1

A DC Stable and Large Time Step Well-BalancedTD-EFIE Based on Quasi-Helmholtz Projectors

Yves Beghein, Kristof Cools, and Francesco P. Andriulli, Senior Member, IEEE

Abstract—The marching-on-in-time solution of the time do-main electric field integral equation (TD-EFIE) has traditionallysuffered from a number of issues, including the emergence ofspurious static currents (DC instability) and ill-conditioning atlarge time steps (low frequencies). In this contribution, a space-time Galerkin discretization of the TD-EFIE is proposed, whichseparates the loop and star components of both the equationand the unknown. Judiciously integrating or differentiating thesecomponents with respect to time leads to an equation which is freefrom DC instability. By choosing the correct temporal basis andtesting functions for each of the components, a stable marching-on-in-time system is obtained. Furthermore, the scaling of thesebasis and testing functions ensure that the system remains well-conditioned for large time steps. The loop-star decomposition isperformed using quasi-Helmholtz projectors in order to avoidthe explicit transformation to the unstable bases of loops andstars (or trees), and to avoid the search for global loops, whichis a computationally expensive operation.

Index Terms—time domain, electric field integral equation, DCinstability, low frequency breakdown.

I. INTRODUCTION

ELECTROMAGNETIC scattering by perfect electricalconductors can be modeled efficiently using boundary

integral equations (BIEs). The two most prominent formula-tions are the electric field integral equation (EFIE) and themagnetic field integral equation (MFIE). This paper focusseson the properties of the EFIE.

The EFIE can be formulated in either the frequency do-main (FD-EFIE, for time-harmonic electromagnetic fields) orthe time domain (TD-EFIE, for general time dependence).Whereas the FD-EFIE is solved at a single point on thefrequency axis, the TD-EFIE requires a discretization of thetime axis. Most often, a causal discretization scheme is chosensuch that the resulting system of equations can be solvedusing the marching-on-in-time (MOT) algorithm [1], [2] (otherapproaches such as marching-on-in-order have been suggested,see e.g. [3]). The stability of the MOT algorithm hinges onboth the accurate evaluation of the interaction integrals [4]–[7] and the choice of temporal discretization scheme [8]–[11].Space-time Galerkin schemes have been found to producegood results in terms of stability, accuracy and extensibilityto higher order in both space and time [10], [12]–[14].

Unfortunately, these schemes suffer from at least one of thefollowing problems.

Y. Beghein is with the Department of Information Technology (INTEC),Ghent University, Belgium, yves.beghein@ugent.be.

K. Cools is with the Electrical Systems and Optics Research Division,University of Nottingham, Nottingham NG7 2RD, U.K.

F.P. Andriulli is with the Microwave Department of Telecom Bretagne –Institut Mines-Telecom, Brest, France.

First, the TD-EFIE allows sourceless harmonic-in-timeregime solutions. When the scatterer is closed, interior res-onances can be excited (resonant instabilities) [15], [16]. Fur-thermore, it supports sourceless constant or linear in time di-vergence free solutions (DC instabilities) [16], [17]. For simplyconnected geometries, DC instabilities can be eliminated byswitching to the Calderon preconditioned TD-EFIE [18] andapplying the so-called dot-trick [16]. However, for multiplyconnected geometries, the dot-trick EFIE still supports staticsolutions [19] and is therefore susceptible to DC instability.

Second, for large time steps ∆t, the scaling of the blocks ofthe TD-EFIE operator that describe the electrostatic and andmagnetostatic problems differs by a factor ∆t2, leading toan ill conditioned system matrix. The resulting system cannotbe solved efficiently (using e.g. iterative techniques), whichdrastically increases the solution time. This phenomenon istermed low frequency breakdown [20], [21], and also occursin the frequency domain, see e.g. [22], [23] and referencestherein.

Finally, the standard TD-EFIE involves the computation ofthe charge as the temporal integral of the current divergenceat every time step. This computationally costly operation isoften avoided by introducing an additional charge variable (atthe cost of greater memory requirements), or by switching tothe time-differentiated TD-EFIE (at the cost of introducinglinear-in-time spurious loop currents to the solution).

Low frequency breakdown can be mitigated by applyinga loop-star or loop-tree decomposition to the EFIE, andrescaling the components with the correct powers of ∆t [21].However, explicitly constructing a basis of loops and stars (ortrees) leads to ill-conditioning [24]. Furthermore, for multiplyconnected surfaces, global loops must be detected, which iscomputationally expensive.

Linear-in-time spurious currents have also been tackled byapplying a loop-tree decomposition to the time-differentiatedTD-EFIE in [25]. While this does result in the elimination ofthe linear-in-time spurious currents, it does not solve constant-in-time DC instability. In [26], a loop-tree decomposition isused to filter out static loop modes after they emerge.

In this contribution, a novel formulation termed the quasi-Helmholtz Projected TD-EFIE (qHP-TDEFIE) is obtained byseparating the quasi-Helmholtz components of both spatialbasis and testing functions using the loop and star projectorsintroduced in AndriulliMultConn, thereby eliminating the needto explicitly construct a loop-star basis. The loop and starparts of both the equation and the unknown are temporallyintegrated or differentiated in such a way that the resultingequation does not possess a static nullspace, and is therefore

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not susceptible to DC instability. Furthermore, it does notrequire the computation of the temporal integral of the current.

Next, the quasi-Helmholtz components of both the unknownand the equation are separately discretized in time usingGalerkin methods. More specifically, the order of regularityof the basis and testing functions is matched to the order ofdifferentiation of each component. This is necessary in orderto obtain a stable MOT scheme. Furthermore, the scaling ofthe basis and testing functions is chosen such that system hasa well-defined and well-conditioned low frequency limit. Inaddition, the interaction matrix elements needed in this schemeare compatible with matrix-vector product accelerators such asthe PWTD method [27], [28].

This paper is organized as follows. In Section II, thestandard TD-EFIE is presented in order to fix the notationsand definitions that will be used throughout the paper. Theproperties of the resulting MOT algorithm are summarily re-viewed. In Section III, the derivation of the new qHP-TDEFIEis presented and discussed. In Section IV, the low frequency(large time step) limit of the resulting numerical schemeis investigated. Finally, a number of numerical experimentsare performed in Section V to demonstrate the favourableproperties of the qHP-TDEFIE formulation, both in terms ofDC stability and independence of the condition number on thetime step size.

II. THE STANDARD EFIE AND ITS PROPERTIES

A. The Time Domain EFIE

Consider a perfectly conducting body Ω, whose boundaryis denoted Γ. When an incident electric field ei(r, t) impingeson it at t > 0, a surface current j(r, t) is induced on Γ, thatsatisfies the time domain EFIE

η (T j) (r, t) = −n× ei(r, t) ∀r ∈ Γ, t > 0, (1)

where the electric field integral operator (EFIO) T is definedas

(T j) (r, t) = Tsj(r, t) + Thj(r, t), (2)

(Tsj) (r, t) = −1

cn×

∫Γ

∂tj(r′, τ)

4πRds′, (3)

(Thj) (r, t) = c n× p.v.∫

Γ

grad∂−1t div′Γj(r′, τ)

4πRds′,(4)

η =√µ0/ε0, c = 1/

√ε0µ0, R = |r − r′|, τ = t − R/c,

and n is the exterior normal vector to Γ. Define ∂−1t f(t) =∫ t

−∞ f(τ)dτ .Note that (1) in itself only defines the current j up to a

constant solenoidal part. Uniqueness is achieved by imposingcausality, i.e., all fields are assumed to vanish for t < 0 in aneighborhood of Ω. Causality also guarantees that ∂−1

t f(t) =∫ t

−∞ f(τ)dτ is well defined.

B. Standard Galerkin Discretization

The surface Γ is now approximated by a triangle mesh withNV vertices, NS edges, and NC cells. On this mesh, NS Rao-Wilton-Glisson (RWG) functions are constructed [29]. Each

rm+

rm−

em

cm−c

m+

Fig. 1. Two adjacent cells on which an RWG function is defined.

RWG function fm(r) is associated with one edge em (seeFig. 1), and is defined on the two adjacent cells c+m and c−m:

fm(r) =

r−r+

m

2Ac+m

for r ∈ c+mr−m−r

2Ac−m

for r ∈ c−m, (5)

where Ac+mand Ac−m

denote the area of cell c+m and c−m,respectively. Note that this definition does not include edgelength normalization, in order to simplify the notation in whatfollows.

The current j(r, t) is approximated as an expansion in theseRWG functions:

j(r, t) =

NS∑m=1

jm(t)fm(r). (6)

Next, (1) is spatially tested with the rotated RWG functionsn× fm(r), m = 1, 2, ..., NS :∫

Γ

(n× fm(r)) · (Eq. (1)) ds. (7)

By defining the following quantities:

Z = Zs + Zh, (8a)

[(Zsj) (t)]m = −∑n

η

c

∫Γ

ds fm ·∫Γ

ds′∂tjn(τ)fn(r′)

4πR, (8b)

[(Zhj) (t)]m = −∑n

ηc

∫Γ

ds (divΓfm(r)) ·∫Γ

ds′∂−1t jn(τ) div′Γfn(r′)

4πR, (8c)

[e(t)]m =

∫Γ

fm(r) · ei(r, t) ds, (8d)

equation (7) can be concisely stated as

Zj(t) = −e(t) ∀t > 0. (9)

This equation is temporally discretized using a Galerkinmethod (alternatively, a collocation method can also be used –see Appendix A). The RWG expansion coefficients jm(t) areapproximated by an expansion in pulse functions p(t − i∆t)(Fig. 2, middle)

j(t) =

NT∑i=1

jip(t− i∆t), (10)

3

−2 −1 0 1 2

0

1

t / ∆t−2 −1 0 1 2

0

1

t / ∆t−2 −1 0 1 2

0

1

t / ∆t

Fig. 2. Temporal basis and testing functions: Dirac delta distrubtion δ(t)(left), pulse p(t) (middle) and hat h(t) (right).

p(t) =

1 t ∈ (−∆t, 0)

0 otherwise, (11)

and (9) is tested with pulses p(t− j∆t):∫Rp(t− j∆t) (Eq. (9)) dt j = 1, 2, 3, ..., NT . (12)

This can be written asj∑

i=0

Zijj−i = −ej j = 1, 2, 3, ..., NT , (13)

whereej =

∫Rp(t− j∆t) e(t)dt, (14)

[Zi]mn

= η

∫Rdt p(t− i∆t)

∫Γ

(n× fm(r)) · T fnp (r, t)ds

= ∆t

∫Γ

(n× fm(r)) · T fnh (r, i∆t)ds, (15)

where h(t) denotes the hat function (Fig. 2, right):

h(t) =

1 + t

∆t t ∈ (−∆t, 0)

1− t∆t t ∈ (0,∆t)

0 otherwise. (16)

In (15), the interaction elements are transformed into the formencountered in traditional collocation-in-time methods. Theseintegrals can be evaluated using techniques outlined in e.g.[4]–[7]. It is also possible to accelerate these computationsusing fast techniques such as PWTD [27], [28]. The systemof linear equations (13) can be solved by recasting it in thefollowing form

−Z0ji =

i∑j=1

Zj ji−j + ei, (17)

which is then successively solved for ji, i = 1, 2, ..., NT . Thisis the marching-on-in-time (MOT) method.

Because of the temporal integral in the hypersingular contri-bution, there are an unlimited number of matrices Zi 6= 0. Thenumber of nonzero terms in the summation in the right handside therefore grows without bound when the MOT algorithmprogresses. The unbounded summation can be avoided byintroducing the charge as an additional variable, see e.g. [11]or [30]. This however leads to overhead in both memoryrequirements and computation time.

C. Null Space of the Discretized EFIO

In Section II-A, it was noted that the sourceless EFIEsupports constant solenoidal regime solutions. This property isconserved by the discretization procedure: if jL is a solenoidalcurrent,

∞∑j=0

Zj jL = 0. (18)

In the continuous case, an energy argument shows that latetime constant signals cannot be part of the solution. Indeed,all energy in the incident wave is reflected during scattering,leaving no energy to sustain a residual magnetostatic field. Inthe discrete case, the finite precision of the numerical schemeallows static loop currents to creep into the solution, afterwhich they persist throughout the simulation [17].

Even though the solution to the discrete EFIE will alwaysbe an approximation of the exact solution, it is possibleto design a scheme that explicitly coerces late time energyconservation and thus cannot support DC signals in the tailof the approximate solution. In practice this property canbe checked by inspecting the expressions for the interactionelements and keeping track of the explicit appearance ofdivergence and differentation operators. In the next section,the EFIE is rewritten and discretised in such a way that theresulting discrete system does not sustain constant-in-timeregime solutions.

III. THE QUASI-HELMHOLTZ PROJECTED TD-EFIE

A. Separation of the Quasi-Helmholtz Components

In [22], the quasi-Helmholtz components (i.e., the diver-gence free and weakly curl free components) of the FD-EFIEare separated not by explicitly constructing a loop-star basis,but using projection matrices PΛH and PΣ.

Consider a triangle mesh consisting of NC cells, on whichNS RWG functions fm(r) are defined. Each RWG functionfm(r) is defined on two cells, c+m and c−m (see Fig. 1), andrepresents a current flowing from c+m to c−m. The NS × NC

star coefficient matrix is given by

Σij =

1 if cell j equals c+i−1 if cell j equals c−i0 otherwise

. (19)

Note that ΣT is the discrete divergence operator in a basis ofRWG functions for the current, and cellwise constant functionsfor the charge. Projection onto the star space is then achievedusing the projection operator

PΣ = Σ(ΣT Σ

)+ΣT , (20)

where(ΣT Σ

)+denotes the pseudoinverse of ΣT Σ. A pseudo-

inverse is required because the constant vector is in the kernelof Σ. Computing this pseudoinverse using standard techniqueswould require O(N3

c ) operations. Section V of [22], however,explains how this computation can be done in linear time usingtechniques developed in [31]. As a result, for any NS×1 vectorc, the matrix-vector product PΣc can be computed in O(NS)operations.

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The operator PΛH projects onto the space of divergence freeexpansion coefficients: ΣTPΛH = 0. Therefore, PΣPΛH =PΛHPΣ = 0. This also means that the loop-star decompositionis coefficient-wise orthogonal. PΛH can easily be found as

PΛH = 1− PΣ. (21)

In this way, the detection of global loops is avoided.

B. Elimination of the Static Null Space

The quasi-Helmholtz projectors PΛH and PΣ are nowapplied to both the test and trial side of the semi-discrete TD-EFIE (9):(

PΛH PΣ)(Zs Zs

Zs Zs + Zh

)(PΛH

PΣ

)j(t) = −e(t), (22)

where ZhPΛH = PΛHZh = 0 has been used. The operator Zh

(Equation (8c)) requires the evaluation of a temporal integral.This is avoided by introducing an auxiliary unknown y(t):

y(t) =(∂−1t PΣ + PΛH

)j(t)

⇐⇒ j(t) =(∂tP

Σ + PΛH)y(t), (23)

satisfying(PΛH PΣ

)(Zs ∂tZs

Zs ∂tZs + ∂tZh

)(PΛH

PΣ

)y(t) = −e(t).

(24)The operator Zs involves a temporal differentiation, whichannihilates constant-in-time currents. Therefore, constant loopcurrents reside in the null space of the operator on the lefthand side of (24). This is resolved by temporally integratingthe loop part of (24):

Z ′y(t) = −(∂−1t PΛH + PΣ

)e(t), (25)

Z ′ =(PΛH PΣ

)(∂−1t Zs Zs

Zs ∂tZs + ∂tZh

)(PΛH

PΣ

)(26)

=(∂−1t PΛH + PΣ

)Z(PΛH + ∂tP

Σ). (27)

Equation (25) is the quasi-Helmholtz projected TD-EFIEor qHP-TDEFIE. Note, however, that it is still continuous intime. A suitable discretization strategy is developed in the nextsection.

The operator Z ′ is constructed in such way that it does notrequire the evaluation of a temporal integral, and it does notannihilate static loop currents. It is, in essence, the inverseFourier transform of the modified EFIE operator proposed in[22], up to irrelevant sign conventions. An alternative methodto obtain this operator is explored in Appendix A.

For slowly varying fields, the off-diagonal componentsas well as ∂tZs in the lower right block of (26) becomenegligible. The remaining dominant contributions (∂−1

t Zs and∂tZh) do neither contain explicit temporal differentiations orintegrations. These diagonal terms have a physical meaning:∂−1t Zs represents the electromagnetic vector potential, and∂tZh the electromagnetic scalar potential.

C. Temporal Discretization

Next, the qHP-TDEFIE (25) is discretized in time. InSection II, j(t) was expanded in pulses. This implies that

y(t) (28)

=(∂−1t PΣ + PΛH

)j(t) (29)

=

NT∑i=1

(p(t− i∆t)PΛH + ∂−1

t p(t− i∆t)PΣ)ji. (30)

The testing coefficients are transformed in a similar way.

ti =

∫Rp(t− i∆t)Zj(t) dt (31)

=

∫Rp(t− i∆t)

(∂tP

ΛH + PΣ)Z ′y(t)dt (32)

=

∫R

(−∂tp(t− i∆t)PΛH

)Z ′y(t)dt

+

∫R

(p(t− i∆t)PΣ

)Z ′y(t)dt. (33)

Applying this discretization scheme to the qHP-TDEFIEwould, however, result in the same system as in Section II.In particular, the resulting MOT algorithm would also sufferfrom DC instability. This is due to the testing functions−∂tp(t − i∆t) = δ (t− (i− 1)∆t) − δ (t− i∆t), which actas discrete derivatives. Furthermore, the number of nonzeroZ-matrices would be infinite due to the infinite support of theexpansion function ∂−1

t p(t − i∆t). Finally also the scalingof the two blocks remains unchanged and leads to a con-dition number that scales like ∆t2 at large time steps (seeSection IV).

All these issues can be solved by directly discretizing (25)rather than inheriting the discretisation of the classic TD-EFIE(9). More specifically:

y(t) =

NT∑i=1

(p(t− i∆t)PΛH + h(t− i∆t)PΣ

)yi, (34)

yi = ∆t

i∑j=1

PΣjj + PΛH ji, (35)

ri =

∫R

(δ(t− i∆t)PΛH +

1

∆tp(t− i∆t)PΣ

)Z ′y(t)dt

=1

∆tPΣti −

i∑j=1

PΛHtj . (36)

In this discretization scheme, the loop part and the star part ofy(t) are expanded in pulse functions p(t−i∆t) (Fig. 2, middle)and hat functions h(t− i∆t) (Fig. 2, right), respectively. Theloop part and the star part of Z ′ are tested with Dirac deltadistributions δ(t− i∆t) (Fig. 2, left) and pulses, respectively.Note that the basis functions of both Helmholtz componentscan represent the constant-in-time function and that the testingfunctions of neither of the Helmholtz components disappearswhen applied to constant-in-time functions. The basis func-tions of both Helmholtz components of y(t) are normalizedto 1. The testing functions of both Helmholtz componentsare also scaled equally in the sense that both

∫R δ(t)dt and

5

∫R

1∆tp(t)dt equal 1. This is the origin of the factor 1

∆t in (36).The global factor ∆t in (35) is not necessary for balancing,but results in a well defined limit for the system matrices aslarge time step, as will be detailed in Section IV.

This expansion and testing scheme is now applied to theqHP-TDEFIE (25).∫

R

(δ(t− j∆t)PΛH +

1

∆tp(t− j∆t)PΣ

)(Eq. (25)) dt,

(37)for j = 1, 2, ..., NT , or

−Z′0yj =

j∑i=1

Z′iyj−i + e′j , (38)

where the matrices Z′i are constructed from four components

Z′i =(PΛH PΣ

)(Z′LLi Z′LS

i

Z′SLi Z′SS

i

)(PΛH

PΣ

), (39)

which are explicitly given in Equations (40a)-(40d). Theexcitation vector e′j is given by

e′j =

∫R

(δ(t− j∆t)∂−1

t PΛH +1

∆tp(t− j∆t)PΣ

)e(t) dt.

(41)Once the expansion coefficients yi are found, the physical

current j(r, t) on Γ can be computed as

j(r, t) =

NS∑m=1

NT∑i=1

[ji]m p(t− j∆t)fm(r), (42)

ji = PΛHyi + PΣ 1

∆t(yi − yi−1) . (43)

The use of different temporal basis and testing functionsfor the loop and star components is necessary to obtain astable MOT scheme. If y(t) is expanded in a single setof basis functions (h(t − i∆t) or p(t − i∆t)), and (25) istested with a single set of testing functions (p(t − i∆t) orδ(t−i∆t)), high frequency instabilities are encountered. Whena stability analysis is conducted as in [16], the eigenvaluesof the companion matrix are not confined to the unit circle,indicating that the scheme is unstable.

In contrast to Z , Z ′ does not contain a temporal integral.As a consequence, the number of nonzero matrices Z′i is finitein this scheme. No further manipulation or auxiliary quantitiesare required.

The integrals (40a)-(40d) can be interpreted as the interac-tions which are also found in traditional collocation-in-timeschemes, meaning that they can be accelerated using fasttechniques such as PWTD [27], [28].

D. Static Null SpaceConsider a constant solenoidal current j(r, t) = jL(r),

divΓj(r) = 0, with RWG expansion coefficients ji = jL =PΛH jL. This current is annihilated by the TD-EFIE operator,in the continuous as well as in the discrete setting:

(T j) (r, t) = 0 ∀t > 0,∀r ∈ Γ, (44)(Zj) (t) = 0 ∀t > 0, (45)

i∑j=0

Zj ji−j = 0 i = 0, 1, 2, ... (46)

This is the origin of the DC instability encountered in standardTD-EFIE simulations.

For solenoidal currents, j(t) = y(t) and ji = yi. Thesefunctions are not annihilated by the qHP-TDEFIE operator:

(Z ′y) (t) 6= 0 ∀t > 0. (47)

Moreover, because the trial functions can resolve constant-in-time functions, this property is conserved upon temporaldiscretization. Therefore, the qHP-TDEFIE does not allowconstant in time solenoidal currents as sourceless regimesolutions. This immediately implies that the qHP-TDEFIE isnot susceptible to DC instabilities.

IV. LOW FREQUENCY LIMIT

In this section, the low frequency limit of the system matrixZ0 (standard TD-EFIE, Section II) and Z′0 (qHP-TDEFIE,Section III) is investigated. For this, the scatterer is assumedto be small, i.e., with diameter D c∆t.

A. Low Frequency Limit of the Standard TD-EFIE

The TD-EFIE system matrix is split into a singular and ahypersingular part

Z0 = Zs0 + Zh

0 , (48)

[Zs0]mn = η

∫Rdt p(t)

∫Γ

(n× fm(r)) · Ts fnp (r, t)ds

= η ∆t

∫Γ

(n× fm(r)) · Ts fnh (r, 0)ds, (49)[Zh

0

]mn

= η

∫Rdt p(t)

∫Γ

(n× fm(r)) · Th fnp (r, t)ds

= η ∆t

∫Γ

(n× fm(r)) · Th fnh (r, 0)ds. (50)

Here we used the fact that a temporal Galerkin scheme isequivalent with a collocation scheme with as effective basisfunction the anti-convolution of the basis and testing functionof the Galerkin scheme (see (15)). Assuming that D c∆t,the integrand of (49) is constant in time:

∂th(0−R/c) =1

∆tfor R < c∆t (51)

Therefore,

[Zs0]mn = −µ ∆t

∫Γ

dsfm(r) ·∫

Γ

ds′fn(r′)∂th(−R/c)

4πR

= −µ∫

Γ

ds fm(r) ·∫

Γ

ds′fn(r′)

4πR

= [Zsstat]mn (52)

where Zsstat is the RWG discretization of the static vector

potential, which is independent of ∆t. For the hypersingularpart, the time dependence of the integrand of (50) can beapproximated by a Taylor series:

∂−1t h(0−R/c) = ∆t

(1

2+O

(R

c∆t

)), (53)

6

[Z′SSi

]mn

=η

∆t

∫Rdt p(t− i∆t)

∫Γ

ds (n× fm(r)) · ∂tT fnh (r, t)

= η

∫Γ

ds (n× fm(r)) · ∂tT fnTph (r, i∆t), (40a)[Z′SLi

]mn

=η

∆t

∫Rdt p(t− i∆t)

∫Γ

ds (n× fm(r)() · Ts fnp (r, t)

= η

∫Γ

ds (n× fm(r)) · Ts fnh (r, i∆t), (40b)[Z′LSi

]mn

= η

∫Rdt δ(t− i∆t)

∫Γ

ds (n× fm(r)) · Ts fnh (r, t)

= η

∫Γ

ds (n× fm(r)) · Ts fnh (r, i∆t), (40c)[Z′LLi

]mn

= η

∫Rdt δ(t− i∆t)

∫Γ

ds (n× fm(r)) · ∂−1t Ts fnp (r, t)

= η

∫Γ

ds (n× fm(r)) · ∂−1t Ts fnp (r, i∆t), (40d)

Tph(t) =1

∆t

∫Rp(τ)h(t+ τ)dτ. (40e)

leading to[Zh

0

]mn

= −∆t

ε

∫Γ

ds divΓfm(r)

p.v.

∫Γ

ds′div′Γfn(r′)∂−1

t h(−R/c)4πR

= ∆t2(

1

2

[Zh

stat

]mn

+O(D

c∆t

)), (54)[

Zhstat

]mn

= −1

ε

∫Γ

ds ∇ · fm(r)

∫Γ

ds′∇′ · fn(r′)

4πR,(55)

where Zhstat is the RWG discretization of the static scalar

potential, which is also independent of ∆t.Thus, for c∆t → +∞, and considering that Zh

0 =PΣZh

0PΣ,

Z0 → Zsstat +O

(D

c∆t

)+∆t2PΣ

(1

2Zh

stat +O(D

c∆t

))PΣ, (56)

or

Z0 →(PΛH PΣ

)(O(1) O(1)O(1) O(∆t2)

)(PΛH

PΣ

), (57)

which leads to a condition number that grows ∝ ∆t2. In [21],this ill-conditioning is resolved for simply connected structuresby scaling the spatial local loop functions proportionally to ∆t.

B. Low Frequency Limit of the qHP-TDEFIEThe same approach is applied to the four components of the

qHP-TDEFIE. First, the loop-loop-part (40d) :[Z′LL

0

]mn

= η

∫Γ

ds (n× fm(r)) · ∂−1t Ts fnp (r, 0)

= −µ∫

Γ

ds fm(r) ·∫

Γ

ds′fn(r′) p(0−R/c)

4πR(58)

Since the scatterer has diameter D < c∆t,[Z′LL

0

]mn

= −µ∫

Γ

ds fm(r) ·∫

Γ

ds′fn(r′)

4πR(59)

= [Zsstat]mn . (60)

The star-loop part (40b) and loop-star part (40c) are, up to afactor ∆t, equal to Zs

0 (52):[Z′LS

0

]mn

=[Z′SL

0

]mn

=1

∆t[Zs

0]mn =1

∆t[Zs

stat]mn .

(61)Finally, the star-star part is split in two contributions[Z′SS

0

]mn

=[Z′SSs,0

]mn

+[Z′SSh,0

]mn

, (62)[Z′SSh,0

]mn

= η

∫Γ

ds (n× fm(r)) · ∂tTh fnTph (r, 0),[Z′SSs,0

]mn

= η

∫Γ

ds (n× fm(r)) · ∂tTs fnTph (r, 0),

The time dependence of these integrands can be approximatedas follows, for R/c ∆t:

Tph(0−R/c) =1

2+O

(R

c∆t

), (63)

∂2t Tph(0−R/c) =

1

∆t2. (64)

Then,[Z′SSs,0

]mn

=1

∆t2[Zs

stat]mn , (65)[Z′SSh,0

]mn

= −ε∫

Γ

ds ∇ · fm(r)∫Γ

ds′∇′ · fn(r′)

4πR

(1

2+O

(R

c ∆t

))=

1

2

[Zh

stat

]mn

+O(∆t−1). (66)

7

Thus, when ∆t→ +∞,

Z′0 → PΛHZsstatP

ΛH +1

2PΣZh

statPΣ +O

(∆t−1

), (67)

or

Z′0 →(PΛH PΣ

)( O(1) O(∆t−1

)O(∆t−1

)O(1)

)(PΛH

PΣ

),

(68)which leads to a condition number that is asymptoticallyconstant. This is the result of rescaling the temporal basis andtesting functions associated with the star part with a factor1

∆t (equations (35) and (36), respectively). This contrasts withthe approach used in [21], where the spatial basis and testingfunctions were rescaled.

V. NUMERICAL RESULTS

A. Torus

As a first example, scattering by a torus with large radius0.8 m and small radius 0.2 m (Fig. 3) is examined. The torusis illuminated by a Gaussian-in-time plane wave

ei(r, t) =4A

w√πp exp

(−(

4

w

(c(t− t0)− k · r

))2),

(69)with amplitude A = 1 V , polarization p = 1x, direction k =1z , width w = 10 m and time of arrival t0 = 100 ns.

The torus is approximated by a triangle mesh on whichNS = 918 RWG functions are defined. The time step ischosen as 0.83 ns (or c∆t = 0.25 m). The scattering problemis solved using the EFIE (Section II), the dot-trick Calderonpreconditioned EFIE [16] and the qHP-TDEFIE (Section III).

The resulting current on the edge indicated by the arrow inFig. 3, is shown in Fig. 4 for the three simulations. At earlytimes (ct < 75 m), the three simulations match very well.However, the EFIE simulation ends in a constant loop current,whereas the dot-trick EFIE simulation exhibits a linearlyincreasing loop current. With the qHP-TDEFIE, the currentexpansion coefficient goes down to 10−14, at which point themachine precision comes into play. It has been verified thatthe resulting current is not a static loop current but randomnumerical noise.

Next, the time step is increased in order to study the lowfrequency limit of the matrix Z0. Its condition number isshown in Fig. 5 for the EFIE, the dot-trick EFIE and theqHP-TDEFIE. Whereas for the EFIE and the dot-trick EFIE,the condition number grows proportional to ∆t2, the qHP-TDEFIE remains constant.

B. Static Nullspace

Consider an MOT system with a finite number NX ofnonzero interaction matrices

−X0ji =

NX∑j=1

Xj ji−j + ei. (70)

The convolution operator allows constant-in-time regime so-lutions if

∃jc :

NX∑i=0

Xijc = 0, (71)

−1−0.5

00.5

1

−0.5

0

0.5

−0.10

0.10.2

xy

z

Fig. 3. Triangle mesh for a torus. The arrow points toward the edge on whichthe current is observed in Fig. 4.

0 50 100 150 200 25010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

c t [m]

j 0(t)

TD−EFIEdot−trick CP−EFIEqHP−TDEFIE

Fig. 4. Current on the torus, obtained using three different formulations. Thestandard TD-EFIE exhibits a constant-in-time DC instablity. The dot-trickCalderon preconditioned EFIE suffers from a linear-in-time DC instability.The qHP-TDEFIE is immune to DC instability.

10−1

100

101

102

100

101

102

103

104

105

106

107

c ∆t [m]

cond

(Z0)

TD−EFIEdot−trick CP−EFIEqHP−TDEFIE

Fig. 5. cond(Z0) for the torus, as a function of the time step ∆t.

8

and linear-in-time regime solutions if

∃jl :

NX∑i=0

Xi(NX − i+ 1)jl = 0. (72)

In other words, the constant and linear nullspaces of theoperator Xi can be investigated by computing the spectrumof

Xc =

NX∑i=0

Xi, (73)

Xl =

NX∑i=0

(NX − i+ 1)Xi. (74)

This approach can readily be applied to the dot-trick EFIEand the qHP-TDEFIE. For the standard EFIE, there are anunlimited number of nonzero interaction matrices. This is onlya technical complication, which is resolved in Appendix B.This type of analysis is much cheaper than a full eigenvalueanalysis on the system’s companion matrix (see [16]).

1) Cuboid: Now consider the cuboid mesh with dimensions2 × 2 × 2/3 m in Fig. 6. On this mesh, NS = 360RWG functions are defined, which can be combined into 121independent loops. The time step is fixed at c∆t = 1 m.

The singular values of Xc are shown in Fig. 7, top, forthe standard TD-EFIE, the dot-trick CP-EFIE and the qHP-TDEFIE. The 121 singular values smaller than 10−14 corre-spond to the constant loop currents that reside in the nullspaceof the EFIE operator. The dot-trick CP-EFIE and the qHP-TDEFIE do not exhibit a constant-in-time nullspace.

The singular values of Xl are shown in Fig. 7, bottom. Theabsence of very small singular values indicates that none ofthe three formulations exhibit a linear-in-time nullspace.

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−0.5

0

0.5

xy

z

Fig. 6. Cuboid (2× 2× 2/3 m) mesh, NS = 360.

2) Rectangular Torus: The experiment is now repeated forthe rectangular torus shown in Fig. 8. On this mesh, 384 RWGfunctions are defined, which can be combined into 127 localand 2 global loops.

Fig. 9 shows that the constant-in-time loops (127 localloops, and 2 global loops) are again in the nullspace of theEFIE. The nullspace of the dot-trick EFIE encompasses bothconstant-in-time and linear-in-time global loops. The qHP-TDEFIE again does not exhibit a static nullspace.

VI. CONCLUSION

The quasi-Helmholtz Projected TD-EFIE developed in thiscontribution, is a novel formulation of the TD-EFIE that

0 200 40010

−20

10−10

100

TD−EFIEconstant

0 200 40010

−20

10−10

100

dot−trick CP−EFIEconstant

0 200 40010

−20

10−10

100

qHP−TDEFIEconstant

0 200 40010

−20

10−10

100

TD−EFIElinear

0 200 40010

−20

10−10

100

dot−trick CP−EFIElinear

0 200 40010

−20

10−10

100

qHP−TDEFIElinear

Fig. 7. Spectral analysis of the static nullspace of the cuboid. Top: constant-in-time currents, bottom: linear-in-time currents.

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−0.5

0

0.5

xyz

Fig. 8. Rectangular torus mesh, NS = 384.

is immune to spurious static currents, on both simply andmultiply connected structures. While it is based on the sepa-ration of quasi-Helmholtz components, it does not require theexplicit construction of a loop-star or a loop-tree basis, nor thedetection of global loops. The qHP-TDEFIE is discretized intime using a Galerkin method, with different basis and testingfunction combinations for each component. This is necessaryin order to obtain a stable marching-on-in-time scheme. Thetemporal basis and testing functions are chosen such that theresulting system of equations is immune to low frequencybreakdown, i.e., the system remains well conditioned for largetime steps.

APPENDIX

A. Alternative Form1) Standard EFIE: In Section II, a temporal Galerkin

discretization of the TD-EFIE was proposed, in which thetemporal testing and trial functions are both pulses p(t− i∆t).This yields a current that is piecewise constant in time.Alternatively, it is also possible to expand the current in hatfunctions:

j(t) =

NT∑i=1

ji h(t− i∆t). (75)

A stable MOT scheme can be obtained by testing the TD-EFIEwith Dirac distributions:∫

Rδ(t− j∆t) (Eq. (9)) dt j = 1, 2, 3, ..., NT . (76)

9

0 200 40010

−20

10−10

100

TD−EFIEconstant

0 200 40010

−20

10−10

100

dot−trick CP−EFIEconstant

0 200 40010

−20

10−10

100

qHP−TDEFIEconstant

0 200 40010

−20

10−10

100

TD−EFIElinear

0 200 40010

−20

10−10

100

dot−trick CP−EFIElinear

0 200 40010

−20

10−10

100

qHP−TDEFIElinear

Fig. 9. Spectral analysis of the static nullspace of the rectangular torus. Top:constant-in-time currents, bottom: linear-in-time currents.

Since

η

∫Rdt δ(t− i∆t)

∫Γ

(n× fm(r)) · T fnh (r, t)ds

=η

∆t

∫Rdt p(t− i∆t)

∫Γ

(n× fm(r)) · T fnp (r, t)ds

=1

∆tZi, (77)

the interaction matrices and therefore the properties concern-ing stability and nullspaces are identical.

2) qHP-TDEFIE: A qHP-TDEFIE similar to this schemecan be developed by defining the auxiliary unknown y(t) as

y(t) =(PΣ + ∂tP

ΛH)j(t)

⇐⇒ j(t) =(PΣ + ∂−1

t PΛH)y(t), (78)

satisfying

Z ′y(t) = −(PΛH + ∂tP

Σ)e(t). (79)

The operator Z ′ is the same as in Section III-B, since

Z ′ =(∂−1t PΛH + PΣ

)Z(PΛH + ∂tP

Σ)

=(PΛH + ∂tP

Σ)Z(∂−1t PΛH + PΣ

). (80)

The temporal discretization of (79) is the similar as in Sec-tion III-C, with only minor modifications:

e′j =

∫R

(δ(t− j∆t)PΛH +

1

∆tp(t− j∆t)∂tPΣ

)e(t) dt,

(81)

j(r, t) =

NS∑m=1

NT∑j=1

[PΣyj + ∆t

j∑i=1

PΛHyi

]m

h(t−j∆t)fm(r).

(82)This results in a current that is also expanded in hats insteadof pulses. The MOT matrices Z′i are the same as in Section III.Therefore, this alternative scheme is also stable, and free ofa static nullspace. Even though the numerical integration ofPΛHy(t) (∆t

∑ji=1 P

ΛHyi in (82)) can result in a nonzeroloop component in j(t), it can never lead to DC instabilities.

B. Spectral Analysis of the Static Null Space for the EFIEThe hypersingular EFIO Th contains a temporal integral,

which corresponds to the integration of the current in order toobtain the electrical charge on Γ. This integral is transformedinto an infinite summation by the discretization procedure, i.e.,an infinite number of matrices Zi 6= 0.

It is therefore convenient to introduce additional unknownsto discretize the temporal integral of j(t)

s(t) = ∂−1t j(t) =

NT∑i=1

si h(t− i∆t), (83)

si = ∆t

i∑j=0

ji. (84)

Then,j∑

i=0

Zijj−i =

j∑i=0

Zsi jj−i +

j∑i=0

Zh

i sj−i, (85)

[Zsi ]mn = η

∫Rdt p(t− i∆t)∫

Γ

(n× fm(r)) · Ts fnp (r, t)ds, (86)[Zh

i

]mn

= η

∫Rdt p(t− i∆t)∫

Γ

(n× fm(r)) · ∂tTh fnh (r, t)ds,(87)

where Zsi = Z

h

i = 0 for i > NZ . Currents that are constantin time satisfy

ji = jc, (88)si = (i+ 1)∆t jc, (89)

and belong to the null space of the EFIE if(∑i

Zsi + ∆t

∑i

(NZ − i+ 1)Zh

i

)jc = 0. (90)

Currents that are linear in time satisfy

ji = (i+ 1)jl, (91)

si = ∆t(i+ 1)(i+ 2)

2jl, (92)

and belong to the null space of the EFIE if(∑i

(NZ − i+ 1)Zsi

)jl

+ ∆t

(∑i

(NZ − i+ 1)(NZ − i+ 2)

2Zh

i

)jl = 0.(93)

Therefore, the static null space of the EFIE operator can beinvestigated by computing the spectrum of

Xc =∑i

Zsi + ∆t

∑i

(NZ − i+ 1)Zh

i , (94)

Xl =∑i

(NZ − i+ 1)Zsi

+∆t∑i

(NZ − i+ 1)(NZ − i+ 2)

2Zh

i . (95)

10

ACKNOWLEDGMENT

The work of Y. Beghein was supported by a doctoral grantfrom the Agency for Innovation by Science and Technologyin Flanders (IWT).

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Yves Beghein received the M.Sc.Eng. degree inEngineering Physics from Ghent University, Bel-gium, in 2011. His master’s dissertation dealt withthe efficient modeling of scattering by chiral media.He is currently pursuing a Ph.D. degree at theElectromagnetics Group at Ghent University, underthe advisorship of Prof. Kristof Cools (Universityof Nottingham, U.K.) and Prof. Daniel De Zutter(Ghent University). His research focusses on bothtime domain and frequency domain boundary ele-ment methods, their stable and accurate discretiza-

tions, and efficient solution methods.

11

Kristof Cools received the MEng degree in AppliedPhysics Engineering from Ghent University, Bel-gium, in 2004. His master’s dissertation dealt withthe full wave simulation of metamaterials using thelow frequency multilevel fast multipole method. Hereceived the Ph.D. degree from Ghent University in2008, under the advisership of Prof. F. Olyslager andProf. Eric Michielssen. In 2008, he was awarded theYoung Scientist Best Paper award at the InternationalConference on Electromagnetics and Advanced Ap-plications. In 2011, he was a visiting research fellow

at TELECOM Bretagne. In November 2011, he took up a position as Lecturerin Computational Electromagnetics at the University of Nottingham. Hiscurrent focuses on the spectral properties of the boundary integral operators ofelectromagnetics, on stable and accurate discretization schemes for frequencyand time domain boundary element methods, on domain decompositiontechniques, and on the implementations of algorithms from computationalphysics for high performance computing.

Francesco P. Andriulli Francesco P. Andriulli (S05, M 09, SM 11) received the Laurea in electricalengineering from the Politecnico di Torino, Italy,in 2004, the MSc in electrical engineering andcomputer science from the University of Illinois atChicago in 2004, and the PhD in electrical engineer-ing from the University of Michigan at Ann Arborin 2008. From 2008 to 2010 he was a ResearchAssociate with the Politecnico di Torino. Since 2010he has been with the Ecole nationale superieuredes telecommunications de Bretagne (TELECOM

Bretagne), Brest, France, where he is currently a Full Professor. His researchinterests are in computational electromagnetics with focus on frequency- andtime-domain integral equation solvers, well-conditioned formulations, fastsolvers, low-frequency electromagnetic analysis, and simulation techniquesfor antennas, wireless components, microwave circuits, and biomedical appli-cations.

Dr. Andriulli was the recipient of the best student paper award at the 2007URSI North American Radio Science Meeting. He received the first placeprize of the student paper context of the 2008 IEEE Antennas and PropagationSociety International Symposium. He was the recipient of the 2009 RMTGAward for junior researchers and was awarded two URSI Young ScientistAwards at the International Symposium on Electromagnetic Theory in 2010and 2013 where he was also awarded the second prize in the best paper contest.In addition, he co-authored another first prize conference paper (ICEAA2009), two honourable mention conference papers (ICEAA 2011, URSI/IEEE-APS 2013), and other three finalist conference papers (URSI/IEEE-APS 2012,URSI/IEEE-APS 2007, URSI/IEEE-APS 2006). Moreover, he received the2014 IEEE AP-S Donald G. Dudley Jr. Undergraduate Teaching Award, the2014 URSI Issac Koga Gold Medal and the 2015 EurAAP Leopold B. FelsenAward for Excellence in Electrodynamics.

Dr. Andriulli is a member of Eta Kappa Nu, Tau Beta Pi, Phi Kappa Phi,and of the International Union of Radio Science (URSI). He serves as anAssociate Editor for the IEEE Transactions on Antennas and Propagation,IEEE Antennas and Wireless Propagation Letters, and IEEE Access.