Page 93 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 77

∆⋅ µµ −∆ + P) = 0 (4.4) ∆⋅µ µ(−∆+P)=0
ϕ
(
0
0
complemented by boundary conditions. The «vortex» vector Р, which meets the
only requirement ∆ × P = j imposed on its curl, must be preset.
By analogy with the magnetic vector potential A, introduced by the
B = ∆ × A equation, the P vector is often referred to as an electric vector poten-
tial, since ∆ × P = j = σE, or E = 1/σ∆ × P. Physically, Р may be interpreted as
a specific magnetic moment of a double magnetic layer that is equivalent, in the
magnetic sense, to closed currents [13,18,19]. To construct P unambiguously,
the domain is partitioned with dummy magnetic shells spatially correspond-
ing to double magnetic layers. The partitions, resting on closed current loops,
form arbitrarily shaped shells. Vector P is co-directed with the partition surface
vector element and is normal to the partition surface. Actually, P is constructed
based on a specific problem formulation, as well as the algorithm optimisation
principles and software functionality.
The approach, in which the magnetic permeability µ is considered as a
constant within each FE and its values are calculated during solving a nonlin-
ear problem is consistent with the ‘linear macro model of a nonlinear-system
magnetic field’ [20]. The magnetisation curve B(H) is assumed to be known
for any material (e.g. derived experimentally). Such model allows the field
vectors to be calculated with a desired accuracy. In addition, it is consistent
with the physical concept of determination of EM forces in nonlinear media.
This concept is based on equations describing a linear connection between
the B and H vectors for a fixed magnetic state, which is determined by a pre-
set distribution of densities of the j macro currents. We choose the effective
diameter (discrete sampling step) of FEs to obtain the desired approximation
accuracy.
This approach is implemented in the KOMPOT code developed in the
Efremov Research Institute. FEs [21–23] employed are hexahedra with tri-
linear shape functions. Galerkin’s method, as a variational method, enables
finding a solution to the magnetostatic problem in the divergence form (4.4)
with homogeneous and non-homogeneous boundary conditions of the first,
second and third kinds, reflecting various types of symmetry in a magnet sys-
tem and arising from the H = −∆ + P, B = µµ H vector definitions. The
0
boundary condition of the third kind, reflecting the asymptotic behaviour of
a field distanced from a source, can be chosen on the external boundary. Al-
ternatively, more ‘simple’ boundary conditions of the first and second kinds
may be considered. In this case, the external boundary should be reasonably
far from the field sources to make the field distribution near independent of
boundary conditions.
This approach allows one to take into account the spatial distribution of
closed currents j, while the use of a variational method makes the calculation
of the P spatial derivatives unnecessary (as P can be treated in that case as a
piecewise linear function of spatial coordinates).