# Numerical Modelling Techniques

Overview, historical development, description

### Overview

Radio frequency devices become increasingly complex, where complicated shapes and various material compositions find widespread use. Also, the interaction among various devices must be considered and modern design processes must be aware of overall system integration aspects.
In order to cope with such requirements, it is no longer sufficient to only understand the basic electromagnetic phenomena such as the resonating behaviour of a wire antenna. It must rather be possible to model electromagnetic fields in complex environments in order to understand and design real-life radio frequency devices. This can only be achieved by the use of advanced numerical modelling techniques.
Over the past years, a variety of powerful commercial modelling packages have become available and are in wide-spread use in the radio frequency community. However, many electromagnetic problems are still beyond the scope of commercial codes and research in novel and more powerful numerical modelling algorithms is of on-going interest.

Numerical modelling techniques are often classified as numerically exact methods derived from a rigorous electromagnetic formulation by an appropriate discretization scheme and as asymptotic (high-frequency) methods derived from a so-called asymptotic representation of the electromagnetic problem which becomes exact for large enough frequencies as often found in optics.

### Historical Development

The asymptotic methods can be separated into field-based (or ray-based) techniques such as Geometrical Optics (GO), Geometrical Theory of Diffraction (GTD), or the more advanced Uniform Geometrical Theory of Diffraction (UTD) and into source (equivalent current)-based methods such as Physical Optics (PO) and the Physical Theory of Diffraction (PTD). The numerically exact methods can be subdivided into local or finite methods (typically derived from differential equations) as Finite Element Methods (FEM) or Finite Difference (FD) techniques and into global methods, which are typically derived from integral equations by the applying the Method of Moments (MoM) (therefore often just termed MoM). Finally, different techniques can be combined with each other leading to the wide class of hybrid methods.

Most basic numerical modelling techniques were known at the beginning of the 1970s and over the years all kinds of hybridizations have been developed. Very important in this process was the development of Fast Integral Methods starting in the 1990s leading to a tremendous increase of computational efficiency. At the Institute of Radio Frequency Technology, a extremely powerful hybrid platform is available, which combines the FEM with a Boundary Integral (BI) formulation and the Uniform Geometrical Theory of Diffraction (UTD), where the MoM solution of the BI is accelerated by an extremely powerful and efficient implementation of the Multilevel Fast Multipole Method (MLFMM).

### Hybrid FEBI-UTD-MLFMM Technique for 3 dimensional Radiation and Scattering Problems

The hybrid method represents a very flexible modelling platform for a variety of radio frequency problems. The discretization model can be imported from various CAD and meshing packages and by many control parameters it is possible to solve an electromagnetic problem with different approaches such as stand-alone BI, FEBI, or even stand-alone UTD.

The core of this hybrid method is the BI, where among Electric Field Integral Equation (EFIE), Magnetic Field Integral Equation (MFIE), and Combined Field Integral Equation (CFIE) can be chosen. Since traditional MoM solutions of BI were applicable only for relatively small problems, considerable effort was spent in the development and optimization of an efficient and accurate MLFMM solver. Based on the concept of the spherical harmonics expansion of the k-space integrals needed in MLFMM, such a solver could be developed. With this MLFMM-BI solver, we were for instance able to solve a CFIE problem with 874179 Rao-Wilton-Glisson (RWG) unknowns on a PC computer using only 1670 MByte of RAM. The largest EFIE problem computed with this method comprised more than 22 Mio. RWG unknowns and was solved on a single AMD Opterion 2.2 GHz processor in about 120 hours with a RAM requirement of 28 GByte.
As further important developments, the UTD was hybridized with MLFMM and the MLFMM acceleration can also be used for the efficient evaluation of near-fields and far-fields once the BI currents are known.

### Hybrid FEBI Technique for Infinite Periodic Configurations (Frequency Selective Surface and Volumes, Antenna Arrays, Metamaterials)

The redundancy inherent to infinite periodic array problems can be eliminated by applying the Floquet theorem leading to a description of the electromagnetic roblem based on a single periodic unit cell of the whole array. Thus, only this single unit cell needs to be discretized in a numerical method and again the hybrid Finite Element (FE) Boundary Integral (BI) method provides for an efficient and accurate modelling approach.
Many antenna array and frequency selective surface problems can be efficiently computed by such a formulation. Current and future applications are found in the field of metamaterials.

Very important for an efficient hybrid algorithm is the availability of a Fast Integral solver. In similarity to the very popular MLFMM for non-periodic problems, we developed the so-called Multilevel Fast Spectral Domain Algorithm (MLFSDA), which starts from the classical spectral domain formulation of the periodic problems and utilizes the fact that this representation directly leads to a diagonal BI operator.