Section 5.2 discussed the decomposed algorithm of HBFEM, which can improve the computation approach and make the harmonic solution converge smoothly. However, while the division on space leads to t parallel processing in the matrix operation, it may encounter some difficulties in handling the material properties on the boundary of the divisions. Since HBFEM is the method of solving multi-frequency problems, we can separate matrix calculations through dividing matrices by each harmonics, instead of dividing it by the space. The detailed approach has been introduced by Lu et al. [74] in 1995. It was the partition of matrices by frequency.

The reluctivity matrix D and harmonic matrix N in HBFEM can be represented by means of the sub-matrices:

Figure 5.69 A sample of (a) Red-black ordering of a 6 x 4 grid and (b) Associated matrix

where

h = 1, 3, 5 ...

p =1,2,3 ...

The vectors {A^{e}k} and {Kcan be rearranged according to harmonics as follows:

Then the matrix equation representing in harmonics is obtained:
where

By using the Gauss iteration method, the system equations for the fundamental component and the h-th harmonics (h = 3, 5, 7, ...) can be obtained as follows:

In this method, the decomposition of the problem is based on harmonic components division instead of space region decomposition. A shared memory execution environment can be utilized in the frequency parallel calculation. Each computation unit has the same space structure for the finite elements, the same number of unknown variables and the same structure of the matrices. The maximum parallelism is M (numbers of

Figure 5.70 Domain decomposition and frequency domain divided for the system matrix

harmonics), and the maximum speed-up will be M time if M processors are available. This parallel computation model for nonlinear EM field analysis can be easily mapped on distributed-memory MIMD parallel computer systems, or on multiple computers connected by LANs.