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
COMPARING LAGRANGIAN GODUNOV AND PSEUDO-VISCOSITY
SCHEMES FOR MULTI-DIMENSIONAL IMPACT SIMULATIONS
Gabi Luttwak
Rafael, P.O.Box 2250, Haifa 31021,Israel
Abstract. Modern Eulerian codes use second order Godunov schemes, while Lagrangian calculations
are routinely carried out with first order pseudo-viscosity codes. We compare these, looking for the
right scheme to be used in a 3D ALE code, which should handle both Eulerian and Lagrangian
meshes. In a Lagrangian Godunov scheme, the grid velocity has to be defined. We first look at
schemes based on interpolating the velocities to the vertices, next we consider the advantages of a new
staggered mesh Godunov (SGM) scheme. These schemes are tested both with the ID Sod shock tube
problem and the 3D normal impact of square rods into a wall.
Therefore, we propose a new staggered mesh
Godunov (SMG) scheme. We show that, carried
out with this new scheme, the results compare
favorably with the standard Lagrangian
simulation. The new SMG scheme is also tested
for the case of the ID Sod shock tube problem.
The numerical simulations were carried out in the
code AUTODYN 3D. The author has participated
in the development of AUTODYN's second order
Eulerian processor(5). The current extensions to
this scheme to handle non-Cartesian and moving
mesh were carried out in external user routines.
INTRODUCTION
Modern Eulerian hydro-codes use second order
Godunov techniques (1"5). They produce more
accurate results than the older, first order, Eulerian
pseudo-viscosity codes. On the other hand, most
Lagrangian codes(6) are only first order accurate
and use pseudo-viscosity(7"8) to handle the shocks.
Still, if the grid remains smooth, the Lagrangian
solutions are superior. We shall compare these
methods both theoretically and numerically. We
look for the best scheme to be used in a multimaterial Arbitrary Lagrangian Eulerian (ALE)
code(4'13) that should handle both Eulerian and
Lagrangian meshes. We shall show the analogy
between the Godunov schemes and the pseudoviscosity defined according to Kuropatenko's rule.
Most Lagrangian schemes use a staggered mesh
with the velocities defined at the vertices.
Godunov schemes have zone-centered variables
with the values at faces obtained from the
Riemann solution. Thus, the vertex velocities have
to be found by interpolation. We experiment with
interpolating either the zone-centered, or the face
centered velocities to the vertices. To check these
schemes, we consider the normal impact with
square rods. We find the simple, first order
Lagrangian calculation to give more robust results.
THEORY
Apart from the material model, the flow regime is
governed by the conservation laws of mass,
momentum and energy. In integral form, these can
be applied over an arbitrary control volume. As the
size of this control volume is made negligibly
small, the differential equations of motions are
obtained. Early finite difference schemes were
deduced from the differential equations, while
later it was found that the best schemes are
obtained directly from the integral conservation
laws, by choosing the finite difference zone as the
control volume. Both the differential and the
difference equations apply only for smooth
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between the zones. Thus to get their values on the
zone faces, a Riemann problem must be solved.
The solution yields the pressure p* and velocity u*
at the face. In many codes(2"4'n) this problem is
solved assuming a two-shock approximation, thus
getting the R-H equations. The resulting pressure
jump at a zone face is given by (1). Moreover, to
get a non-iterative solution it is customary to
assume locally the linear relationship^"4'n) (2),
getting again (3). The pressure p* exerts an
impulse on the neighboring zones. This impulse is
summed up, to update the zone momentum and
cell centered velocity. For an Eulerian mesh, the
face velocity u* is used to compute the advection
terms. For a Lagrangian scheme, this face and
mid-time step centered velocity might be used to
interpolate for the vertex velocities. For onedimensional schemes, the face center coincides
with the zone vertex and no interpolation is
needed. In CAVEAT(4) a least square fit of the face
velocities u* was used, to get the vertex velocities.
This implies finding the inverse of a 3x3 matrix at
each point and multiplying by it. Instead we tried
simple weighted averaging. Following CAVEAT,
the weights were chosen inversely proportional to
the face area. The averaging was done separately
for the faces around the node in each mesh
direction. The resulting vector components were
summed up:
regions. However, as the equations of motions are
hyperbolic, discontinuities may emerge during the
flow. The conservation laws also yield the
Rankine-Hugoniot (R-H) relations, which hold
across a shock.
In the presence of discontinuities, the difference
equations lead to instabilities. VonNeuman(7)
solved this problem by adding to the pressure a
pseudo-viscosity, which captures the shock over 23 zones. With a term quadratic in the velocity
gradient Vi/ , this width remains constant as the
shock travels over the mesh and the R-H relations
are satisfied over the shock. Kuropatenko(9)
reverted this procedure and obtained the pseudoviscosity term directly from the pressure jump
resulting from the R-H equations:
(1)
where pQ is the density before the shock, U is
the shock speed and 5u is the velocity jump
across the shock. Wilkins(8) considered (2) for the
cases of an ideal gas and of an elastic solid.
Instead we note, that most solids over an wide
range of pressures obey a linear relationship
between the shock speed and particle velocity:
U = C +sdu
(2)
_* \
2JOaU a
^-i
Thus inserting (2) into (1), we get directly:
2
Q = sp0Su + p0c0du
a=l,4
a=ij,k
(3)
>®« =
(4)
In the past(14), we have incorporated an ALE
capability into the first order Euler processor of
the 3D code MSC/DYTRAN. The Euler velocities
were defined at zone center, and we took a massweighted average to define the nodal velocities.
We have also tried to use a similar approach here,
thus, instead of (4):
This is the form of pseudo-viscosity, chosen by
Wilkins(8) and used in most Lagrangian codes. He
also extended the pseudo-viscosity technique to
multi-dimensions. His choice of directional
pseudo-viscosity works well for oblique shocks
and zones with large aspect ratios.
The integral form of the conservation laws does
hold for regions containing discontinuities. When
passing from them to the difference equations, the
surface integral terms are computed. Evaluating
the variables on the surface by simple averaging is
valid only if there exist a continuous solution
there. Godunov(1) considered the variables constant
over each zone, with a jump at the interface
(5)
7=1,8
Once the vertex velocity is known, the grid
points can be moved. Thus, the geometry of the
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problem changes and the zone volumes and face
vectors have to be calculated.
In second order Godunov(2"4) schemes, a piecewise linear variable distribution is assumed. The
zone-centered values are projected to the faces at
mid-time step, using the known slopes. These
slopes and the projected, face-centered values are
limited(2"3) to prevent instabilities. Due to the jump
at the faces, at Tn+1/2, a Riemann problem has to be
solved there. Using the Gauss theorem, the slopes
at Tn+1 are computed as surface integrals over the
zone. The surface values are found again from the
solution at Tn+1 of a Riemann problem at the face.
Before applying (5), the slopes can be used to
project the zonal velocities, ut, to the vertices
andtor +7/2 .
Figure 1. The staggered mesh
We add the impulse ApASt , (A is the face area
vector), to the neighboring zones and split it into
its vertices. The contribution of the face impulse
on its four vertices cancels out. After the vertex
momentum is updated, the new vertex velocity is
found by dividing it by the node mass. The zonecentered velocity is found by averaging over the
vertices, taking into account the velocity gradient
in the zone:
THE SMG SCHEME
First order, Lagrangian difference schemes use a
staggered mesh. In smooth regions, at fixed time
step, this is a "cheap" way to get second order
accuracy. These schemes are robust, as the
pressures have an immediate effect on the zone
nodes, without any interpolation. A vertex defined
mass and velocity is also required for slidingimpacting interface calculations. Still, for an ALE
scheme, which should also work with an Eulerian
mesh, we would like to use a second order scheme.
Therefore, we propose a new staggered mesh
Godunov (SMG) scheme. In a Godunov scheme
the impulses on the zone faces are summed up and
used to update the zone momentum. The simplest
solution would be to split the momentum influx
into the zone to its 8 vertices. Yet for the impulse,
this would effectively mean the use overlapping
control volumes. Therefore we look at the
staggered mesh formed by connecting the zone
and face centers. A zone is split into 4 quads in 2D
(see in figure 1) or or 8 hexahedrons in 3D. We
use the zone-centered pressure to find the impulse
over the faces AF and FB of the staggered mesh.
This impulse is added to the vertex momentum at
O. However the contribution of the face pressures,
p*, along OA,OB,etc... cancel out and there
would be no terms to handle the shocks. Let p be
the average pressure at the face and p* the
(6)
THE CALCULATIONS
The normal impact of a square rod into a wall is a
3D flow problem modeled with a simple and
regular mesh. For low speed, it is the square rod
Taylor impact(12) , but here we varied the speed,
from 0.5 to 1 mm/us , looking at the Lagrangian
grid motion for each difference scheme. We
considered rods made of copper and steel and we
also turned off the strength calculation, to check
the numerical scheme. In Fig. 2, we can see the
results at for the 0.5 mm/us impact of the copper
rod at T= 8 us. A shock EOS with c0=:3.94mm/us,
s=1.49, Gruneisen T=2.0 and failure on pressure
at pnm^-3.0 GPa was used in the calculations. In
Fig.2a, we can see the grid distortion for the
Godunov scheme. Using (6) instead of (5) did not
improve the results. Figs. 2b, 2c show the same
results for the SMG and the first order Lagrangian
scheme.
The Sod problem(14) is a test which shows how
well a given scheme resolves shocks and contact
surfaces in ID motion. It is a shock tube filled
with a y=1.4 ideal gas, with the density, p=l, and
pressure, p=l on the left and p=0.125, p=0.1, on
the right half of the tube. In Fig. 3 we can see the
density profiles at T= 25 for the respective cases.
Riemann solution there. Ap = p — p , has a
similar effect to a face defined pseudo-viscosity.
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Figure 2. Normal impact of square rod in a Lagrange mesh (a) Godunov (b) SMG (c) Lagrangian. (This figure is in color
on the CD.)
Figure 3. Density profile for Sod problem with Lagrange mesh ( a) Godunov (b) SMG (c)Lagrangian. (This figure is in
color on the CD.)
AUTODYN3D", Proc. 9th Int. Symposium on
"Interaction of the Effects of Munitions with
Structures", Berlin, May 1999
CONCLUSIONS
Lagrangian Godunov schemes, with zone centered
velocities, are less robust than first order,
staggered, pseudo-viscosity schemes. This does
not exclude their use, coupled with an ALE grid
motion . The new staggered mesh Godunov, SMG
scheme, seems to behave well, both in the 3D
impact and for the Sod problem and thus it seems
a better candidate for use in an ALE code.
6. Wilkins M. L. /'Calculation of Elastic-Plastic Flow",
Meth. Comp. Phys. ,3, p211, B. Alder et al. eds.,
Academic Press (1964)
7. Von Neumann, Richtmyer R., D., J. Appl. Phys, 21,
p232,(1950)
8. Wilkins, M. L. ,J.Comp.Phys,36,p281,(1980)
9. Kuropatenko,V.,F.,pll6 in "Difference Methods for
Solutions of Problems of Mathematical Physics",
Janenko N.,N. ed., Am. Math. Soc. Providence,
RI,(1967)
10. SodG.,A, J.Comp.Phys., 27, pi, (1978)
AKNOWLEDGEMENTS
11. Dukowicz, J.K. , J.Comp.Phys., 61, pi 19, (1985)
12. Luttwak G., Rosenberg Z., Falcovitz J.,
I want to thank Prof. J. Falcovitz, for helpful
discussions on the Godunov schemes.
"Experimental and Computational Study of
Taylor Impact with Square Rods", p291, 15th Int.
Symposium on Ballistics, Jerusalem,(1995),
Mayseless M., Bodner S.,R. eds.
13. Hirt, C, W., Amsden, A. A., Cook J. L.,
J.Comp.Phys. 14, p227, (1974)
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