Page 99 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 83

where N is the number of facet edges, j is the edge number for a respective facet
(j = 1,….,N ); ξ, η are the local coordinates on the facet plane, ζ is the coordinate
normal to the facet; A, B, C,  and ψ are coefficients for a respective edge that
are rational functions of the coordinates of the edge vertices and the observation
point.
Analytical expressions combined with high-precision numerical integration
procedures allow the field outside the ferromagnetics to be smooth. That is im-
portant for precision field calculation.
KLONDIKE is also able to calculate mean values of the magnetisation vec-
tor M inside the elements exposed to an external magnetic field. So, nonlinear
magnetic properties can be accounted for, if necessary.
The field H can alternatively be calculated with the KOMPOT code,
ext
which employs the FE approach. A difficulty may be encountered with the
KOMPOT computations, however, related to the need to interpolate results
using various meshes. This code is quite efficient in cases where it is difficult
to describe coils as simple (close to rectangular or ring-shaped) current-car-
rying configurations.
The combined differential/integral approach improves the reliability of nu-
merical results. It is also effective in solving the problems of magnet system
synthesis, in which influence matrix coefficients, as a rule, have to be com-
puted very accurately. The combined technique is also well suited for problems,
in which a magnetisation vector is obtained using the differential formulation,
while the field in a region of interest is calculated using the integral formulation.
Then, the total field is the sum of vacuum field H and magnetised medium
ext
field H . The stray field of devices constitutes the case.
M
In addition to solving static problems, KLONDIKE is integrated into a com-
putational algorithm for simulation of EM transients to provide a high-precision
evaluation of fields from given field sources.
These computational tools allow a magnetostatic field simulation for the
whole reactor complex. They are flexible enough to provide data processing and
transfer between models of different complexity and detail.

4.3.2 Stationary Field Synthesis

Synthesis problems are a reverse to analysis problems. In the general case, they
fall in the class of ill-posed inverse problems from the mathematical point of
view. Their ill-posedness is shown, particularly, by the instability of a solution
at low perturbations of input data.
Let us consider any system of linear algebraic equations (SLAE)

Az = u, (4.11) Az=u,
where A = {a } is a rectangular discretisation problem matrix of size m × n,
i,j
n
z = {z } ∈ R is the vector of solutions in the form of a set of design variables
i