Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of
electromagnetic fields with physical objects and the environment.
It typically involves using
computer program
A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components.
A computer program ...
s to compute approximate solutions to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
to calculate
antenna performance,
electromagnetic compatibility,
radar cross section
Radar cross-section (RCS), also called radar signature, is a measure of how detectable an object is by radar. A larger RCS indicates that an object is more easily detected.
An object reflects a limited amount of radar energy back to the source. ...
and electromagnetic
wave propagation
Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation.
With respect to the direction of the oscillation relative to ...
when not in free space. A large subfield is ''antenna modeling'' computer programs, which calculate the
radiation pattern
In the field of antenna design the term radiation pattern (or antenna pattern or far-field pattern) refers to the ''directional'' (angular) dependence of the strength of the radio waves from the antenna or other source.Constantine A. Balanis: “A ...
and electrical properties of radio antennas, and are widely used to design antennas for specific applications.
Background
Several real-world electromagnetic problems like
electromagnetic scattering
Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
,
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
, modeling of
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
s etc., are not analytically calculable, for the multitude of irregular geometries found in actual devices. Computational numerical techniques can overcome the inability to derive closed form solutions of Maxwell's equations under various
constitutive relation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
s of media, and
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s. This makes ''computational electromagnetics'' (CEM) important to the design, and modeling of antenna, radar,
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioi ...
and other communication systems,
nanophotonic devices and high speed
silicon
Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ta ...
electronics,
medical imaging, cell-phone antenna design, among other applications.
CEM typically solves the problem of computing the ''E'' (electric) and ''H'' (magnetic) fields across the problem domain (e.g., to calculate antenna
radiation pattern
In the field of antenna design the term radiation pattern (or antenna pattern or far-field pattern) refers to the ''directional'' (angular) dependence of the strength of the radio waves from the antenna or other source.Constantine A. Balanis: “A ...
for an arbitrarily shaped antenna structure). 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. CEM models may or may not assume
symmetry, simplifying real world structures to idealized
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
s,
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
s, and other regular geometrical objects. CEM models extensively make use of symmetry, and solve for reduced dimensionality from 3 spatial dimensions to 2D and even 1D.
An
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
problem formulation of CEM allows us to calculate steady state normal modes in a structure.
Transient response
In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. The transient response is not necessarily tied to abrupt events but to any event that affec ...
and impulse field effects are more accurately modeled by CEM in time domain, by
FDTD
Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to ...
. Curved geometrical objects are treated more accurately as finite elements
FEM, or non-orthogonal grids.
Beam propagation method The beam propagation method (BPM) is an approximation technique for simulating the propagation of light in slowly varying optical waveguides. It is essentially the same as the so-called parabolic equation (PE) method in underwater acoustics. Bot ...
(BPM) can solve for the power flow in waveguides. CEM is application specific, even if different techniques converge to the same field and power distributions in the modeled domain.
Overview of methods
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. Discretization consumes computer memory, and solving the equations takes significant time. Large-scale CEM problems face memory and CPU limitations. As of 2007, CEM problems require supercomputers, high performance clusters, vector processors and/or
parallelism. Typical formulations involve either time-stepping through the equations over the whole domain for each time instant; or through banded
matrix inversion
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
to calculate the weights of basis functions, when modeled by finite element methods; or matrix products when using transfer matrix methods; or calculating
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s when using
method of moments (MoM); or using
fast Fourier transforms, and time iterations when calculating by the split-step method or by BPM.
Choice of methods
Choosing 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. 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.
Davidson
gives two tables comparing the FEM, MoM and FDTD techniques in the way they are normally implemented. One table is for both open region (radiation and scattering problems) and another table is for guided wave problems.
Maxwell's equations in hyperbolic PDE form
Maxwell's equations can be formulated as a
hyperbolic system
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
of
partial differential equations. This gives access to powerful techniques for numerical solutions.
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 magnetic (TM) 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
:
:
:
:
In this representation,
is the
forcing function, and is in the same space as
. It can be used to express an externally applied field or to describe an optimization
constraint. As formulated above:
:
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.
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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
. 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
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system ...
approximation. The resulting linear system of equations is commonly solved using
conjugate gradient
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterat ...
iterations. The discretization matrix has symmetries (the integral form of Maxwell equations has form of convolution) enabling
fast Fourier transform to multiply matrix times vector during conjugate gradient iterations.
Method of moments element method
The
method of moments (MoM) or
boundary element method
The 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), including fluid mechanics, acoustics, el ...
(BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
s (i.e. in ''boundary integral'' form). It can be applied in many areas of engineering and science including
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
,
acoustics,
electromagnetics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
,
fracture mechanics, and
plasticity.
BEM has become more popular since the 1980s. Because it requires calculating only boundary values, rather than values throughout the space, it is significantly more efficient in terms of computational resources for problems with a small surface/volume ratio. Conceptually, it works by constructing a "mesh" over the modeled surface. However, for many problems, BEM are significantly computationally less efficient than volume-discretization methods (
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
,
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are ...
,
finite volume method
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergenc ...
). 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 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 and geometry of the problem.
BEM is applicable to problems for which
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s can be calculated. These usually involve fields in
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
homogeneous media. This places considerable restrictions on the range and generality of problems suitable for boundary elements. Nonlinearities can be included in the formulation, although they generally introduce volume integrals which require the volume to be discretized before solution, removing an oft-cited advantage of BEM.
Fast multipole method
The
fast multipole method __NOTOC__
The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the ''n''-body problem. It does this by expanding the system Green's function using a multipole expansion, w ...
(FMM) is an alternative to MoM or Ewald summation. It is an accurate simulation technique and requires less memory and processor power than MoM. The FMM was first introduced by
Greengard and
Rokhlin
Rokhlin is a Slavic language-influenced Jewish surname of matronymic derivation. It literally means "Rokhl's", where "Rokhl" is a transcription of ''Rochl'', a Yiddish form of the name Rachel. Variants include Rohlin, Rochlin and (via French) Roch ...
and is based on the
multipole expansion technique. The first application of the FMM in computational electromagnetics was by Engheta et al.(1992). FMM can also be used to accelerate MoM.
Plane wave time-domain
While the fast multipole method is useful for accelerating MoM solutions of integral equations with static or frequency-domain oscillatory kernels, the plane wave time-domain (PWTD) algorithm employs similar ideas to accelerate the MoM solution of time-domain integral equations involving the
retarded potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so th ...
. The PWTD algorithm was introduced in 1998 by Ergin, Shanker, and Michielssen.
Partial element equivalent circuit method
The
partial element equivalent circuit (PEEC) is a 3D full-wave modeling method suitable for combined
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and
circuit analysis. Unlike MoM, PEEC is a full
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
method valid from
dc to the maximum
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
determined by the meshing. In the PEEC method, the
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
is interpreted as
Kirchhoff's voltage law
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
applied to a basic PEEC cell which results in a complete circuit solution for 3D geometries. The equivalent circuit formulation allows for additional
SPICE
A spice is a seed, fruit, root, bark, or other plant substance primarily used for flavoring or coloring food. Spices are distinguished from herbs, which are the leaves, flowers, or stems of plants used for flavoring or as a garnish. Spice ...
type circuit elements to be easily included. Further, the models and the analysis apply to both the time and the frequency domains. 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 direct current 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
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
and
hexahedral
A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.
Ther ...
elements. This helps in keeping the number of unknowns at a minimum and thus reduces computational time for nonorthogonal geometries.
Cagniard-deHoop method of moments
The Cagniard-deHoop method of moments (CdH-MoM) is a 3-D full-wave time-domain integral-equation technique that is formulated via the
Lorentz reciprocity theorem. Since the CdH-MoM heavily relies on the
Cagniard-deHoop method, a joint-transform approach originally developed for the analytical analysis of seismic wave propagation in the crustal model of the Earth, this approach is well suited for the TD EM analysis of planarly-layered structures. The CdH-MoM has been originally applied to time-domain performance studies of cylindrical and planar antennas and, more recently, to the TD EM scattering analysis of transmission lines in the presence of thin sheets and electromagnetic metasurfaces, for example.
Differential equation solvers
Finite-difference time-domain
Finite-difference time-domain
Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born 1934) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to ...
(FDTD) is a popular CEM technique. It is easy to understand. It has an exceptionally simple implementation for a full wave solver. It is at least an order of magnitude less work to implement a basic FDTD solver than either an FEM or MoM solver. FDTD is the only technique where one person can realistically implement oneself in a reasonable time frame, but even then, this will be for a quite specific problem.
[David B. Davidson, ''Computational Electromagnetics for RF and Microwave Engineering'', Second Edition, Cambridge University Press, 2010] Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run, provided the time step is small enough to satisfy the
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that per ...
for the desired highest frequency.
FDTD belongs in the general class of grid-based differential time-domain numerical modeling methods.
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
(in
partial differential
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
form) are modified to central-difference equations, discretized, and implemented in software. The equations are solved in a cyclic 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
''IEEE Transactions on Antennas and Propagation'' is a peer-reviewed scientific journal published by the IEEE Antennas & Propagation Society. It covers research on and applications of all aspects of antenna technology and the propagation of elec ...
.
Allen Taflove
Allen Taflove (June 14, 1949 - April 25, 2021) was a full professor in the Department of Electrical and Computer Engineering of Northwestern's McCormick School of Engineering, since 1988. Since 1972, he pioneered basic theoretical approaches, n ...
originated the descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym in a 1980 paper in
IEEE Trans. Electromagn. Compat. Since about 1990, FDTD techniques have emerged as the primary means to model many scientific and engineering problems addressing electromagnetic wave interactions with material structures. An effective technique based on a time-domain finite-volume discretization procedure was introduced by Mohammadian et al. in 1991. Current FDTD modeling applications range from near-DC (ultralow-frequency geophysics involving the entire Earth-
ionosphere waveguide) through
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
s (radar signature technology, antennas, wireless communications devices, digital interconnects, biomedical imaging/treatment) to visible light (
photonic crystal
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic ...
s, nanoplasmonics,
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medi ...
s, and
biophotonics
The term biophotonics denotes a combination of biology and photonics, with photonics being the science and technology of generation, manipulation, and detection of photons, quantum units of light. Photonics is related to electronics and photons. P ...
). Approximately 30 commercial and university-developed software suites are available.
Discontinuous time-domain method
Among many time domain methods, discontinuous Galerkin time domain (DGTD) method has become popular recently since it integrates advantages of both the finite volume time domain (FVTD) method and the finite element time domain (FETD) method. Like FVTD, the numerical flux is used to exchange information between neighboring elements, thus all operations of DGTD are local and easily parallelizable. Similar to FETD, DGTD employs unstructured mesh and is capable of high-order accuracy if the high-order hierarchical basis function is adopted. With the above merits, DGTD method is widely implemented for the transient analysis of multiscale problems involving large number of unknowns.
Multiresolution time-domain
MRTD is an adaptive alternative to the finite difference time domain method (FDTD) based on
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
analysis.
Finite element method
The
finite element method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEM) is used to find approximate solution of
partial differential equations (PDE) and
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
s. The solution approach is based either on eliminating the time derivatives completely (steady state problems), or rendering the PDE into an equivalent
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
, which is then solved using standard techniques such as
finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
s, 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
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorit ...
, meaning that errors in the input data and intermediate calculations do not accumulate and destroy the meaning of the resulting output. There are many ways of doing this, with various 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.
Finite integration technique
The finite integration technique (FIT) is a spatial discretization scheme to numerically solve electromagnetic field problems in time and frequency domain. It preserves basic
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
properties of the continuous equations such as conservation of charge and energy. FIT was proposed in 1977 by
Thomas Weiland
Thomas Weiland (born October 24, 1951) is a German physicist, engineer and university lecturer. He is Professor of Electrical Engineering at the Department of Electrical Engineering and Information Technology of the Technische Universität Darmst ...
and has been enhanced continually over the years. This method covers the full range of electromagnetics (from static up to high frequency) and optic applications and is the basis for commercial simulation tools: CST Studio Suite developed by
Computer Simulation Technology (CST AG) and
Electromagnetic Simulation solutions developed by
Nimbic.
The basic idea of this approach is to apply the Maxwell equations in integral form to a set of staggered grids. This method stands out due to high flexibility in geometric modeling and boundary handling as well as incorporation of arbitrary material distributions and material properties such as
anisotropy
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
, non-linearity and dispersion. Furthermore, the use of a consistent dual orthogonal grid (e.g.
Cartesian grid
A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks).
Its opposite is irregular grid.
Grids of this type appear on graph paper and may be used in finite element analysis, finite vol ...
) in conjunction with an explicit time integration scheme (e.g. leap-frog-scheme) leads to compute and memory-efficient algorithms, which are especially adapted for transient field analysis in
radio frequency
Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the ...
(RF) applications.
Pseudo-spectral time domain
This class of marching-in-time computational techniques for Maxwell's equations uses either discrete Fourier or
discrete Chebyshev transform
In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind T_n (x) a ...
s 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.
Pseudo-spectral spatial domain
PSSD 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 FFTs. 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. However, the choice to propagate forward in space (rather than in time) brings with it some subtleties, particularly if reflections are important.
Transmission line matrix
Transmission line matrix (TLM) can be formulated in several means as a direct set of lumped elements solvable directly by a circuit solver (ala SPICE,
HSPICE
A spice is a seed, fruit, root, bark, or other plant substance primarily used for flavoring or coloring food. Spices are distinguished from herbs, which are the leaves, flowers, or stems of plants used for flavoring or as a garnish. Spices are ...
, et al.), as a custom network of elements or via a
scattering matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
approach. TLM is a very flexible analysis strategy akin to FDTD in capabilities, though more codes tend to be available with FDTD engines.
Locally one-dimensional
This is an implicit method. In this method, in two-dimensional case, Maxwell equations are computed in two steps, whereas in three-dimensional case Maxwell equations are divided into three spatial coordinate directions. Stability and dispersion analysis of the three-dimensional LOD-FDTD method have been discussed in detail.
Other methods
Eigenmode expansion
Eigenmode expansion (EME) is a rigorous bi-directional technique to simulate electromagnetic propagation which relies on the decomposition of the electromagnetic fields into a basis set of local eigenmodes. The eigenmodes are found by solving Maxwell's equations in each local cross-section. Eigenmode expansion can solve Maxwell's equations in 2D and 3D and can provide a fully vectorial solution provided that the mode solvers are vectorial. It offers very strong benefits compared with the FDTD method for the modelling of optical waveguides, and it is a popular tool for the modelling of
fiber optics and
silicon photonics
Silicon photonics is the study and application of photonic systems which use silicon as an optical medium. The silicon is usually patterned with sub-micrometre precision, into microphotonic components. These operate in the infrared, most comm ...
devices.
Physical optics
Physical optics (PO) is the name of a
high frequency approximation (short-
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
approximation) commonly used in optics,
electrical engineering and
applied physics
Applied physics is the application of physics to solve scientific or engineering problems. It is usually considered to be a bridge or a connection between physics and engineering.
"Applied" is distinguished from "pure" by a subtle combination ...
. 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 geometrical 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 integral, 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 theory, perturbation.
Uniform theory of diffraction
The uniform theory of diffraction (UTD) is a high frequency method for solving Electromagnetic radiation, electromagnetic scattering problems from electrically small discontinuities or discontinuities in more than one dimension at the same point.
The uniform theory of diffraction approximates near and far field, 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 (waves), 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.
Validation
Validation is one of the key issues facing electromagnetic simulation users. The user must understand and master the validity domain of its simulation. The measure is, "how far from the reality are the results?"
Answering this question involves three steps: comparison between simulation results and analytical formulation, cross-comparison between codes, and comparison of simulation results with measurement.
Comparison between simulation results and analytical formulation
For example, assessing the value of the
radar cross section
Radar cross-section (RCS), also called radar signature, is a measure of how detectable an object is by radar. A larger RCS indicates that an object is more easily detected.
An object reflects a limited amount of radar energy back to the source. ...
of a plate with the analytical formula:
where A is the surface of the plate and
is the wavelength. The next curve presenting the RCS of a plate computed at 35 GHz can be used as reference example.
Cross-comparison between codes
One example is the cross comparison of results from method of moments and asymptotic methods in their validity domains.
Comparison of simulation results with measurement
The final validation step is made by comparison between measurements and simulation. For example, the RCS calculation and the measurement of a complex metallic object at 35 GHz. The computation implements GO, PO and PTD for the edges.
Validation processes can clearly reveal that some differences can be explained by the differences between the experimental setup and its reproduction in the simulation environment.
full article
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Light scattering codes
There are now many efficient codes for solving electromagnetic scattering problems. They are listed as:
* discrete dipole approximation codes,
* codes for electromagnetic scattering by cylinders,
* codes for electromagnetic scattering by spheres.
Solutions which are analytical, such as Mie solution for scattering by spheres or cylinders, can be used to validate more involved techniques.
See also
* EM simulation software
* Analytical regularization
* Computational physics
* Electromagnetic field solver
* Electromagnetic wave equation
* Finite-difference time-domain method
* Finite-difference frequency-domain
* Mie theory
* Physical optics
* Rigorous coupled-wave analysis
* Space mapping
* Uniform theory of diffraction
* Shooting and bouncing rays
References
Further reading
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External links
Computational electromagnetics at the Open Directory Project
{{DEFAULTSORT:Computational Electromagnetics
Computational electromagnetics,
Electrodynamics
Partial differential equations
Computational fields of study