PNNL-13501 - Pacific Northwest National Laboratory

equires fixed electromagnetic boundary conditions and
prescribed deformation on rectangular domains has been
achieved. The element formulation is a standard trilinear,
8-node hexahedron.
A series of test problems was simulated to aid in
debugging the code and to verify satisfaction of boundary
conditions and fundamental divergence conditions
required by the theory. A three-dimensional rectangular
mesh is shown in Figure 1 that represents an aluminum
plate. This aluminum plate is assumed initially
motionless and bathed in a uniform magnetic field of unit
strength. Since the ability to capture the effects of
deformation on electromagnetic fields is a key feature, a
series of simple deformation fields was prescribed in this
model so that the effects on electromagnetic fields could
be observed. One of those deformation modes is
indicated in Figure 1 as biaxial stretching of the plate with
the initial magnetic field oriented normal to the plate.
B
Figure 1. Finite element mesh of an aluminum plate with
biaxial stretching and initially uniform magnetic field
The symmetry of the problem allows the mesh to
represent only one-quarter of an actual plate so that two
edges are fixed while the other two edges stretch and
displace in accordance with the prescribed uniform strain
field. An example of the spatial distribution of
disturbance to the magnetic field (B) within the plane of
the quarter plate is shown in Figure 2. This result
indicates the correct symmetry of the solution and shows
that the main disturbance occurs at the fast moving edges
of the plate and propagates inward toward the center of
the plate. In Figure 3, a predicted distribution of Lorentz
190 FY 2000 Laboratory Directed Research and Development Annual Report
B 1 -B 1 (0), T
(b)
t = 0.4µs
0.002
0.000
-0.002
-0.004
-0.006
-0.008
-0.010
0.00
0.02
0.02
0.04
0.04
0.06
0.06
0.08
0.08
0.10 0.10
X 3 , m
force (interaction of electric current and magnetic field,
JxB) is shown. Again the required symmetry is preserved
in the simulation and strong coupling is indicated by the
significant forces generated by the deformation. Finally,
results of divergence error calculations are shown in
Figure 4. In each of the three curves found in Figure 4;
for the electric displacement, magnetic induction, and
conduction current, the divergence of the field is very
small compared to the field strength within the plate.
These divergence conditions are required to satisfy
Maxwell’s equations and are a strong indicator of the
correct numerical implementation of those equations.
Summary and Conclusions
X 2 , m
Figure 2. Snapshot of the magnetic field (B) distribution in
the plate
|b|, N/m 3
1e+6
8e+5
6e+5
4e+5
2e+5
0
0.10
(d)
t = 0.8µs
0.08
X 3 , m
0.06
0.04
0.02
0.00
Building on the theory developments, this year’s project
produced a numerical analysis code capable of solving
0.00
0.02
0.04
X 2 , m
0.06
0.08
0.10
Figure 3. Snapshot of the Lorentz (body) force field
distribution in the plate