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
DISCRETE ELEMENT METHOD MODELING OF GAS
Wenqiang Wang, Zhiping Tang, Ping Gong*, and Y. Horie+
* University of Science and Technology of China,Hefei,Anhui 230027 China
+ Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA
Abstract. Gaseous elements for the Discrete Element Method (DEM) modeling are presented. One
distinct feature is that they have interactions even if they are separated over a large distance. The
long-range interaction presents a problem in searching neighbors and calculating interaction forces. An
approach is proposed to deal with this difficulty. Using a modified DM2 code[l], several problems
involving gas have been investigated. The most meaningful simulation is the problem of laser-induced
explosion damage in optical film. Results are in quantitative agreement with experimental measurements,
demonstrating a unique way of studying such problems.
INTRODUCTION
INTERACTION FORCES BETWEEN
Discrete Element Method is a relatively new
concept numerical simulation technique in
mechanical engineering. Since Cundall[2] conducted
the first practical DEM simulation in 1971, it has
become the most popular method of solving
problems in granular materials. However, its
application to continuum mechanics problems has
not yet been fully explored.
The basic idea of DEM simulation is to divide a
real material into separate elements and solve the
evolution of the discrete system that is governed by
Newton's second law. This idea is similar to that of
molecular dynamics method (MD), however,
elements herein are not real atoms or molecules.
In previous DEM studies, elements are either in
solid or liquid state. It seems there is no report on
how to conduct DEM modeling of problems
involving gas. In this paper a "gaseous discrete
element" model is presented which can be used to
simulate such problems. And a simulation of
laser-induced explosion damage in optical film is
presented to illustrate the efficiency of the model.
GASEOUS ELEMENTS
In DEM simulations, it is crucial to correctly
calculate the interaction forces between neighboring
elements. Different interaction force models have to
be used to represent different material properties
such as elasticity, plasticity, and viscoelasticity, etc.
In this study, interaction force F between two
neighboring elements (at least one is in gaseous
state), e.g. elements A and B, is calculated as
follows:
r^
_
AB
jf
o
J AB AB
,* \
^ '
where f AB and SAB are the interaction force per
unit area and the interaction area (between A and B),
respectively.
f AB is taken as the local gas pressure/? between
elements A and B, which can be obtained from the
equation-of-state (EOS) of gas.
For instance, for a polytropic gas [3],
, — A ^J
259
(2)
illustrated in figure 1 where the hollow circles
where p is gas density, A is a constant and y is the
isentropic exponent.
For an explosive product [4],
p = a(l - ——) exp(-^j v) +
——
R2v
(3)
where v is relative volume (i.e., the ratio of current
volume to initial volume), £0 is the specific internal
energy, the others are non-dimensional parameters.
In using the EOS, v and p are defined as local
quantities as follows:
v=
FIGURE 1. Neighbor searching scheme
represent gaseous elements.
Element C is taken as an example to present the
scheme, which can be expressed as follows: an
element is a neighbor of element C if and only if: (1)
It is inside the "influence region" of C;
(2)
There are no other elements locating inside the
"interaction region" between these two elements.
The so-called "influence region" of element C
is the region enclosed by a circle centered at the
center of element C. The radius R of the circle is
determined according to the type of gas, e.g., for the
polytropic gas with y = 3, R is set to be 2.2d. All the
elements in the "influence region" are potential
neighbors of element C.
The "interaction region" between two elements,
e.g. C and D, is the region enclosed by four lines, two
of which are perpendicular to and the others parallel
to line CD. The first two lines pass through points C
and D respectively. The other two locate
symmetrically at the two sides of line CD. The
distance between them and CD is an adjustable
parameter. However, a value of 1.05 times of the
element diameter seems reasonable according to our
simulation experience.
It can be seen from figure 1 that, although
element D is in the "influence region" of element C,
there is an element, I, locating inside the "interaction
region" between C and D. The interaction between C
and D is shielded, hence C and D are not neighbors.
(4)
where dAB is the distance between the centers of
elements A and B. d is the initial diameter of element
A or B (supposing they have the same size).
(5)
where pQ is the initial gas density.
NEIGHBORHOOD RELATIONSHIP FOR
GASEOUS ELEMENTS
Neighborhood relationship is an important
concept in DEM simulations. An element can
interact with its neighbors only. Hence, only when
neighborhood relationship has been established,
could interaction forces be calculated. For solid or
liquid elements, neighborhood relationship is
determined simply through distance judgement.
However, this is not the case for gaseous elements,
due to the fact that they may have interactions even if
they are separated over a large distance. For instance,
for a polytropic gas with y — 3, it can be seen from
equations (2) and (4) that, when dAB=2.2d,p reduces
to 0.8% of its initial value. This means that,
interaction forces can not be ignored within the range
of 2.2 d.
The long-range interaction makes it more
cumbersome to search neighbors. A scheme is
proposed in this study to tackle this difficulty. It is
SIMULATION RESULTS
Several problems involving gas have been
investigated to test the above gaseous element model.
260
laser-induced damage in optical film, of which the
experimental records [5] are shown in figure 2.
All the results are in quantitative or qualitative
agreement with experimental measurements. The
most meaningful simulation is the problem of
FIGURE 2. Laser-induced damage and surface wave phenomenon in optical film
Free Surface
PeriodicalBoundary^
^Periodical
•Boundary
Rigid Wall
FIGURE 3. The DEM model
FIGURE 4. Enlargement of the central part of the model
EOS for the explosion product, in which the
parameters are: a=26l.7GPa, b=Q.85%GPa,
Eo=l.98kJ/cm2, R{=5.9, R2=IA, w=\A. It is assumed
that the film has melted under laser irradiation, so its
elastic modulus decreases considerably compared
with the corresponding solid state value. In the
simulation, 7.76GPa is used to give a reasonable
result. This equals to one tenth of the solid state
value. The assumption of melting was originally
inspired by the water-wave-like surface wave
phenomenon, considering the fact that the surface
wave phenomenon here is very similar to what
occurs in water. This assumption was confirmed by
further experimental observation and temperature
calculation [5].
The phenomenon is believed being caused by
the explosion of defects inside the film. The defects
explode due to the large amount of deposited laser
energy. The explosion event induces a complex
interaction of stress waves (the longitudinal pressure
wave, the longitudinal rarefaction wave, and the
surface Rayleigh wave, etc.), and finally cuts holes
and forms ripples on the surface of the film.
Figure 3 shows the DEM model of a SiO2 film
that is Sum thick and 200um long. The model
contains 9 layers of elements in triangular dense
packing. The element radius is 0.3um. Total number
of elements is 2993, among which 34 are defect
elements in gaseous state. The material of the defect
is some kind of silicate. Equation (3) is taken as the
261
clearly. The simulated deformation of the film is
shown in Figure 5. It clearly shows that the explosion
cuts a hole with almost the same size as the defect.
Figure 3 is not displayed according to the real X
to Y ratio, as the thickness of the film is much
smaller than its length. In Figure 4 the central part of
the model is enlarged so that it can be seen more
(a) The central part of the film at 6ns
(b) The whole film at 20ns
FIGURE 5. Explosion of the defect and formation of the hole
flM
-20.0
20,0
40.0
60.0
0.0
20.0
40.0
60.0
X(um)
(b) Simulated result
FIGURE 6. Surface wave profiles
observations, showing that the approach is practical.
The details of the surface wave profile are
shown in figure 6. The amplitude given by the
ACKNOWLEDGEMENTS
simulation is about lOum, close to the experimental
We gratefully thank professor Fan Zhenxiu and Mr. Hu
value, lO.lum. However, the simulated wavelength,
Haiyang
for their experimental data and valuable discussions.
18.6um, is considerably smaller than the
experimental result, 30.9um. It is believed the
REFERENCES
discrepancy is due to two reasons: (1) The simulation
1. Tang, Z.P., Horie, Y, and Psakhie, S. G, in High Pressure
is two-dimensional, whereas the experiment is
Shock Compression of Solids IV, edited by L Davidson, Y.
three-dimensional; (2) The DEM model is much
Horie, and M. Shahinpoor (Springer,New York, 1997),
smaller than the real film in experiment, hence
p.143.
2. Cundall, PA. and Strack, O.D.L., Geotechnique, 29(1979),
boundaries have greater influence on the result.
(a) Experimental result
CONCLUSIONS
3.
A gaseous element model for discrete element
method simulation of gas is presented in this paper.
The model is validated through a simulation of
laser-induced explosion damage of optical film.
Simulated results are in agreement with experimental
4.
5.
262
47.
Henrych, J., The Dynamics of Explosion and Its
(/se,Elsevier Scientific Publishing Company, New
York, 1979.
Liu, Yunjian, et al., Explosion and Shock Waves (in
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Fan, Zhenxiu, and Hu Haiyang, private communications,
2000.