Finite-difference time-domain (FDTD) or Yee's method (named after the Chinese American applied mathematician

Kane S. Yee Kane Shee-Gong Yee (born March 26, 1934) is a Chinese-American electrical engineer and mathematician. He is best known for introducing the finite-difference time-domain method (FDTD) in 1966. His research interests include numerical electroma ...

, born 1934) is a

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

technique used for modeling

computational electrodynamics 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 ...

(finding approximate solutions to the associated system of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s). Since it is a

time-domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the ca ...

method, FDTD solutions can cover a wide

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 ...

range with a single

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the s ...

run, and treat nonlinear material properties in a natural way. The FDTD method belongs in the general class of

grid Grid, The Grid, or GRID may refer to: Common usage * Cattle grid or stock grid, a type of obstacle is used to prevent livestock from crossing the road * Grid reference, used to define a location on a map Arts, entertainment, and media * News ...

-based differential numerical modeling methods ( finite difference methods). The time-dependent

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 discretized using central-difference approximations to the space and time partial derivatives. The resulting finite-difference equations are solved in either software or hardware in a

leapfrog Leapfrog is a children's game in which players vault over each other's stooped backs. History Games of this sort have been called by this name since at least the late sixteenth century.electric field

vector component In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...

s in a volume of space are solved at a given instant in time; then the magnetic field vector components in the same spatial volume are solved at the next instant in time; and the process is repeated over and over again until the desired transient or steady-state electromagnetic field behavior is fully evolved.

History

Finite difference schemes for time-dependent partial differential equations (PDEs) have been employed for many years in

computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...

problems, including the idea of using centered finite difference operators on staggered grids in space and time to achieve second-order accuracy. The novelty of Kane Yee's FDTD scheme, presented in his seminal 1966 paper, was to apply centered finite difference operators on staggered grids in space and time for each electric and magnetic vector field component in Maxwell's curl equations. The descriptor "Finite-difference time-domain" and its corresponding "FDTD" acronym were originated by

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 ...

in 1980. Since about 1990, FDTD techniques have emerged as primary means to computationally model many scientific and engineering problems dealing with

electromagnetic wave 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, (visib ...

interactions with material structures. Current FDTD modeling applications range from near- DC (ultralow-frequency

geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...

involving the entire Earth- ionosphere waveguide) through

microwaves 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 rang ...

(radar signature technology,

antennas In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an ...

, wireless communications devices, digital interconnects, biomedical imaging/treatment) to

visible light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...

(

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, nano

plasmon In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantiz ...

ics,

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 ...

). In 2006, an estimated 2,000 FDTD-related publications appeared in the science and engineering literature (see

Popularity In sociology, popularity is how much a person, idea, place, item or other concept is either liked or accorded status by other people. Liking can be due to reciprocal liking, interpersonal attraction, and similar factors. Social status can be ...

). As of 2013, there are at least 25 commercial/proprietary FDTD software vendors; 13 free-software/ open-source-software FDTD projects; and 2 freeware/closed-source FDTD projects, some not for commercial use (see

External links An internal link is a type of hyperlink on a web page to another page or resource, such as an image or document, on the same website or domain. Hyperlinks are considered either "external" or "internal" depending on their target or destinatio ...

Development of FDTD and Maxwell's equations

An appreciation of the basis, technical development, and possible future of FDTD numerical techniques for Maxwell's equations can be developed by first considering their history. The following lists some of the key publications in this area.

FDTD models and methods

When Maxwell's differential equations are examined, it can be seen that the change in the E-field in time (the time derivative) is dependent on the change in the H-field across space (the curl). This results in the basic FDTD time-stepping relation that, at any point in space, the updated value of the E-field in time is dependent on the stored value of the E-field and the numerical curl of the local distribution of the H-field in space. The H-field is time-stepped in a similar manner. At any point in space, the updated value of the H-field in time is dependent on the stored value of the H-field and the numerical curl of the local distribution of the E-field in space. Iterating the E-field and H-field updates results in a marching-in-time process wherein sampled-data analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory. This description holds true for 1-D, 2-D, and 3-D FDTD techniques. When multiple dimensions are considered, calculating the numerical curl can become complicated. Kane Yee's seminal 1966 paper proposed spatially staggering the vector components of the E-field and H-field about rectangular unit cells of a Cartesian computational grid so that each E-field vector component is located midway between a pair of H-field vector components, and conversely. This scheme, now known as a Yee lattice, has proven to be very robust, and remains at the core of many current FDTD software constructs. Furthermore, Yee proposed a leapfrog scheme for marching in time wherein the E-field and H-field updates are staggered so that E-field updates are conducted midway during each time-step between successive H-field updates, and conversely. On the plus side, this explicit time-stepping scheme avoids the need to solve simultaneous equations, and furthermore yields dissipation-free numerical wave propagation. On the minus side, this scheme mandates an upper bound on the time-step to ensure numerical stability. As a result, certain classes of simulations can require many thousands of time-steps for completion.

Using the FDTD method

To implement an FDTD solution of Maxwell's equations, a computational domain must first be established. The computational domain is simply the physical region over which the simulation will be performed. The E and H fields are determined at every point in space within that computational domain. The material of each cell within the computational domain must be specified. Typically, the material is either free-space (air),

metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...

, or

dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...

. Any material can be used as long as the permeability,

permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...

, and

conductivity Conductivity may refer to: *Electrical conductivity, a measure of a material's ability to conduct an electric current **Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution ** Ionic conductivity (solid state), ele ...

are specified. The permittivity of dispersive materials in tabular form cannot be directly substituted into the FDTD scheme. Instead, it can be approximated using multiple Debye, Drude, Lorentz or critical point terms. This approximation can be obtained using open fitting programs and does not necessarily have physical meaning. Once the computational domain and the grid materials are established, a source is specified. The source can be current on a wire, applied electric field or impinging plane wave. In the last case FDTD can be used to simulate light scattering from arbitrary shaped objects, planar periodic structures at various incident angles, and photonic band structure of infinite periodic structures. Since the E and H fields are determined directly, the output of the simulation is usually the E or H field at a point or a series of points within the computational domain. The simulation evolves the E and H fields forward in time. Processing may be done on the E and H fields returned by the simulation. Data processing may also occur while the simulation is ongoing. While the FDTD technique computes electromagnetic fields within a compact spatial region, scattered and/or radiated far fields can be obtained via near-to-far-field transformations.

Strengths of FDTD modeling

Every modeling technique has strengths and weaknesses, and the FDTD method is no different. * FDTD is a versatile modeling technique used to solve Maxwell's equations. It is intuitive, so users can easily understand how to use it and know what to expect from a given model. * FDTD is a time-domain technique, and when a broadband pulse (such as a Gaussian pulse) is used as the source, then the response of the system over a wide range of frequencies can be obtained with a single simulation. This is useful in applications where resonant frequencies are not exactly known, or anytime that a broadband result is desired. * Since FDTD calculates the E and H fields everywhere in the computational domain as they evolve in time, it lends itself to providing animated displays of the electromagnetic field movement through the model. This type of display is useful in understanding what is going on in the model, and to help ensure that the model is working correctly. * The FDTD technique allows the user to specify the material at all points within the computational domain. A wide variety of linear and nonlinear dielectric and magnetic materials can be naturally and easily modeled. * FDTD allows the effects of apertures to be determined directly. Shielding effects can be found, and the fields both inside and outside a structure can be found directly or indirectly. * FDTD uses the E and H fields directly. Since most EMI/EMC modeling applications are interested in the E and H fields, it is convenient that no conversions must be made after the simulation has run to get these values.

Weaknesses of FDTD modeling

* Since FDTD requires that the entire computational domain be gridded, and the grid spatial discretization must be sufficiently fine to resolve both the smallest electromagnetic wavelength and the smallest geometrical feature in the model, very large computational domains can be developed, which results in very long solution times. Models with long, thin features, (like wires) are difficult to model in FDTD because of the excessively large computational domain required. Methods such as eigenmode expansion can offer a more efficient alternative as they do not require a fine grid along the z-direction. * There is no way to determine unique values for permittivity and permeability at a material interface. * Space and time steps must satisfy the CFL condition, or the

leapfrog integration In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form \ddot x = \frac = A(x), or equivalently of the form \dot v = \frac = A(x), \;\dot x = \frac = v, particularly in the case of a d ...

used to solve the partial differential equation is likely to become unstable. * FDTD finds the E/H fields directly everywhere in the computational domain. If the field values at some distance are desired, it is likely that this distance will force the computational domain to be excessively large. Far-field extensions are available for FDTD, but require some amount of postprocessing. * Since FDTD simulations calculate the E and H fields at all points within the computational domain, the computational domain must be finite to permit its residence in the computer memory. In many cases this is achieved by inserting artificial boundaries into the simulation space. Care must be taken to minimize errors introduced by such boundaries. There are a number of available highly effective absorbing boundary conditions (ABCs) to simulate an infinite unbounded computational domain. Most modern FDTD implementations instead use a special absorbing "material", called a perfectly matched layer (PML) to implement absorbing boundaries. * Because FDTD is solved by propagating the fields forward in the time domain, the electromagnetic time response of the medium must be modeled explicitly. For an arbitrary response, this involves a computationally expensive time convolution, although in most cases the time response of the medium (or Dispersion (optics)) can be adequately and simply modeled using either the recursive convolution (RC) technique, the auxiliary differential equation (ADE) technique, or the Z-transform technique. An alternative way of solving

that can treat arbitrary dispersion easily is the pseudo-spectral spatial domain (PSSD), which instead propagates the fields forward in space.

Grid truncation techniques

The most commonly used grid truncation techniques for open-region FDTD modeling problems are the Mur absorbing boundary condition (ABC), the Liao ABC, and various perfectly matched layer (PML) formulations. The Mur and Liao techniques are simpler than PML. However, PML (which is technically an absorbing region rather than a boundary condition ''per se'') can provide orders-of-magnitude lower reflections. The PML concept was introduced by J.-P. Berenger in a seminal 1994 paper in the Journal of Computational Physics. Since 1994, Berenger's original split-field implementation has been modified and extended to the uniaxial PML (UPML), the convolutional PML (CPML), and the higher-order PML. The latter two PML formulations have increased ability to absorb evanescent waves, and therefore can in principle be placed closer to a simulated scattering or radiating structure than Berenger's original formulation. To reduce undesired numerical reflection from the PML additional back absorbing layers technique can be used.

Popularity

Notwithstanding both the general increase in academic publication throughput during the same period and the overall expansion of interest in all Computational electromagnetics (CEM) techniques, there are seven primary reasons for the tremendous expansion of interest in FDTD computational solution approaches for Maxwell's equations: # FDTD does not require a matrix inversion. Being a fully explicit computation, FDTD avoids the difficulties with matrix inversions that limit the size of frequency-domain integral-equation and finite-element electromagnetics models to generally fewer than 10⁹ electromagnetic field unknowns. FDTD models with as many as 10⁹ field unknowns have been run; there is no intrinsic upper bound to this number. # FDTD is accurate and robust. The sources of error in FDTD calculations are well understood, and can be bounded to permit accurate models for a very large variety of electromagnetic wave interaction problems. # FDTD treats impulsive behavior naturally. Being a time-domain technique, FDTD directly calculates the impulse response of an electromagnetic system. Therefore, a single FDTD simulation can provide either ultrawideband temporal waveforms or the sinusoidal steady-state response at any frequency within the excitation spectrum. # FDTD treats nonlinear behavior naturally. Being a time-domain technique, FDTD directly calculates the nonlinear response of an electromagnetic system. This allows natural hybriding of FDTD with sets of auxiliary differential equations that describe nonlinearities from either the classical or semi-classical standpoint. One research frontier is the development of hybrid algorithms which join FDTD classical electrodynamics models with phenomena arising from quantum electrodynamics, especially vacuum fluctuations, such as the

Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pr ...

.S. G. Johnson,
Numerical methods for computing Casimir interactions
" in Casimir Physics (D. Dalvit, P. Milonni, D. Roberts, and F. da Rosa, eds.), vol. 834 of ''Lecture Notes in Physics'', ch. 6, pp. 175–218, Berlin: Springer, June 2011. # FDTD is a systematic approach. With FDTD, specifying a new structure to be modeled is reduced to a problem of mesh generation rather than the potentially complex reformulation of an integral equation. For example, FDTD requires no calculation of structure-dependent Green functions. # Parallel-processing computer architectures have come to dominate supercomputing. FDTD scales with high efficiency on parallel-processing CPU-based computers, and extremely well on recently developed GPU-based accelerator technology. # Computer visualization capabilities are increasing rapidly. While this trend positively influences all numerical techniques, it is of particular advantage to FDTD methods, which generate time-marched arrays of field quantities suitable for use in color videos to illustrate the field dynamics. Taflove has argued that these factors combine to suggest that FDTD will remain one of the dominant computational electrodynamics techniques (as well as potentially other

multiphysics In computational modelling, multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the interactions among them. For example, simultaneous ...

problems).

Implementations

There are hundreds of simulation tools (e.g. OmniSim, XFdtd, Lumerical, CST Studio Suite, OptiFDTD etc.) that implement FDTD algorithms, many optimized to run on parallel-processing clusters.

References

External links

Free software Free software or libre software is computer software distributed under terms that allow users to run the software for any purpose as well as to study, change, and distribute it and any adapted versions. Free software is a matter of liberty, no ...

Open-source software Open-source software (OSS) is computer software that is released under a license in which the copyright holder grants users the rights to use, study, change, and distribute the software and its source code to anyone and for any purpose. Op ...

FDTD projects:
FDTD++
advanced, fully featured FDTD software, along with sophisticated material models and predefined fits as well as discussion/support forums and email support
openEMS
(Fully 3D Cartesian & Cylindrical graded mesh EC-FDTD Solver, written in C++, using a

Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...

/ Octave-Interface)
pFDTD
(3D C++ FDTD codes developed by Se-Heon Kim)
JFDTD
(2D/3D C++ FDTD codes developed for nanophotonics by Jeffrey M. McMahon)
WOLFSIM
(NCSU) (2-D)
Meep
(

MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...

, 2D/3D/cylindrical parallel FDTD)
(Geo-) Radar FDTD

bigboy
(unmaintained, no release files. must get source from cvs)
Parallel (MPI&OpenMP) FDTD codes in C++
(developed by Zs. Szabó)
FDTD code in Fortran 90

FDTD code in C for 2D EM Wave simulation

Angora
(3D parallel FDTD software package, maintained by Ilker R. Capoglu)
GSvit
(3D FDTD solver with graphics card computing support, written in C, graphical user interface XSvit available)
gprMax
(Open Source (GPLv3), 3D/2D FDTD modelling code in Python/Cython developed for GPR but can be used for general EM modelling.)

Freeware Freeware is software, most often proprietary, that is distributed at no monetary cost to the end user. There is no agreed-upon set of rights, license, or EULA that defines ''freeware'' unambiguously; every publisher defines its own rules for t ...

Closed source Proprietary software is software that is deemed within the free and open-source software to be non-free because its creator, publisher, or other rightsholder or rightsholder partner exercises a legal monopoly afforded by modern copyright and inte ...

FDTD projects (some not for commercial use):
EMTL (Electromagnetic Template Library)
(Free С++ library for electromagnetic simulations. The current version implements mainly the FDTD). {{DEFAULTSORT:Finite-Difference Time-Domain Method Numerical software Simulation software Electromagnetic radiation Numerical differential equations Computational science Computational electromagnetics Electromagnetism Electrodynamics Scattering, absorption and radiative transfer (optics)