### Efield^{®} Frequency-Domain Solvers

The Efield® frequency domain solvers (EfieldFD) are based on the Method of Moments (MoM) technique applied to surface integral formulation of the electric and magnetic field of an arbitrarily shaped object consisting of conductors, dielectrics and thin wires. In MoM the analysis is accurately carried out by representing the conductors, dielectrics and thin wires using surface triangular elements and wire segments. Since only the surfaces are modeled in MoM, the size of the model in terms of unknowns is much smaller than for differential equation methods. The technique leads to a matrix system that has to be solved where the matrix is dense. Also the mesh generation is much faster and simpler than for example finite element methods (FEM) using tetrahedral or hexahedral elements.

Different solver modes and integral formulations are available in the EfieldFD solvers to be able to cover a very wide range of applications including antenna design, antenna integration, waveguides and radar cross section analysis. While standard MoM is the preferable choice for electrically small antenna or waveguide design problems the Multilevel Fast Multipole Method (MLFMM) or the Physical Optics method (PO) is the preferable choice for applications involving electrically large structures. Examples of applications where MLFMM and PO is the preferred choice are radar cross section analysis or antenna installation analysis. In addition to these three basic solver modes the EfieldFD solvers also include hybrid MoM-PO and hybrid MLFMM-PO solver modes.

#### Solver Modes

Different solver modes and integral formulations are available in EfieldFD to be able to cover a very wide range of applications including antenna design, antenna integration, waveguides and radar cross section analysis. Solver modes available are MoM, MLFMM, PO, MoM-PO and MLFMM-PO.

#### The MoM mode

The standard MoM is limited to structures that are electrically small and applications include antennas and waveguides. Typically structures of the size up to five or ten wavelengths can be treated.

In the MoM mode the classical Gaussian elimination process is used to solve the matrix system. This requires
solution time proportional to N^{3} and memory usage proportional to N^{2}, where N is the number
of unknowns in the system. However, since the Efield® MoM solver is parallelized and the resulting matrix and right
hand side can be stored on disc the size of the problem that can be treated can be increased quite a lot using a
cluster of computers.

#### The MLFMM mode

For applications involving electrically large structures such as antenna installation analysis, radar cross section analysis or reflector antenna design the MLFMM approach can be used to reduce the numerical complexity. In many cases the MLFMM is the only method available on the market that can solve these problems with sufficient accuracy.

In the MLFMM mode an iterative solver (GMRES, QMR or BiCGstab) is used to solve the matrix system.

The MLFMM considerably reduces both the solution time and memory need compared to MoM. The basic idea of the
MLFMM is to diagonalize the Green's function in order to expedite a fast matrix-vector multiplication, which is
the main operation in the iterative solver used for the MLFMM mode. When using a MLFMM technique the solution
time of an iterative method is proportional to N_{iter }N log(N) and the memory requirement is proportional to N log(N).

#### The PO mode

PO is an asymptotic high frequency technique applicable to electrically very large metallic structures where the accuracy in the method increases with the size of the object and where the curvature of the object should be small compared to the wavelength. Typical example where standalone PO is suitable is for radar cross section analysis when MLFMM is not longer applicable because of the size of the object is too large. For PO solution time and memory are proportional to N where N is the number of unknowns in the system.

#### The MoM/MLFMM-PO mode

The hybrid MoM-PO and MLFMM-PO are used for large scale problems where standalone MoM or MLFMM are not longer applicable because the size of the object is to large and where standalone PO is no longer valid. Applications where MoM-PO or MLFMM-PO are useful are for example in the analysis of reflector antennas where the reflector is treated using PO and the feed structure (for example a horn antenna) is treated using MoM or MLFMM.

#### Integral equation formulations

In EfieldFD MoM or MLFMM solvers different integral formulations are available that improves accuracy or decrease solution time. There are two different issues that are linked to the integral equation formulation that are very important when using MoM or MLFMM based solvers. These issues are the internal resonance problem for closed PEC structures and the convergence rate (and solution time) of the iterative solver when using the MLFMM solver mode. MoM based solvers are generally a form of the electric field integral equation (EFIE), the magnetic field integral equation (MFIE) or a combination of these (CFIE). In the EfieldFD solvers a CFIE formulation is used avoid the internal resonance problem. The CFIE formulation also have better convergence properties than pure EFIE based solvers and is a key feature in the MLFMM solver to get fast convergence.

In most commercial MoM based solvers CFIE formulations are only available for pure PEC structures. In the EfieldFD solvers a new an innovative integral formulation has been developed for problems involving both perfectly electric conductors and dielectric or magnetic bodies. Most commercial solvers use the so called PMCHWT (Poggio-Miller-Chang-Harrington-Wu-Tsai) formulation which has very poor convergence properties. The formulation used in the EfieldFD solver decrease the number of iterations to reach convergence dramatically with large savings in solution time.

#### Material and Boundary conditions

The EfieldFD solvers can handle lossy and loss free dielectrics and magnetic materials, perfect electric and magnetic conductors as well as imperfectly conducting conductors. Boundary conditions that can be used are perfect electric and magnetic conductors (PEC/PMC) as well as imperfect conductors which are modeled using impedance boundary conditions (IBCs) or resistive boundary conditions (RBCs). Lumped elements (RLC) can be used on both wires nodes and surface edges.

- Dielectric and magnetic materials, lossy and loss free
- PEC
- PMC
- IBC
- RBC
- Lumped elements (RLC)

#### Thin Wires

The EfieldFD solvers incorporate a wire method that stays stable even for small wire segment discretization. Wire-wire junctions as well as surface-wire junctions can be handled.

#### Excitations

Available excitations in the EfieldFD solvers are

- Plane wave
- Dipole
- Voltage excitations on wire nodes and surface edges
- Waveguide mode excitations using 2D numerical or analytical eigenmode solver
- General field distributions

#### Post-processing

Output from the EfieldFD solvers includes

- S-parameters
- Input impedance
- SVWR
- Reflection loss
- Far fields (2D, 3D, directivity, gain, field pattern and polarisation)
- Radar Cross section (RCS) calculation, bistatic and monostatic
- Near fields
- Surface and wire currents
- Far field power
- Power through user defined surfaces

#### Parallelization and out-of-core

The MoM formulation leads to a linear system of equations and depending on the problem size, the memory need to store the system matrix can exceed the available main memory on the computer system. There are two different options in this case. One is to run the EfieldFD solver on a parallel computer and use the memory on the PCs or workstations. The other is to store the system matrix (and right hand side) on disc rather then in the main memory using the EfieldFD out-of-core solver.

All EfieldFD solvers are parallelized and the solvers will run effectively on both shared memory and distributed memory machines. The solvers can be run either in out-of-core mode or in in-core mode depending on available memory. In the case of out-of-core mode the storage is on files on disc whereas for the in-core mode storage is only in the internal memory.