 # Parallel Computing of HBFEM in Multi-Frequency Domain

## HBFEM in Multi-Frequency Domain

The finite element method (FEM) is a discrete approach to solve the partial different equations (PDEs) for complex problems in many regions. It requires the entire volume to be meshed, compared with integral techniques, which only need the surface to be gridded. FEM can model a complex inhomogeneous medium where each element may have completely different material properties from neighbor elements. FEM discretizes the computing domain into a group of small sub-regions with boundary conditions.

In computational electromagnetics (CEMs), a typical electromagnetic model contains the device geometry information, material constants, excitation, and boundary constraints. In each element, a simple variation of the field quantity is assumed. The vertices of the elements are called nodes, and are determined after the FEM calculation. CEM problems can be solved through FEM, either in frequency domain or time domain.

Harmonic balance method is a numerical solution technique which has been used for nonlinear high-frequency circuits in the periodic and steady-state regime for two decades. It was first introduced to analyze the low-frequency EM field problem, as briefly introduced in Chapter 1.3. HBFEM is a combination of harmonic balance and FEM, to numerically solve time-periodic steady-state nonlinear problems. HBFEM uses the linear combination of sinusoids to build and represent the results. A generalized HBFEM matrix equation can be written as follows: The coefficients of the matrix D are determined by only the Fourier coefficients. Matrix D acts as a reluctivity, and is called the Reluctivity Matrix. On the other hand, matrix N is a constant concerned with harmonic orders, and is called the Harmonic Matrix.

The system matrix equation for current source excitation can then be written in a compact form: where  is the system matrix and [M] is the harmonic-related matrix. All harmonic components of magnetic vector potential A can be directly obtained by solving this system matrix equation.

A significant difference between the HBFEM and conventional FEM is the size of the matrices and vectors, as shown in Figure 5.66. The conventional simulation vector contains one value for each node, while there are K values in one vector for each node in HB - that is, K x K diagonal sub-matrix for each node in the stiffness matrices. Figure 5.66 Size of the equation matrices for HBFEM method (b) compared with the traditional FEM (a)

The harmonic matrix can include DC, as well as the even and odd harmonics, depending on the application problems.

Suppose there are M harmonics considered in the application, then the size of harmonic balance matrix is 2M larger than that of the traditional FEM. The ratio of matrix size between HBFEM FEM is 2M : 1.