Page 92 - Fundamentals of Magnetic Thermonuclear Reactor Design

76 Fundamentals of Magnetic Thermonuclear Reactor Design

different ε, µ and σ values, or due to surface (infinitely thin) current/free charge
layers. Boundary conditions can be derived by passing to the limit relations be-
tween the field vectors on medium interfaces and the Ostrogradsky–Gauss and
Stokes integral laws [11].

4.3 STATIONARY FIELD ANALYSIS AND SYNTHESIS

The magnetostatic field obeys the two Maxwell equations:
∆×H=j ∆× H = j (4.1)

∆⋅B=0 ∆⋅ B = 0 (4.2)
that are complemented by boundary conditions and material equation B = B(H)
(magnetisation curve). For an anhysteretic isotropic medium, the latter takes
the form

B=µµ H=µ (H+M), B = µµ 0 H = µ ( H + M), (4.3)
0
0
0
where the magnetic permeability µ and the magnetisation vector M are preset
functions of H for any material. The current density j distribution is known ini-
e
tially and can formally be taken as a field of the extrinsic current j .
4.3.1 Stationary Field Analysis
A variety of methods exists to calculate in detail the magnetostatic field. The
magnet system computational model can be built using the differential, integral,
or integral–differential formulations [14]. The reactor design, as well as engi-
neering and physical features, suggest that a combination of these approaches
would be most effective.
Variational methods, often in the form of the finite element method, for the
differential formulation of the problem, are well suited for numerical solution
and calculation of the attendant characteristics of geometrically complex mag-
net systems. However, in practice, it is often difficult to cope with the number
of unknowns on a large finite element (FE) mesh that may have 40–100 million
nodes. To reduce the problem dimension, one can use the reduced magnetic sca-
lar potential [15–17]. The associated approach is known as the Т–Ω method or
the electric vector potential technique. In this case, the number of variables for
a potential is minimal and is practically the same as the number of mesh nodes.
Let us discuss this method in more detail.
We introduce vector P, satisfying equation ∆ × P = j with ∆ × H = j,
∆ × H = ∆ × P, or ∆ × (H − P) = 0. Then, a reduced magnetic scalar potential
 (also denoted as Ω) can be introduced to describe the field (H − P). Hence,
H − P= −∆, or H = − ∆ + P. The potential distribution is derived from a
solution of the boundary value problem for the equation