Page 105 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 89

ments of the mathematical programming theory and methods have been de-
veloped for solving well-posed extremum problems. On the other hand, we
α
can introduce the smoothing functional M [z, u ] formally, without relating
δ
it to the variational conditional extremum problem. Thus, the regularising
operator α = α(δ) can be developed using the unconditional optimisation
algorithm.
2
2
The lower bound of the residual functional γ = inf Az − u 2 E z , ∈F , which γ =inf Az−u¯E2, z∈F ,
1
1
reflects the inconsistency of the equation Az = u , corresponds to u¯
z * = lim z .
α
α→0 z*=limα→0zα.
A synthesis problem has no solution at γ > δ max . In this case, we should
modify the synthesised system configuration to achieve the desired approxima-
tion accuracy. This entails changes in the operator A.
The efficiency of the described procedure depends on such factors as the
consistency of functionals involved in the synthesis problem statement with the
initial equation, a priori information about the solution, and the minimisation
problem complexity.
To estimate how close u¯ and Az are, we use a quadratic metric evasion.
α
The latter reflects an integral quality criterion for a magnet system under con-
sideration. We introduce a weight function in the residual functional to ob-
tain a better approximation for some ‘regions of interest’. The same function
is introduced in the stabilising functional to take into account the potential
design constraints on some system components. The stabilising functional
may be taken in any positive quadratic form z 2 1 = (Lzz, ) (where L is a lin- z12=Lz,z
ear operator) that does not affect the relation (4.21) between the δ, d and α
parameters.
To find a solution, a set of functions given by a problem statement is usu-
ally employed. A certain set of functions with a finite number of coefficients
is taken as a basis. For instance, the chosen set may contain functions in the
form of a linear combination ∑ n = j 1 zg r (), where () are basic functions. In gjr
r
g
∑j=1nzjgjr
j
jj
g
the context of the problem, () are preset influence functions of the design gjr
r
j
variables z .
j
We assume that these functions are given in a discrete set of points r , i = 1,
i
2,..., m, such that () ≡g j r i a . We further assume a vector of the desired values gjri≡aij
ij
of the field u¯ to be also given in the same set of points. Then the residual func-
i
tional and the stabilising functional take the form
m  n  2
δ = ∑∑ jij − u i  , (4.22) δ =∑i=1mρi∑j=1nzjaij−u¯i ,
ρ
2
2
i 
2
za
= i 1 j  =1 
n
Ω () =z ∑ ω j z , j 2 (4.23) Ωz=∑j=1nwjzj2,
j=1