View source for Method of moments (electromagnetics)

{{short description|Numerical method in computational electromagnetics}}
{{too technical|date=September 2021}}
{{for|the general integral equation method|Boundary element method}}
[[File:Negative reflection from meta-mirrors.jpg|thumb|325px|Simulation of [[negative refraction]] from a [[Electromagnetic metasurface|metasurface]] at 15 GHz for different angles of incidence. The simulations are performed through the method of moments.]]

The '''method of moments''' ('''MoM'''), also known as the '''moment method''' and '''method of weighted residuals''',{{sfn|Davidson|2005|p=7}} is a [[numerical analysis|numerical method]] in [[computational electromagnetics]]. It is used in computer programs that simulate the interaction of [[electromagnetic field]]s such as [[radio wave]]s with matter, for example [[computational electromagnetics|antenna simulation programs]] like [[Numerical Electromagnetics Code|NEC]] that calculate the [[radiation pattern]] of an antenna.  Generally being a [[frequency domain|frequency-domain]] method,{{efn|While the method is commonly formulated in frequency domain, time domain formulations (MoM-TD) have been reported in the literature.<ref name="orlandi-tdmom"/><ref name="Bretones-tdmom"/><ref name="firouzeh-tdmom"/>}} it involves the projection of an [[integral equation]] into a [[system of linear equations]] by the application of appropriate [[Boundary value problem|boundary conditions]]. This is done by using discrete [[Partition of an interval|meshes]] as in [[finite difference method|finite difference]] and [[finite element methods]], often for the surface. The solutions are represented with the linear combination of pre-defined [[basis function]]s; generally, the coefficients of these basis functions are the sought unknowns. [[Green's function]]s and [[Galerkin method]] play a central role in the method of moments.
 
For many applications, the method of moments is identical to the [[boundary element method]].{{efn|For surface-integral formulations, the method of moments and boundary element method are synonymous: the name "method of moments" is particularly used by the electromagnetics community. Nevertheless, certain volumetric formulations are also present in MoM.{{sfn|Davidson|2005|pages=7, 197–200}}}} It is one of the most common methods in [[microwave engineering|microwave]] and [[antenna (radio)|antenna]] engineering.

==History==
Development of [[boundary element method]] and other similar methods for different engineering applications is associated with the advent of [[History of computing hardware (1960s–present)|digital computing in the 1960s]].<ref name="cheng-history-bem"/> Prior to this, [[Calculus of variations|variational methods]] were applied to engineering problems at [[microwave]] frequencies by the time of [[World War II]].<ref name="mom-history-rh"/> While [[Julian Schwinger]] and [[Nathan Marcuvitz]] have respectively compiled these works into lecture notes and textbooks,<ref name="schwinger-notes"/><ref name="wg-handbook"/> [[Victor H. Rumsey|Victor Rumsey]] has formulated these methods into the "reaction concept" in 1954.<ref name="rumsey-reaction"/> The concept was later shown to be equivalent to the [[Galerkin method]].<ref name="mom-history-rh"/> In the late 1950s, an early version of the method of moments was introduced by [[Yuen Tze Lo|Yuen Lo]] at a course on mathematical methods in electromagnetic theory at [[University of Illinois Urbana-Champaign|University of Illinois]].<ref name="ytlo-memoriam"/>
[[File:Antena espiral con diagrama de radiación.png|thumb|A scheme and radiation pattern of a log-[[spiral antenna]], designed with a [[Numerical Electromagnetics Code|NEC]]-based modeling software]] 
In the 1960s, early research work on the method was published by [[Kenneth K. Mei|Kenneth Mei]], [[Jean van Bladel]]<ref name="mei-1963"/> and Jack Richmond.<ref name="richmond-1965"/> In the same decade, the systematic theory for the method of moments in electromagnetics was largely formalized by [[Roger F. Harrington|Roger Harrington]].<ref name="harrington-awardee"/> While the term "the method of moments" was coined earlier by [[Leonid Kantorovich]] and Gleb Akilov for analogous numerical applications,<ref name="medgyesi-94"/> Harrington has adapted the term for the electromagnetic formulation.<ref name="mom-history-rh"/> Harrington published the seminal textbook ''Field Computation by Moment Methods'' on the moment method in 1968.<ref name="harrington-awardee"/> The development of the method and its indications in [[radar]] and antenna engineering attracted interest; MoM research was subsequently supported [[Federal government of the United States|United States government]]. The method was further popularized by the introduction of generalized antenna modeling codes such as [[Numerical Electromagnetics Code]], which was released into [[public domain]] by the United States government in the late 1980s.{{sfn|Davidson|2005|p=8}}<ref name="nec-history"/> In the 1990s, introduction of [[fast multipole method|fast multipole]] and [[multilevel fast multipole method]]s enabled efficient MoM solutions to problems with millions of unknowns.{{sfn|Chew|Michielssen|Song|Jin|2001|pp=21–22}}<ref name="song97"/><ref name="fisc"/>

Being one of the most common simulation techniques in [[Radio-frequency engineering|RF]] and [[microwave engineering]], the method of moments forms the basis of many commercial design software such as [[FEKO]].{{sfn|Davidson|2005|pp=7–8}} Many non-commercial and public domain codes of different sophistications are also available.{{sfn|Balanis|2012|p=732}} In addition to its use in electrical engineering, the method of moments has been applied to light [[scattering]]<ref name="Lakhtakia1993"/> and [[Plasmonics|plasmonic]] problems.<ref name="kern2009"/><ref name="Taboada2011"/><ref name="Hohenester2012"/>

==Background==
===Basic concepts===
{{see also|Boundary element method}}
An inhomogeneous integral equation can be expressed as:
<math display="block">L(f) = g</math>
where {{math|''L''}} denotes a [[linear operator]], {{math|''g''}} denotes the known forcing function and {{math|''f''}} denotes the unknown function. {{math|''f''}} can be approximated by a finite number of [[basis function]]s (<math>f_n</math>):
<math display="block">f \approx \sum_n^N a_n f_n.</math>

By [[linearity]], substitution of this expression into the equation yields:
<math display="block">\sum_n^N a_n L(f_n) \approx g .</math>

We can also define a [[Residual (numerical analysis)|residual]] for this expression, which denotes the difference between the actual and the approximate solution:
<math display="block">R = \sum_n^N a_n L(f_n) - g</math>

The aim of the method of moments is to minimize this residual, which can be done by using appropriate weighting or testing functions, hence the name method of weighted residuals.{{sfn|Davidson|2005|pp=139–140}} After the determination of a suitable [[inner product]] for the problem, the expression then becomes:
<math display="block">\sum_n^N a_n \langle w_m, L(f_n) \rangle \approx \langle w_m, g \rangle</math>

Thus, the expression can be represented in the matrix form:
<math display="block">\left[\ell_{mn}\right] \left[\alpha_m\right] = [g_n]</math>

The resulting matrix is often referred as the impedance matrix.<ref name="yla2003"/> The coefficients of the basis functions can be obtained through [[Invertible matrix|inverting the matrix]].{{sfn|Harrington|1993|pp=5–9}} For large matrices with a large number of unknowns, [[iterative method]]s such as [[conjugate gradient method]] can be used for [[Series acceleration|acceleration]].{{sfn|Gibson|2021|pp=68-77}} The actual field distributions can be obtained from the coefficients and the associated integrals.{{sfn|Balanis|2012|p=679}} The interactions between each basis function in MoM is ensured by [[Green's function]] of the system.{{sfn|Gibson|2021|p=18}}

===Basis and testing functions===
[[File:Rooftop basis function for MoM.png|thumb|Interpolation of function with rooftop basis functions|270px]]
Different basis functions can be chosen to model the expected behavior of the unknown function in the domain; these functions can either be subsectional or global.{{sfn|Gibson|2021|pp=43–44}} Choice of [[Dirac delta function]] as basis function is known as point-matching or [[collocation method|collocation]]. This corresponds to enforcing the boundary conditions on <math>N</math> discrete points and is often used to obtain approximate solutions when the inner product operation is cumbersome to perform.{{sfn|Harrington|1993|pp=9–10}}{{sfn|Davidson|2005|p=123}} Other subsectional basis functions include [[Rectangular function|pulse]], piecewise triangular, piecewise sinusoidal and rooftop functions.{{sfn|Gibson|2021|pp=43–44}} Triangular patches, introduced by S. Rao, [[Donald R. Wilton|D. Wilton]] and A. Glisson in 1982,<ref name="rwg"/> are known as RWG basis functions and are widely used in MoM.{{sfn|Davidson|2005|pp=186–187}} Characteristic basis functions were also introduced to accelerate computation and reduce the matrix equation.<ref name="prakash03"/><ref name="maaskant08"/>

The testing and basis functions are often chosen to be the same; this is known as the [[Galerkin method]].{{sfn|Harrington|1993|pp=5–9}} Depending on the application and studied structure, the testing and basis functions should be chosen appropriately to ensure convergence and accuracy, as well as to prevent possible high order algebraic [[Singularity (mathematics)|singularities]].<ref name="aksun93"/>

==Integral equations==
Depending on the application and sought variables, different integral or [[integro-differential equation]]s are used in MoM. Radiation and scattering by thin wire structures, such as many types of antennas, can be modeled by specialized equations.{{sfn|Gibson|2021|p=81}} For surface problems, common integral equation formulations include electric field integral equation (EFIE), magnetic field integral equation (MFIE){{sfn|Davidson|2005|pp=184–186}} and mixed-potential integral equation (MPIE).{{sfn|Kinayman|Aksun|2005|p=311}}

===Thin-wire equations===
As many antenna structures can be approximated as wires, thin wire equations are of interest in MoM applications. Two commonly used thin-wire equations are Pocklington and Hallén integro-differential equations.{{sfn|Gibson|2021|p=86-93}} Pocklington's equation precedes the computational techniques, having been introduced in 1897 by [[Henry Cabourn Pocklington]].<ref name="pocklington1897"/> For a linear wire that is centered on the origin and aligned with the z-axis, the equation can be written as:
<math display="block">\int^{l/2}_{-l/2} I_z(z') \left[ \left(\frac{d^2}{dz^2}+\beta^2 \right) G(z,z') \right]\,dz'=-j \omega \varepsilon E^\text{inc}_z(p=a)</math>
where <math>l</math> and <math>a</math> denote the total length and thickness, respectively. <math>G(z,z')</math> is the Green's function for free space. The equation can be generalized to different excitation schemes, including [[Magnetic current|magnetic frills]].{{sfn|Balanis|2012|p=442}}

Hallén integral equation, published by E. Hallén in 1938,<ref name="hallen1938"/> can be given as:
<math display="block"> \left(\frac{d^2}{dz^2} + \beta^2 \right) \int^{l/2}_{-l/2} I_z(z') G(z,z')\,dz' = -j \omega \varepsilon E^\text{inc}_z(p=a)</math>

This equation, despite being more well-behaved than the Pocklington's equation,{{sfn|Gibson|2021|p=91-93}} is generally restricted to the delta-gap voltage excitations at the [[antenna feed]] point, which can be represented as an impressed electric field.{{sfn|Balanis|2012|p=442}}

===Electric field integral equation (EFIE)===
{{main|Electric field integral equation}}
The general form of electric field integral equation (EFIE) can be written as:
<math display="block">\hat\mathbf{n} \times \mathbf{E}^\text{inc}(\mathbf{r}) = \hat\mathbf{n} \times \int_S \left[ \eta j k \, \mathbf{J}(\mathbf{r}') G(\mathbf{r},\mathbf{r}') + \frac{\eta}{jk} \left\{\boldsymbol\nabla_s' \cdot \mathbf{J}(\mathbf{r}') \right\} \boldsymbol\nabla' G(\mathbf{r},\mathbf{r}') \right] \, dS'</math>
where <math>\mathbf{E}_\text{inc}</math> is the incident or impressed electric field. <math>G(r,r')</math> is the Green function for [[Helmholtz equation]] and <math>\eta</math> represents the [[wave impedance]]. The boundary conditions are met at a defined [[perfect conductor|PEC]] surface. EFIE is a [[Fredholm integral equation]] of the first kind.{{sfn|Davidson|2005|pp=184–186}}

===Magnetic field integral equation (MFIE)===
Another commonly used integral equation in MoM is the magnetic field integral equation (MFIE), which can be written as:
<math display="block">-\frac{1}{2} \mathbf{J}(r) + \hat\mathbf{n} \times \oint_S \mathbf{J}(r') \times \boldsymbol\nabla' G(r,r')\,dS' = \hat\mathbf{n} \times \mathbf{H}_\text{inc}(r)</math>

MFIE is often formulated to be a Fredholm integral equation of the second kind and is generally [[Well-posed problem|well-posed]]. Nevertheless, the formulation necessitates the use of closed surfaces, which limits its applications.{{sfn|Davidson|2005|pp=184–186}}

===Other formulations===
Many different surface and volume integral formulations for MoM exist. In many cases, EFIEs are converted to mixed potential integral equations (MFIE) through the use of [[Lorenz gauge condition]]; this aims to reduce the orders of singularities through the use of [[Magnetic vector potential|magnetic vector]] and [[Electric potential|scalar electric potentials]].{{sfn|Kinayman|Aksun|2005|p=281}}<ref name="michalski97"/> In order to bypass the internal resonance problem in dielectric scattering calculations, combined-field integral equation (CFIE) and Poggio—Miller—Chang—Harrington—Wu—Tsai (PMCHWT) formulations are also used.{{sfn|Chew|Michielssen|Song|Jin|2001|pp=15–18}} Another approach, the volumetric integral equation, necessitates the discretization of the volume elements and is often computationally expensive.{{sfn|Davidson|2005|pp=197–199}}

MoM can also be integrated with [[physical optics]] theory{{sfn|Davidson|2005|pp=202–208}} and [[finite element method]].<ref name="ilic-2009"/>

==Green's functions==
{{main|Green's function}}
[[File:Microstrip scheme.svg|thumb|275px|A [[microstrip]] scheme. MoM analysis of such layered structures necessitates the derivation of appropriate Green's functions.]]
Appropriate Green's function for the studied structure must be known to formulate MoM matrices: automatic incorporation of the [[Sommerfeld radiation condition|radiation condition]] into the Green's function makes MoM particularly useful for radiation and scattering problems. Even though the Green function can be derived in closed form for very simple cases, more complex structures necessitate numerical derivation of these functions.{{sfn|Davidson|2005|p=8-10}} 

Full wave analysis of planarly-stratified structures in particular, such as [[microstrip]]s or [[patch antenna]]s, necessitate the derivation of Green's functions that are peculiar to these geometries.<ref name="michalski97"/>{{sfn|Kinayman|Aksun|2005|p=278}} This can be achieved in two different methods. In the first method, known as spectral-domain approach, the inner products and convolution operation for MoM matrix entries are evaluated in the [[Fourier transform|Fourier space]] with analytically-derived spectral-domain Green's functions through [[Parseval's theorem]].<ref name="itoh1973"/><ref name="itoh1980"/><ref name="das1987"/> The other approach is based on the use of spatial-domain Green's functions. This involves the inverse [[Hankel transform]] of the spectral-domain Green's function, which is defined on the Sommerfeld integration path. Nevertheless, this integral cannot be evaluated analytically, and its [[Numerical analysis|numerical evaluation]] is often computationally expensive due to the oscillatory [[integral kernel|kernels]] and slowly-converging nature of the integral.{{sfn|Kinayman|Aksun|2005|p=274}} Common approaches for evaluating these integrals include tail extrapolation approaches such as weighted-averages method.<ref name="michalski2016"/> 

Other approaches include the approximation of the integral kernel. Following the extraction of [[Quasistatic approximation|quasi-static]] and [[surface wave|surface pole]] components, these integrals can be approximated as closed-form complex exponentials through [[Prony's method]] or [[generalized pencil-of-function method]]; thus, the spatial Green's functions can be derived through the use of appropriate identities such as [[Sommerfeld identity]].<ref name="chow91"/><ref name="aksun95"/><ref name="alparslan10"/> This method is known in the computational electromagnetics literature as the discrete complex image method (DCIM), since the Green's function is effectively approximated with a discrete number of [[Method of image charges|image dipoles]] that are located within a [[Complex number|complex]] distance from the origin.<ref name="ling2000"/> The associated Green's functions are referred as closed-form Green's functions.<ref name="aksun95"/><ref name="alparslan10"/> The method has also been extended for cylindrically-layered structures.<ref name="tokgoz2000"/> 

Rational-function fitting method,<ref name="okhmatovski04"/><ref name="kourkoulos06"/> as well as its combinations with DCIM,<ref name="alparslan10"/> can also be used to approximate closed-form Green's functions. Alternatively, the closed-form Green's function can be evaluated through [[method of steepest descent]].<ref name="cui1999"/> For the [[Periodic function|periodic]] structures such as [[phased array]]s and [[frequency selective surface]]s, [[series acceleration]] methods such as [[Kummer's transformation of series|Kummer's transformation]] and [[Ewald summation]] is often used to accelerate the computation of the periodic Green's function.<ref name="capolino2005"/><ref name="kinayman1995"/>

==See also==
*[[Boundary element method]]
*[[Characteristic mode analysis]]
*[[Discrete dipole approximation]]
*[[Fast multipole method]]
*[[Finite element method]]
*[[Multilevel fast multipole method]]

==Notes==
{{notelist}}

==References==
{{Reflist|30em|refs=

<ref name="aksun93">{{cite journal |last1=Aksun |first1=M. I. |last2=Mittra |first2=R. |title=Choices of expansion and testing functions for the method of moments applied to a class of electromagnetic problems |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=March 1993 |volume=41 |issue=3 |pages=503–509 |doi=10.1109/22.223752 |bibcode=1993ITMTT..41..503A |hdl=11693/10851 |author-link1=İrşadi Aksun |author-link2=Raj Mittra|hdl-access=free }}</ref>

<ref name="aksun95">{{cite journal |last1=Aksun |first1=M. I. |title=A robust approach for the derivation of closed-form Green's functions |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=May 1996 |volume=44 |issue=5 |pages=651–658 |doi=10.1109/22.493917 |bibcode=1996ITMTT..44..651A |hdl=11693/10779 |author-link1=İrşadi Aksun|hdl-access=free }}</ref>

<ref name="alparslan10">{{cite journal |last1=Alparslan |first1=A. |last2=Aksun |first2=M. I. |last3=Michalski |first3=K. A. |title=Closed-Form Green's Functions in Planar Layered Media for All Ranges and Materials |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=February 2010 |volume=58 |issue=3 |pages=602–613 |doi=10.1109/TMTT.2010.2040354 |bibcode=2010ITMTT..58..602A |s2cid=15018213 |author-link2=İrşadi Aksun}}</ref>

<ref name="Bretones-tdmom">{{cite journal |last1=Bretones |first1=A.R. |last2=Mittra |first2=R. |last3=Martin |first3=R. G. |title=A hybrid technique combining the method of moments in the time domain and FDTD |journal=IEEE Microwave and Guided Wave Letters |date=August 1998 |volume=8 |issue=8 |pages=281–283 |doi=10.1109/75.704414 |author-link2=Raj Mittra}}</ref>

<ref name="capolino2005">{{cite journal |last1=Capolino |first1=F. Capolino |last2=Wilton |first2=D. R. |last3=Johnson |first3=W. A. |title=Efficient computation of the 2-D Green's function for 1-D periodic structures using the Ewald method |journal=[[IEEE Transactions on Antennas and Propagation ]] |date=September 2005 |volume=53 |issue=9 |pages=2977–2984 |doi=10.1109/TAP.2005.854556|bibcode=2005ITAP...53.2977C |s2cid=32502117 |url=https://escholarship.org/uc/item/6vd3x1c2 |author-link2=Donald R. Wilton}}</ref>

<ref name="cheng-history-bem">{{cite journal |last1=-D. Cheng |first1=Alexander H. |last2=Cheng |first2=Daisy T. |title=Heritage and early history of the boundary element method |journal=Engineering Analysis with Boundary Elements |date=March 2005 |volume=29 |issue=3 |pages=268–302 |doi=10.1016/j.enganabound.2004.12.001}}</ref>

<ref name="chow91">{{cite journal |last1=Chow |first1=Y. L. |last2=Yang |first2=J. J. |last3=Fang |first3=D. G. |last4=Howard |first4=G. E. |title=A closed-form spatial Green's function for the thick microstrip substrate |journal=[[ IEEE Transactions on Microwave Theory and Techniques]] |date=March 1991 |volume=39 |issue=3 |pages=588–592 |doi=10.1109/22.75309|bibcode=1991ITMTT..39..588C }}</ref>

<ref name="cui1999">{{cite journal |last1=Cui |first1=Tie Jun |last2=Chew |first2=Weng Cho |title=Fast evaluation of Sommerfeld integrals for EM scattering and radiation by three-dimensional buried objects |journal=IEEE Transactions on Geoscience and Remote Sensing |date=March 1999 |volume=37 |issue=2 |pages=887–900 |doi=10.1109/36.752208 |bibcode=1999ITGRS..37..887C |author-link2=Weng Cho Chew}}</ref>

<ref name="das1987">{{cite journal |last1=Das |first1=N. K. |last2=Pozar |first2=D. M. |author-link1=David Pozar |title=A Generalized Spectral-Domain Green's Function for Multilayer Dielectric Substrates with Application to Multilayer Transmission Lines |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=March 1987 |volume=35 |issue=3 |pages=326-335 |doi=10.1109/TMTT.1987.1133646}}</ref>

<ref name="firouzeh-tdmom">{{cite journal |last1=Firouzeh |first1=Z. H. |last2=Moini |first2=R. |last3=Sadeghi |first3=S. H. H. |last4=Faraji-Dana |first4=R. |last5=Vandenbosch |first5=G. A. E. |display-authors=3 |title=A new robust technique for transient analysis of conducting cylinders – TM case |journal=Proceedings of the 5th European Conference on Antennas and Propagation |date=April 2011}}</ref>

<ref name="fisc">{{cite journal |last1=Song |first1=J. M. |last2=Lu |first2=C. C. |last3=Chew |first3=W. C. |last4=Lee |first4=S. W. |title=Fast Illinois solver code (FISC) |journal=[[IEEE Antennas and Propagation Magazine]] |date=1998 |volume=40 |issue=3 |pages=27–34 |doi=10.1109/74.706067 |author-link3=Weng Cho Chew}}</ref>

<ref name="hallen1938">{{cite journal |last1=Hallén |first1=E. |title=Theoretical investigations into the transmitting and receiving qualities of antennae |journal=Nova Acta Upsal. |year=1938 |volume=4 |issue=11 |pages=1–44}}</ref>

<ref name="harrington-awardee">{{cite book|first1=Donald R. |last1=Wilton |first2=Ercument |last2=Arvas |first3=Chalmers M. |last3=Butler |first4=Joseph R. |last4=Mautz|title=2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting |chapter=Roger F. Harrington, 1989 IEEE AP-S Distinguished Achievement awardee |doi=10.1109/APUSNCURSINRSM.2017.8072371|date=19 October 2017|pages=657–658 |isbn=978-1-5386-3284-0 |s2cid=1484406 |author-link1=Donald R. Wilton}}</ref>

<ref name="Hohenester2012">{{cite journal |last1=Hohenester |first1=Ulrich |last2=Trügler |first2=Andreas |title=MNPBEM – A Matlab toolbox for the simulation of plasmonic nanoparticles |journal=[[Computer Physics Communications]] |date=February 2012 |volume=183 |issue=2 |pages=370–381 |doi=10.1016/j.cpc.2011.09.009|arxiv=1109.5783 |bibcode=2012CoPhC.183..370H |s2cid=17489164 }}</ref>

<ref name="ilic-2009">{{cite journal |last1=Ilic |first1=Milan M. |last2=Djordjevic |first2=Miroslav |last3=Ilic |first3=Andjelija Z. |last4=Notaros |first4=Branislav M. | author-link4=Branislav Notaros |title=Higher Order Hybrid FEM-MoM Technique for Analysis of Antennas and Scatterers |journal=[[IEEE Transactions on Antennas and Propagation]] |date=May 2009 |volume=57 |issue=5 |pages=1452–1460 |doi=10.1109/TAP.2009.2016725|bibcode=2009ITAP...57.1452I |s2cid=31273380 }}</ref>

<ref name="itoh1973">{{cite journal |last1=Itoh |first1=T. |last2=Mittra |first2=R. |title=Spectral-Domain Approach for Calculating the Dispersion Characteristics of Microstrip Lines |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=July 1973 |volume=21 |issue=7 |pages=496-499 |doi=10.1109/TMTT.1973.1128044 |author-link1=Tatsuo Itoh |author-link2=Raj Mittra}}</ref>

<ref name="itoh1980">{{cite journal |last1=Itoh |first1=T. |title=Spectral Domain Immitance Approach for Dispersion Characteristics of Generalized Printed Transmission Lines |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=July 1980 |volume=28 |issue=7 |pages=733-736 |doi=10.1109/TMTT.1980.1130158 |author-link1=Tatsuo Itoh }}</ref>

<ref name="kern2009">{{cite journal |last1=Kern |first1=Andreas M. |last2=Martin |first2=Olivier J. F. |title=Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures |journal=[[Journal of the Optical Society of America A]] |year=2009 |volume=26 |issue=4 |pages=732–740 |doi=10.1364/JOSAA.26.000732|pmid=19340246 |bibcode=2009JOSAA..26..732K |url=http://infoscience.epfl.ch/record/164808 }}</ref>

<ref name="kinayman1995">{{cite journal |last1=Kinayman |first1=Noyan |last2=Aksun |first2=M. I. |title=Comparative study of acceleration techniques for integrals and series in electromagnetic problems |journal=[[Radio Science]] |date=1995 |volume=30 |issue=6 |pages=1713-1722 |doi=10.1029/95RS02060|hdl=11693/48408 |hdl-access=free }}</ref>

<ref name="kourkoulos06">{{cite journal |last1=Kourkoulos |first1=V. N. |last2=Cangellaris |first2=A. C. |title=Accurate approximation of Green's functions in planar stratified media in terms of a finite sum of spherical and cylindrical waves |journal=[[IEEE Transactions on Antennas and Propagation]] |date=May 2006 |volume=54 |issue=5 |pages=1568–1576 |doi=10.1109/TAP.2006.874329|bibcode=2006ITAP...54.1568K |s2cid=24691332 }}</ref>

<ref name="Lakhtakia1993">{{cite journal |last1=Lakhtakia |first1=Akhlesh |last2=Mulholland |first2=George W. |author-link1=Akhlesh Lakhtakia |title=On Two Numerical Techniques for Light Scattering by Dielectric Agglomerated Structures |journal=[[Journal of Research of the National Institute of Standards and Technology]] |year=1993 |volume=98 |issue=6 |pages=699–716 |doi=10.6028/jres.098.046|pmid=28053494 |pmc=4922401 }}</ref>

<ref name="ling2000">{{cite journal |last1=Ling |first1=Feng |last2=Jin |first2=Jian-Ming |title=Discrete complex image method for Green's functions of general multilayer media |journal=[[IEEE Microwave and Wireless Components Letters|IEEE Microwave and Guided Wave Letters]] |date=October 2000 |volume=10 |issue=10 |pages=400–402 |doi=10.1109/75.877225}}</ref>

<ref name="maaskant08">{{cite journal |last1=Maaskant |first1=Rob |last2=Mittra |first2=Raj |last3=Tijhuis |first3=Anton |title=Fast Analysis of Large Antenna Arrays Using the Characteristic Basis Function Method and the Adaptive Cross Approximation Algorithm |journal=[[IEEE Transactions on Antennas and Propagation]] |date=November 2008 |volume=56 |issue=11 |pages=3440–3451 |doi=10.1109/TAP.2008.2005471 |bibcode=2008ITAP...56.3440M |s2cid=8654158 |author-link2=Raj Mittra}}</ref>

<ref name="medgyesi-94">{{cite journal |last1=Medgyesi-Mitschang |first1=L. N. |last2=Putnam |first2=J. M. |last3=Gedera |first3=M. B. |title=Generalized the method of moments for three-dimensional penetrable scatterers |journal=[[Journal of the Optical Society of America A]] |year=1994 |volume=11 |issue=4 |pages=1383–1398 |doi=10.1364/JOSAA.11.001383|bibcode=1994JOSAA..11.1383M }}</ref>

<ref name="mei-1963">{{cite journal |last1=Mei |first1=K. |last2=Van Bladel |first2=J. |title=Scattering by perfectly-conducting rectangular cylinders |journal=[[IEEE Transactions on Antennas and Propagation]] |date=March 1963 |volume=11 |issue=2 |pages=185–192 |doi=10.1109/TAP.1963.1137996|bibcode=1963ITAP...11..185M }}</ref>

<ref name="michalski97">{{cite journal |last1=Michalski |first1=K. A. |last2=Mosig |first2=J. R. |title=Multilayered Media Green's Functions in Integral Equation Formulations |journal=[[IEEE Transactions on Antennas and Propagation]] |date=March 1997 |volume=45 |issue=3 |pages=508–519 |doi=10.1109/8.558666|bibcode=1997ITAP...45..508M }}</ref>

<ref name="michalski2016">{{cite journal |last1=Michalski |first1=Krzysztof A. |last2=Mosig |first2=Juan R. |title=Efficient computation of Sommerfeld integral tails – methods and algorithms |journal=[[Journal of Electromagnetic Waves and Applications]] |date=2016 |volume=30 |issue=3 |pages=281-317 |doi=10.1080/09205071.2015.1129915|doi-access=free }}</ref>

<ref name="mom-history-rh">{{cite journal |last1=Harrington |first1=R. |author-link=Roger F. Harrington |title=Origin and development of the method of moments for field computation |journal=IEEE Antennas and Propagation Magazine |date=June 1990 |volume=32 |issue=3 |pages=31–35 |doi=10.1109/74.80522|bibcode=1990IAPM...32...31H |s2cid=46584485 }}</ref>

<ref name="nec-history">{{cite book |last1=Burke |first1=G. J. |last2=Miller |first2=E. K. |last3=Poggio |first3=A. J. |title=IEEE Antennas and Propagation Society Symposium, 2004 |chapter=The Numerical Electromagnetics Code (NEC) - a brief history |date=June 2004 |pages=2871-2874 Vol.3 |doi=10.1109/APS.2004.1331976|osti=891397 |isbn=0-7803-8302-8 |s2cid=24849672 |chapter-url=https://digital.library.unt.edu/ark:/67531/metadc882837/ }}</ref>

<ref name="okhmatovski04">{{cite journal |last1=Okhmatovski |first1=V. I. |last2=Cangellaris |first2=A. C. |title=Evaluation of layered media Green's functions via rational function fitting |journal=[[IEEE Microwave and Wireless Components Letters]] |date=February 2004 |volume=14 |issue=1 |pages=22–24 |doi=10.1109/LMWC.2003.821492|s2cid=12104884 }}</ref>

<ref name="orlandi-tdmom">{{cite journal |last1=Orlandi |first1=A. |title=Lightning induced transient voltages in presence of complex structures and nonlinear loads |journal=[[IEEE Transactions on Electromagnetic Compatibility]] |date=May 1996 |volume=38 |issue=2 |pages=150–155 |doi=10.1109/15.494617}}</ref>

<ref name="pocklington1897">{{cite journal
  | last1  = Pocklington | first1 = Henry C. | authorlink1=H. C. Pocklington
  | title = Electrical oscillations in wires
  | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
  | volume = 9 
  | pages = 324–332
  | date = October 25, 1897
  | url = http://www.teslacollection.com/tesla_articles/1897/cambridge_philosophical_society_proceedings/h_c_pocklington/electrical_oscillations_in_wires
  | accessdate = 7 December 2024}}</ref>

<ref name="prakash03">{{cite journal |last1=Prakash |first1=V. V. S. |last2=Mittra |first2=Raj |title=Characteristic basis function method: A new technique for efficient solution of the method of moments matrix equations |journal=[[Microwave and Optical Technology Letters]] |year=2003 |volume=36 |issue=2 |pages=95–100 |doi=10.1002/mop.10685 |s2cid=121106998 |author-link2=Raj Mittra}}</ref>

<ref name="richmond-1965">{{cite journal |last1=Richmond |first1=J. H. |title=Digital computer solutions of the rigorous equations for scattering problems |journal=[[Proceedings of the IEEE]] |date=August 1965 |volume=53 |issue=8 |pages=796–804 |doi=10.1109/PROC.1965.4057}}</ref>

<ref name="rumsey-reaction">{{cite journal |last1=Rumsey |first1=V. H. |author1-link=Victor H. Rumsey |title=Reaction Concept in Electromagnetic Theory |journal=[[Physical Review]] |date=June 1954 |volume=94 |issue=6 |page=1483 |doi=10.1103/PhysRev.94.1483|bibcode=1954PhRv...94.1483R }}</ref>

<ref name="rwg">{{cite journal |last1=Rao |first1=S. |last2=Wilton |first2=D. |last3=Glisson |first3=A. |title=Electromagnetic scattering by surfaces of arbitrary shape |journal=[[IEEE Transactions on Antennas and Propagation ]] |date=May 1982 |volume=30 |issue=3 |pages=409–418 |doi=10.1109/TAP.1982.1142818|bibcode=1982ITAP...30..409R |author-link2=Donald R. Wilton}}</ref>

<ref name="schwinger-notes">{{cite book |last1=Saxon |first1=David S. |author1-link=David S. Saxon |title=Notes on Lectures by Julian Schwinger: Discontinuities in Waveguides |date=1945 |publisher=[[Massachusetts Institute of Technology]]}}</ref>

<ref name="song97">{{cite journal |last1=Song |first1=J. |last2=Lu |first2=Cai-Cheng |last3=Chew |first3=Weng Cho |title=Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects |journal=[[IEEE Transactions on Antennas and Propagation]] |date=October 1997 |volume=45 |issue=10 |pages=1488–1493 |doi=10.1109/8.633855 |bibcode=1997ITAP...45.1488S |author-link3=Weng Cho Chew}}</ref>

<ref name="Taboada2011">{{cite journal |last1=Taboada |first1=José M. |last2=Rivero |first2=Javier |last3=Obelleiro |first3=Fernando |last4=Araújo |first4=Marta G. |last5=Landesa |first5=Luis |title=Method-of-moments formulation for the analysis of plasmonic nano-optical antennas |journal=[[Journal of the Optical Society of America A]] |year=2011 |volume=28 |issue=7 |pages=1341–1348 |doi=10.1364/JOSAA.28.001341|pmid=21734731 |bibcode=2011JOSAA..28.1341T }}</ref>

<ref name="tokgoz2000">{{cite journal |last1=Tokgoz |first1=C. |last2=Dural |first2=G. |title=Closed-form Green's functions for cylindrically stratified media |journal=[[IEEE Transactions on Microwave Theory and Techniques]] |date=January 2000 |volume=48 |issue=1 |pages=40–49 |doi=10.1109/22.817470|bibcode=2000ITMTT..48...40T }}</ref>

<ref name="wg-handbook">{{cite book |last1=Marcuvitz |first1=Nathan |author1-link=Nathan Marcuvitz |title=Waveguide Handbook |date=1951 |publisher=McGraw-Hill |isbn=978-0863410581}}</ref>

<ref name="yla2003">{{cite journal |last1=Yla-Oijala |first1=P. |last2=Taskinen |first2=M. |title=Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions |journal=[[IEEE Transactions on Antennas and Propagation]] |date=August 2003 |volume=51 |issue=8 |pages=1837–1846 |doi=10.1109/TAP.2003.814745}}</ref>

<ref name="ytlo-memoriam">{{cite journal |first1=Weng Cho |last1=Chew |author-link1=Weng Cho Chew |first2=Shun-Lien |last2=Chuang |author-link2=Shun Lien Chuang|first3= Jian-Ming |last3=Jin  |first4=Shung-Wu |last4=Lee |display-authors=3 |title=In memoriam: Yuen-Tze Lo |journal=IEEE Antennas and Propagation Magazine |date=August 2002 |volume=44 |issue=4 |pages=82–83 |doi=10.1109/MAP.2002.1043152|bibcode=2002IAPM...44...82. }}</ref>

}}
;Bibliography
*{{cite book |last1=Balanis |first1=Constantine A. |author1-link=Constantine A. Balanis |title=Advanced Engineering Electromagnetics |date=2012 |publisher=Wiley |isbn=978-0-470-58948-9 |edition=2}}
*{{cite book |title=Fast and Efficient Algorithms in Computational Electromagnetics |date=2001 |publisher=Artech House |isbn=9781580531528 |editor1-last=Chew |editor1-first=W. C. |editor2-last=Michielssen |editor2-first=E.|editor3-last=Song |editor3-first=J. M.|editor4-last=Jin |editor4-first=J. M.|editor1-link=Weng Cho Chew }}
*{{cite book |last1=Davidson |first1=David B. |title=Computational Electromagnetics for RF and Microwave Engineering |date=2005 |publisher=Cambridge University Press |isbn=978-0-521-83859-7 | author-link=David Bruce Davidson}}
*{{cite book |last1=Gibson |first1=Walton C. |title=The Method of Moments in Electromagnetics |date=2021 |publisher=Chapman & Hall |isbn= 9780367365066|edition=3rd}}
*{{cite book |last=Harrington |first=Roger F. |date=1993 |title= Field Computation by Moment Methods |publisher=IEEE Press |isbn= 9780470544631 |author-link=Roger F. Harrington}}
* {{cite book |last1=Kinayman |first1=Noyan |last2=Aksun |first2=M. I. |author-link2=İrşadi Aksun |date= 2005 |title= Modern Microwave Circuits |location=Norwood |publisher=[[Artech House]] |isbn= 9781844073832}}

{{Numerical PDE}}
[[Category:Computational electromagnetics]]
[[Category:Numerical differential equations]]