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
SIMULATION OF SHAPED-CHARGE WITH
SPH REZONE METHOD
Yao, J., Gunger, M.E., Matuska, D. A.
General Dynamics-OTS, Niceville, FL 32578
Abstract. This paper documents our development of two dimensional SPH rezoning. The
MLS functions we have chosen naturally provide a second order accurate rezone scheme. It
satisfies the conservation laws accurately. It also conserves geometry. Huygens
construction is employed to pack particles with an intrinsic level-set method that
automatically deals with the change of connectivity. The initial smoothness of arbitrary
boundaries is well preserved. With an SPH rezoner, the stability, particle size/shape effect,
boundary smoothness, and the symmetry of neighbor searching can be considerably
improved. We have applied the rezone method to simulate a variety of problems involving
energetic materials, include a shaped charge jet problem.
INTRODUCTION
Moving Least-Squared interpolation. It was
concluded that with the MLSPH method, the
stability of the numerical system is the same as
the stability of the dynamic system itself to the
leading order. It is the most satisfying theoretical
treatment of the stability issue of SPH. But it
should be noted that Dilt's derivation of the
stability growth factor did not cover the
boundary particles. Instability may still occur
easily on boundaries.
The SPH (Smooth Particle Hydrodynamics)
method, because of its flexibility, is an ideal
candidate for the link between Euler and
Lagrange methods. However major difficulties
with SPH have been identified that limited its
usage. The one most well known among them is
the tension instability. Another, which has been
less noticed, is the particle size effect. The
tension instability rendered the approach
unsuitable for general hyperbolic boundary value
problems. The particle size/shape effect, which
occurs when the density of the material change,
or the material is highly distorted, directly
reduces the accuracy of the SPH method. A third
draw back of the SPH method is that the physical
volume of a particle is not equal to its SPH
volume because the mass equation does not
provide volume conservation exactly. It will,
after a large number of cycles, make the density
uncertain.
GD-OTS has employed the MLSPH method to
solve general problems of material interactions.
We have spent a significant effort on reducing
these difficulties. Primarily, we have developed a
general rezone method that effectively deals with
these difficulties. With the use of our rezoner,
the OTI-MLSPH code is stable, has avoided
inaccuracy caused by the changing of particle
size/shape, and ensures the particle volume is
equal to its physical volume. We have used our
code to simulate shaped-charge jet problems.
The results are encouraging .
In Dilt's 1999 paper [1], the instability
problem of the SPH method was treated with the
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A SPH rezoner cannot allow misidentification of boundary particles in order to
determine accurate boundary geometry. All the
boundary detecting techniques identified in
Dilt's 2000 paper [2] can easily result in
misidentifications. GD-OTS uses a simpler
method to naturally identify boundary particles
and the method seems to avoid misidentification
in all the well defined cases we have tested. The
local uniformity of rezoned particles is achieved
with an intrinsic level-set Huygens construction
method. One determines the desired level curves,
then arranges particles nearly uniformly on the
level curves. The spacing between level-curves
can be predetermined.
ESSENTIAL POINTS REGARDING
SPH METHODOLOGY
Through application, we developed some rules
associated with the SPH methodology. We list
some of them here we believe are essential:
a). The boundary particles need to be packed
carefully. Boundary smoothness is not optional if
local accuracy (which involves the proper
determination of the normal vector) is a
consideration.
b). Particles should be nearly uniformly
distributed and should have nearly uniform
diameters to eliminate the particle size effect.
After initial packing of new particles, it is
important to assign a volume to each particle.
This can be achieved by finding the Voronoi cell
of each interior particle. We use a geometric
method to determine the vertices of the Voronoi
cell with the known neighbors of a particle.
Therefore the cost is of O(n). The corresponding
particle position is then moved to the geometric
center of its cell to conserve the volume.
c). If A is found to be B's neighbor, then B
should also be A's neighbor for their shapefunction to be symmetric and to satisfy the
global conservation laws.
THE GD-OTS MLSPH REZONER
The conservation of mass, momentum and
energy can be achieved by collecting the partial
distribution from the unrezoned particles that
overlap with a new particle. We define the
fraction of overlap between a newly defined
particle and its 'old neighbors' with an integral of
the MLSPH interpolants. We prove that the
conservation laws are satisfied exactly if the
integral is evaluated exactly. In practice, we
evaluate the integral with a one point quadrature
and obtain second order accuracy. With the
advantage of knowing the MLS interpolant, our
SPH rezoner is accurate to second order
automatically.
The last step of SPH rezoning is to assign the
mass, momentum, energy and other material
properties for new particles. This is
accomplished by an interpolation using the MLS
interpolants of the old particles. The accuracy is
of the second order and consistent with global
conservation.
CONSERVATION LAWS
Since the SPH particle is not assumed to have
a definite geometry except for its volume and
center of mass, the definition of overlap fraction
between the 'old' and 'new' particles cannot be a
geometric one. We can, however, define a
function that serves as a numerical 'overlap'
fraction. The conservation of mass, momentum,
energy can be proven trivially by summation
over all particles.
A major difference between a SPH method
and a Lagrange method is that a SPH particle
has only a numerical volume, but a Lagrange
cell has a definite geometry. The 'overlap' of
particles in SPH can not be as easily defined
geometrically as for a Lagrange method.
However we have a way to assign a new particle
a definite geometry, which conserves volume
exactly, therefore the calculation of density and
pressure is more realistic.
The overlap of the volume of new particle i
and its old neighbor j for all i and j can be
defined with the following MLS integrals,
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The only problem left is how to calculate
Py. We make the choice similar to what is done
in [1] by using a single quadrature to evaluate the
integrals centered at particle z, we find that
^=<(*r-**>.
This allows us to interpolate any linear
function (or quadratic with more cost) exactly.
Assuming that every particle has local support,
we conclude that the error is at most of the
second order for differentiable functions. Thus,
dissipation is well controlled.
th
where (ft is the i MLS interpolant of new
particle at xt and <p$ is the/'1 MLS interpolant at
old particle jc,- . py can be considered as the
portion of new particle i that overlaps with the
old particley'.
The first observation is Zj Py = 1, so the
definition is consistent with the meaning of a
partial fraction. It will be used to calculate the
properties of the new particles / by summation
over partial distributions from all old particles j
in order to ensure conservation. We calculate the
particle properties in the following way:
BENEFITS OF REZONING
The introduction of rezoning in a SPH code
not only solves particle size/shape effects, it also
improves the stability of an SPH solver. When
particles get too close or too far away, a rezoning
will reset the position of particles, therefore
tension instability cannot easily develop if the
rezone is done frequently. One can confidently
run this SPH code without artificial viscosity, but
with the use of a rezoner. Also because the
spacing of particles is locally 'uniform', the CFL
condition is easier to satisfy and one can expect
his SPH code to run faster after rezoning.
Furthermore, one obtains the optimized total
number of particles and has the choice of higher
boundary accuracy with the use of Huygens
construction.
Symbolically, the mass, momentum and the
energy are conserved exactly, for example
V
»™ = V
V™p™
L-ii m i
L^i i ri
m°/d
__
y old Pyy ~
m°ld
From the description of the rezoning scheme,
it is easily seen that the transformation between
an Euler or a Lagrange system and a SPH
system can be done naturally through the use of
the MLS rezoner. The one-point quadrature
evaluation of the MLS integrals makes an
MLSPH method the second order accurate in
space, therefore the rezone will not affect the
accuracy of the method.
y old
however by definition
this amount is exactly V°°
since the MLS
function (p"*™ (x — xt ) exactly interpolates the
number 7. Therefore
With the help of the rezoner, GD-OTS is able
to calculate problems such as shaped-charge
(Fig.l) and explosion clouds with our SPH code,
and to obtain accurate and visually reasonable
results. The original SPH method will most
likely fail in these cases. With the SPH rezone
scheme developed at GD-OTS, the stability and
accuracy of the smooth particle hydrodynamics
old
The mass is conserved exactly. Similarly one
can prove that momentum and energy are also
conserved exactly.
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With the level-set technique, a) can be
achieved provided a method to identify isosurfaces is available. We are quite sure such
resources exist. However, to reduce the
calculation burden, a 3D intrinsic fast-tube levelset method is desired. It is not currently available
but we are working on it. Task b) can also be
done with existing software of three dimensional
triangulation. c) can be done with a Huygens
construction-like technique. Since a general
mesh is not available for an arbitrary surface, we
are working on a marked particle technique to
accomplish the task.
is substantially improved. Our MLSPH simulator
with a rezoner is capable of simulating a widerange of problems.
The SPH method, because of its extreme
flexibility, has the potential to become a
universal solver for all types of problems
involving material interactions at high rates. The
original SPH algorithm had major problems with
tension instability and low accuracy and
therefore could not have served as a substitution
for either Euler or Lagrange codes. The
development of the MLSPH technique by Dilts
provided a solution of the instability issue and
improved the accuracy of SPH method (MLSPH
is second order accurate in space). However
Dilt's treatment does not completely eliminate
these problems. The rezone methodology we
presented in this paper further extends Dilt's
result and successfully addresses all the major
numerical drawbacks of the original SPH
method. This technique has been used to
simulate a variety of problems including shapedcharge jets and the results are promising. We
expect our MLSPH code to eventually become
an effective link between or a substitute for our
Lagrange and Euler methods.
FIGURE 1. A shaped charge Calculation.
FUTURE DEVELOPMENTS
It is not too costly to do SPH rezone in two
dimensional and the algorithm is not hard to
implement. One may rezone quite frequently if
necessary. In the extreme case, it is practical to
use a Voronoi cell to replace the mass equation
so every interior particle is associated with a
definite geometry. However, it is much more
costly to do three dimensional SPH rezoning,
and it is not practical to rezone frequently. By
following the same methodology we employed
for two dimensional axisymmetric geometry, a
three dimensional SPH rezoner can be expected
to work in a similar way.
REFERENCES
1. Dilts, G. A. " Moving Least Squares Particle
Hydrodynamics ~ I. Consistency and Stability",
Int. J. Numer. Mech. Eng. 44, 1115-1155 (1999).
One should obtain the following numerical
resources for coding a three dimensional SPH
rezoner:
2. Dilts, G. A. " Moving Least Squares Particle
Hydrodynamics - II. Conservation and
Boundaries", Int. J. Numer. Mech. Eng. 48,
1503-1524(2000).
a). A 3D Huygens construction method.
b). A 3D Voronoi cell solver.
c). An algorithm to distribute particles
nearly uniformly on a level surface.
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