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In this section we perform two sets of calculations. The first set demonstrates that
our support-operators method converges with second-order accuracy for a problem with
a material discontinuity and a non-smooth mesh. The second set demonstrates the
effectiveness of our preconditioner as a function of mesh skewness. There are
three types of meshes used in all of the calculations: orthogonal, random, and
Kershaw-squared. Every mesh geometrically models a unit cube, and the outer surface of
each mesh conforms exactly to the outer surface of that cube. Each orthogonal mesh is
composed of uniform cubic cells having a characteristic length, lc
. The random
meshes represent randomly distorted orthogonal grids. In particular, each vertex on
the mesh interior is randomly relocated within a sphere of radius r0
, where
r0 = 0.25lc
. These random meshes are both non-smooth and skewed, but these
properties are approximately constant independent of the mesh size. The
Kershaw-squared meshes are a 3-D variation on the 2-D Kershaw meshes that first
appeared in
[13]. An example of a
20 x 20 x 20
Kershaw-squared mesh is
shown in Fig. 7.
Figure 7:
A
20 x 20 x 20
Kershaw-squared mesh.
|
This mesh becomes increasingly non-smooth and skewed as the mesh size
is increased.
The problem associated with the first set of calculations can be described as follows:
- D(z) = Qz2 ,
|
(70) |
for
z
[0, 1]
, where
D(z) = ![$\displaystyle \left\{\vphantom{
\begin{array}{ll}
D_1 & \hspace{0.25in},\mbox{...
...}, \\
D_2 & \hspace{0.25in},\mbox{for $z \in [0.5,1]$},
\end{array} }\right.$](img342.gif)
|
(71) |
with a reflective boundary condition at z = 0
, a Marshak vacuum boundary
condition at z = 1
, and where
D1 =
,
D2 =
, and Q = 1
.
We refer to this problem as the two-material problem. The exact solution to the
two-material problem is:
where
a = , b = , c1 = - , c2 = - .
|
(73) |
This problem is solved in 3-D on a unit cube having the vacuum boundary condition on
one side of the cube together with reflecting conditions on the remaining five sides.
We have performed several calculations for the two-material problem with meshes of
various sizes. Each calculation uses a mesh with an average cell width
that is half that of the preceding calculation. The relative L2
intensity error
was computed for each calculation. This error is defined as the L2
norm of the
difference between the vector of exact cell-center intensities and the vector of
computed cell-center intensities divided by the L2
norm of the vector of exact
cell-center intensities, i.e.,
|
-
|2
|
|2
. The errors are plotted as a function of
average cell length in Fig. 8 together with a linear fit to the logarithm of the
error as a function of the logarithm of the average cell length.
Figure 8:
Plot of convergence data and least-squares fit to data.
|
The slope of this
linear function is 1.98. Perfect second-order convergence corresponds to a slope of
2.0. Thus our support operators diffusion scheme converges with second-order
accuracy for the two-material problem on random meshes.
The problem associated with the second set of calculations can be described as
follows:
- D = Qz2 ,
|
(74) |
for
z
[0, 1]
, with Marshak vacuum boundary conditions at z = 0
and z = 1
, and
where
D =
, and Q = 1
. We refer to this problem as the homogeneous
problem. The homogeneous problem is solved in 3-D on a unit cube by having the
vacuum boundary conditions on two opposing sides of the cube with reflecting
conditions on the remaining four sides. We have performed calculations for this
problem using both random and Kershaw-squared meshes in conjunction with two
different solution techniques. The first is to apply row and column scaling to the
coefficient matrix and then solve the resulting system using the conjugate-gradient
method in conjunction with symmetric successive over-relaxation (SSOR) for
preconditioning. We refer to this as the one-level solution technique. The second
is to apply row and column scaling to the coefficient matrix and then solve the
resulting system using the conjugate-gradient method in conjunction with the
low-order 7-point cell-center diffusion scheme for preconditioning. We refer to this
as the two-level solution technique. The low-order equations are solved by first
applying row and column scaling to the low-order coefficient matrix and then using
the conjugate-gradient method in conjunction with SSOR preconditioning. Note that
the low-order system is solved once per full-system conjugate gradient iteration.
The total conjugate-gradient iterations required for the full system, the maximum
iterations required for the low-order system, and the total CPU time is given for each
calculation in Table I.
Table I:
Comparison of One-Level and Two-Level Solution Techniques.
Technique |
Mesh Type |
FS |
Max LO |
CPU Time |
|
|
Iterations |
Iterations |
(Sec) |
One-Level |
Random |
97 |
- |
143.24 |
Two-Level |
Random |
7 |
32 |
61.53 |
One-Level |
Kershaw2
|
175 |
- |
247.17 |
Two Level |
Kershaw2
|
46 |
42 |
352.91 |
It can be seen from Table I that the two-level solution
technique takes 14 times fewer full-system iterations than the one-level solution
technique on the random mesh, but it takes only about 3.5 times fewer full-system
iterations on the Kershaw-squared mesh. This is expected since the low-order scheme
becomes increasingly inaccurate relative to the full scheme as the mesh becomes
increasingly skewed. The reduction in iterations observed for the random mesh is
indicative of the reduction seen in well-formed meshes. Note that the two-level
scheme is faster than the one-level scheme on the random mesh, but it is slower than
the one-level scheme on the Kershaw-squared mesh. The decrease in CPU time for the
two-level scheme will be very dependent upon the method used to solve the low-order
system. For instance, rather than solve the low-order system to a high level of
precision using a Krylov method, one might simply perform a fixed number of multigrid
V-cycles. This would greatly reduce the cost of the preconditioning step and thereby
reduce the total CPU time as well. Such a strategy was employed with great benefit
in [1]. It is important to realize that the structure of the low-order
cell-center system on structured meshes is compatible with standard multigrid methods
such as Dendy's method [14], whereas the full system has a structure that is
incompatible with standard methods. Thus the low-order preconditioning approach
enables highly efficient solution techniques to be used in an indirect manner when
they cannot be directly applied to the full system.
Next: Appendix
Up: Morel99a
Previous: Solution of the Equations
Michael L. Hall