Computational electromagnetics

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Computational electromagnetics, computational electrodynamics or electromagnetic modeling refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computationally efficient approximations to Maxwell's Equations and is used to calculate things such as antenna performance, electromagnetic compatibility, radar cross section and electromagnetic wave propagation when not in free space.

Background

Why do electromagnetic modeling? What is being modeled? What are the computational issues?

Choice of methods

Discussion of which method is chosen when should go here. Integral equation solvers versus differential equation solvers.

Maxwell's equations in hyperbolic PDE form

Maxwell's equations can be formulated as a hyperbolic system of partial differential equations. This gives access to powerful mathematical theories for the numerial solutions of hyperbolic PDE's.

It is assumed that the waves propagate in the (x,y) plane and restrict the direction of the magnetic field to be parallel to the z-axis and thus the electric field to be parallel to the (x,y) plane. The wave is called a Transverse Electric (TE) wave. In 2D and no polarization present Maxwell's equations can then be formulated as:

${\frac {\partial }{\partial t}}{\bar {u}}+A{\frac {\partial }{\partial x}}{\bar {u}}+B{\frac {\partial }{\partial y}}{\bar {u}}+C{\bar {u}}=g$

where u, A, B and C are defined as:

${\bar {u}}=\left({\begin{matrix}E_{x}\\E_{y}\\H_{z}\end{matrix}}\right)$

$A=\left({\begin{matrix}0&0&0\\0&0&{\frac {1}{\epsilon }}\\0&{\frac {1}{\mu }}&0\end{matrix}}\right)$

$B=\left({\begin{matrix}0&0&{\frac {-1}{\epsilon }}\\0&0&0\\{\frac {-1}{\mu }}&0&0\end{matrix}}\right)$

$C=\left({\begin{matrix}{\frac {\sigma }{\epsilon }}&0&0\\0&{\frac {\sigma }{\epsilon }}&0\\0&0&0\end{matrix}}\right)$

Integral equation solvers

Method of moments (MOM) or boundary element method (BEM)

The Method of moments (MOM) or boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form). It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, fracture mechanics, and plasticity.

It has become more and more popular since the 1980s. Because it requires calculating only boundary values, rather than values throughout the space defined by a partial differential equation, it is significantly more efficient in terms of computatioal resources for problems where there is a small surface/volume ratio. Conceptually, it works by constructing a "mesh" over the modelled surface. However, for many problems boundary element methods are significantly less efficient than volume-discretisation methods (Finite element method, Finite difference method, Finite volume method). Boundary element formulations typically give rise to fully populated matrices. This means that the storage requirements and computational time will tend to grow according to the square of the problem size. By contrast, finite element matrices are typically banded (elements are only locally connected) and the storage requirements for the system matrices typically grow quite linearly with the problem size. Compression techniques (e.g. multipole expansions or adaptive cross approximation/hierarchical matrices) can be used to ameliorate these problems, though at the cost of added complexity and with a success-rate that depends heavily on the nature of the problem being solved and the geometry involved.

BEM is applicable to problems for which green's functions can be calculated. These usually involve fields in linear homogeneous media. This places considerable restrictions on the range and generality of problems to which boundary elements can usefully be applied. Nonlinearities can be included in the formulation, although they will generally introduce volume integrals which then require the volume to be discretised before solution can be attempted, removing one of the most often cited advantages of BEM.

Computer codes

Example computer codes using the MOM are:

• Numerical Electromagnetic Code (NEC)

Fast multipole method (FMM)

See also Multipole expansion.

Differential equation solvers

Finite-difference time-domain (FDTD)

Finite-difference time-domain (FDTD) is a popular computational electrodynamics modeling technique. It is considered easy to understand and easy to implement in software. Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run.

The FDTD method belongs in the general class of differential time-domain numerical modeling methods. Maxwell's equations (in partial differential form) are modified to central-difference equations, discretized, and implemented in software. The equations are solved in a leapfrog manner: the electric field is solved at a given instant in time, then the magnetic field is solved at the next instant in time, and the process is repeated over and over again.

The term FDTD was coined by Allen Taflove in the year 1980 [1]. His work on FDTD modeling helped solve problems in, for instance, stealth technology for the US military, high speed data transmissions and with cellphone antenna designs.

Computer codes

• One-dimensional FDTF applet

Finite element method (FEM)

The finite element method (FEM) is used for finding approximate solution of partial differential equations (PDE) and integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an equivalent ordinary differential equation, which is then solved using standard techniques such as finite differences, etc.

In solving partial differential equations, the primary challenge is to create an equation which approximates the equation to be studied, but which is numerically stable, meaning that errors in the input data and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complex domains or when the desired precision varies over the entire domain.

Other methods

Physical optics (PO)

Physical optics (PO) is the name of a high frequency approximation (short-wavelength approximation) commonly used in optics, electrical engineering and applied physics. It is an intermediate method between geometric optics, which ignores wave effects, and full wave electromagnetism, which is a precise theory. The word "physical" means that it is more physical than geometric or ray optics and not that it is an exact physical theory.

The approximation consists of using ray optics to estimate the field on a surface and then integrating that field over the surface to calculate the transmitted or scattered field. This resembles the Born approximation, in that the details of the problem are treated as a perturbation.

Uniform theory of diffraction (UTD)

The uniform theory of diffraction (UTD) is a high frequency method for solving electromagnetic scattering problems from electrically small discontinuities or discontinuities in more that one dimension at the same point.

The uniform theory of diffraction approximates near field electromagnetic fields as quasi optical and uses ray diffraction to determine diffraction coefficients for each diffracting object-source combination. These coefficients are then used to calculate the field strength and phase for each direction away from the diffracting point. These fields are then added to the incident fields and reflected fields to obtain a total solution.

Computer codes

• UTD edge diffraction applet

See also

• Born approximation
• Boundary element method
• Electromagnetism
• Electromagnetic wave equation
• Finite-difference time-domain method
• Finite-difference frequency-domain
• Finite element analysis
• Finite element method
• High frequency approximation
• Physical optics
• Uniform theory of diffraction

References

• Computational electromagnetics: a review

Computer codes

Free or shareware codes

• Unofficial NEC homepage
• Unoffical NEC archives

External links

• The Applied Computational Electromagnetics Society Website
• The Center for Computational Electromagnetics and Electromagnetics Laboratory
• Computational Electromagnetics at Bilkent University