A Efficient Integral Formulation for Electric Thin Region
Résumé
Different works have shown the interest of using Volume Integral Method
(VIM) for 3D magnetic field analysis the main advantage being that the air
region does not need to be meshed. Moreover solving Maxwell's equations by
Green's Function Integral Equation Method has known a renewed interest in
recent years, with the development of matrix compression algorithms (FMM,
ACA) which greatly improves storage and resolution of full matrix systems. In
this work we propose a specific formulation dedicated to thin region, based on
a VIM circuit approach with the use of current meshes as unknowns.
The formulation proposed allows solving electric thin regions problems or
volume regions treated by surface impedance boundary conditions. Dynamic
effects along the thickness of the thin region can be taking into account thanks
to the use of an equivalent complex conductivity. Non simply connected
regions are naturally taken into account thanks to a circuit approach, by the
determination of a fundamental branch independent loop matrix Mc.
For obtaining a quick and efficient matrix Mc, an unusual procedure is
developed, with the help of the curl connectivity matrix Cc. The Cc matrix,
which rely nodes and edges on the surface mesh, can be easily obtained from
the finite element Whitney representation. In practice this incidence Cc matrix
gives directly an independent branch loop matrix in the case of simply
connected thin regions. In presence of non-simply connected regions, the
matrix Cc is adapted thanks to a specific algorithm which allows determining
the missing loops. Moreover, this approach allows obtaining a good
convergence of the system when using an iterative solver.
The figure below illustrates an example of a multiply connected conducting
problem, with the use of a surface impedance boundary condition. Starting
from the Cc matrix, two supplementary loops are found, corresponding to the
eddy currents rotatable along the torus and the section.
An example of a non-simply connected conducting problem (eddy currents on
a torus)