Page 117 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 101

Possible approaches to building global models for ITER’s conducting struc-
tures will be discussed in detail. The first approach utilises a concept of multi-
connected magnetic shells arbitrarily located in space. The second one is based
on a ‘totally’ 3D model and may reflect the skin effect where necessary. The
discretisation over the depth of conducting materials is the chosen reasoning
from the characteristic depth of the field penetration. These approaches have
been implemented using the TYPHOON and TORNADO computer codes. Both
software tools utilise the electric vector potential and magnetic scalar potential
and the FE method. Relevant time problems are described by systems of ordi-
nary differential equations (ODEs).
By introducing the potentials, we spare our computational resources to a
considerable extent, which is important in situations where a large amount
of conductive structures has to be modelled comprehensively. For the in-
tegral–differential formulation of the problem involving conductive shells,
the solution of an unknown parameter at the mesh nodes is provided using
the only component of an electric vector potential normal to a shell sur-
face. It can only be determined on the shell. Such an approach offers the
advantage of easy shell adding/removal, reducing problem dimensionality
requirements.
An alternative 3D model based on the differential approach involves the
generation of a mesh over the entire space surrounding the machine. Math-
ematically, this requires the solution of several unknowns at each mesh node
within a conducting structure. Evidently, a spatial mesh model with approxi-
mation properties equivalent to those of the shell model will give rise to prob-
lems with several orders of magnitude higher dimensionality. However, we
note that the integral–differential problem formulation corresponds to a sys-
tem of ODEs with dense matrices, while the differential formulation leads to
band matrices.
A mathematical modelling of a thin-wall structure, such as a vacuum vessel,
may be performed using a simplified quasi-3D shell model. In such a model,
conducting structures are approximated using piecewise linear or piecewise
2
quadratic interpolations for FE meshing. In the time interval t > τ = µ d /ρ ,
0
1
0
this approximation provides a reasonable numerical error. However, the error
gets too large at t < τ . To reduce it, we place two or three shells over the
1
structure’s thickness. From the expression for τ it follows that two 0.5 d-thick
1
shells, associated with the structure outer surfaces, enable a fourfold reduction
of the interval τ . The thickness is taken into account by the introduction of the
1
shell’s effective resistance. The estimate for τ assumes a one-way diffusion of
1
the field through an infinitely long thin conductive sheet. In reality, the MFR’s
conducting structures have finite dimensions, and the field penetrates through
them from all sides. For this reason, actual characteristic time τ is shorter than
1
the estimated one.
To analyse the transient initial stage, (t « τ ), and account for a pronounced
1
skin effect, we use a perfect conductor approximation, in which shells are as-