PNNL-13501 - Pacific Northwest National Laboratory

Global divergence error
1 10 -14
0
-1 10 -14
-2 10 -14
-3 10 -14
-4 10 -14
Magnetic induction
Electric displacement
(d)
-6 10 -14
-5 10 -14
0 5 10 15
0.5
Conduction current
0
20 25 30
Time step number
Figure 4. Global divergence errors
Maxwell’s equations in a Material (Lagrangian) frame.
Our work was motivated by the need for a simulation tool
to study advanced (three-dimensional) applications of the
electromagnetic forming process. The code was based on
the finite element method and employs the least squares
variational principle to solve the first order system of
equations directly. The code is presently limited to 8node
hexahedral elements.
For numerical efficiency, a preconditioned conjugate
gradient solver was employed along with optimized
skyline matrix storage. The code was tested on a series of
simple problems to verify the numerical implementation
of the theory. This software has proven to be robust and
will serve as a basis for further development into a very
general numerical capability for solving Maxwell’s
equations in deforming media. This is a first of its kind
numerical capability that will have very broad application.
3
2.5
2
1.5
1
Future work under programmatic funding will involve
developing the code to efficiently implement general
boundary conditions and systems with multiple
electromagnetically interacting bodies. The code must
also be coupled to a mechanics code that solves systems
involving three-dimensional elasto-viscoplasticity and
general three-dimensional contact.
References
Bochev PB and MD Gunzburger. 1998. “Finite-element
methods of least-squares type.” SIAM Reviews
40:789-837.
Jiang B-N, J Wu, and LA Povinelli. 1996. “The origin of
spurious solutions in computational electromagnetics.”
Journal of Computational Physics 125:104-123.
Publications and Presentations
El-Azab A and MR Garnich. “Lagrangian treatment of
the initial-boundary-value problem of electromagnetics in
deforming media.” Journal of Computational Physics
(submitted).
El-Azab A and MR Garnich. 2000. “On the numerical
treatment of Maxwell’s equations in deforming
electromagnetic media” Presented at the Int. Conf.
Computational Engineering Sciences, August 2000,
Anaheim California. In Advances in Computational
Engineering & Sciences, Vol. II, 1687-1692, Eds.
SN Atluri and FW Brust, Tech Science Press, Palmdale,
California.
Design and Manufacturing Engineering 191