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 antenna performance, electromagnetic compatibility, radar cross section and electromagnetic wave propagation when not in free space.

Specific part of computational electromagnetics deals with electromagnetic radiation scattered and absorbed by small particles.


Contents

[edit] Background

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

[edit] Choice of methods

Discussion of which method is chosen when should go here. Integral equation solvers versus differential equation solvers. When and why to use high-frequency approximations.

[edit] 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 numerical 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 terms 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)

[edit] Integral equation solvers

[edit] The discrete dipole approximation

The discrete dipole approximation is a flexible technique for computing scattering and absorption by targets of arbitrary geometry. The formulation is based on integral form of Maxwell equations. The DDA is an approximation of the continuum target by a finite array of polarizable points. The points acquire dipole moments in response to the local electric field. The dipoles of course interact with one another via their electric fields, so the DDA is also sometimes referred to as the coupled dipole approximation. Resulting linear system of equations is commonly solved commonly using the conjugate gradient iterations. Because discretization matrix has symmetries (the intgeral form of Maxwell equations has form of convolution) it is possible to use Fast Fourier Transform to multiply matrix times vector during the conjugate gradient iterations.

[edit] 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 computational resources for problems where there is a small surface/volume ratio. Conceptually, it works by constructing a "mesh" over the modeled surface. However, for many problems boundary element methods are significantly less efficient than volume-discretization 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 discretized before solution can be attempted, removing one of the most often cited advantages of BEM.

[edit] Computer codes

Example computer codes using the MOM are:

[edit] Fast multipole method (FMM)

See also Multipole expansion.

[edit] Recursive T-Matrix Algorithms (RTMA)

[edit] Partial Element Equivalent Circuit (PEEC)


[edit] Adaptive Integral Method (AIM)

[edit] Differential equation solvers

[edit] 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 grid-based 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 basic FDTD algorithm traces back to a seminal 1966 paper by Kane Yee in IEEE Transactions on Antennas and Propagation. The descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym were originated by Allen Taflove in a 1980 paper in IEEE Transactions on Electromagnetic Compatibility. Since about 1990, FDTD techniques have emerged as primary means to model many scientific and engineering problems dealing with electromagnetic wave interactions with material structures. Current FDTD modeling applications range from near-DC (ultralow-frequency geophysics involving the entire Earth-ionosphere waveguide) through microwaves (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (photonic crystals, nanoplasmonics, solitons, and biophotonics). Approximately 30 commercial and university-developed FDTD software suites are available for use (see below).

[edit] External links

[edit] Multiresolution time-domain (MRTD)

An adaptive alternative to the finite difference time domain method (FDTD) based on wavelet analysis.

[edit] 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.

[edit] Computer codes

Finite Element Programs with Online Free/Demo Version:

Other Finite Element Programs with Websites:

[edit] Pseudospectral Time Domain (PSTD)

This class of marching-in-time computational techniques for Maxwell's equations uses either discrete Fourier or Chebyshev transforms to calculate the spatial derivatives of the electric and magnetic field vector components that are arranged in either a 2-D grid or 3-D lattice of unit cells. PSTD causes negligible numerical phase velocity anisotropy errors relative to FDTD, and therefore allows problems of much greater electrical size to be modeled. For a recent comprehensive summary of PSTD techniques for Maxwell's equations, see Q. Liu and G. Zhao, "Advances in PSTD Techniques," Chapter 17 in Computational Electrodynamics: The Finite-Difference Time-Domain Method, A. Taflove and S. C. Hagness, eds., Boston: Artech House, 2005.

[edit] Pseudo Spectral Spatial Domain (PSSD)

This approach solves Maxwell's equations by propagating them forward in a chosen spatial direction. The fields are therefore held as a function of time, and (possibly) any transverse spatial dimensions. The method is pseudo spectral because temporal derivatives are calculated in the frequency domain with the aid of fast Fourier transforms. Because the fields are held as functions of time, this enables arbitrary dispersion in the propagation medium to be rapidly and accurately modelled with minimal effort. See J.C.A. Tyrrell et al, J.Mod.Opt. 52, 973 (2005).

[edit] Transmission line matrix (TLM)

[edit] Other methods

[edit] 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.

[edit] Geometric theory of diffraction (GTD)

[edit] Physical theory of diffraction (PTD)

[edit] 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.

[edit] Computer codes

[edit] Method of lines


[edit] See also

[edit] Researchers/Scientists

[edit] References

[edit] Computer codes

[edit] Free or shareware codes

[edit] Commercial codes

  • EpsilonTM from Roke Manor Research Ltd. High frequency RCS prediction.
  • FEKO from EM Software & Systems. A hybridized MoM computational electromagnetics simulation software suite. Additional techniques include FEM, MLFMM, PO, UTD and special integral equation formulations.
  • General Electric Research in Niskayuna, NY
  • HFSS from Ansoft
  • IBM - Electronic Interconnect and Packaging sells software (to Juniper Networks) for chip design.
  • Infolytica for modeling electrical motors.
  • Microwave Studio from CST
  • QuickField from Tera Analysis Ltd. FEA software for EM, Heat Transfer and Stress simulations. No training required!
  • SAIC (Champaigne-Urbana, IL) has specialized software to model radars, stealth technology, and DoD applications.
  • SEMCAD X by SPEAG
  • Vector Fields Unltd.


[edit] Pages with lists of codes


[edit] External links

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