Page 123 - Fundamentals of Magnetic Thermonuclear Reactor Design

Simulation of Electromagnetic Fields Chapter | 4 107

NI MFE
(
)(
B(t,, i () ,( ) = ∑ BI r I t) + ∑ BIr I t)
r I tI t)
(
)(
i
j
j
j
i
i j Bt,r,Ii(t),Ij(t)=∑iNIBIi(r)Ii(t)+∑jM
FEBIj(r)Ij(t)
We assume the methods for computing those values to be known. For in-
stance, their numerical values can be obtained for all current sources.
Let w (t), w (t) be partial solutions to a system of linear ODEs with right-
1
2
hand sides f(t) = f (t) and f(t) = f (t), respectively. If we present the right-hand
2
1
side in terms of a sum f(t) = αf (t) + βf (t), then, according to the superposi-
1
2
tion principle, the partial solution takes the form w(t) = αw (t) + βw (t). Sub-
2
1
stituting the set of independent parameters I (t) for their decomposition with
i
a predetermined basis, we will find a solution of the Cauchy problem in the
form of a linear combination of all sources. Among possible basic functions
for I (t) approximation, we chose those that enable a high efficiency of the
i
algorithm.
ITER can operate over a wide range of reference scenarios for normal and
abnormal conditions. A predictive analysis of electro-mechanical and thermal
loads enables selection of principal and nonprincipal scenarios. However, an a
priori selection of scenarios to be analysed is not always justified and may not
be comprehensive enough. Alternatively, one can analyse all of the reference
scenarios using the superposition principle.
th
Let δI (t) be a known pulse change of current in each i independent
t
current source with current I (t), (I = 1, 2,…, NI). We solve the NI bound-
i
ary value problems for the equation’s right sides, defined by different δI t
(t) values and generate a multi-dimensional table that contains distributions
of potentials and eddy currents over the FE mesh at given time points of
the observation interval t ∈ (0,T). We call it the Impact Pulse Test Table
(IPTT).
The I (t) approximation as a linear combination of pulse functions can be
i
found for any operating mode. Varying the number of these functions and the
time increment, the best approximation is obtained that ensures the accuracy
criteria. Selecting elements from the IPTT and using superposition, a partial
solution to a linear system of ODEs can be assembled for any required scenario
without integration. This approach is even more efficient and promising when
we use high-performance parallel computing techniques and multiprocessor
computers. In principle, it allows the synthesis of operational boundaries or
optimal operating conditions. Numerical experiments have demonstrated that
−4
the computation error as low as 10 can be achieved for characteristic times
typical for ITER EM transients.
The described approaches and their implementation in software codes
enable an effective simulation of EM transients taking place in MFR struc-
tures and the reactor as a whole. They complement one another and provide
an independent validation of computational models and results. It is there-