CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
CONVERSION OF FINITE ELEMENTS INTO MESHLESS
PARTICLES FOR PENETRATION COMPUTATIONS
INVOLVING CERAMIC TARGETS
Gordon R. Johnson1'2, Robert A. Stryk1, Stephen R. Beissel2
and Timothy J. Holmquist2
l
Mliant Techsystems, 600 2nd St. N.E., Hopkins, MN 55343
Network CS, 1200 Washington Ave. S., Minneapolis, MN 55415
2
Abstract. This paper presents a new computational algorithm to automatically convert distorted finite
elements into meshless particles during dynamic deformation. It also presents computed results for
projectiles impacting ceramic targets. The new computational algorithm, together with an appropriate
ceramic model, provides computed results that are in good agreement with test data. Included are
problems involving dwell and penetration. This computational approach is especially well-suited for
brittle materials such as ceramics, because the conversion from elements into particles generally occurs
after the material has failed. The result is that the particles represent only failed material, which does not
produce any tensile stresses. For some particle algorithms it is possible to introduce tensile instabilities,
but this is not a concern if the particles represent only failed material.
work [1], where the general approach was
demonstrated with a simplified algorithm. For brittle
ceramic materials the conversion occurs after failure
such that there are no tensile stresses (and therefore
no tensile instabilities) in the particle nodes. The
finite element formulation is provided in Reference
2 and the Generalized Particle Algorithm (GPA) is
presented in References 3 and 4.
INTRODUCTION
Lagrangian meshless methods (or particle
methods) have been recently developed and applied
to solid mechanics problems. An important
characteristic of meshless methods is that they can
be used to represent severe distortions in a
Lagrangian framework. Although the accuracy and
efficiency of meshless methods are not generally as
good as finite element methods for dynamics
problems with mild distortions, the meshless
particle methods can be more accurate, more
efficient and more robust for dynamics problems
involving severe distortions. Therefore, a logical
computational approach would be to use finite
elements for the mildly distorted regions and
meshless particles for the highly distorted regions.
This paper presents an explicit Lagrangian
algorithm to automatically convert distorted
elements to meshless particles during the course of
the computation. This is an extension of previous
CONVERSION ALGORITHM
Figure 1 shows a finite element grid with a
surface defined by nodes a ... j. Three elements on
the surface (A, B, C) are designated as candidates
for conversion to particles. An element is converted
to a particle when the element has at least one side
on a surface or interface, and the equivalent plastic
strain exceeds a user-specified value (in the range of
0.3 to 0.6). Criteria other than plastic strain could
also be used.
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EXAMPLES
The first example is shown in Fig. 2. It is an
Armo iron cylinder that is impacted against a rigid
surface at a velocity of 305 m/s. This is not a
problem that requires conversion, but it
demonstrates the algorithm for a problem that can
be compared to a finite element computation.
Interface (nodes a... y) before conversion
Interface after conversion of elements to particles
FIGURE 1. Conversion of Finite Elements Into Particles
Plastic Strains
If element A exceeds the plastic strain criterion
the following steps are taken:
- Element A is removed from the finite element
grid.
- Particle A is added as a GPA node.
- All of the element variables (stress, strain,
internal energy, damage, etc.) are transferred
to the GPA node.
- The mass, velocity and center of gravity of the
GPA node are set to those of the replaced
elemeni.
- The GPA nodal diameters, (initial and current)
Conversion
FIGURE 2. Impact of an Armco Iron Cylinder onto a Rigid
Surface at 305 m/s
The upper position of Fig. 2 shows the remaining
finite elements and the generated GPA nodes at 40
jus after impact. The lower left position shows the
plastic strain contours for the finite element/GPA
solution and the lower right provides the plastic
strain distributions for a finite element solution
without any conversion to particles. There is good
general agreement between the strain distributions.
The deformed lengths and diameters of the two
results are essentially identical.
The next example involves a tungsten rod
impacting a steel target at 1500 m/s, as shown in
Fig. 3. For this problem both the projectile and
target materials are converted to elements. The
interfaces between the particles of different
materials are represented by springs and dashpots,
determined from the characteristics of the materials
and the artificial viscosity coefficients [4]. With this
interface treatment the GPA algorithm includes only
those neighbors which are of the same material.
are determined from d - V A , where A is the
cross-sectional area of the element.
- The masses of nodes b, c and k are reduced by
the removal of element A.
- Line segment b-c is removed from the list of
interface (master surface) segments, and line
segments b-k and k-c are added to the list.
The conversion of element B (with two sides on
the surface) to GPA node B, and the conversion of
element C (with three sides on the surface) to GPA
node C, follows in a similar manner.
When a finite element is converted into a GPA
node (under the conditions described for GPA nodes
A and B in Fig. 1) the newly generated GPA node
must be attached to the adjacent master line
segment. Unlike a sliding algorithm, the attached
algorithm does not allow the GPA node to separate
from the master line segment or to slide along the
master line segment. The details of this algorithm
are provided in Reference 5.
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higher impact velocities penetrate in an expected
manner. It can be seen that the computed results are
in excellent agreement with the test data.
;:;! i, s* 'IS:.? f?WI
10 % 7J? mm
I
1
Tungsten rod
• mi
L = 150 mm
D = 2 mm
4mm
V
————— Computations
• - - - - - - • Test data
Lundberg et al, 1998
7T
Boron
Carbide
I: as SH |g
47.6 mm
4
$t
Steel case
|<— 19mm —=>)
<——— o ———>
20
30
40
50
time (ms)
FIGURE 4. Impact of a Tungsten Rod onto a Boron Carbide
Target at Various Velocities
FIGURE 3. Impact of a Tungsten Rod onto a Steel Target at
1500/n/s
The lower portion of Fig. 3 shows the
penetration process at 50 and 125 jus after impact.
Here the tungsten is represented by darkened
elements and particles, but the steel target is
represented by light gray elements and particles.
The black line in the target represents the outline of
the elements such that all of the gray target material
on the highly strained side of the line has been
converted to particles. The final penetration is very
close to comparable experimental data [6].
The final example is shown in Figs. 4 and 5. It
involves a very long tungsten rod impacting a
complex target composed of boron carbide ceramic
surrounded by a steel case. The left side of Fig. 4
shows the configuration as provided by Lundberg et
al [7]. The penetration in Fig. 4 is measured from
the top surface of the boron carbide prior to impact.
The right side of Fig. 4 shows computed results
and selected test data (penetration versus time) for
four impact velocities. The test data are provided by
Lundberg et al [7]. For the lowest velocity
(1427 rn/s) the tungsten rod does not penetrate the
ceramic. When the velocity is increased slightly
(1480 m/s), the tungsten rod dwells on the top
surface of the ceramic until 25 jus after impact, and
then it penetrates at a significant velocity. The two
FIGURE 5. Computed Results for Impact of a Tungsten Rod
onto a Boron Carbide Target
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Figure 5 shows the computed results. For the
four geometry plots (not including the damage plot
in the upper right), the tungsten rod is represented
by the darkened elements and particles, while the
steel and ceramic are represented by light gray
elements and particles. The black lines define the
outlines of the elements. In some instances the lines
represent the outlines of the two materials (case and
ceramic), and in other instances they represent an
interface between the elements and the particles.
The upper left of Fig. 5 shows the response for
an impact velocity of Vs = 1427 m/s. Here the
ceramic remains intact, while the defeated rod
moves radially outward along the top surface of the
ceramic until it is contained by the steel case. The
distribution of damage is shown in the upper right
and it can be seen that there is a thin region of
undamaged ceramic directly under the impacting
rod, and this enables the ceramic to remain intact
and to defeat the rod.
The lower left of Fig. 5 is for a slightly higher
impact velocity of Vs = 1480 m/s. The response for
this case is similar to the slightly lower impact
velocity (Vs = 1427 m/s) for the first 25 jus, as
shown in Fig. 4. The higher impact velocity causes
the defeated rod to push the case wall outward more
than the lower velocity, and this in turn reduces the
confining pressure in the ceramic thus allowing it to
be more damaged and to fail. After the ceramic
directly under the rod is fully damaged the rod
penetrates the ceramic. For the two higher velocities
the dwell is limited and the rod penetrates in a
normal manner.
The boron carbide ceramic model used for these
computations is similar to that presented by Johnson
and Holmquist [8], although some changes have
been made for damaged material.
A final comment concerns an important
advantage of converting distorted elements into
particles rather than simply eroding (or removing)
the distorted elements. When an element is eroded it
introduces a void which allows surrounding
material to expand into the void and to lose pressure
as it expands. If the material strength or failure
characteristics are pressure dependent (as they are
for brittle materials) then the pressure drop can lead
to lower strength and/or more damage. If an erosion
algorithm (rather than a conversion algorithm) is
used for the impact conditions (Vs = 1427 m/s) in
the upper left of Figure 1, premature failure occurs.
The highly distorted rod elements erode and the
adjacent ceramic elements expand and fail, thus
allowing the rod to penetrate the failed ceramic
rather than being defeated by the intact ceramic.
Replacing the distorted elements with particles does
not introduce a void and/or pressure drop.
ACKNOWLEDGEMENTS
This work was funded by the U.S. Army TankAutomotive
Research,
Development
and
Engineering Center (TARDEC), the U.S. Army
Soldier and Biological Chemical Command
(SBCCOM), the Defense Advanced Research
Projects Agency (DARPA), the Army High
Performance
Computing
Research
Center
(AHPCRC), and Southwest Research Institute
(SwRI), under contracts DAAN02-98-C-4039,
D AAD16-00-C-9260, and DAS WO 1 -01 -C-0015.
The content does not necessarily reflect the position
or policy of the government, and no official
endorsement should be inferred. The authors would
also like to thank D.W. Templeton (TARDEC),
P.M. Cunniff (SBCCOM), J.E. Ward (SBCCOM),
and C.E. Anderson (SwRI) for their contributions.
REFERENCES
1. Johnson, G.R., Peterson, E.H., and Stryk, R.A.,
International Journal of Impact Engineering, 14, 373383 (1992).
2. Johnson, G.R., Stryk, R.A., Holmquist, T.J., and
Beissel, S.R., Numerical Algorithms in a Lagrangian
Hydrocode, Report WL-TR-1997-7039, Wright
Laboratory, U.S. Air Force, July 1997.
3. Johnson, G.R., Beissel, S.R., and Stryk, R.A.,
Computational Mechanics, 25, 245-256 (2000).
4. Johnson, G.R., Beissel, S.R., and Stryk, R.A.,
International Journal for Numerical Methods in
Engineering, to appear (2001).
5. Johnson, G.R., Stryk, R.A., Beissel, S.R., and
Holmquist, T.J., "An Algorithm to Automatically
Convert Distorted Finite Elements into Meshless
Particles During Dynamic Deformation," Submitted
for publication.
6 Anderson, C.E. and Walker, J.D., International
Journal of Impact Engineering, 11, 481-501 (1991).
7. Lundberg, P., Holmberg, L., and Janson, B., "An
Experimental Study of Long Rod Penetration into
Boron Carbide at Ordnance and Hypervelocities,"
Proceedings of the 17th International Symposium on
Ballistics, Midrand, South Africa, 1998.
8. Johnson, G.R. and Holmquist, T.J., Journal of Applied
Physics, 85, 8060-8073 (1999).
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