In a closed-loop mathematical system, it often is impossible to represent the complex geometries and boundary conditions of RF/microwave devices and applications. Due to these devices’ diverse material properties and colliding mechanical-electrical properties, however, there is a growing need to grasp, through simulation, their behavior in their actual environments. Computational electromagnetics (CEM) delves into the array of techniques used to efficiently compute approximations to Maxwell’s equations. In doing so, it enables these complex electrodynamic systems to be modeled. While many numerical methods are known to CEM, this article will delve into the numerical methods that are used most commonly in solutions for EM simulation with the latest technologies.

Among the methods used for CEM, for example, are integral-equation solvers, differential-equation (DE) solvers, asymptotic techniques, and other numerical methods (Fig. 1). In the top EM simulation software, which uses integral-equation solving techniques, the methods of moments (MOM) and multilevel fast multipole method (MLFMM) are commonly used. Typically, the numerical methods that use DE solving techniques include the finite-difference time-domain (FDTD), finite-element-method (FEM), and transmission-line-matrix (TLM) methods. In contrast, asymptotic techniques include physical optics (PO), geometric optics (GO), and the uniform theory of diffraction (UTD). The eigen-mode expansion (EME) method is often used for closed resonant systems (Fig. 2).

### Integral-Equation Solvers

The MOM is performed by solving linear partial differential equations (PDEs), which are formulated as integral equations (IEs). Called boundary integral form, this form gives rise to MOM’s other name—boundary element method (BEM). Many industries, ranging from fluid mechanics to acoustics, use this method. When dealing with problems with low surface-to-volume ratios, the computation using MOM is relatively straightforward and efficient. MOM also can be used for problems in which the free-space Green’s function is calculable. Generally speaking, this means that nonlinearities and non-linearly homogeneous media are difficult to simulate as a function of the boundary conditions’ restrictions. As a result, MOM is most useful for problems with currents on metallic objects, dielectric structures, and radiation in free space (Fig. 3). In addition, it is suited for electrically small dimensions that are predominantly composed of metallic objects.

As an alternative to MOM for large electrical scenarios, MLFMM is an integral-equation solver based upon the multipole expansion technique. MOM and MLFMM are similar in that they both model the interaction between triangles with basis functions. In contrast to MOM’s individual interaction-based computation, however, MLFMM computes the interactions of the basis functions in groups (Fig. 4).

Thus, MOM produces a matrix with elements that number as the square of the quantity of basis functions—or impedance matrices—required for the solution. Computation time for MOM scales as the cube of the number of basis functions. In contrast, MLFMM’s logarithmically subdivided groups of basis functions reduce the matrix elements to the number of basis functions multiplied by the logarithm of the number of basis functions. This method reduces computation time to the square of the logarithm of the number of basis functions multiplied by the number of those functions.

These differences enable MLFMM to tackle electrically large structures, such as antennas placed on large vehicles/ships. It also can be used to compute specific-absorption-rate (SAR) levels for possible health hazards near large radiators. For many situations, these BEMs may be drastically less computationally efficient than volume-discretization methods (VDM), such as FEM. After all, the computed matrices for BEMs tend to be fully populated and the computational resources increase as a square of the problem space. With VDMs, however, the matrices are usually just locally connected and the system matrices tend to rise linearly with the increase in problem size.