COMPUTATIONAL ELECTROMAGNETICS NUMERICAL METHODS
Computation is an integral part of modern science and engineering, and the ability to exploit effectively the power offered by computers is therefore essential
to an engineer or researcher.
Unfortunately the ability to solve problems with the computer is not cultivated by the standard university courses,
as requires an integration of three disciplines (electromagnetic field theory, numerical analysis, and computer programming)
covered in usually unrelated courses.
Computational electromagnetics has become an essential tool in modern electrical engineering,
starting from classical applications like the modeling of electrical machines, and going until high technology applications
like microwave device modeling, semiconductor device design, the design of the modern particle
accelerators and plasma fusion reactors, superconductor devices, very high speed suspended trains, etc.
The analytical methods of solving electromagnetic field problems are very useful
for understanding the behavior of electromagnetic systems, but there are only special cases when they can be used in
real engineering applications.
That is why the approximate methods appeared long before the computers, and nowadays with the explosive development of the computer
capabilities the numerical methods and techniques are extremely numerous and attracted a great deal of attention in the engineering
and scientific communities, as evidenced both by the great amount of published material and by the increasing number of conferences,
workshops, and short courses specifically devoted to this subject.
Modern methods of solving electromagnetic field problems involve a judicious
mixture of analysis and computation. The analysis occurs in the mathematical
formulation and in establishing that it has the requisite properties. Conversion to a form
suitable for the computer entails numerical analysis, whose justification may also rest on considerable body of analysis.
The electromagnetic field problems are a part of the general chapter of mathematics: boundary value problems
for partial differential equations, studied and developed by the great mathematicians of the classical mathematics.
The partial differential equations of the second order play a primordial role in mathematical
physics, and can be classified into equations of the elliptic, parabolic, and hyperbolic
type, with solutions of each type displaying specific properties. Partial differential
equations for the EM fields enter in this classification as following: elliptic type
for low frequencies, hyperbolic type (wave like) for high frequencies, and parabolic
type (diffusion like) for intermediate frequencies and lossy media.
The boundary conditions also range from simple Dirichlet and
Neumann conditions to complicated impedance and radiation conditions, and to even
more complicated higher order conditions.
The most important numerical techniques used in commercial EM software
Despite the advertising claims of some software companies, there is no perfect numerical technique that is most efficient and accurate
in every possible situation. Each approach
has its own advantages in some areas, there is no method that is best everywhere.
Many times one's opinion on which technique is "the best" depends on the flavor preferred at his/her university.
We are here to help you make an objective optimal choice for your applications.
We have a much more objective view since we are not the manufacturers of the software, and we sell various type of software, hopefully
covering soon all methods.
A first classification of the computational electromagnetics methods would be into:
- differential methods (FEM, FD, TLM), are based on the discretization of Maxwell's equations over the entire domain
and compute the unknown variables over the entire domain. Note that the discretized Maxwell's equations can be in differential
or integral form (like in FIM) since they are basically equivalent.
- integral methods (MOM), are based on the discretization of certain specific integral equations involving the
Green's function of the structure (the Green's function can be described as the generalization of the impulse response,
familiar from linear linear systems theory, to 3D EM and space or space-time variables. As Green's functions are related to sources of the field
the unknowns are, in most cases, discretized currents - and thus the discretization takes place only for the metallic surfaces or sometimes volumes
There are less unknowns with the integral methods, however the matrix to be solved is a full matrix while for differential methods it is sparse.
All methods, in principle, have these following versions:
- static, time variation completely ignored. This situation is of little interest for microwave or high-speed digital engineering,
sometimes is used as low frequency approximation or to compute the DC bias field configuration.
- time domain, the unknowns are "real" time varying quantities with a source that is some sort of time domain function (usually a pulse).
Frequency domain results all over the bandwidth of the input pulse are obtained by Fourier transforms, however very low frequencies can
be difficult to analyze.
- frequency domain, the excitation is considered sinusoidal, the time derivatives are replaced by jω and each round of solving
gives the results for one single frequency and have to be repeated and interpolated to cover the desired frequency range. Special interpolation
methods have been developed. Broadband results can be transformed into time domain by inverse Fourier transform, and causality errors can be present.
- eingensolver, or modal solver. This version is used to determine the resonant frequencies and modes of cavities or other type of resonators (3D)
and also to find the propagation modes of waveguides, cutoff frequencies, characteristic impedance and propagation constants (2D).
The first three situations can be grouped under the name of driven problems as opposed to eigenproblems where there is no particular source, but
the results are the field configurations that would be preferentially excited by sources if placed in that environment.
Frequency domain, static and eigenvalue programs generate a matrix system (sometimes of complex numbers) while time domain methods generally don't.
Commercial programs in general favor one of the two main approaches - time or frequency domain:
- FEM and MOM programs available are frequency domain versions, some packages including an eigensolver also.
- Finite difference and TLM programs are time domain versions, some with eigensolver.
Another way to classify/characterize the methods is the dimensionality of the problems solved.
While all methods can be made to solve complicated 3D problems, sometimes a version
that has certain restrictions on the geometry can work much faster and solve much bigger problems.
Thus for applications like PCBs or planar microwave components it might not be needed to use a general solver
but a solver specifically formulated for layered media. These solvers are typically based on the method of moments.
When the geometry that can be analyzed is restricted to arbitrary 2D shapes in multiple layers (possibly with vertical vias between them) and the
metallic layers are considered infinitesimally thin we call that a 2.5D solver. The E and H fields are actually 3D
however the currents are mostly two dimensional and only a few of them are vertical (vias).
2D solvers are used when the problem can be approximated as being uniform in one direction
(planar structures with arbitrary shaped layers are not 2D) or have rotational symmetry.
They are used as fast approximations of real problems of for finding the modes of a waveguide.
The Finite Element Method FEM
One of the most important method in electromagnetic field computation is the finite
element method. In this case the actual discretization of the geometry is the essence
of the method, each part of the discretized domain being called finite element. Each
finite element is considered to have constant properties inside, and that the unknown
has a well defined behavior in the element, behavior that depends upon a number of
unknown parameters to be determined.
The division of the domain can be made with triangular or quadrilateral cells in
two dimensions or with tetrahedral, hexahedral, prismatic or pyramidal cells in three
dimensions. This cells can also have curved faces when we need to discretize domains
with curved boundaries. The most often used cells are those triangular in 2D and
tetrahedral in 3D.
Thus the finite element method is an approximation technique to discretize the
geometric area or volume of a physical problem into small elements, reducing the
operator equation to a finite matrix equation by a certain form of averaging/integrating
over the volume (area) domain, i.e. sum of integrals over the cells.
A huge, but sparse matrix is generated and solving this system of equations with thousands - millions of unknowns is the
most time consuming part of a FEM code. This also gives the main memory limitation of the method.
The FEM is usually used in frequency domain and each solving of the matrix system gives the solution for one frequency.
Repeated runs and interpolation are used to obtain the systems response over a frequency band.
This might give problems for resonant systems, especially those with high Q, where it is not easy to get the resonant frequency
unless you sweep the frequency carefully.
The Finite Difference Time Domain Method FDTD
In this case the discretization grid is rectangular in general (although FEM type of meshes are possible for some versions) and the Maxwell equations
are discretized by relatively simple finite difference equations. There is no huge system of equations to be solved (in most cases) and the solution is
found via a step by step propagation on the discretized grid. In most implementations there are two separate grids, one for the unknown electric field E and
one for the unknown magnetic field H components. The two grids are displaced by a half a cell, such that the nodes of one are in the centers
of the other's cells.
Each FDTD time step can be divided into two substeps:
- solve Faraday's induction law (differential or integral form) on E field contours and find the H fields
- solve Ampere's law (differential or integral form) on H field contours and find the E fields
The input is usually a gaussian pulse with spectral components centered in the band of interest. After solving the time domain pulse response, the frequency response of the system can be found by
inverse Fourier transform.
This way the method practically cannot miss a resonance and the user doesn't have to be concerned about selecting the correct frequency.
However, when high Q resonances are present the computation is longer because we have to wait for the pulse response ringing to decay to a low enough level.
Some versions of FDTD discretize the integral form of Maxwell's equations and thus are more flexible in choosing the mesh, that can,
in principle, contain tetragonal, or curved cells(Finite Integrals Method, Conformal FDTD).
FDTD can have stability problems if the parameters are not chosen well. One important fact is that the time domain step size has to
be related to the minimum spatial discretization distance.
This can slow down two fold the simulation when large objects with fine features are analyzed.
The Transmission Line Matrix Method TLM
In many ways similar to FDTD. One of the advantages of these two methods is that their core algorithms are very straightforward and in the
case of TLM it has a physical interpretation based on the transmission line theory.
The grid lines in the TLM case represent mini transmission lines that intersect. Wave amplitude pulses are
transmitted on the lines and scattered at each intersection.
Like for FDTD each time step can be divided into two substeps, and for TLM they are: scatter and connect.
During the scatter, the wave pulses incident on the node are scattered to produce a new set of outgoing wave pulses.
During the connect, wave pulses are transferred to the adjacent nodes.
It is possible to combine both processes together but it is simpler to consider them separately.
TLM shares many of the advantages and disadvantages of FDTD and it was proven that certain formulations of the two are equivalent.
The Method of Moments MOM
This method is based on integral equations (IE) and Green's functions and it is also known as the Boundary Elements Method (BEM).
Since the Green's functions describe the effect the sources from one point have on the field, the method is concerned with
the discretization of the field sources i.e. the currents inside conductors or fields at the boundaries of the domain.
Thus only the metallic parts or the boundaries are discretized and a 3D problem generally needs only 2D discretization. The matrix obtained by discretizing the integral
equations is smaller than that of FEM but it is not sparse.
Some MOM versions can discretize the volume of the conductors (Volumetric IE), not only the surface like most versions (Surface IE),
this way metallization thickness and finite conductivity effects are considered. Note that even the surface MOM can be a full 3D method, when the surface
doesn't have to be planar.
One classification of MOM types is:
- unbounded (or open)
- bounded or partially bounded (or close, or shielded)
The unbounded MOM applies naturally for open problems, while the differential methods from above need special absorbing boundary conditions in that case.
The bounded (or partially bounded) version of MOM on the other hand can be very fast an efficient - for the suitable problems. This version considers
the simulated devices as enclosed in a metallic box (bounded) or metallic waveguide infinite in the top-bottom direction (partially bounded)
MOM is usually used in frequency domain and has the advantage of dealing easily with long thin wires or thin patches, layered configurations,
situation that can present serious challenges to the differential methods since they have to discretize the entire domain and the presence of thin
structures require a very fine mesh.
For the applications that can be analyzed with it, MOM is the best choice. It is faster and more accurate and also because does not need
complicated full domain discretization MOM packages have a much lower price than of those based on differential methods.