The DDA is an approximation of the continuum target by a finite array of polarizable points. The formulation is based on integral form of Maxwell equations. The discrete dipole approximation is a flexible technique for computing scattering and absorption by targets of arbitrary geometry. Integral equation solvers The discrete dipole approximation ∂ ∂ t u ¯ + A ∂ ∂ x u ¯ + B ∂ ∂ y u ¯ + C u ¯ = g ¯ may also be explicitly defined equal to zero to simplify certain problems, or to find a characteristic solution, which is often the first step in a method to find the particular inhomogeneous solution. In 2D and no polarization terms present, Maxwell's equations can then be formulated as: The wave is called a transverse magnetic (TM) wave. 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. This gives access to powerful techniques for numerical solutions. Maxwell's equations can be formulated as a hyperbolic system of partial differential equations. Maxwell's equations in hyperbolic PDE form One table is for both open region (radiation and scattering problems) and another table is for guided wave problems. However, the name of a technique does not always tell one how it is implemented, especially for commercial tools, which will often have more than one solver.ĭavidson gives two tables comparing the FEM, MoM and FDTD techniques in the way they are normally implemented. Typical formulations involve either time-stepping through the equations over the 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 fast Fourier transforms, and time iterations when calculating by the split-step method or by BPM.Ĭhoosing the right technique for solving a problem is important, as choosing the wrong one can either result in incorrect results, or results which take excessively long to compute. As of 2007, CEM problems require supercomputers, high performance clusters, vector processors and/or parallelism. Large-scale CEM problems face memory and CPU limitations. Discretization consumes computer memory, and solving the equations takes significant time. One approach is to discretize the space in terms of grids (both orthogonal, and non-orthogonal) and solving Maxwell's equations at each point in the grid. CEM is application specific, even if different techniques converge to the same field and power distributions in the modeled domain. Beam propagation method (BPM) can solve for the power flow in waveguides. Curved geometrical objects are treated more accurately as finite elements FEM, or non-orthogonal grids. Transient response and impulse field effects are more accurately modeled by CEM in time domain, by FDTD. CEM models extensively make use of symmetry, and solve for reduced dimensionality from 3 spatial dimensions to 2D and even 1D.Īn eigenvalue problem formulation of CEM allows us to calculate steady state normal modes in a structure. CEM models may or may not assume symmetry, simplifying real world structures to idealized cylinders, spheres, and other regular geometrical objects. Also calculating power flow direction ( Poynting vector), a waveguide's normal modes, media-generated wave dispersion, and scattering can be computed from the E and H fields.
This makes computational electromagnetics (CEM) important to the design, and modeling of antenna, radar, satellite and other communication systems, nanophotonic devices and high speed silicon electronics, medical imaging, cell-phone antenna design, among other applications.ĬEM typically solves the problem of computing the E (electric) and H (magnetic) fields across the problem domain (e.g., to calculate antenna radiation pattern for an arbitrarily shaped antenna structure). Computational numerical techniques can overcome the inability to derive closed form solutions of Maxwell's equations under various constitutive relations of media, and boundary conditions. Several real-world electromagnetic problems like electromagnetic scattering, electromagnetic radiation, modeling of waveguides etc., are not analytically calculable, for the multitude of irregular geometries found in actual devices. 8.3 Comparison of simulation results with measurement.8.1 Comparison between simulation results and analytical formulation.5.5 Partial element equivalent circuit method.4 Maxwell's equations in hyperbolic PDE form.