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Several real-world electromagnetic problems like scattering, radiation, waveguiding etc, are not analytically calculable, for the multitude of irregular geometries designed and used. The inability to derive closed form solutions of Maxwell's equations under various constitutive relations of media, and boundary conditions, is overcome by computational numerical techniques. This makes computational electromagnetics (CEM), an important field in the design, and modeling of antenna, radar, satellite and other such communication systems, nanophotonic devices and high speed silicon electronics, medical imaging, cell-phone antenna design, among other applications. Outline of Electromagnetic
Theory, Including Problems With Step-By-Step Solutions
CEM problems typically solve for the problem of computing the E (Electric), and H (Magnetic) fields across the domain of the problem (i.e to calculate antenna radiation pattern, for an arbitrarily shaped antenna structure is solved by CEM). Also, power flow direction (Poynting vector), normal modes of a waveguide, dispersion of wave due to media, and scattering are quantities of interest, that can be computed from the knowledge of the E and H fields. CEM models may or may not assume symmetry, simplify real world structures to cylinders, spheres, and other regular geometrical objects. CEM models extensively make use of symmetry, and solve for reduced dimensions of the system from 3 spatial dimensions, to 2D and even 1D. CEM can be formulated into a various problems
Wing-section optimization for supersonic viscous flow [abstract] (SuDoc NAS 1.26:199746)
depending on any of the several quantities of interest mentioned previously. An eigenvalue problem formulation of CEM allows us to calculate steady state normal modes in a structure. Transient response and impulse field effects are more accurately modeled by CEM in time domain, by FDTD. Treating curved geometrical objects is done more accurately by using finite elements FEM, or non-orthogonal grids. Beam propagation methods like BPM, solve for the power flow in waveguides. So, CEM model used is application specific, even if different techniques converge to the same field and power distributions in the modeled domain.
Overview of methods
CEM can be used to model the domain generally by discretizing the space in terms of grids (both orthogonal, and non-orthogonal), and then solve the Maxwell's equations at each point in the grid. Naturally, such discretization of the computational space consumes computer memory, and solving the equations takes a longer time. Large scale CEM problems place computational limitations in terms of memory space, and CPU time on the computer. Generally CEM problems, as of 2007, are run on supercomputers, high performance clusters, vector processors and parallel computers; see article on, parallel computing for more computer/machine specific details. Typical formulations involve either time-stepping through the Maxwell's equations over whole domain for each time instant; or through banded matrix inversion to calculate the weights of basis functions, when modeled by finite element methods; or matrix products when using transfer matrix methods; or calculating integrals when using method of moments (MoM); or using FFT, and time iterations when calculating by the split-step method or by BPM.
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
where u, A, B, and C are defined as
Integral equation solvers
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 using the conjugate gradient iterations. Because discretization matrix has symmetries (the integral 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.
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.
Fast multipole method (FMM)
The fast multipole method (FMM) is a computational electromagnetic technique that may be applied instead of techniques like the method of moments (MoM) or Ewald summation. It is an accurate simulation technique and is computationally more efficient than the MoM. Both memory and processor runtime requirements are greatly reduced over the MoM. The FMM was first introduced by Greengard and Rokhlin and is based on the multipole expansion technique. Can be used to accelerate MOM.
Partial element equivalent circuit (PEEC) method
The partial element equivalent circuit (PEEC) is a 3D full-wave modeling method suitable for combined electromagnetic and circuit analysis. Unlike the method of moments (MoM), PEEC is a full spectrum method valid from dc to the maximum frequency determined by the meshing. In the PEEC method, the integral equation is interpreted as Kirchhoff's voltage law applied to a basic PEEC cell which results in a complete circuit solution for 3D geometries. The equivalent circuit formulation allows for additional SPICE type circuit elements to be easily included. Further, the models and the analysis apply to both the time and the frequency domain. The circuit equations resulting from the PEEC model are easily constructed using a modified loop analysis (MLA) or modified nodal analysis (MNA) formulation. Besides providing a dc solution, it has several other advantages over a MoM analysis for this class of problems since any type of circuit element can be included in a straightforward way with appropriate matrix stamps. The PEEC method has recently been extended to include nonorthogonal geometries.[1] This model extension, which is consistent with the classical orthogonal formulation, includes the Manhattan representation of the geometries in addition to the more general quadrilateral and hexahedral elements. This helps in keeping the number of unknowns at a minimum and thus reduces computational time for nonorthogonal geometries.
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).
