Page 104 - Fundamentals of Magnetic Thermonuclear Reactor Design

88 Fundamentals of Magnetic Thermonuclear Reactor Design
u¯

account. In R&D activities, physical and constructional feasibility are often dif-
u¯ ficult to separate.
The synthesis problems require us to better define the correctness criteria.
First, solution u = Az should approximate a given parameter u to a reasonable
δ
uδ−u¯≤δ accuracy: u δ − u ≤ δ. Usually, in magnet systems u is represented by the
desired field distribution. The strict requirements for approximation lead inevi-
tably to the design complication and, hence, to higher costs and weaker perfor-
mance. Moreover, the approximation requirements often need to be adjusted in
the process of design. Therefore, there is no need to get the exact solution of the
synthesis problem. For this sort of problems, we often have situations, where it
is feasible to get a set of possible solutions and select the final one by the design
optimisation criteria.
u¯ Let vector z be a set of design variables of a magnet system with specified
output parameters. The variables may include geometrical dimensions, electro-
physical properties, electrical characteristics, and so on.
It is important to note that one cannot ensure that an adequate magnet sys-
tem is synthesisable at any given field distribution u . This is because (1) there
Az=u¯ is no z ∈ F fitting Az = u and (2) the synthesised parameters are impossible or
hardly possible to implement to achieve the lower bound of residual functional
Az−u¯E2 Az − u 2 E .
Mathematically, the feasibility condition in many cases can be represent-
2
zE2≤d 2 ed as z 2 E ≤ d , where d is a known value serving as the complexity factor
for a synthesised configuration. Then vector z, which allows the lower bound
F ∈F of the residual functional to be reached, belongs to the subdomain F 1 ∈F,
1
2
2
{:
F ≡{z:zE2≤d }. F 1 ≡ z z 2 E ≤ d }.
1
α
To find vector Z , which minimises the smoothing functional M [z, u ], we
δ
α
2
zE2≤d 2 apply the Lagrange multiplier method; while α can be derived from z 2 E ≤ d .
2
We note that in practice, the stabilising functional z reflects the design feasi-
E
zE2 bility and mathematically provides the solution stability. Therefore, vector z is
α
Az=−u¯ a regularised solution of Az =−u over domain F .
1
Error δ, complexity factor d and regularisation parameter α are related as
−
,
zαE=d, Az−u¯E=δ α z E = d Az u E = δ (4.21)
Setting one of these parameters, we can unambiguously derive the other two
from Eq. (4.21). Hence, α can act as a parameter governing the synthesis pro-
cess. Setting different values for α and using the  (α) and ψ (α) strict mono-
u¯ tonicity, we can obtain a set of solutions z α that fit the function u with different
accuracies. The best solution can be chosen as a reasonable trade-off between
accuracy and feasibility.
Minimisation of functions of a large number of variables over a con-
strained domain may be very difficult, as is often the case in practice. Even
in the simplest cases, an extremum is achieved at the boundary points of the
domain where the function is nondifferentiable. In addition, the key state-