6261 * z5

 - 42.871 * z4 + 838.0 * z3 - 7904.5 * z2 + 93790.0 * z + 182688 (2) 

where, z represents the distance from the tail as defined in eq 1 and Vz is the corresponding 

velocity in the positive z direction, along the axis of symmetry. Note that, variation in velocities 

in the radial direction is ignored. Besides being negligible as compared to the velocities in the 

z-direction, the radial variation presently cannot be modeled in ALEGRA. The modified 

velocity distribution at 35 μs, as compared to the original output from ALEGRA, is shown in 

figure 7. 

11 
0.0E+00

1.0E+05

2.0E+05

3.0E+05

4.0E+05

5.0E+05

6.0E+05

7.0E+05

8.0E+05

9.0E+05

1.0E+06

0 2 4 6 8 10 12 14 16

z (cm)

vz

(cm/s)

computed velocity modified velocity

Figure 7. Computationally obtained velocity distribution, modified based on experimental results, 

to represent the axial velocity distribution along the length of the Viper SCJ at 35 μs

following detonation. 

4. Stretching Jet Model Baseline Results and Sensitivity Studies 

The jet shape and the modified velocity distribution obtained (as described in sec. 3) represent a 

stretching Viper SCJ at 35 μs after detonation. The input file that contains the stretching jet 

baseline model and uses the APREPRO preprocessing utility (Sjaardema, 1992b) is listed in 

appendix C-1. APREPRO is an algebraic preprocessor that reads a file that contains general text, 

as well as algebraic, string, or conditional expressions. It interprets the expressions and outputs 

them to a file, along with the general text. A mesh input that also uses the APREPRO utility is 

provided in appendix C-2. A sample of parameters used in both files listed in appendices C-1 

and C-2 are listed in appendix C-3. The initial configuration of the new computational model is 

shown in figure 8. The jet length was 13.2 cm with a minimum tip diameter of 0.14 cm. This 

configuration ignores the enlarged diameter of the tip of the actual jet. Consequently, if 

penetration of this jet against a target is considered, the diameter of the crater due to the impact 

of the tip of the jet would be less than the diameter observed experimentally. The tip velocity 

was 9.2 km/s and the tail velocity was 1.95 km/s, i.e. the same as observed experimentally. 

12 
Figure 8. Stretching jet model initial configuration, representing Viper SCJ description at 35 μs. 

Once the baseline model was run, various meshes of different aspect ratio and refinement were 

studied, to determine their impact on jet tip velocity and particulation characteristics. The 

particulation of SCJs is discussed in detail in Walters and Zukas (1989) by stating the 

contributions of many authors toward developing a better understanding of the phenomenon. 

The length of the stretching jet directly influences the penetration depth achieved once it hits a 

target; however, penetration starts to decrease significantly when the jet starts to particulate. 

Hence, it is important for a computational model to represent jet particulation characteristics. 

Currently, computational jet particulation is largely, if not completely, due to numerical error 

propagation. Whether the analyst intends to study a problem involving a particulating jet or not, 

breakup may be observed, depending on the mesh selected. Since the real jet eventually 

particulates, and since the physics in currently available codes does not enable the prediction of 

jet particulation, the computational analyst is faced with a major obstacle, especially if having a 

particulated jet is an important aspect of the problem being studied. One approach is to achieve 

particulation using numerical effects by carefully designing the mesh. The sensitivity study 

conducted during this study, and summarized in this section aims at developing a better 

understanding of mesh effects on this numerically observed phenomenon, so that analysts can 

pick the properties of their mesh based on the desired accuracy of various jet characteristics in 

the end. 

13 
Table 2 provides a detailed summary of the meshes studied, along with a schematic 

representation of the cell geometries and the jet characteristics computed at 85 and 120 μs. The 

mesh input listed in appendix C-2, using parameters listed in appendix C-3, corresponds to 

Mesh 1 in table 2. Note that the tip velocity was not preserved in any of the cases. Mesh 3 

provided the best preservation of tip speed; however, it produced the worst result for jet breakup 

time. The best estimate for jet breakup time was achieved using Mesh 1. All the meshes, except 

for Meshes 6 and 7, resulted in jet particulation. 

Table 2. Summary of mesh sensitivity investigations using the stretching jet model in ALEGRA. 

Results at 85 μs Results at 120 μs

Mesh ∆y 

(cm)

∆x 

(cm)

Aspect 

Ratio

∆y/∆x 

Cell 

Geometry 

Min # 

of Cells 

per Radius 

(at the tip)

Max # 

of Cells 

per Radius

(at the tail)

Jet 

Breakup

Time

(μs) 

Tip 

Velocity

(km/s)

Jet 

Length 

(cm)

# of 

Particles 

Tip 

Velocity

(km/s)

Jet 

Length 

(cm)

# of 

Particles

1 0.10 0.05 2.00 3 11 85 8.93 47.8 2 8.92 71.2 18

2 0.05 0.05 1.00 3 11 70 8.98 47.7 20 8.98 71.4 26

3 0.02 0.03 0.50 4 18 65 9.04 47.9 25 9.04 71.8 41

4 0.02 0.02 1.00 7 29 84 8.99 47.9 4 8.98 71.5 33

5 0.01 0.03 0.44 5 23 50 8.99 47.8 21 n/a n/a n/a

6 0.10 0.03 4.00 5 22

No 

particulation

up to 150 μs

8.96 47.9 0 8.89 71.2 0

7 0.10 0.01 8.00 10 45

No 

particulation 

up to 150 μs

8.97 47.9 0 8.92 71.4 0

14 
Considering the meshes that produced jet particulation, the number of particles numerically 

formed was less than the experimentally observed number of particles. According to the ARL 

experimental results shown in figure 1, first particulation was observed at approximately 90 μs. 

Hence, the number of particles at 85 μs should be 0. At 120 μs, however, there should be 

55 particles. Mesh 3 came closest to matching this number, with 41 particles; whereas, Mesh 2 

produced only 18 particles. 

The mesh sensitivity study summarized in table 2 can be used to form some trends and develop 

guidelines for selecting cell size and aspect ratio in future studies. However, these trends should 

not be regarded as conclusive, but only as indicative, since the number of meshes studied was 

limited. The validity of these trends should be further investigated through additional numerical 

tests. 

Note that there are two square meshes considered. The coarser mesh was 0.05 by 0.05 cm and 

the refined mesh was 0.02 by 0.02 cm. Based on these two results alone, the jet particulation 

seemed to be delayed when using refined square mesh, as indicated in figure 9(a). Once the jet 

started to particulate, the error in the number of particles, calculated at both 85 and 120 μs, was 

smaller for the refined square mesh, as shown in figure 9(b). Considering the jet breakup times 

for each mesh, 33 particles at the end of a duration of 36 μs were obtained for the fine mesh; 

whereas, 26 particles at the end of a duration of 50 μs were obtained for the coarser mesh. In 

other words, after a late start, more particles were obtained in a shorter amount of time using a 

finer square mesh; similar to what is observed experimentally. There is no significant difference 

in the loss in jet tip velocity between the two meshes. Hence, not surprisingly, the more refined 

square mesh produced better results overall. 

15 
68

70

72

74

0.00 0.01 0.02 0.03 0.04 0.05 0.06

cell size (cm)

umber of particles

0

10

20

30

40

50

60

0.00 0.01 0.02 0.03 0.04 0.05 0.06

at 85 μs at 120 μs

Experimental at 85 μs Experimental at 120 μs

(b) N

(a) Jet break-up time (μs)

76

78

80

82

84

86

Figure 9. Effect of refinement of a square mesh on (a) jet break-up time and 

(b) number of particles. 

The results summarized in table 2 indicate that as the cell aspect ratio was increased, the jet 

breakup time also increased, as shown in figure 10. Particulation never occurred for the two 

highest aspect ratios investigated, i.e. for aspect ratio = 4 and aspect ratio = 8. The earliest jet 

breakup time occurred for the lowest aspect ratio investigated, 0.44. Note that the cell size was 

the same in radial direction for the mesh that produced the earliest jet breakup time, and for 

Mesh 6, with aspect ratio 4 that produced no particulations at all. Approximately the same

breakup time was obtained by Meshes 1 and 4. Although Mesh 4 was more refined than Mesh 1, 

Mesh 1 had twice the aspect ratio of Mesh 4. As the aspect ratio increased, the number of 

particles obtained decreased, as shown in figure 11. Despite the various different cell sizes, the 

results were consistent. 

16 
Jet break-up time (μs)

0

20

40

60

80

100

120

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Cell aspect ratio

Figure 10. Effect of cell aspect ratio (∆y/∆x) on jet break-up time. 

ber of particles at 120 μs

0

10

20

30

40

50

0.00 2.00 4.00 6.00 8.00 10.00

Cell Aspect Ratio

∆x=0.05

∆x=0.03

∆y=0.02

∆y=0.1

mber of particles at 85 μs

0

5

10

15

20

25

30

35

0.00 2.00 4.00 6.00 8.00 10.00

Cell Aspect Ratio

∆x=0.05

∆x=0.03

∆y=0.02

∆y=0.1

(a) nu

(b) num

Figure 11. Effect of cell aspect ratio (∆y/∆x ) on number of particles 

formed (a) at 85 μs and (b) at 120 μs. 

17 
The difference between the jet tip speeds at 120 μs, obtained using various computational grids, 

was insignificant; however, the jet tip speed was reduced for all the meshes considered. The 

worst result, in terms of maintaining the initially specified jet tip speed, was obtained for aspect 

ratio = 2 and the best result obtained was for aspect ratio = 0.5, as shown in figure 12. However, 

the best result corresponded to a 1.7% drop in tip speed, whereas the worst result corresponded 

to a 2.9% drop in tip speed. Therefore, the comparative trends cannot be considered reliable. 

Jet Tip Speed ( km/s)

8.92

8.94

8.96

8.98

9.00

9.02

9.04

9.06

0.00 2.00 4.00 6.00 8.00 10.00

Cell Aspect Ratio

Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 5 Mesh 6 Mesh 7

Figure 12. Effect of cell aspect ratio (∆y/∆x) on preserving the jet tip speed. 

In terms of matching the experimentally observed jet characteristics, the square mesh labeled 

Mesh 4 (0.02 by 0.02 cm cells) in table 2 produced the best results. Initially, there were a 

minimum of 7 cells across the radius at the tip, and 29 cells across the radius at the tail. For this 

mesh, the error in jet tip speed was 2%, the error in jet breakup time was approximately 

7%, the error in jet length at 85 μs was 11%, and the error in number of particles 

at 120 μs was 40%. The results at 120 μs using this mesh are shown in figure 13. If matching a 

specific jet characteristic is more important than achieving the overall best response, then 

changing the cell geometry may reduce the error for one characteristic, while possibly increasing 

it for another. Note that, the results reported herein are based on ALEGRA version 4.2. 

18 
Figure 13. Axial velocity distribution for the particulated jet at 120 μs. 

5. Conclusions 

An analytical representation of the geometric shape and axial velocity distribution of a fully 

formed, stretching Viper SCJ was presented. Computational results were obtained using 

ALEGRA and were compared to experimental results. The mesh dependency of the jet tip 

velocity, jet length, breakup time, and the number of particles formed were investigated. 

Among various jet characteristics, the jet tip speed and length were found to be the least mesh 

dependent; whereas, breakup time and number of particles were found to be highly mesh 

sensitive. The error in jet tip speed was corrected at 35 μs, which helped reduce the error at 

85 μs from 7% to 2%. Depending on cell geometry, the difference between the experimentally 

observed and computationally obtained breakup times varied between 6% and 44%. For cells of 

a higher aspect ratio in the axial direction, the breakup time was delayed and the difference was 

reduced. However, the number of particles at later times was also reduced, resulting in a greater 

difference between the experimental data and computational results. For cells with an aspect 

ratio of 5 and higher, jet particulation did not occur for the duration of computations. The largest 

error observed for all meshes studied was in estimating the number of particles at 120 μs; this 

19 
value varied between 25% and 67%. Nevertheless, the large errors associated with jet 

particulation was not a surprising conclusion, since the particulation is due to numerical effects 

rather than to the physics implemented in the code, as discussed within this report. The results of 

the mesh sensitivity studies are applicable to SCJ simulations using ALEGRA, and are not 

confined to the stretching jet model presented in this report. 

20 
References 

Blacker, T. D. FASTQ Users Manual Version 1.2; SAND88-1326; Sandia National 

Laboratories: Albuquerque, NM, 1988. 

Boucheron, E. A.; Brown, K. H.; Budge. K. G.; Burns, S. P.; Carroll, D. E.; Carroll, S. K.; 

Christon, M. A.; Drake, R. R.; Garasi, C. G.; Haill, T. A.; Peery, J. S.; Petney, S. V.; 

Robbins, J.; Robinson, A. C.; Summers, R. M.; Voth, T. E.; Wong, M. K. ALEGRA: User 

Input and Physics Descriptions Version 4.2; SAND2002-2775; Sandia National 

Laboratories: Albuquerque, NM, October 2002. 

EnSight User Manual for Version 7.6. Computational Engineering International, Inc.: 2166 N. 

Salem St., Suite 101, Apex, NC, 27523, May 2003. 

Johnson, G. R.; Cook, W. H. A Constitutive Model and Data Subjected to Large Strains, High 

Strain Rates and High Temperatures. Proceedings of the 7th International Symposium on 

Ballistics, The Hague, The Netherlands, 1983. 

G. I. Kerley, G. I. CTH Reference Manual: The Equation of State Package; SAND98-0947; 

Sandia National Laboratories: Albuquerque, NM, 1998. 

Kmetyk, L.N.; Yarrington, P.; Vigil, M.G. CTH Analyses of Viper Conical Shaped Charges with 

Comparisons to Experimental Data; SAND90-2604, Sandia National Laboratories: 

Albuquerque, NM, January, 1991. 

Lee, E. L.; Hornig, H. C.; Kury, J. W. Adiabatic Expansion of High Explosive Detonation 

Products; UCRL-50422, Lawrence Livermore National Laboratory: Livermore, CA, 1968. 

MATLAB: The Language of Technical Computing, Version 5, The Mathworks Inc., June 1997. 

McGlaun, J. M.; Thompson, S. L.; Elrick, M. G. CTH: A Three-Dimensional Shock Wave 

Physics Code. I. J. of Impact Engineering 1990, 10, 351–360. 

Sjaardema, G. D. GJOIN: A Program for Merging Two or More GENESIS Databases; 

SAND92-2290; Sandia National Laboratories: Albuquerque, NM, December, 1992a. 

Sjaardema, G. D. APREPRO: An Algebraic Preprocessor for Parameterizing Finite Element 

Analyses; SAND92-2291, Sandia National Laboratories: Albuquerque, NM, 1992b. 

Thompson, S. L. Improvements in the CHARTD Radiation-Hydrodynamics Code III: Revised 

Analytic Equation of State; SC-RR-710714; Sandia National Laboratories: Albuquerque, 

New Mexico, March 1972 (ANEOS models). 

21 
Thompson, S. L. Improvements in the CHARTD Energy Flow–Hydrodynamics Code V: 

1972/1973 Modifications; SLA-73-0477; Sandia National Laboratories: Albuquerque, New 

Mexico, October 1973 (ANEOS extensions).

Walters, W. P.; Zukas, J. A. Fundamentals of Shaped Charges, soft cover ed.; CMC Press: 

Baltimore, MD, 1998. 

22 
Acronyms

2-D two-dimensional 

ALE Arbitrary Lagrangian Eulerian 

ALEGRA Arbitrary Lagrangian Eulerian General Research Application

ANEOS Analytic Equation of State package 

ARL U.S. Army Research Laboratory 

EOS equation of state 

HRIT High-Resolution Interface Tracker 

JWL Jones-Wilkins-Lee 

KEOS Kerley Equation of State 

LANL Los Alamos National Laboratory 

MHD Magneto-Hydrodynamics 

MMALE Multi-Material Arbitrary Lagrangian Eulerian 

SCJ shaped charge jet 

SEACAS Sandia National Laboratories Engineering Analysis Code Access System

23 
INTENTIONALLY LEFT BLANK. 
Appendix A-1. ALEGRA (Version 4.2) Input File Used for Viper SCJ 

Formation 

$ 

$ Title: 2D axisymmetric VIPER shaped charge simulation 

$ 

title 

 viper 2d 

exodus version two 

$ 

termination time 90.0e-6 

emit output: time = 1.0E-6, from 0.0 to 1.0E-6 

emit output: time = 1.0E-5, from 10.0E-6 to 100.0E-6 

emit output: time = 2.5E-5, from 100.0E-6 to 250.0E-6 

emit plot: time = 1.0E-6, from 0.0 to 20.0E-6 

emit plot: time = 2.5E-6, from 20.0E-6 to 100.0E-6 

emit plot: time = 2.5E-5, from 100.0E-6 to 250.0E-6 

emit hisplt: time = 1.0e-6, from 0.0 to 9.0e-5 

emit restart: time = 1.0e-6, from 2.0e-5 to 2.3e-5 

$ PHYSICS OPTIONS 

solid dynamics 

cylindrical 

$ 

 programmed burn 

 material 2 

 detonation point 

 x=0.0, y=0.01 at time 0.0 

 burn radius 20.0 

 end 

$ 

 domain 

 smyra interface tracker 

 voided sideset 2 

 remesh iterations 10 

 end 

$ 

 block 1 

 add diatom input 

25 
 eulerian mesh 

 remesh frequency 1 

 end 

$ BOUNDARY CONDITIONS 

 no displacement: nodeset 1 x 

$ MATERIAL INSERTION 

 diatoms 

 package 'CU LINER - 2' 

 material 1 

 numsub 50 

 insert circle 

 center 0.0 5.77550 

 radius 0.64516 

 endinsert 

 delete circle 

 center 0.0 5.77550 

 radius 0.52578 

 enddelete 

 delete box 

 p1 0.0 5.588 

 p2 0.59817 9.0 

 enddelete 

 endpackage 

$ 

 package 'CU LINER - 1' 

 material 1 

 numsub 50 

 insert uds 

 p1 0.0 12.31392 

 p2 0.0 5.58800 

 p3 0.59817 5.58800 

 p4 3.26009 12.11580 

 p5 3.26009 12.31392 

 endinsert 

 delete uds 

 p1 0.0 12.21486 

 p2 0.0 5.58800 

 p3 0.483607 5.58800 

 p4 3.16103 12.21486 

 enddelete 

 delete box 

 p1 0.0 12.21486 

 p2 3.16103 12.50000 

 enddelete 

 endpackage 

$ 

 package 'EXPLOSIVE' 

 material 2 

 numsub 50 

 insert uds 

 p1 0.0 0.0 

 p2 1.59385 0.0 

26 
 p3 3.22834 5.40233 

 p4 3.26009 12.11580 

 p5 0.0 12.11580 

 endinsert 

 delete uds 

 p1 0.0 5.588 

 p2 0.55880 5.5880 

 p3 3.26009 12.11580 

 p4 0.0 12.11580 

 enddelete 

 delete circle 

 center 0.0 5.77550 

 radius 0.64516 

 enddelete 

 endpackage 

 enddiatoms 

end 

$ EXECUTION CONTROL 

plot variable 

 artificial viscosity 

 pressure 

 stress 

 strain 

 velocity 

 density 

 temperature 

 detonation time 

 eqps 

 yield stress 

end 

$ MATERIAL MODELS 

$ MATERIAL 1 

material 1 "COPPER" 

 model = 1 $ copper eos 

 model = 2 $ constitutive 

 model = 3 $ fracture 

 density 8.94 $ g/cm3 

 temperature 298.0 $ Kelvin 

end 

model 1 keos sesame 

 neos 3331 

 feos 'aneos' 

end 

model 2 elastic plastic 

 youngs modulus 1.2e+12 $ dyne/cm^2 

 poissons ratio 0.33 

 yield stress 3.5e+9 $ dyne/cm^2 

 hardening modulus 0.0 

 beta 1.0 

27 
end 

model 3 frac presdep 

 init frac pres = -15.0e9 

end 

$ MATERIAL 2 

material 2 "LX-14" 

 model = 5 

 density 1.835 $ g/cm3 

 temperature 298.0 $ Kelvin 

end 

model 5 programmed burn jwl 

 rho ref 1.835 

 tref 298.0 

 a 11.65e12 

 b 0.5572e12 

 c 0.01844e12 

 omega 0.45 

 r1 5.40 

 r2 2.0 

 pcj 0.360e12 

 dcj 0.88e6 

 tcj 3970.0 

end 

$ 

exit 

28 
Appendix A-2. Part 1 of Mesh Used to Simulate Shaped Charge Detonation, 

Liner Collapse, and Jet Formation 

$ APREPRO ($Revision: 1.69 $) 

$ 

$ Part 1 of mesh for 

$ shaped charge detonation jet formation 

$ 

title 

meshpart1forshapchardetandjetformtn 

$ 

$ point definitions 

$ 

$ 

point 1 0 -10 

point 2 1 -10 

point 3 10 -10 

point 4 10 20 

point 5 1 20 

point 6 0 20 

$ 

$ line definitions 

$ 

line 1 str 1 2 0 25 1.0 

line 2 str 2 3 0 180 1.0019 

line 3 str 3 4 0 750 1.0 

line 4 str 5 4 0 180 1.0019 

line 5 str 6 5 0 25 1.0 

line 6 str 1 6 0 750 1.0 

$ 

$ region definitions 

$ 

$region 1 1 -1 -2 -3 -4 -5 -6 

$region 2 1 -5 -7 -8 -9 

$ 

region 1 1 -1 -2 -3 -4 -5 -6 

$ 

$ boundary conditions 

$ 

nodebc 1 6 

elembc 2 1 2 3 4 

$ 

exit 

29 
INTENTIONALLY LEFT BLANK. 
Appendix A-3. Part 2 of Mesh Used to Simulate Shaped Charge Detonation, 

Liner Collapse, and Jet Formation 

$ APREPRO ($Revision: 1.69 $) 

$ 

$ Part 2 of mesh for 

$ shaped charge detonation jet formation 

$ 

title 

meshpart2forshapchardetandjetformtn 

$ 

$ point definitions 

$ 

$ 

$point 1 0 -10 

$point 2 2 -10 

$point 3 10 -10 

$point 4 10 20 

point 5 1 20 

point 6 0 20 

point 7 1 120 

point 8 0 120 

$ 

$ line definitions 

$ 

line 5 str 6 5 0 25 1.0 

line 7 str 5 7 0 3000 1.0 

line 8 str 8 7 0 25 1.0 

line 9 str 6 8 0 3000 1.0 

$ 

$ region definitions 

$ 

$region 1 1 -1 -2 -3 -4 -5 -6 

region 2 1 -5 -7 -8 -9 

$ 

$ 

$ boundary conditions 

$ 

nodebc 1 9 

elembc 2 7 8 

$ 

exit 

31 
INTENTIONALLY LEFT BLANK. 
Appendix B-1. MATLAB Program to Determine the Jet Shape at 35 μs, Using 

Extracted Data from ALEGRA Calculations 

% MATLAB 

% 

% Program to extract material boundary 

% Example : boundary of a shaped charge jet 

% To use this program, following is needed: 

% Series of numbered files (Ex: velgeo1, velgeo2, ...,velgeo25) 

% that contain 5 columns of data, preceded by 5 comment lines to be skipped 

% based on an output from ensight7. 

% 

% 5 columns of data are: volumefraction of copper, velocity_z, x, y, z 

% where x is constant in each file, since the files are generated 

% using a line tool along the y axis at constant x. z is always zero. 

% 

% Input filename without the number as a string 

 filenamebase=input('What is the filename (without the number) ? ','s'); 

% 

% Input number of files 

 n=input('How many files ? '); 

% 

 k=0; 

 bk=0; 

% 

 for i=1:n 

 istr=num2str(i); 

 filename=strcat(filenamebase,istr); 

 f1=fopen(filename); 

% 

 for j=1:6 

 fgets(f1); 

 end 

% 

% arrayname=strcat('a',istr) 

 a=fscanf(f1,'%e %e %e %e %e',[5 inf]); 

 a=a'; 

 size_a=size(a); 

 size_a=size_a(1,1); 

% 

 for j=1:size_a 

 if a(j,1) >= 0.5 

 k=k+1; 

 c(k,1)=a(j,3); 

 c(k,2)=a(j,4); 

 c(k,3)=a(j,2); 

 if j > 1 j < size_a 

 if (a(j-1,1) < 0.5 a(j+1,1) >= 0.5) | (a(j-1,1) >= 0.5 

a(j+1,1)< 0.5) 

 bk=bk+1; 

 B(bk,1)=a(j,3); 

 B(bk,2)=a(j,4); 

 end 

33 
 else 

 bk=bk+1; 

 B(bk,1)=a(j,3); 

 B(bk,2)=a(j,4); 

 end 

 end 

 end 

% 

% 

% 

% 

% Check to see if any rows are skipped in matrix a. 

% If skipped then it means that the max and min values above 

% may not be the boundaries of the shape of the jet. 

% i.e., there may be gaps between cells occupied by the jet 

% for the given x location. 

% If noneskipped is 1 or true, then it means none of the rows are skipped 

% and there are no gaps 

% 

 nonzeroindices=find(a(:,1)); 

 noneskipped=all(diff(diff(nonzeroindices))==0); 

 if ~noneskipped 

 disp('skipped at least one row for i=') 

 disp(istr) 

 end 

% 

% end of program 

 end 

34 
Appendix B-2. The Viper Jet Boundary at 35 μs 

Radial Coordinate x 

(cm) 

Axial Coordinatey 

(cm) 

0 13.28

0 27.8636

0.04 13.28

0.04 27.8631

0.08 13.28

0.08 27.8925

0.12 13.28

0.12 23.497

0.12 26.4077

0.12 27.8925

0.16 13.28

0.16 23.497

0.16 26.4077

0.16 27.8925

0.2 13.28

0.2 17.0728

0.2 27.7014

0.2 27.8631

0.24 13.28

0.24 17.0728

0.24 27.7014

0.24 27.8631

0.28 13.28

0.28 15.2793

0.28 27.7014

0.28 27.7308

0.32 13.28

0.32 14.6766

0.32 27.7014

0.32 27.7602

0.36 13.28

0.36 14.6766

0.36 27.7014

0.36 27.7602

0.4 13.28

0.4 13.8386

0.44 13.28

0.44 13.8386

0.48 13.28

0.48 13.5593

0.52 13.28

0.52 13.427

0.56 13.28

0.56 13.3094

0.6 13.28

35 
INTENTIONALLY LEFT BLANK. 
Appendix C-1. ALEGRA Problem Specification Deck for the Stretching Jet 

Model 

{include(parameters)} 

$ 

$ ---------------------------------------- 

$ begin execution control section of input 

$ ---------------------------------------- 

$ 

title 

2D axisymm viper shaped charge jet representation at 35 μs 

with an axial velocity function defined along its length 

$ 

$ define stop time/cycle 

$ 

$termination cycle = 1 

termination time = 85.0e-06 

$ 

$ output control 

$ 

emit output: time = 1.0e-06, from 0.0 to 1.0 

emit plot: time = 1.0e-06, from 0.0 to 1.0 

emit restart: time = 10.0e-06, from 0.0 to 1.0 

emit hisplt: time = 1.0e-06, from 0.0 to 1.0 

$ 

$ define variables written to exodus file(s) 

$ 

plot variable 

 pressure: avg 

 density: avg 

 temperature: avg 

 velocity 

 stress 

 strain 

 yield stress 

 eqps 

 stretch 

 sound speed 

 mass 

 el mass 

end 

$ 

$ ------------------------------ 

$ begin physics section of input 

$ ------------------------------ 

$ 

solid dynamics 

 $ 

 $ define the geometry 

 $ 

 cylindrical 

 $ 

 $ controls applied to the entire computational domain 

37 
 $ 

 domain 

 smyra interface tracker 

 voided sideset {bc_exterior} 

 end 

 $ 

 $ boundary conditions 

 $ 

 no displacement: nodeset {bc_symmetry} x 

 $ 

 $ tracers 

 $ 

 tracer points 

 eulerian tracer 1 x=0.0 y={jet_tail_coordinate}. 

 eulerian tracer 2 x=0.0 y={jet_tail_coordinate+0.25*jet_length} 

 eulerian tracer 3 x=0.0 y={jet_tail_coordinate+0.50*jet_length} 

 eulerian tracer 4 x=0.0 y={jet_tail_coordinate+0.75*jet_length} 

 eulerian tracer 5 x=0.0 y={jet_tip_coordinate} 

 end 

 $ 

 $ controls for the mesh blocks 

 $ 

 block 1 

 eulerian mesh 

 add diatom input 

 end 

 $ 

 $ material insertion (diatom) inputs 

 $ 

 diatoms 

 package 'copper_jet' 

 material={copper*100} 

 agraded p1=0.0 {jet_tail_coordinate}. p2=0.0 {jet_tip_coordinate} 

 mvelocity=0.0 T1 

 iteration=3 

 insert uds 

p{i}= 0.0 {jet_tail_coordinate} 

p{i+=1}= {jet_radius} {jet_tail_coordinate} 

{Loop(no_of_segments)} 

 p{i+=1}= {jet_radius-=delta_radius} {-4644.2*(jet_radius**5) + 

10582.*(jet_radius**4) - 9473.5*(jet_radius**3) + 4176.8*(jet_radius**2) - 

915.87*(jet_radius) + 81.852} 

{EndLoop} 

p{i+=1}= 0.0 {jet_tip_coordinate} 

 endi 

 endp 

 enddiatom 

 $ 

 $ function 1: axial velocity along length of jet 

 $ 

 function 1 

{Loop(no_of_segments)} 

{dist_fr_tail+=delta_length} {0.6261*(dist_fr_tail**5) - 

42.871*(dist_fr_tail**4) + 838.*(dist_fr_tail**3) - 7904.5*(dist_fr_tail**2) 

+ 93790.*(dist_fr_tail) + 182688. } 

{EndLoop} 

 end 

38 
$ 

$ this end completes the physics section of the input 

$ 

end 

$ 

$ ---------------------------------------- 

$ begin material modeling section of input 

$ ---------------------------------------- 

$ 

$ material definition for copper jet 

$ 

material {copper*100} "COPPER in EOS_data file" 

 model {copper*100} 

end 

$ 

model {copper*100} cth elastic plastic 

 eos model = {copper*100+1} 

 yield model = {copper*100+2} 

 poissons ratio = 0.3 

end 

$ 

model {copper*100+1} keos sesame 

 datafile = 'EOS_data' 

 matlabel = 'COPPER' 

end 

$ 

model {copper*100+2} johnson cook ep 

 ajo = 8.970000E+08 

 bjo = 2.918700E+09 

 cjo = 2.500000E-02 

 mjo = 1.090000E+00 

 njo = 3.100000E-01 

 tjo = 1380.718 $1.189813E-01 ev 

end 

$ 

$ constitutive model 

$ elastic plastic (Copper) 

$model,{copper*100},elastic plastic 

$youngs modulus = 1.076e+12 

$poissons ratio = 0.355 

$yield stress = 6.0e+09 

$hardening modulus = 2.0e+09 

$beta=0.5 

$end 

$ 

$ EOS Mie-Gruneisen 

$model {copper*100+1} keos miegruneisen 

$ matlabel = 'COPPER' 

$end 

$ 

$ pressure dependent fracture 

$ model {copper*100+2} frac presdep 

$ init frac pres = -5.e10 

$ density tolerance = 1.e-6 

$ pressure tolerance = 1.e+2 

$ end 

$ 

39 
$ 

exit 

40 
Appendix C-2. APREPRO File to Create a Mesh with the FASTQ Tool 

{include(parameters)} 

$ 

$ simulation of a copper jet with an initial velocity gradient imposed 

$ along its length 

$ 

title 

copper jet with velocity gradient along its length 

$ 

$ point definitions 

$ 

point 1 {domain_xmin} {domain_ymin} 

point 2 {domain_xmax} {domain_ymin} 

point 3 {domain_xmax} {domain_ymax} 

point 4 {domain_xmin} {domain_ymax} 

$ 

$ line definitions 

$ 

line 1 str 1 2 0 {nx} 1.0 

line 2 str 2 3 0 {ny} 1.0 

line 3 str 3 4 0 {nx} 1.0 

line 4 str 4 1 0 {ny} 1.0 

$ 

$ region definitions 

$ 

region 1 1 -1 -2 -3 -4 

$ 

$ boundary conditions 

$ 

nodebc {bc_symmetry} 4 

elembc {bc_exterior} 1 2 3 

$ 

exit 

41 
INTENTIONALLY LEFT BLANK. 
Appendix C-3. Parameters File Used in Conjunction with the Files Listed in 

Appendices C-1 and C-2 

{ECHO(OFF)} 

Parameters for simulation of stretching jet 

{i=1} 

Geometry Parameters 

------------------- 

{jet_tip_radius=0.14} 

{jet_tail_radius=0.6} 

{jet_length=-4644.2*(jet_tip_radius**5) + 10582.*(jet_tip_radius**4) - 

9473.5*(jet_tip_radius**3) + 4176.8*(jet_tip_radius**2) - 

915.87*(jet_tip_radius) + 81.852} 

{jet_radius=jet_tail_radius} 

{no_of_segments=100} 

{delta_radius=(jet_tail_radius-jet_tip_radius)/no_of_segments} 

{jet_tip_coordinate=jet_length} 

{jet_tail_coordinate=0.0} 

{dist_fr_tail=jet_tail_coordinate} 

{delta_length=jet_length/no_of_segments} 

Initial Conditions 

------------------ 

{jet_tip_velocity=0.6261*(jet_length**5) - 42.871*(jet_length**4) + 

838.*(jet_length**3) - 7904.5*(jet_length**2) + 93790.*(jet_length) + 

182688.} 

{jet_tail_velocity=182688.} 

Boundary Conditions 

------------------- 

{bc=int(0)} 

{bc_symmetry=++bc} 

{bc_exterior=++bc} 

Material Parameters 

------------------- 

{material=int(0)} 

{copper=++material} 

Grid Parameters 

--------------- 

{domain_xmin=0.0} 

{domain_xmax=1.0} 

{domain_ymin=-1.0} 

{domain_ymax=100.0} 

{deltax=0.05} 

{deltay=0.1} 

{domain_x_length=domain_xmax-domain_xmin} 

{domain_y_length=domain_ymax-domain_ymin} 

43 
{nx=int(domain_x_length/deltax)} 

{ny=int(domain_y_length/deltay)} 

{ECHO(ON)} 

44 
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