Page 386 - Fundamentals of Magnetic Thermonuclear Reactor Design

364 Fundamentals of Magnetic Thermonuclear Reactor Design

12.4.2 Reaction to Off-Normal Current Combinations
in Windings

Deviations of electric current combinations in different windings from design
values may cause a mechanical damage of the MS. Therefore, one of the main
problems of the MFR optimisation design is to determine the allowable current
combinations in windings.
A large tokamak has quite a lot of independent coils creating non-stationary
poloidal fields. For example, ITER has six poloidal field coils and five CS sections
powered independently, making the total number N = 11. Generation of an array of
independent variables in an 11-dimensional space is not a trivial design problem.
In addition, stress calculations for a magnet system generally involve high-dimen-
sional finite-element models. To solve the discussed problem, considerable com-
putational resources are required even with the state-of-the-art computers at hand.
A well-designed MS is expected to behave like a single linear mechanical
system after the tightening of bolts, structure reinforcement with pre-compres-
sion rings and feeding current to the TFC. As to the overturn forces, they are
linearly dependent on the PFC and the CS currents. This allows the superposi-
tion method to be employed when determining the MS mechanical response to
varying current combinations [7].
The method algorithm is as follows. Global and local modelling is used to
perform N calculations of the MS mechanical response to the overturn forces due
to unit currents flowing in each of the PFC and CS sections. The components of
displacement (u ), stresses (σ ) and strains (ε ) are derived, saved and used to cre-
ij
i
ij
ate response matrices. Each matrix has N columns showing responses to the unit
currents. The number of matrix lines is governed by the number of selected nodes
(K). For example, in a response matrix for displacements along the i axis, the n-th
column shows the nodes displacement under the action of ponderomotive forces
due to a unit current in the n-th coil with zero currents in the other coils. Response
(ij)
(ij)
matrices for the components of stress tensor (S ), deformation tensor (E ), and
so on, are calculated in a similar fashion. With response matrices
 u i () u i () … u i () 
 11 i () 12 i () 1 i () N 
U i () = u i ()  =   u … u … … u …  
22
21
2
N

 kn
…
 i () i () i () 
U(i)=ukn(i)=u11(i)u12(i)…u1N(i  u K1 u K 2 … u KN 
)u21(i)u22(i)…u2N(i)…………uK
1(i)uK2(i)…uKN(i) at hand, the desired displacements and stresses in all nodes of the model can be
obtained by multiplying a matrix times the vector (column) of currents (I = [i ],
n
where i is a given current in the n-th coil)
n
I
I
u(i)=U(i)⋅I, σ(ij)=S(ij)⋅I, ε(ij) u i () = U i () ⋅ , σ ij () = S ij () ⋅ I, ε ij () = E ij () ⋅ ⋅
=E(ij)⋅I⋅
To sum it up, an MS stress–strain analysis involving arbitrary current
combinations is reduced to an algebra matrix multiplication. Obtained stress