Page 96 - Fundamentals of Magnetic Thermonuclear Reactor Design

80 Fundamentals of Magnetic Thermonuclear Reactor Design

In terms of surface coordinates ξ and η a radius-vector of any point on the quad-
rilateral surface can be taken in the form

4
r ξη = ∑ N ( ξ n,) r
)
(,
r(ξ,η)=∑i=14Ni(ξ,n)ri−1≤ξ, η≤1, i i
= i 1
−≤ ξη ≤1,
,
1
Ni(ξ,η)=1/4(1+ξiξ)(1+ηiη),
ξη =±1.
ξη =)1 /4(1
ξi,ηi=±1. where N (, + ξξ)(1 + ηη), i , i
i
i
i
We use the same shape functions N (,
ξη) to approximate the B distribution
i
Ni(ξ,η) on the quadrilateral surface. Then the final equation for the magnetic flux will
take the form
Φ=1/2{(r14×r23)⋅B +1/3(r14×∆ Φ = 1/ 2{( r × r ) ⋅ B +1/ 3( r ×∆ B + r × ∆ B ) r ⋅ ξη },
14
14
23
0
23
23
14
0
B23+r23×∆B14)⋅rξη},
r
,
i
B
B =1/4(B +B +B rij=rj−ri, where r = r − , B∆ ij = B − B , i B = 1/ 4( B + B + B + B ),
=+B ),
j
2
4
1
3
0
ij
j
j
i
−
B 4
∆Bij 3
1
0
2
r
r = 1/ 4 ξη r = 1 /4( r − r − r + ).
rξη=1/4∑ξiηiri=1/4(r −r −r + ξη ∑ i i i 1 2 3 4
3
2
1
r ). Software modules enabling the implementation of this approach have
4
been incorporated, as post-processing tools, in the computer codes intended
to simulate 3D magnetostatic fields. The user can control the computation ac-
curacy determining the balance of magnetic fluxes coming through any closed
surfaces.
A complex system design has to account for the interaction of different
functional components. As to the ITER design, the stray field of the reactor,
affected by the building steel rebar and ferromagnetic surroundings, influ-
ence the operation of the neutral beam injectors, turbomolecular pumps, and
diagnostic equipment. The modelling of such problems is often complicated
by the fact that each MS would be well described by a local FE mesh. How-
ever, common meshing is a very laborious task. The common mesh gets
so sizable, that the problem cannot be solved in principle or in reasonable
time, while the mutual effect of the functional components is often specified
analytically or in a tabulated form, rather than described in terms of the FE
approach.
To solve the problem, field H , generated by external sources, should be
ext
applied and adjusted at the next stage to find a self-consistent solution in the
regions of interest.
Let a static magnetic field be described by vectors B, H, M and j. The vec-
tors are related by Eqs. (4.1)–(4.3). We introduce the field H that may be
ext
determined, particularly, as a field generated by current sources j in vacuum.
This field fits
∇×Hext=j, ∇⋅µ Hext=0. ∇× H ext = j, ∇ µ ⋅ 0 H ext = 0.
0