This part describes concisely algorithms which are used in the EMC2020 software-series for People who are not very aware of numerical tecniques in Electrostatics & Electromagnetics.  They will find here:

• some details to clarify things,
• the Pros & Cons of the numerical methods,
• suggestions  for their use in engineering .

Find here some details about:

FD Solvers: FD means Finite Difference of equations. They can be applied on differential equations in 1d, 2d & 3d geometrical domain for analysis in the frequency or in the time domain. This is a volumic method where Electric & Magnetic Fields are computed in the space around the scattering object(s), not directly on the object itself.

• The most well known is the FDTD (Finite Difference Time Domain) method in the time domain applied on 3d Maxwell’s Equations  under their differential form (See YEE algorithm in the litterature).
• The main avantage of the FDTD is an explicit computation of fields, i.e no linear system to solve: New values at a next time step are computed directly from old values & local terms.
• The major difficulties are: the generation of a volumic stair-case model (done with rectangular bricks) and the treatment of the external boundaries of the domain space which equations must mimic the propagation of waves in free-space (In order to ensure waves are not reflected back to the model at the boundary).
• The equivalent equations in the frequency domain may be used as well. The main drawback is the solution of a very large, but very sparse linear system. As before boundary conditions must be carefully treated.

Finite Difference Solver may be found for the solution of electromagnetic problems such as case involving absorbing media (dielectrics, …) where the FD excells: IC, Micro-Electronics, Patch-Antennas, Microwave Circuits, …..

For the EMC2020 software-series, FD are used only for solving the propagation of signals on transmission lines (1d problem in the time domain).

Other popular volumic methods not used here are: the FV (Finite Volumes), the FIT (Finite Integral Technique) and the FE (Finite Elements).

BIE Solvers:  BIE means Boundary Integral Equation. They are mainly used in the frequency domain. Their time domain equivalent, i.e TDIE is rarely used and found since it is prone to suffer instabilities and it is very complex to implement (Find here the fundations for the TDIE).

The most well known IE’s in Electromagnetics are the EFIE (Electric Field Integral Equation) and the MFIE (Magnetic Field Integral Equation) in the frequency domain. The CFIE is a combination of both and is used for improving the convergence with iterative solvers or removing dummy resonances. The PMCHW uses both with Electric & Magnetic currents via the equivalence principle to represent apertures & bulk homogeneous dielectric regions (via their external surfaces only).

Such integro-differential equations are solved via what’s is named in the dedicated litterature the Method of Moments (in short the MoM). See here for details about the method. This method transforms an integro-differential equation in a linear system well adapted to be solved on a computer. The MoM solves currents and not directly ElectroMagnetic Fields. These last quantities are computed in a second step, when currents are known. This method is known as leading to an “exact solver”. Results are very accurate with a high fidelity in the representation of the geometrical model.

These kinds of solvers are quite popular since most are based on the “Surface Equations” (EFIE, MFIE, …). They describe the behaviour of Electromagnetic Fields around the surface of interest subjected to some stimuli. This means a volumic mesh of the object is not required as in FD, Finite Elements or Finite Volume methods. Only the representation of surfaces is required, which is provided by CAD and standard meshers under the form of a collection of Trias and eventually Quads. Linear & Surface objects may cohabit in the model thus allowing to represent complex models made of wires & surfaces.

• The advantage of such a method is its simplicity in its use since it is directly interfaced with outputs of meshers. The geometry is quite easy to manage. Furthermore, the radiation condition outside the scattering objects is automatically handled inside the equations.
• The main drawback lies in the linear solver (Standard LU solver). Things remain OK so long as the size of the linear system don’t exceed the RAM memory size (up to 40.000 unknowns) assuming 16Gb of RAM. Since the meshing and as a consequence the number of unknows is conditionned with the frequency of operation, the problem size grows with the chosen frequency. The problem can be remedied by means of the  “Out of Core Method” where the matrix associated to the linear system is stored on the disk in place of the RAM memory. This works for problem size not exceeding  some thenth of thousands of  unknowns (up to 200.000 unknowns) assuming 320Gb available on the disk. After this limit, one has to rely with  the solution of the linear system based on iterative techniques. Most of these solvers make use of the Krylov method + a matrix preconditionner to  garanty & to improve the convergence of the solution. For effciency and to do not store the matrix,  this technique is coupled with an accelerating method (FMM, pFFT, …) .

Free-space applications  (Surrounding media is Free-Space with no dissipative materials) allow to solve very complex shape geometries for most engineering situations like CEM, Antenna problems, …  at system & sub-system level  on structure like cars, ships, missile, satellite & aircraft, ….

• This method can be used for other configurations like close media: Change of the Free-Space Green’s function, i.e exp(-j.k.R)/R  to the Green’s dyadic of some predefined geometries .
• It can be hybrided with others method (i.e PTD, GTD, …) in a “natural way”.
• Note “Dyadics” are a mathematical tool to adapt equations when the wave propagation is not uniform in the surrounding media . Furthermore, they allow to adapt in a consistent & automatic manner the external boundary condition in some predefined situations: cavity, … , dielectric interface, … without meshing them.
• The MPIE (Mixed Potential Integral Equation) allows to solve complex situations at equipment and component level. It is a re-definition of the integral equation inside absorbing media of a certain shape under the form of a dyadic representation of potentials. It can be specialized to situations like planar/spheric stratified dielectric/magnetic media to solve radiating problems for microelectronic, IC , patch-antenna, … problems.

In the EMC2020 software series, The 3d MoM is planed for free problems with an acceleration of the solution via the pFFT. The MPIE will be used in the µRad software. BCap is a 2d softwares in the present electrostatic-series making use of the MoM with nested dielectric regions and the concept of equivalent charges. In the same spirit, the µCap electrostatic software use the MoM with the MPIE equation adapted for the static case for a planar stratified dielectric media.

Ray-Solvers: These methods make use of the asymptotic form (at high frequency) of Maxwell’s Equations. Among the well known, one finds the GTD/UTD (Geometric/Unified Theory of Diffraction) and the PTD (Pysical Theory of Diffraction)

GTD : The GTD & its uniform version (the UTD) are Ray-Methods. This class of solvers is based on the construction of a high-frequency solution of the diffraction of an electromagnetic-wave incident on some target (a canonical shape in the GTD terminology): Flat/curved surface, Edge/Corner, Tip, …. The theory is based on the treatment at high frequency  of the E-field dyadic for some specific canonical shapes. As an example, the edge/wedge problem is obtained as the asymptotic form of the wave diffraction by a perfectly conducting corner. The result is a ray-representation for which the diffracted field is obtained via a constrainted path depending on the observer, source location and edge direction (Minimal path principle, i.e Keller cone). The diffracted field is expressed under the form of a diffraction coefficient multiplied by a propagator function depending on the path of the ray (See here for more details). In the same spirit, the illumination of a large surface leads to the well-known Reflection-Path (The snell law).

• Suitable selections of diffracting processes (Reflection: R, Diffraction: D, Tip: T, …) for flat as well as for curved surfaces (Reflection: R, Diffraction: D, Creeping: C, …) can be implemented . Therefore, high order mutual combinations (RR, RD, RC, …) allow to improve the solution.
• Note that GTD assumes the notion of path (minimal path) for which a ray starts from some source, interacts at some point (R, D, …) & hits the observer point. In-between, the full path length may be intercepted and shadowed by some object, thus nullifying such interaction. This is why the algorith must handle this situation. The Path is found using the “mimimal-path” principle. Then Shadowing is treated by means of a Ray-Tracer to detect in a fast manner if some part of the launched-RayPath is intercepted by an object.
• Compared to the Integral methods, the GTD is less accurate, particularly in deep shadow zones, except in some simple situations. It is well adaped for very large surfaces at high frequency. Its advantage is no meshing is required thus allowing a very high frequency solution with minimal computer resources. The geometry is exploited directly from CAD information: Polygonal geometries are represented as a collection of trias, quads, polygons while curved surfaces needs to exploit their NURBS representation.
• In the EMC2020 software series, The GTD solver handles at the present time polygonal models. A ray-tracer is integrated with the search of rays. Various interaction terms till the 3rd order are handled (R,D, RR,RD,…,RRR,…). Curved geometries are planed in a future version.

PTD: The PTD solves scattering problems for electrically large objects similar to the GTD. It is a numerical method which combines the PO (Physical Optics) with the ECM (Equivalent Current Method) to take into account the presence of geometric discontinuities .

• The PO uses Ray Optics. This consists of integrating the surface currents via the MFIE in the Far-Field approximation to derive the scattered E-Field. The currents induced on the scatterer are approximated as if each illuminated part of the discretized surface was infinite in its tangent plane, hence the current is directly related to the incident EM field vectors.  The PO integral can be computed directly using a brute force algorithm. In that case, the method is known as the  “PO method”. As the integral is highly oscillatory, other techniques have to be applied for accuracy & fast convergence: Analytic integration on flat surfaces (Ludwig‘s method, Gordon‘s method),  minimal phase/Stationary point methods for curved surfaces which makes the physical interpretation of the involved components apparent (Flash-Point, …). A major difficulty encountered in determining the PO currents on the scatterer is the identification of the lit and shadowed regions. This can be solved as in the GTD via ray-tracing techniques with surface sub-division during the computation (Tesselation).
• The ECM method: The principle is to set fictive equivalent Electric & Magnetic Currents along discontinuities ( Edge/Wedges) such that the ECMdiffracted & PO field match the physical behaviour of waves. Via this method, the PO+ECM field matches exactly the R+D GTD field
• Multiple interactions like GOPO, GOECM, ECM-PO, ECM-ECM, GO-GO-PO, …  can be defined as well to handle complex interactions.
• Compared to the Integral methods, the PTD is less accurate except in some simple situations, but well adaped for very large surfaces at high frequency. Its advantage is no consistent meshing is required, only tesselation of surfaces (for shadowing) thus allowing a very high frequency solution with minimal computer resources. The geometry is exploited directly from CAD information: Polygonal geometries are represented as a collection of trias, quads, polygons while curved surfaces needs to exploit their NURBS representation to find “Flash-points”.
• This software is planed to be available in the future: the kernel is ready, but the graphical interface of the PTD solver has to be renovated to handle at the same time antenna pattern evaluation & RCS calculation for static and moving objects.

The GTD is mostly used for antenna pattern applications . The specific field of antenna pattern perturbation by the antenna structure or any other perturbating structure is well treated by the GTD providing all kind of interactions are well treated by the solver (multiple interactions, creeping rays for curved surfaces, …).

The PTD is mostly used for RCS computation although it can find applications in antenna pattern.

For the EMC2020 software-series, The GTD is actually available for polyhedral structures while the PTD is planed in the future.