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 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. 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.Adapted from the Wikipedia article Computational electromagnetics, under the G. N. U. Free Documentation License. Please also see http://en.wikipedia.org/wiki
