Page 118 - Fundamentals of Magnetic Thermonuclear Reactor Design

102 Fundamentals of Magnetic Thermonuclear Reactor Design

sociated with the conductor’s external surfaces. The boundary conditions follow
from ∇ × H = j and express the relationship between field intensity H at the
e
boundary region and surface current density J [28]:

J = ∫ ∞ j() xdx = n × H J = H ,
,
J=∫0∞jxdx=ne×He,J=He, 0 e e e
where n is normal to the boundary. The integration is performed along the co-
e
ordinate directed from the conductor’s surface inwards.
Shells for modelling thin-wall structures may logically be aligned in space
with mid-plane and/or external surfaces of the conducting structures.
The TYPHOON code employs triangular FEs to discretise integral–
differential equations. The distribution of eddy currents on the shells is ob-
j
∇×P=j tained by approximating relationship ∇× P = within each element. ‘Local’
computation accuracy governs the degree of irregularity of the FE mesh over
a shell.
Turning back to the shell vs. 3D modelling, let us discuss the technique
for estimating eddy currents in thin-walled components. The requirements for
mesh fines over conducting structures are, in principle, the same. A shell model
involves the surface meshing only, while a 3D model requires FE meshing
inside and outside the conducting structures. This may divert a large share
of computational cost. With a large amount of thin-walled conducting struc-
tures arbitrarily arranged in space, a relatively coarse mesh may be heavily
deformed. This weakens the approximation quality and may lead to substantial
errors. One solution is to refine the mesh, but this would critically affect the
problem’s dimensionality.
The solution of certain problems involved in an MS design optimisation
often requires the spatial distribution of a field to be ‘smooth’ enough. This
is required, for example, for the computation of field derivatives. The desired
‘smoothness’ sometimes cannot be achieved with the FE approach. In this
case, integrals based on the Biot–Savart law [30] may be useful. For example,
the TYPHOON code, based methodologically on the integral–differential for-
mulation, uses the KLONDIKE computation procedures to calculate a field.
Likewise, the TORNADO code, based on the differential formulation, em-
ploys integral relations to simulate a smooth field distribution at the post-
processing stage.
The analysis of EM force distributions is more difficult. Let an elec-
tric current of density j flow through a small-volume dV of a conductor
located in an external field B. The volume is exposed to a Lorentz force,
dF = (j × B) dV of density f = dF/dV = j × B. If there is a pronounced
skin effect, the current density and induction decrease exponentially with
distance from the conductor surface, making the force density distribution
over the conductor thickness very non-uniform. This results in increased