Page 103 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 87

l there is functional α(u, δ) such that for any ε > 0 there exists δ(ε) ≤ δ , such
0
that if ρ (u, u ) ≤ δ(ε), then ρ (z , z ) ≤ ε, where z = R(u, α(u, δ)).
α
α
T
F
T
u
Solution z is said to be a regularised solution of equation Az = u and the
α
numerical parameter α is said to be a regularisation parameter. From the op-
erator definition it follows that at δ ≤ δ one can take a regularised solution
0
as an approximate one, if α is consistent with the given input error δ. The
regulariser dictates a robust method for constructing approximate solutions
to Eq. (4.11), converging to the exact solution z , that is, ρ (z (u ), z ) → 0
α
δ
t
T
F
at δ → 0.
The variational regularisation method applied requires the introduction of a
smoothing regularising functional:
2
−
zu ]
[, δ = Az u δ E + α z 2 E (4.17) [z,uδ]=Az−uδE2+αzE2
Note that the choice of functional (4.17) is dictated by a mathematical
statement of the synthesis problem for magnet systems. The solution z is an
α
extremum of the functional
α
α
M [, δ u ] = min M [, δ u ], z ∈U. (4.18) Mα[za,uδ]=min Mα[z,uδ], z∈U.
z
z
a
Hence, z may be considered as a result of the application of an α-dependent
α
operator R, and therefore z = R (u , α).
δ
α
For the Euclidean norm of vectors ⋅ and α > 0, functional (4.17) has a ⋅
single minimum point z . A regularised solution of (4.11) can be also found by
α
solving a respective Euler equation
*
*
( AA + αEZ α = Au δ (4.19) (A*A+αE)Zα=A*uδ
)
The stability of the minimisation problem (4.17) is achieved due to a stabi-
lising functional z 2 E . This functional determines a set of stable solutions for zE2
Eq. (4.11).
We denote
ψ α = z
m() α α zu, δ ], ϕ α = AZ a − u δ E 2 ,( ) α E 2 (4.20) m(α)=Mα[zα,uδ],(α)=AZa−uδE
α = M [
()
2,ψ(α)=zαE2
It is important that functions m (α),  (α) and ψ (α) are strictly monoto-
nous, with m (α) and  (α) being increasing functions and ψ (α) a decreasing
one.
Particular attention should be paid to an adequate choice of the regularisa-
tion parameter. In practice, a mathematical statement for design optimisation,
as well as the optimisation methods, depends upon the feasibility of magnet
system construction.
First, physical conditions crucial for the design implementation include the
smoothness of a spatial field distribution in a domain of regularity, where the
medium and magnetic field characteristics are finite, continuous and have con-
tinuous derivatives. Second, relevant design constraints should be taken into