Methods for the solution of open-boundaryelectromagnetic-field problems

C.R.I. Emson, BSc, DipEng, MSc, PhD

Indexing terms: Electromagnetic theory, Mathematical techniques, Algorithms, Magnetic fields

Abstract: A review of methods suitable for thesolution of open-boundary field problems is given.Emphasis is placed on algorithms that can be usedin conjunction with the finite-element method.The requirements for any algorithm to model anexterior region is outlined, as well as a descriptionof some of the implications of the possiblemethods. It is hoped that users of such models willthen appreciate the assumptions made in anymethod and thus its limitations, and so be able tochoose the most appropriate one for a particularapplication.

1 Introduction

In the analysis of electromagnetic-field problems, thetechnique of finite elements has become increasinglypopular, and is probably the most commonly usedmethod for solving all but the most trivial problems.However, the technique does suffer the disadvantage that,in general, only a finite number of elements can be usedto discretise a given problem, owing to the finite amountof computer storage available.

Unfortunately, it is a feature of electromagnetic fieldsthat the extent of the problem being solved can be infin-ite. There is then a contradiction in requirements whensolving such problems — the need for an infinite numberof finite elements to model an infinite space, but only theresources to allow a finite (although these days, onmodern computers, a very large) number of finite ele-ments.

It has therefore become common practice, when usingthe finite-element method, to truncate the mesh at somefinite distance from the centre of the problem. As thisartificial boundary is made more remote, its effect dimin-ishes, until the error thus created can be regarded asbeing part of the 'noise' in the solution. Although thisrequires a large number, of elements, it does mean that asolution can be obtained, theoretically to within anydesired accuracy.

The aim of this paper is to describe alternative pro-cedures currently available for dealing with exteriorregions, all of which could prove more efficient thansimple truncation. A description will also be given ofsome of the implications of using such procedures.

Paper 5822A (S8), first received 4th September 1986 and in revised form28th July 1987The author is with the SERC, Rutherford Appleton Laboratory,Chilton, Didcot, Oxfordshire OX 11 OQX, United Kingdom

2 Types of problem to be modelled

Before attempting to deal with models for the infiniteexterior domain, it is first necessary to examine the typesof problem that can be solved. The two cases of staticfields and travelling waves are quite different, and will bediscussed separately. Transient problems will also bebriefly mentioned.

2.1 Static fieldsFor any well posed problem, the fields associated withany sources in the problem should decay to zero withdistance from those sources. This implies that the energyof the system is finite, in other words there are no sourcespositioned at infinity. If there were to be sources at infin-ity, the field would not decay to zero, the value beingunknown, and the energy of the system would be infinite.The problem would then not be well posed, i.e. not aphysically possible problem.

For 3-dimensional problems, there is usually no diffi-culty in identifying well posed problems — all sourcesshould be within the finite-element mesh, leaving theexterior source free (including values at infinity).

In two dimensions, the situation is less straightfor-ward. Considering the field due to a single current-carrying conductor, it would at first sight be thoughtreasonable to assume the field decays to zero away fromthe conductor in the x- and ^-directions in Fig. 1.

Fig. 1 Current carrying conductor (current in z-direction)

However, by stating that the problem is 2-dimensional (inthe xy-plane), there is the assumption of no variation inthe z-direction. There is therefore a current existing at

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z = + oo (the current return path). From the above argu-ment, this problem is not well posed, and as such theexterior cannot be modelled correctly.

Consider next the situation in Fig. 2, with a pair ofconductors with currents in opposing directions. This

Fig. 2 Pair of current carrying conductors (currents in z-directiori)

time the currents are balanced, with no return pathrequired at infinity. With no sources at infinity, theproblem is well posed, and the exterior can be modelledby one of the techniques to be described.

In a 2-dimensional analysis of an infinite problemtherefore, there must only appear balanced sources (nonet charge or current), in order that the exterior be mod-elled correctly by some special technique.

It should be mentioned however that a uniform inci-dent field is an exceptuon to this statement, as there is aninfinite energy implied in the system (there must besources at infinity to create the incident field). In thiscase, since the field at infinity is known, it can be imposedas a boundary condition; although a nonphysicalproblem it can be solved.

2.2 Travelling wavesThe situation is somewhat different with travelling-waveproblems, as the problem to be solved can often beseparated into two parts. First, it is assumed that there issome form of wave incident on the region of interest(within the finite-element mesh). This implies sources atinfinity to generate the incoming wave. Secondly there isa scattered wave travelling outwards, and it is this wavethat can be modelled by some special technique, becauseits amplitude must decay to zero at infinity.

2.3 Transient fieldsThe same arguments as above apply equally well to tran-sient problems. For example, it is common practice tomove the outer boundary far enough away, such that anywaves reflected back into the interior will not reach theregion of interest until after the time scale being exam-ined.

tages. The following is a list of many of these methods,and some of the implications of using them. The list is anextension of that presented in a previous review paper[72], including the many techniques that have beendeveloped since that time.

The list is divided into global and elemental methods:global methods deal with the exterior as a whole, whereaselemental methods subdivide the exterior into a finitenumber of subregions (or elements).

3.1 Global methods

3.1.1 Truncation: The most commonly used method todate has been that of truncation, being simple to applyand not needing any modifications to an existing com-puter program. By regarding the application of boundaryconditions as equivalent to adding extra sources, someuseful consequences arise.

If, in a magnetostatic problem a boundary condition(f> = 0 is applied, Ht = 0 is imposed on the outer bound-ary. This can also be regarded as equivalent to placing asource of equal strength, but opposite sign, an equal dis-tance beyond the boundary. In Fig. 3, the added source

R

P

R

r

L .

1

1I+JZ I

0 = 0

Fig. 3 Implied sources due to Dirichlet boundary conditions

will generate a field at point P of strength 1/4R2 timesthat of the original source. Similarly, if d<j)/dn = 0, Bn = 0is imposed, which can be regarded as a second source ofequal strength and same sign placed an equal distancebeyond the boundary.

It is therefore possible to predict the likely effect of anytruncated boundary on the results by considering howsignificant the implied secondary source field is, com-pared with that due to the original source. As a furthercheck, the boundary can be moved further away and anew solution obtained. If this is repeated a few times, it ispossible to extrapolate the solution to give a better esti-mate of the solution when the boundary is lying at infin-ity [63].

Difficulties arise when simple truncation is applied towave-type problems however, as any outgoing wave willbe reflected by the artificial boundary. For a Neumannboundary condition (normal derivative constrained) thereflected wave is in phase with the incident (outgoing)wave, and, converely, for a Dirichlet (constrainedpotential) condition, the reflected wave is out of phase.Unless there is damping present in the system, this tech-nique is likely to lead to totally erroneous results.

3 Algorithms for exterior regions

There are many methods currently available for exteriorproblems, each having its own advantages and disadvan-

3.1.2 Mapping: Simple truncation described above willgenerally lead to a large number of elements to model agiven problem, many of which will be used to define the'exterior' region. Mapping can therefore be used as a

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means of moving the outer boundary farther away,without requiring such a large number of elements.

As well as using a finite co-ordinate mapping, it is alsopossible to use a singular mapping function to map theoutside of a unit circle back onto its interior using l/R[48] (in fact the exterior of any polygon can be mappedonto the exterior of a unit circle, and hence onto the inte-rior of the circle). This works well when the regionoutside the polygon is source-free and contains linearmaterials, but leads to complications otherwise.

Care must be taken when using singular mappings toensure that the mapped region is adequately discretised,especially towards the centre point (which maps toinfinity) where the solution can be changing very rapidly.When applying singular mappings to wave problems[49], an outgoing wave of a given wavelength will maponto a wave whose wavelength is decreasing to zerotowards the centre point. This could also cause problemswhen performing the necessary integrals.

3.1.3 Analytic solutions: For many problems, suitableboundary conditions can be applied to a finite-elementmesh based on the far-field analytic solution to the gov-erning equations. This technique has been applied inboth static (originally with the finite-difference method[54]) and wave-type problems [38].

As the outer boundary of the finite-element mesh ismoved further away from the region containing thesources in the problem, the solution over the outerboundary will resemble that due to point sources at thecentre of the problem [58, 60]. The exterior can thereforebe approximated by applying, as boundary conditions,the analytic solution due to appropriate point sources atthe centre of the problem, either in terms of Dirichlet orNeumann conditions [63]. If, for example, the field dueto a pair of current-carrying bars is required (as in Fig.2), an approximate boundary condition could involve theanalytic solution due to a dipole positioned at the pointP. If more bars were added, higher order multipolescould then be added to generate the correct boundarycondition.

Alternatively, the analytic solution on the boundarycan be used to generate an equivalent 'stiffness' matrix forthe exterior region, as if it were a single element. Thisinvolves all the boundary nodes being connected,although if the magnitude of the coupling were sufficient-ly small for a pair of nodes, that term could be removedfrom the matrix, slightly decreasing the storage required[51].

The advantage of these techniques over truncation isthat the outer boundary can be placed much closer to theregion of interest, thus requiring fewer elements for agiven accuracy of solution. Of course, the far-field solu-tion for the given problem needs to be known before-hand.

3.1.4 Approximate solutions: It is sometimes not pos-sible to use an analytic solution to approximate theboundary conditions, for example when the analytic solu-tion is not known to sufficient accuracy on the proposedboundary. In these circumstances, the exterior field canbe expressed in terms of eigenvectors over the boundary[30, 34]. For example, in terms of cylindrical [19, 44, 64]or spherical [14, 45] harmonics, or Fourier series [27,59]. If the boundaries are parallel to the co-ordinatesystem axis, separation of variables may be applied, andappropriate boundary conditions thus imposed [55]. Theconstants in all of these approximations would then

appear as unknowns in the final equations to be solved,usually corresponding in number to the nodes on theexterior boundary.

In the case of wave problems, it is also required thatany outgoing wave is not reflected, in other words theyare completely damped by the boundary. A techniqueknown as 'boundary dampers' is an attempt to do justthat [37, 65]. The method is an approximate one becausean arbitrary wave cannot be completely damped. Instead,the outgoing wave is assumed to be a plane wave, andappropriate boundary conditions can then be imposed tocompletely damp out plane waves. An improvement onthis was to assume the outgoing wave was cylindrical[68], and this has then led to a series of higher orderdampers being developed [4, 3]. The higher the order themore the harmonics present are damped, but inevitablyat a higher cost computationally.

Iterative methods for approximating the boundaryconditions have also been applied [18]. If the boundarycondition can be written as a function of the solutionderivative, the boundary conditions can be continuouslyupdated as a new estimate for the solution derivative isfound. For example, in electrostatics the exterior can berepresented by an equivalent charge density on the outerboundary. The charge strengths can be written in termsof the solution derivatives on the boundary, and hencethis technique can be applied.

3.1.5 Global exterior elements: There are also a seriesof methods that rely on treating the exterior region as asingle 'super' element, with the obvious result that allnodes on the boundary are coupled to each other, thusincreasing the bandwidth of the final matrices. However,some of the methods to be described have been found togive very good results, and for that reason are still verypopular.

One of the simplest of these global methods is prob-ably that known as ballooning [57, 35, 15]. The tech-nique involves generating a single 'super' element outsidethe finite-element mesh (Fig. 4). This super element is

Fig. 4 Concentric rings of elements around a finite-element mesh, usedin the method of 'ballooning'

constructed by merging successive concentric rings of ele-ments (Qu Q2 etc. in Fig. 4) in an iterative manner, untilthe outer boundary has a sufficiently large radius. All

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nodes between adjacent rings are removed so that thereare only nodes on the boundary with the finite-elementmesh, and at infinity. Boundary conditions at infinity canthen be imposed.

The radius of the outer boundary created increases ina geometric series (hence, therefore, it never becomesinfinite), but, after some seven iterations, the radius ofthis boundary is measured in light years, and so it can beregarded as an extremely effective method of finitemapping.

In the same way as other forms of mapping, balloon-ing models the exterior region using progressively largerelements away from a predefined 'star point'. Care mustbe taken therefore that the solution is not changing toorapidly for these constructed elements to model satisfac-torily.

Because the method of ballooning effectively deletesnodes in the exterior region, it becomes quite difficult toimpose symmetry conditions on a plane extending toinfinity, or to recover solutions in the exterior regionitself. It must also be remembered that, as with othermethods of mapping, the technique cannot be applied totravelling-wave-type problems if Dirichlet or Neumannboundary conditions are imposed at infinity (but couldbe used to great advantage if, for example, analytic orapproximate boundary conditions from the precedingSections were employed).

There are a few other methods also making use of iter-ative techniques to construct a model of an infiniteexterior region. It is possible, for example, to generate aninfinite number of elements in the exterior, created in asystematic manner, and then to reduce this to a finite setof equations that give a best minimum for the energyfunctional in the exterior [61].

Another technique involves applying a similar methodto ballooning an infinite number of times (infinite sub-structuring [23]). It is then possible to generate the'stiffness' matrix for the exterior (as well as provide ameans for recovering the solution at points outside thefinite-element mesh) by solving the resulting quadraticeigenvalue matrix equation. It should be mentioned thatbecause an infinite number of iterations are made, thesizes of the elements generated in the exterior are van-ishingly small, so a rapidly varying solution can be mod-elled with no difficulty (cf. ballooning). Also, an infinitenumber of elements implies the outer boundary lies atinfinity, and this technique is therefore also suitable forwave-type applications [22].

A related method is 'infinitesimal scaling' [28], whichalso generates an expression for the energy functional forthe exterior. In this method, a solution to a nonlinearordinary differential matrix equation is required.

In fact, identical results to infinitesimal scaling can beobtained by employing another well known method inthe exterior, namely the boundary element method.Boundary elements are often used on their own to solvelinear problems, both finite and infinite in extent, and theliterature on this subject is vast (see Reference 16 for ageneral description of the technique). More recently,boundary elements have been used in conjunction withfinite elements to solve infinite-domain problems [8, 56,69, 70, 17]. In this way, the finite elements can modelnonlinear interiors, and the boundary elements used togive a very accurate model of the exterior region.

The boundary element method itself can be visualisedas being a method of replacing a region by a set of equiv-alent sources on its boundary (for example, chargedensity or currents) (Fig. 5A). In general, there will be an

equivalent source at each node on the boundary(corresponding to the nodes of the finite-element mesh onthe same boundary). The field anywhere due to these

"equivalent sources"

Fig. 5A Distribution of equivalent sources on exterior boundary offinite-element mesh, to model the exterior space (boundary elementmethod)

sources can then be expressed in terms of appropriateGreen's functions (the solution to the governing equationdue to a point source). The strengths of these equivalentsources are then solved, either directly or indirectly [47].The technique does, however, involve the evaluation ofsome singular integrals, requiring special consideration.This has been overcome to some extent by regarding theequivalent sources as not lying on the boundary itself,but slightly inside the boundary (Fig. 5B) [39]. It has also

"equivalent sources'

Fig. 5B Equivalent sources placed within the finite-element meshrather than on the boundary itself, to avoid singular integrals (modifiedboundary element method)

been proposed that it is not even necessary to have asmany sources as nodes on the boundary; and, by usingless equivalent sources, a best set of source strengths canbe determined [46].

In the boundary element method the outer boundaryis implied at infinity, and is therefore ideally suited tosolving propagating-wave problems [43, 46].

3.2 Elemental methods (infinite elements)The idea of dividing the exterior into a series of 'infiniteelements' (Fig. 6) was first proposed in 1973 [62], and,since then, many variations have been published. Most ofthe methods can, in fact, be separated into two broad

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categories, one that modifies the functions representingthe solution in the exterior (using decay functions), andthe second that modifies the way in which the infinite

Fig. 6 Exterior region subdivided into semi-infinite regions or elements(often called 'infinite elements')

element is mapped onto a finite dimensionless region(over which the necessary integrals can be performedmore readily).

3.2.1 Decay functions: In the first application of infiniteelements to static problems, the solution variation in theinfinite direction was modified by the introduction ofsome form of decay function (Fig. 7). In this way, the

decay function inradial (infinite)

direction 3 4 5decay

function ininfinite(local u)directionpolynomial

mapping

Fig. 7 Decay function included in infinite (radial) direction, maps ontodecay in local u-direction (with this orientation of element).

solution was forced to decay either reciprocally [62, 1, 11,41] or exponentially [66, 9, 24, 20] with distance.

The choice of reciprocal decay seems obvious if thefar-field solution is also known to be reciprocal (in three-dimensional statics problems), but the use of exponentialdecay functions may not appear quite so obvious andwill need some elaboration.

It is, in fact, not necessary for an infinite element to tryto impose an analytic solution in the exterior, rather it issufficient to ensure that the solution decays to zero withdistance to ensure a reasonable solution inside the regionof interest, the finite-element mesh (see Section 4 forfurther details). It is then to be expected that the solutionin the exterior itself will not be particularly good.

In some cases, however, it may be important to beable to generate good approximations to the solution,even at large distances from the finite part of theproblem. In these cases, the use of more specialised decayfunctions may be necessary.

Reciprocal decay functions have already been men-tioned, and at large distances they may be an adequaterepresentation of the solution. They can also be extendedto include higher orders by the introduction of more

nodes in the element (in the same way as higher orderfinite elements). As more nodes are added, more terms inthe series l/R" are introduced, thus improving the accu-racy of the solution. Another form of higher order decayfunction has been suggested [36] using (a/R)n, but, in fact,as R => 0 the solution becomes l/R again, and so it is nottruly a higher order method.

Of a more specialised nature, it is possible to use func-tions very much problem-dependent [2], or indeedalmost any function that may be wanted [53].

3.2.2 Co-ordinate mapping: It is usual in the finite-element method to employ some form of co-ordinatemapping between the global system and a local one.Typically, Lagrange polynomial mappings could be used,leading to the usual set of parametric finite elements [29].However, it is also feasible to use some form of singularmapping function (e.g. based on l/R) [26, 5, 6], whereone of the nodes in the local element is mapped onto apoint lying at infinity (Fig. 8).

singularmapping

poles of singularmapping function

Fig. 8 Semi-infinite region mapped onto finite local element using sin-gular mapping function (including implied poles of mapping), (nodes 7, 8and 9 map onto infinity)

A variation of this was later developed [72, 73, 50]such that normal Lagrange polynomials could be used inthe local element to represent the solution variation. Thiswas shown to be equivalent (after performing themapping) to a global variation of the form 1/R". As ordi-nary polynomial functions are used for the solutionvariation in the local element, standard numerical inte-gration methods could be used without modification (forstatic problems). By varying the order of the singularmapping function used, different powers of decay of theglobal solution can be obtained [21].

A form of reciprocal decay element has also beendevised [52] using techniques from both of the categorieslisted above. The solution variation over the localelement is modified, and a nonpolynomial mapping isperformed (based on l/R again). The advantage of thistechnique is that a numerical integration rule can bedeveloped to integrate the necessary term exactly.

3.2.3 Extension to wave problems: Many of themethods mentioned so far have been extended to includeperiodic time variation. The first method to implementthis [10, 67] included the term exp (icot) • exp (ikR) in thedecay functions representing the solution in the exterior.The first part, exp (icot), can be neglected (being presentin all terms in the equations it can be cancelled out). Thesecond part is used to indicate that, in the infinite region,the wave is propagating in a radial direction, with wavenumber k. Inevitably, this requires a new set of integra-tion routines to be developed, to deal with the new termsarising.

As before, reciprocal and exponential decay functionscan be used in the infinite direction, with the amount of

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decay being determined by comparing the decay functionwith the expected far-field solution. (In three dimensionsthis is l/R, and in two dimensions this is typically azeroth-order Hankel function, decaying as approximatelyR~1'2 for large/?.)

The same technique has been applied to the method ofco-ordinate mapping [72, 13], again with great success.Whereas, with the static version of the method, standardnumerical integration routines could be used, on intro-ducing the wave term new integration rules had to bedeveloped. A modified form of mapping was also used toallow for the fact that, in two dimensions, the far fielddoes not decay as l/R, but as R~112, giving improvedaccuracy for such problems [74, 25].

As the wave number of the outgoing wave is includedin the description of the solution, these methods can onlybe applied to problems where a single wave of knownfrequency is propagating. Problems involving more thana single wave have been tackled [20, 40], but thesemethods have not, to date, been applied to transientproblems, where, in principle, an infinite number of fre-quencies are propagating.

One further method that should be mentioned is onein which the boundary element method is applied in theexterior, but over the boundaries of a finite number ofinfinite elements [31, 32]. This technique does not sufferthe same disadvantage of the normal boundary elementmethod of coupling all boundary nodes to every other.The method has also been found to be more accuratethan ordinary infinite elements, although not as accurateas the full implementation of boundary elements.

4 Requirements of a valid algorithm

A list of requirements for a valid infinite element hasbeen previously published [42], and a few of the moreimportant features that apply to all exterior domain algo-rithms are expanded here.

In Section 2, the types of problem that could be solvedusing special techniques for the exterior region were dis-cussed. It was stated that only well posed problems couldbe solved, and these were described as problems wherethe solution decays to zero with distance away from thefinite part of the problem. This is a rather loose state-ment, and will need elaboration.

When applying the finite-element method, it iscommon practice to reduce the order of the Laplacianoperator using Green's theorem (integration by parts)[67]. This results in first-order differential terms over thevolume and the boundaries. Within the finite part of theproblem, the volume terms are modelled with finite ele-ments. In the exterior, the volume term can be modelledusing some form of infinite element, or, alternatively, theycan be transformed into a boundary term, which is thenmodelled using boundary elements, analytic or approx-imate methods, or some form of iterative technique.

Most of the methods mentioned therefore rely, in someway, on the application of Green's theorem in theexterior. However, care must be taken as there are certainrestrictions on the use of Green's theorem. In infiniteregions, the theorem can only be applied if the solution isdecaying to zero at a certain rate [33] (the solution mustbe regular at infinity). In other words, for a solution </>and radial distance R,

must be bounded for all sufficiently large R. Therefore,the solution must decay to zero at least as fast as l/R (inthree dimensions, and R ~1/2 in two dimensions).

This is inherent in all the methods mentioned that insome way approximate the solution in the exterior(rather than extend the finite-element mesh in some wayas in the mapping techniques). In particular, infinite ele-ments choose decay functions that decay at least asrapidly as this, or employ mappings such that this isimplied [12]. Boundary element methods also satisfy thisconstraint, because, although the boundary is representedby equivalent sources (e.g. charges), the interior has zeronet charge; all surface charges will therefore balance, and,in the far field, the effect is equivalent to that of a dis-tribution of half the number of dipoles.

If wave problems are being solved, there is a furtherconstraint to be imposed, the so called 'Sommerfeld radi-ation condition' [68]. This requires that, for an n-dimension problem:

2 )

dx dy dz

that is, the magnitude of the solution decays as for thestatic problem, and the wave is always propagating out-wards.

If finite elements are being used, the usual constraintof compatibility applies. The solution in adjacent ele-ments must be continuous, both between adjacent infiniteelements (if being used), and between whatever techniqueis employed and the finite-element mesh. (In fact, onlyone of the methods mentioned above [62, 1] does notsatisfy this constraint.)

5 Summary

In Section 3 of this paper, many algorithms suitable forinfinite domains have been mentioned. For convenience,they were divided into two main groups: global and ele-mental methods.

Many of the algorithms presented under the heading'global' methods treat the exterior region in terms of asingle element, with the natural consequence that allboundary nodes become coupled to every other. If someform of banded matrix solver is to be employed, this willgreatly increase the cost of solution. Even when usingsparse solvers, some increase in cost is to be expectedbecause the matrix associated with the exterior will befully populated.

The advantage, then, of using elemental methods(infinite elements) is that there is not the drastic increasein bandwidth of the matrix equation, and, indeed, fewextra degrees of freedom are usually needed.

What must be remembered is that, for all the methodsdescribed (with the exception of the methods based onmappings), some knowledge of the far-field solution isrequired. For analytic and approximate methods this willappear reasonable, but maybe less so for some of theother techniques. Infnite elements, for example, requirethe evaluation of some form of decay parameter so thatthe chosen decay function best matches the far-field solu-tion (albeit over part of the element only). Boundaryelement methods require a suitably chosen Green's func-tion, an analytic solution to the governing equation.

The infinite elements that make use of singular co-ordinate mappings will also require the choice of some

156 IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988

star point to define the mapping. The position of thismust be chosen with some care, as the far field willappear to originate from a point source at this position(the same situation arises when applying analytic bound-ary conditions).

As far as relative accuracy is concerned, it is generallyaccepted [69, 71, 31, 32, 7] that boundary elements aremore accurate in modelling an exterior region thaninfinite-element methods. The reason for this is that theboundary element method is employing an analytic solu-tion to the governing equation, to represent the solutionin the exterior, whereas with infinite elements it is moreusual to simply require that the solution deays to zerowith distance. However, there is the added complexitywith boundary elements, not only due to the more fullypopulated matrices mentioned here, but also with regardto the evaluation of singular integrals inherent in the for-mulation.

It is interesting to note that, for all those globalmethods which require the evaluation of coefficients, gen-erally corresponding to the nodes on the boundary, it isnot possible to increase the accuracy of the exterior solu-tion without increasing the number of these boundarynodes. The infinite-element methods, however, allow anincrease in the order of the solution in the infinite direc-tion, quite independently of the number of nodes or ele-ments on the boundary itself. In principle then, theaccuracy of the infinite-element methods can be increasedto any desired level, at the expense of more nodes beingrequired in the exterior.

In general, it is not possible to say which of themethods mentioned is the best, as the choice of methoddepends on many factors, many of which may beproblem-specific. For example, if accuracy is more impor-tant than speed of solution, or the size of the model, thenboundary element methods are probably the best choice.On the other hand, if size and speed are to be kept to aminimum, some form of infinite element may be moresuitable. The type of approach to be used may thendepend on whether the solution is required in the exteriorregion or not.

6 Conclusions

In the preceding Sections, the types of physical problemsthat can be solved have been described as well as a fewguidelines concerning the requirements for such methods.In these Sections, it has been emphasised that, to solve anexterior problem, the solution must decay to zero withdistance; physically interpreted as requiring all sources tobalance, and necessary numerically in order to applyGreen's theorem in the infinite region.

A list of many of the algorithms currently available forexterior problems has been given. It is not particularlyeasy to draw any conclusions concerning the many tech-niques, and it is probably not appropriate to try. Instead,a few points of comparison have been given, particularlyconcerning the relative accuracy of some of the tech-niques.

The use of the methods for wave-type problems hasbeen discussed, although some have been found to becompletely inappropriate. Most of the methods men-tioned have been frequency-dependent and, as such, arenot suitable for transient types of problem (where aninfinite number of frequencies can be expected). Althoughinfinite-element methods have been applied to problems

with more than one frequency, it is still only the bound-ary element method that has been applied to transientproblems, using a time dependence in the Green's func-tion.

7 Acknowledgments

I would like to thank SERC for financial support duringmy period of research at University College Swansea,during which time most of this work was carried out. Iwould also like to thank Professor P. Bettess for his helpand guidance during that time.

8 References

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10 BETTESS, P., and ZIENKIEWICZ, O.C.: 'Diffraction and refrac-tion of surface waves using finite and infinite elements', ibid., 1977,11, pp. 1271-1290

11 BETTESS, P.: 'More on infinite elements', ibid., 1980, 15, pp. 1613-1626

12 BETTESS, P., EMSON, C , and BANDO, K.: 'Some useful tech-niques for the testing of infinite elements', Appl. Math. Modelling,1982, 6, pp. 436-440

13 BETTESS, P., EMSON, C , and CHI AM, T.C.: 'A new mappedinfinite element for exterior wave problems', in LEWIS, R.W. (Ed.):'Numerical methods in coupled systems' (Wiley, 1984), p. 489

14 BRAMBLE, J.H., and PASCIAK, J.E.: 'A new computationalapproach for the linearised scalar potential formulation of the mag-netostatic field problem', IEEE Trans., 1982, MAG-18, (2), pp. 357-361

15 BRAUER, J.R.: 'Open boundary finite elements for axisymmetricmagnetic and skin effect problems', J. Appl. Phys., 1982, 53, (11),pp. 8366-8368

16 BREBBIA, C.A.: 'The boundary element method for engineers'(Pentech Press, 1980)

17 BRYANT, C : 'A mixed integral differential method for low fre-quency electromagnetic fields'. PhD thesis, Dept. Electrical Engin-eering, Imperial College of Science and Technology, London, 1981

18 CERMAK, I.A., and SILVESTER, P.: 'Solution of 2-dimensionalfield problems by boundary relaxation', Proc. IEE, 1968, 115, (9),p.1341

19 CHEN, H.S., and MEI, C.C.: 'Oscillations and wave forces in a manmade harbor in the open sea'. Proc. 10th Symp. Naval Hydrodyn.,Cambridge Mass., USA, 1974

20 CHOW, Y.K., and SMITH, I.M.: 'Static and periodic infinite solidelements', Int. J. Numer. Methods Eng., 1981,17, pp. 503-526

21 CURNIER, A.: 'A static infinite element', ibid., 1983, 19, pp. 1479-1488

22 DASGUPTA, G.: 'A finite element formulation for unboundedhomogeneous continua', Trans. ASME. J. Appl. Mech., 1982, 49,pp. 132-140

23 DASGUPTA, G.: 'Computation of exterior potential fields by infin-ite substructuring', Comput. Methods Appl. Mech. & Eng., 1984, 46,pp. 295-305

IEE PROCEEDINGS, Vol. 135, Pt. A, No. 3, MARCH 1988 157

24 EMSON, C , and BETTESS, P.: 'Application of infinite elements toexternal electromagnetic field problems'. Paper presented at Int.Conf. Coupled Problems, Swansea, UK, 1980

25 EMSON, C.R.I.: 'A new infinite element for static and dynamic elec-tromagnetic field problems'. PhD thesis, Dept. Civil Engineering,University College Swansea, University of Wales, 1985, C/Ph/88/85(Also Rutherford Appleton Laboratory report RAL-T-019)

26 GARTLING, D.K., and BECKER, E.B.: 'Finite element analysis ofviscous, incompressible fluid flow, Part I: Basic methodology',Comput. Methods Appl. Mech. & Eng., 1976, 8, pp. 51-60

27 GUPTA, S., PENZIEN, J. LIN, T.W., and YEH, C.S.: 'Threedimensional hybrid modelling of soil-structure interaction', Earth-quake Eng. & Struct. Dyn., 1982,10, pp. 69-87

28 HURWITZ, H. (Jr.): 'Infinitesimal scaling, a new procedure formodelling exterior field problems'. Invited paper, INTERMAG'84Conference, Hamburg, 1984

29 IRONS, B., and AHMAD, S.: 'Techniques of finite elements' (EllisHorwood, 1980)

30 JUNG, H.K., LEE, G.S., and HAHN, S.Y.: '3-D magnetic field com-putations by infinite elements', J. Appl. Phys., 1984, 55, (6), Part IIB,pp. 2201-2203

31 KAGAWA, Y., YAMABUCHI, T., and KITAGAMI, S.: 'The infin-ite boundary element and its application to a combined finiteboundary element technique for unbounded field problems',COMPEL, 1983, 2, (4), pp. 179-193

32 KAGAWA, Y., YAMABUCHI, T., and ARAKI, Y.: 'The infiniteboundary element and its application to the unbounded Helholtzproblem', ibid., 1985, 4, (1), pp. 29-41

33 KELLOG, O.D.: 'Foundations of potential theory' (Dover Pub-lications, 1953)

34 KIM, I.H., JUNG, H.K., LEE, G.S., and HAHN, S.Y.: 'Magneticfield computations by infinite elements', J. Appl. Phys., 1982, 53,(11), Part II, pp. 8372-8374

35 LOWTHER, D.A., and SILVESTER, P.P.: 'Finite element modelsof exterior regions containing moving conductors'. COMPUMAGConference Paper 3.4, Grenoble, 4-6 September 1978

36 LYNN, P.P., and HADID, H.A.: 'Infinite elements with 1/r" typedecay', Int. J. Numer. Methods Eng., 1981,17, pp. 347-355

37 LYSMER, J., and KUHLMEYER, R.L.: 'Finite dynamic model forinfinite media', J. Eng. Mech. Div., Proc. ASCE, 1969, EM-4,pp. 859-877

38 LYSMER, J., and WAAS, G.: 'Shear waves in plane infinite struc-tures', ibid., 1972, EM-1, pp. 85-105

39 McDONALD, B.H., and WEXLER, A.: 'Finite element solution ofunbounded field problems', IEEE Trans., 1972, MTT-20, (12),pp. 841-847

40 MEDINA-MELO, F.J.: 'Modelling of soil structure interaction byfinite and infinite elements'. PhD thesis, University California,Berkeley, 1981

41 MEDINA, F.: 'An axisymmetric infinite element', Int. J. Numer.Methods Eng., 1981,17, pp. 1177-1185

42 MEDINA, F., and Taylor, R.L.: 'Problems of unbounded domains',Int. J. Numer. Methods Eng., 1983,19, pp. 1209-1226

43 MEI, C.C.: 'Numerical methods in water wave diffraction and radi-ation', Ann. Rev. Fluid Mech., 1978,10, pp. 393-416

44 MEI, K.K.: 'Unimoment method of solving antenna and scatteringproblems', IEEE Trans., 1974, AP-22, pp. 760-766

45 MIYOSHI, T., and MAEDA, G.: 'Finite element analysis of leakagemagnetic flux from an induction heating system', ibid., 1982,MAG-18, pp. 917-920

46 MURAKAMI, H., SHIOYA, S., and YAMADA, R.: 'Transmittingboundaries for time harmonic elastodynamics on infinite domains',Int. J. Numer. Methods Eng., 1981, 17, pp. 1697-1716

47 MUSTOE, G.G.W.: 'Symmetric variational boundary integral andfinite element procedure in continuum mechanics'. PhD thesis, Dept.Civil Engineering, University College Swansea, 1981

48 NATH, B., and JAMSHADI, J.: 'The w-plane finite element methodfor the solution of scalar field problems in two dimensions', Int. J.Numer. Methods Eng., 1980, 15, pp. 361-379

49 NATH, B., and JAMSHADI, J.: The w-plane finite element methodfor the solution of the steady state scalar wave equation in 2-D',Earthquake Eng. & Struct. Dyn., 1980, 8, pp. 181-191

50 OKABE, M.: 'Generalized Lagrange family for the cube with specialreference to infinite-domain interpolations', Comput. Methods Appl.Mech. & Eng., 1983, 38, pp. 153-168

51 PIRCHER, H., and BEER, G.: 'On the treatment of infinite bound-aries in the finite element method', Int. J. Numer. Methods Eng.,1977,11, pp.1194-1197

52 PISSANETZKY, S.: 'An infinite element and a formula for numeri-cal quadrature over an infinite interval', ibid., 1983,19, pp. 913-927

53 PISSANETZKY, S.: 'A simple infinite element', COMPEL, 1984, 3,(2), pp. 107-114

54 RICHARDSON, L.F.: 'The approximate arithmetical solution byfinite differences of physical problems involving differential equa-tions, with an application to the stresses in a masonry dam', Trans.Roy. Soc. London Ser. A, 1911, 210, pp. 307-357

55 ROLICZ, P.: 'Use of Bubnov-Galerkin method to skin effect prob-lems for the case of unknown boundary conditions', IEEE Trans.,1982, MAG-18, pp. 527-530

56 SILVESTER, P., and HSIEH, M.S.: 'Finite element solution of 2-dimensional exterior field problems', Proc. IEE, 1971, 118, pp. 1743—1747

57 SILVESTER, P.P., LOWTHER, DA., CARPENTER, C.J., andWYATT, E.A.: 'Exterior finite elements for 2-dimensional fieldproblems with open boundaries', Proc. IEE, 1977,124, p. 1267

58 SMYTHE, W.R.: 'Static and dynamic electricity' (McGraw-Hill,1968, 3rd edn.)

59 STEELE, C.W.: 'Iterative algorithm for magnetostatic problemswith saturable media', IEEE Trans., 1982, MAG-18, pp. 393-396

60 STRATTON, J.A.: 'Electromagnetic theory' (McGraw-Hill, 1941)61 THATCHER, R.W.: 'On the finite element method for unbounded

regions', SI AM J. Numer. Anal., 1978,15, pp. 466-47762 UNGLESS, R.F.: 'An infinite finite element'. MASc thesis, Uni-

versity of British Columbia, 197363 WOOD, W.L.: 'On the finite element solution of an exterior bound-

ary value problem', Int. J. Numer. Methods Eng., 1976, 10, pp. 885-891

64 YUE, D.K.P., CHEN, H.S., and MEI, C.C: 'A hybrid method fordiffraction of water waves by three dimensional bodies', Int. J.Numer. Methods Eng., 1978,12, pp. 245-266

65 ZIENKIEWICZ, O.C, and NEWTON, R.E.: 'Coupled vibration ofa structure immersed in a compressible fluid'. Proc. Symp. FiniteElement Techniques, Stuttgart, 1969

66 ZIENKIEWICZ, O.C, and BETTESS, P.: 'Infinite elements in thestudy of fluid structure interaction problems'. Proc. 2nd Int. Symp.Comput. Methods Appl. Sci., Versailles, 1975

67 ZIENKIEWICZ, O.C: 'The finite element method' (McGraw-Hill,1977, 3rd edn.)

68 ZIENKIEWICZ, O.C, KELLY, D.W., and BETTESS, P.: 'TheSommerfeld (radiation) condition on infinite domains and its model-ling in numerical procedures'. Proc. 3rd Int. Symp. Comput. MethodsAppl. Sci., IRIA Laboria, 1977

69 ZIENKIEWICZ, O.C, KELLY, D.W., and BETTESS, P.: 'Mar-riage a la mode — the best of both worlds (finite elements andboundary integrals)'. Int. Symp. Innovative Numer. Anal. Appl. Eng.Sci., Versailles, 1977

70 ZIENKIEWICZ, O.C, KELLY, D.W., and BETTESS, P.: 'Thecoupling of the finite element method and boundary solution pro-cedure', Int. J. Numer. Methods Eng., 1977,11, pp. 355-375

71 ZIENKIEWICZ, O.C, KELLY, D.W, and BETTESS, P.: 'Thefinite element method for determining fluid loadings on rigid struc-tures', in LEWIS, R.W. (Ed.): 'Numerical methods in offshore engin-eering' (Wiley, 1978), Chap. 4

72 ZIENKIEWICZ, O.C, BETTESS, P., CHIAM, T.C, and EMSON,C: 'Numerical methods for unbounded field problems and a newinfinite element formulation'. Winter Annual Meeting, ASME,Washington, AMD-46, pp. 115-148 (& University College Swanseareport C/R/384/81)

73 ZIENKIEWICZ, O.C, EMSON, C, and BETTESS, P.: 'A novelboundary infinite element', Int. J. Numer. Methods Eng., 1983, 19,pp. 393-404

74 ZIENKIEWICZ, O.C, BANDO, K., BETTESS, P., EMSON, C,and CHIAM, T.C: 'Mapped infinite elements for exterior waveproblems', ibid., 1985, 21, pp. 1229-1251

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