Page 122 - Fundamentals of Magnetic Thermonuclear Reactor Design

106 Fundamentals of Magnetic Thermonuclear Reactor Design

∫ P (, r ⋅dl = ∫∫ j ( t rd S = I t(),
⋅
,)
)
t
∫asPsrc(t,r)⋅dl=∬Sjsrc(t,r)⋅dS= as src S src
I(t),
where I(t) is a given time dependence of the total current flowing through a
selected cross-section S within the boundary ∂S. Substituting P for P in the
tot
system of equations and summing the right-side terms that describe the ex-
ternal sources, the final system of equations is obtained to describe eddy and
halo currents in tokamaks. Examples of generated 3D meshes are given in Figs.
A.4.2.2–A.4.2.4, Appendix A.4.2.
4.4.5 Computation of Potentials: Global and
Local Model Integration
Let us discuss the procedure for computing the time-dependent distribution of
potentials, eddy currents, fields, EM forces and heat releases in detail. A set of
independent parameters I (t, r ), (i = 1, 2,…, NI) which, together with their de-
i
i
rivatives, are assumed to be smooth continuous functions and can be introduced
to describe external currents. These parameters give the spatial and temporal
distributions of external magnetic fields.
At the beginning of the simulation, we assume that the design and structural
materials for a magnet system are pre-determined. This allows material proper-
ties to be taken into account and the calculated domain to be partitioned into
a set of appropriate FEs. Every FE (MFE is the total number of finite element
mesh) is characterised by eddy current density vector I (t). To find it, we can use
j
computation algorithms and procedures that employ either the potential P de-
scribed earlier, or EM field vectors B and D, or the magnetic vector and electric
scalar potentials.
In any case, interpolation on FE meshes leads to a system of linear (or lin-
earised) ODEs with constant coefficients. For vector P it takes the form

{(
P t)}
P t)}
L{P˙(t)}+R{P(t)}=F{Ii(t)}, L{( + R{( = F It)},
i
{P(t)} where P t{( )} is a column vector of the unknowns, the point above the vector
denotes differentiation over time, and the L, R and F matrices are known. The
system includes the initial and boundary conditions. We therefore have a linear
Cauchy problem, a partial solution to which is obtained numerically.
The methods for solving linear ODEs are well known. Time distributions are
obtained by numerical integration of the system with the right side defined by
a given current scenario, I (t). Sets of relevant modules are incorporated in the
j
TYPHOON and TORNADO codes.
Let us now discuss another approach based on the use of a linear ODE with
the right side having the form of a superposition of known functions. The total
field, determined by the instantaneous values of currents I (t) and eddy currents
i
I (t) can be written as
j