Page 98 - Fundamentals of Magnetic Thermonuclear Reactor Design

82 Fundamentals of Magnetic Thermonuclear Reactor Design

The KLONDIKE code intended for a 3D field simulation is used to calculate
H at any point of the domain. The software development was guided by the
ext
following considerations: (1) automated data input, (2) data input should be
limited and checked, (3) enhanced visualisation and (4) software should be easy
to use. KLONDIKE enables analytical computation of the magnetic induction
vector for the following cases:

l any uniformly magnetised polyhedron,
l straight and arc-shaped current filaments,
l straight and arc-shaped current conductors with rectangular cross-sections
and uniform current density distribution.
KLONDIKE uses a number of dedicated modules for field computation. The
field of a polyhedral conductor with preset current density j may be described as



1  N f N i 
⋅
H =  j × ∑ η A ⋅ln B + d tan −1 C   ,

,
,
4 ∑ i  i j, i j i ij
H=14πj×∑i=1Nf∑j=1NiηiAi,j⋅lnBi,j   = i 1 = j 1  
+di⋅tan−1Ci,j, where N is the number of facets; N is the number of edges of the i facet
th
f
i
th
(i = 1,….N ); η is the vector normal to the i facet; and A , B , C and d are
i,j
f
i,j
i
i
i,j
th
th
the coefficients for the i edge of the j facet that are rational functions of the
coordinates of the edge vertices and the observation point.
Using the volume integral equation method [24,25] as a basic one, a re-
gion occupied by ferromagnetics is discretised with a set of convex elements
bounded by flat facets. Assuming the magnetisation vector M inside an element
to be constant, we can write
1 K rS
d
⋅
H =− ∑ ( n M) ∫ 3 ,
3
H=−14π∑k=1K(n⋅M)∫skrdSr , 4 k =1 s k r
th
where S is the surface of the k facet, n is the outer normal to the facet, and r is
k
the radius vector directed from the surface element dS to the observation point.
Then, an analytical solution for the field produced by a uniformly mag-
netised element is reduced to the vector summation of the field produced by
the magnetic surface ‘charge density’. The latter field is calculated by a direct
surface integration and takes the form
N 1  η + B  | M |
−
H ξ = ∑ ∑ (1) k  A sh −1 + jk ξ j , 
ξ j,
= k
j 1 =0  C ξ j ,  4
N 1  ξ + jk + B ,  | M |
H η = ∑ ∑ − (1) k  η A sh −1 η j 
j ,
j 1 =0  C η j ,  4
= k

N 1  sin( ϕ + jk − ψ ) | M |
H ζ = ∑ ∑ − (1) k −  ϕ + jk + sin −1 j  ,
= k
Hξ=∑j=1N∑k=01(−1)kAξ,jsh−1ηj+k+B j 1 =0   ζ C , j   4
ξ,jCξ,j|M|4πHη=∑j=1N∑k=01(−1)kA
η,jsh−1ξj+k+Bη,jCη,j|M|4πHζ=∑j=1N∑k=01(−1)k−j+k+sin−
1sin(j+k−ψj)Cζ,j|M|4π,