Page 106 - Fundamentals of Magnetic Thermonuclear Reactor Design

90 Fundamentals of Magnetic Thermonuclear Reactor Design

ρi,wj>0 where ρω >, j 0 are the weight functions. For the chosen set, the functional
i
α
M becomes a function of a finite number of variables z , z ,…, z and can be
1
2
n
minimised by different methods. For our practical problems involving as low as
several dozens of design variables, the best methods are direct techniques for
Zα solving the respective Euler matrix equation, such as the square root method.
α
Zjα In this case, the components Z of the solution vector Z are obtained from
α
j
SLAE
n
… n
az α + αω z = b , k = 1,2,,
∑j=1na¯kjzjα+αwkzk=bk, k=1,2,…,n ∑ kj j kk k (4.24)
= j 1
α
applying the minimum condition for the functional M [z ,u]
α
α
∂M / ∂z α = 0, j = 1,2, … n,
,
∂Mα/∂zjα=0, j=1,2,…,n, j
,
r
a¯kj=∑i=1mρiaikaij, bk=∑i=1 where a kj =∑ m = i 1 ρ a ab k =∑ m = i 1 ρ au . If the influence functions g () are
gj(r)
iikij
j
iiki
mρiaiku¯i. linearly independent, then the system (4.24) must have a unique solution, even
at α = 0. In this case, however, the initial system of Eq. (4.11) may have no solu-
2
2
inf Az−u¯E2=γ ≥0. tion, which means that inf Az − u 2 E = γ ≥ 0.
max
d
zαE γ>δmax Different system configurations are needed, if γ > δ max or z α E > d max ,
>
δmax where δ max and d max are given design criteria.
dmax
Using the pre-set sequence of values α , α ,…,α for the regularisation pa-
p
1
2
z
zαp rameter, a solution α p can be found for each α , which minimises the func-
p
Mαpz,uδu¯=δ tional M α p [z u, δ ] and allows Az α p − u = δ to be calculated. An optimum
Azαp−
zαp solution is found by the trial-and-error method. It corresponds to z α p , at which
δ = δ max is achieved with a reasonable accuracy. At the same time, the condition
d ≤ d max must, of course, be fulfilled.
In the ITER application, this approach may be useful for a special design
problem of optimising the system of ferromagnetic inserts to reduce non-uni-
formity (ripple) of the toroidal field.
4.3.3 Ripple of the Tokamak Toroidal Field
Physically, the ripple is associated with a discrete arrangement of the tokamak
toroidal field coils (Figs. 4.1 and 4.2). The ripple is observed as local magnetic
‘wells’ along the toroidal circumference. The number of wells is the same as
the number of TFCs, that is, 18 for ITER. The ripple is particularly pronounced
at the plasma periphery. The ripple is practically unaffected by the poloidal
fields produced by the PFCs and the plasma current. Ferromagnetic inserts are
placed inside the toroidal magnet structures to decrease the ripple. The ferro-
magnetic material is saturated. The saturation magnetisation, M , fulfils condi-
S
tion µ M = 1.47 T.
0
S
Blocks of ferromagnetic inserts are installed in the shadow of each TFC
between the vacuum vessel walls along the major circumference. Each in-