Page 109 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 93

8  8 
min ∑ max δ  TFC () +i C δ ( ) (4.31) minCJ∑i=18maxiδTFCi+∑j=18C
,
ij
C J i ∑ jFe
= i 1   = j 1   jδFei,j
that can be treated as minimisation of a smoothing regularisation functional
M α [C, δ ] = δ 2 E + α C 2 E , (4.32) MαC,δ=δE2+αCE2,
{ } j,
where C = C j ∈1,8 is the column vector of the field ripple at the character- C=Cj, j∈1,8
istic points of the plasma separatrix.
An approximate regularisation solution of (4.31) is the extremum of func-
tional (4.32), meaning that
M α [C, δ ] = min c M α [C, δ ] (4.33) MαC,δ=mincMαC,δ

and regularisation parameter α must be chosen such that conditions (4.29) and
(4.30) are fulfilled.
The relations in Fig. 4.5 allow one to find an optimal solution

C opt = C opt j ∈1,8, Copt=Cjopt, j∈1,8,
{ } ,
j
−5
consistent with the regularisation parameter α = 7.5·10 , optimal in terms of
a root-mean-square norm and meeting the inserts feasibility criterion (4.29).
Values C ≤ 1 correspond to the physical implementability of the in-
j
serts. The curves show how a solution numerical stability deteriorates with
decreasing α.
One of the candidate ferromagnetic inserts for ITER is described quantita-
tively in Appendix 4.1 (see Tables A.4.1.1–A.4.1.3). As one can see from Figs.
A.4.1.1 and A.4.1.2, showing the ripple isoamplitude curves, optimised ferro-
magnetic inserts are able to efficiently suppress the ripple. For example, peak
ripple on the major radius of the plasma separatrix falls from 1.1% to 0.9%. The
share of areas with a ripple below 0.1% increases by around 20%. δ (r, z) > 0
throughout the plasma column, meaning that the no-overcompensation require-
ment (4.30) is met. The ripple is within 0.8% when the TFCs are operating at
half the nominal current.
Ripple modes for a toroidal field distributed along a circle with coordinates
r = 7.25 m; z = 3.0 m (the plasma periphery) have been subjected to harmonic
analysis under the assumption that the spatial distribution of fast α-particle flows
followed the most likely pattern. The analysis discovered the following mode
distribution: δ (fundamental mode) = 0.19%; δ = 0.017%; δ = 0.0067%. This
3
2
1
suggests that the loss of particles because of a ripple of the ITER toroidal field
is due mostly to the ripple fundamental mode.
To sum it up, the algorithm and its software implementation comprise an
effective tool for the MS optimisation. An extensive practical experience has
been gained that enables their employment for the synthesis of current-carrying
configurations and MSs containing constructional and functional materials with