Andriulli, Francesco P

Bogaert, Ignace

Cools, Kristof

Michielssen, Eric

Ghent University Academic Bibliography

A Well-Conditioned Combined Field IntegralEquation Based on Quasi-Helmholtz ProjectorsFrancesco P. Andriulli∗ Ignace Bogaert† Kristof Cools‡ Eric Michielssen§Abstract — We introduce a novel combined ﬁeldintegral equation that does not suﬀer from inter-nal resonances and solves several drawbacks of ex-isting resonance-free formulations. The new equa-tion is obtained by combining a regularized electrictype operator with a new magnetic type operatorthat exhibits uniform frequency scaling when actingon, or being tested within, the harmonic Helmholtzsubspace for surface currents. With an appropriateuse of quasi-Helmholtz projectors, the equation isstable for arbitrarily low-frequencies. Numerical re-sults conﬁrm the theoretical developments and showthe eﬀectiveness of the scheme.1 INTRODUCTIONAll known integral equation techniques for sim-ulating scattering and radiation from arbitrarilyshaped, perfect electrically conducting objects suf-fer from one or more of the following shortcomings:(i) they give rise to ill-conditioned systems when thefrequency is low (ii) and/or when the discretizationdensity is high; (iii) their applicability is limited tothe quasi-static regime; (iv) they require a searchfor global topological loops; (v) they suﬀer from nu-merical cancelations in the solution when the fre-quency is very low. A recent paper [1] presented anew integral operator of the electric type that doesnot suﬀer from any of the above drawbacks.When the scatterer under consideration is closed,integral operators of the electric and magnetic typesuﬀer from interior resonances, i.e. null-spaces forwavenumbers that correspond to a resonance of aninterior problem. The presence of these resonancesoften negatively impacts the accuracy and solutiontime of solvers leveraging these operators. Whenstandard electric and magnetic ﬁeld operators areused, this problem can be remedied by using acombined ﬁeld operator, viz. a linear combinationof the electric and magnetic ﬁeld operator that isprovably resonance-free. Unfortunately, standardcombined ﬁeld operators inherit all the drawbacks(i)-(v).This contribution extends the recently developed∗Microwave Departement, E´cole Nationale Supe´rieure desTe´le´communications de Bretagne (TELECOM Bretagne), e-mail:francesco.andriulli@telecom-bretagne.eu†University of Ghent, e-mail:ibogaert@ugent.be‡University of Nottingham,e-mail:kristof.cools@nottingham.ac.uk§University of Michigan, e-mail:emichiel@umich.eduelectric type operator immune from (i)-(v) by de-veloping a new magnetic type operator that can beused to construct a combined ﬁeld operator thatdoes not suﬀer from (i)-(v) and is immune from in-terior resonances, i.e. that is uniquely solvable forall frequencies. The new formulation is obtainedstarting from a Helmholtz decomposition of twodiscretizations of the electric ﬁeld integral operatorand from a suitably symmetrized mixed discretiza-tion of the magnetic ﬁeld integral operator obtainedby using RWGs and dual bases functions, respec-tively. The new decomposition does not leverageLoop and Star/Tree basis functions; rather, it em-ploys projectors that derive from them and doesnot require the explicit detection of global topo-logical loops. The theoretical developments will becorroborated by numerical results, conﬁrming theeﬀectiveness of the newly developed method.2 BACKGROUND AND FORMULA-TIONConsider the surface Γ of an orientable PEC objectresiding in a space of permittivity  and permeabil-ity μ. Denote with nˆr its outward pointing unitnormal at r. Denote with Ω+ and Ω− the exteriorand interior regions of Γ, respectively. An incidentelectromagnetic ﬁeld(Ei(r),Hi(r))impinges onΓ, inducing a surface current density J(r). Thecurrent density J(r) can be retrieved by solvingthe Electric Field Integral Equation (EFIE)Tk (J) = −nˆr ×Ei (1)where Tk(J) = ik Ts(J) + 1ik Th(J) withTs(J) = nˆr ×∫Γeik|r−r′|4π |r − r′|J(r′) dr′, (2)Th(J) = −nˆr ×∇∫Γeik|r−r′|4π |r − r′|∇s · J(r′) dr′,(3)and k = 2π/λ = ω√μ. Alternatively, if Γ is closed,J(r) can be retrieved by solving the Magnetic FieldIntegral Equation (MFIE)(I2+Kk)(J) = η(nˆr ×Hi)(4)whereKk(J) = −nˆr ×∫Γ∇ eik|r−r′|4π |r − r′| × J(r′) dr′ (5)and η =√μ/. Note that with these deﬁnitionsthe current J(r) represents the jump over Γ of thetotal magnetic ﬁeld multiplied by the medium char-acteristic impedance.When the surface Γ is closed and when k corre-sponds to an interior resonance, i.e. to an eigen-value of the Laplacian in Ω−, both the EFIE andMFIE are not solvable [2]. When k approachessuch a resonant value, the linear systems arizingfrom the discretization of the EFIE and MFIE be-come almost singular and diﬃcult to solve. A sim-ilar problem plagues the Caldero´n preconditionedEFIE. A classical solution to this problem lever-ages the Combined Field Integral Equation (CFIE)[3], viz. a linear combination of EFIE and MFIEαTk (J) +(I2+Kk)(J) == −αnˆr ×Ei + η(nˆr ×Hi)(6)where α = 0 and real. The CFIE can be proven tobe uniquely solvable for all values of k. On the otherhand, the presence of the EFIE operator T resultsin ill-conditioning that, although milder than theill-conditioning of the EFIE, still renders the equa-tion diﬃcult to solve in many real case scenarios.One could attempt to obtain a resonance-free andwell-conditioned equation by linearly combining theMFIE and the Caldero´n preconditioned EFIE asαT 2k (J) +(I2+Kk)(J) == −αTk(nˆr ×Ei)+ η(nˆr ×Hi). (7)Unfortunately this choice does not lead to aresonance-free equation sinceT 2k = −I2+K2k = −(I2−Kk)(I2+Kk)(8)i.e. the null space of T 2k contains that of(I2 +Kk)and so does (7). A way to solve this problem isto precondition the EFIE operator T not with theoperator T itself, but with its localized counter-part. This localization can be obtained either by aspace windowing the Green’s function [4] or by us-ing, in the leftmost operator T , a purely complexwavenumber, as was proposed in [5], obtainingC (J) = −αTik(nˆr ×Ei)+ η(nˆr ×Hi)(9)whereC (J) = αTikTk (J) +(I2+Kk)(J) . (10)Although (9) is immune from both low-frequencyand dense discretization breakdown, it can beshown to suﬀer from very low-frequency solutioncancelations and to have a static null-space when-ever the geometry under consideration containshandles or holes (i.e. for non-simply connected ge-ometries). The reader is referred to [1] for a detaileddiscussion of these phenomena. Recently, an elec-tric operator that is immune from the above men-tioned problems has been proposed [1]. This opera-tor is obtained by using Helmholtz projectors thatproperly alter the low frequency behavior of thestandard Calderon EFIE. The resulting equationhas been shown to be stable till very low frequencyeven for non-simply connected geometries. Beingof electric type, the operator deﬁned in [1] still suf-fers from internal resonances and hence needs to beincluded in a suitable combined ﬁeld formulation.This is the objective of this contribution.The reader should note that a key property ofthe EFIE operator, fundamental for realizing theproperties obtained in [1] is that, if we consider theharmonic subspace of the Helmholtz decompositionof the surface current, the EFIE operator has thesame frequency scaling when being applied to orbeing tested with any element of the harmonic sub-space. This key property, unfortunately, does nothold for MFIE operator. This is the reason whyit is not possible to obtain an eﬀective combinedﬁeld formulations by simply using the electric typeoperator in [1] in an equation like (9). Instead, anew magnetic type operator should be deﬁned thatscales in a uniform way when acting on or testedwith the harmonic Helmholtz subspaces. This willbe the subject of next session where a new com-bined ﬁeld formulation will be proposed togetherwith the proof of its resonance-free behavior.3 A NEW COMBINED FIELD EQUA-TIONThe new combined ﬁeld equation we propose reads(η2(I2−Kik)(I2+Kk)(k) + TikTk)(J) =(I2−Kik)(nˆr ×H) + Tik (nˆr ×E) (11)We now prove that then the operator((I2−Kik)(I2+Kk)(k) + TikTk)is always invertible ∀k. Given that (I2 −K) (ik) isalways invertible, we will study the operator(I2+Kk)+((I2−Kik))−1TikTk.The following commutationT −1K +KT −1 = 0 (12)is a straightforward consequence of the sec-ond Caldero´n identity T −1K = T −1KT T −1 =−T −1T KT −1 = −KT −1. Thus, deﬁningA =((I2−Kik))−1Tikwe obtain(nˆr ×A)T =(nˆr ×((I2−Kik))−1Tik))T=(nˆr ×(T −1ik(I2−Kik))−1)T=(nˆr ×((I2+Kik))T −1ik)−1)T=(nˆr × Tik((I2+Kik))−1)T=(((I2+Kik))−1)Tnˆr × Tik= −nˆr ×((I2−Kik))−1nˆr × nˆr × Tik= nˆr ×((I2−Kik))−1Tik= nˆr ×A.Finally, given that nˆr × A = nˆr ×((I2 −Kik))−1 Tik is a real operator, thesymmetry implies it being hermitian, so thatx†(nˆr ×((I2 −Kik)(ik))−1 Tik)x is real andnonzero. By leveraging a straightforward extensionof Theorem 3.1 in [6], it follows that((I2−Kik)(I2+Kk)+ TikTk)is always invertible.4 DISCRETIZATION STRATEGYThe new operator will be discretized by adopting amixed-discretization strategy [7, 8] where the mag-netic operators are tested with Buﬀa-Christiansen(BC) basis functions. For the sake of brevity weconsider a discretization for the case of simply con-nected structures; minor modiﬁcations are requiredfor the non-simply connected case. The discretiza-tion reads:(η2M(G¯Tmix2 − K¯ikmix)(G¯Tmix)−1··(G¯Tmix2 + K¯kmix)M¯+MT¯ikBCMG¯−1mixM¯T¯kRWGM¯)I¯ = (13)η2M(G¯Tmix2 − K¯ikmix)(G¯Tmix)−1V¯H+MT¯ikBCMG¯−1mixM¯V¯where (T¯kRWG)i,j =〈nˆr × f i, Tk(f j)〉and(T¯kBC)i,j =〈nˆr × fBCi , Tk(fBCj )〉are the EFIEoperators discretized with standard RWG f[9] and BC basis functions fBC [10] respec-tively,(K¯kmix)i,j=〈nˆr × fBCi ,Kk(f j)〉is themixed discretized MFIE operator, (G¯mix)i,j =〈nˆr × f i,fBCj〉is the mixed Gram matrix,(V¯)i = −〈nˆr × f i, nˆr ×Ei〉and (V¯H)i =−〈nˆr × fBCi , η(nˆr ×H i)〉are the electric andmagnetic right hand sides respectively, and (I¯)j =Ij is the unknown current vector such that J(r) ≈∑Nn=1 Infn(r). Finally, the deﬁnition of the projec-torsM and M¯ is omitted here for space limitations,but it can be found in [1] for low frequency simu-lations. They are set equal to the identity, instead,for high frequency ones.5 NUMERICAL RESULTSThe numerical tests involve a sphere of unit ra-dius that is excited by a plane wave. The fact thatthe new equation is immune from the very low fre-quency current cancelation is conﬁrmed by Fig. 1(a)which show the far ﬁeld calculated using (13) at10−40Hz. From Fig. 1(a) it is clear that although astandard Caldero´n equation can provide a stable so-lution till relatively low frequencies, the new equa-tion is immune from the very low frequency cur-rent cancelation and provides stable solutions evenwhen the frequency is arbitrarily low. The reso-nance free behavior of the new equation is tested inFig. 1(b) where the new formulation clearly showsto be resonance-free.AcknowledgmentsThis work has been partially funded by the ANR(Agence Nationale de la Recherche) under the0 50 100 150 20010−20010−18010−16010−14010−120Angle (degrees)r2(|Eθ|2+|E φ|2 )Frequency: 10−40 Hz  Mie SeriesStandard CalderonThis Work(a) Sphere: far field calculated when the frequencyequals 10−40Hz.2 2.5 3 3.5 4x 108100101102103FrequencyCondition number  EFIEMixed MFIEThis work(b) Sphere: condition number as a function of frequency.Figure 1: Numerical resultsproject FASTEEG-ANR-12-JS09-0010. One of theauthors also thank GENCI (Grand Equipement Na-tional de Calcul Intensif) for providing the neces-sary computing facilities for this work.References[1] F. P. Andriulli, K. Cools, I. Bogaert, andE. Michielssen, “On a well-conditioned electricﬁeld integral operator for multiply connectedgeometries,” IEEE Trans. Antennas Propaga-tion, vol. 61, no. 4, pp. 2077–2087, 2013.[2] J.-C. Nede´le´c, Acoustic and ElectromagneticEquations. Springer, 2000.[3] J. R. Mautz and R. F. Harrington, “H-ﬁeld, E-ﬁeld, and combined ﬁeld solution for conduct-ing bodies of revolution,” Archiv. f. Elektroniku. Ubertragungstechnik, vol. 32, pp. 159–164,1978.[4] R. J. Adams, “Physical and analytical proper-ties of a stabilized electric ﬁeld integral equa-tion,” IEEE Trans. Antennas Propagation,vol. 52, no. 2, pp. 362–372, Feb. 2004.[5] H. Contopanagos, B. Dembart, M. Epton,J. Ottusch, V. Rokhlin, J. Visher, and S. M.Wandzura, “Well-Conditioned Boundary Inte-gral Equations for Three-Dimensional Electro-magnetic Scattering,” IEEE Trans. AntennasPropagation, vol. 50, no. 12, pp. 1824–1930,Dec. 2002.[6] O. Bruno, T. Elling, R. Paﬀenroth, andC. Turc, “Electromagnetic Integral EquationsRequiring Small Numbers of Krylov-subspaceIterations,” Journal of Computational Physics,vol. 228, pp. 6169–6183, 2009.[7] K. Cools, F. P. Andriulli, D. D. Zutter,and E. Michielssen, “Accurate and conformingmixed discretization of the mﬁe,” IEEE Anten-nas and Wireless Propagation Letters, vol. 10,pp. 528–531, 2011.[8] I. Bogaert, K. Cools, F. Andriulli, andD. De Zutter, “Low frequency scaling of themixed mﬁe for scatterers with a non-simplyconnected surface,” in International Confer-ence on Electromagnetics in Advanced Appli-cations (ICEAA), 2011, sept. 2011, pp. 951 –954.[9] S. M. Rao, D. R. Wilton, and A. W. Glisson,“Elettromagnetic scattering by surfaces of ar-bitrary shape,” IEEE Trans. Antennas Prop-agation, vol. AP-30, no. 3, pp. 409–418, May1982.[10] A. Buﬀa and S. H. Christiansen, “A dual ﬁ-nite element complex on the barycentric re-ﬁnement,” Math. Comp., vol. 76, no. 260, pp.1743–1769, 2007.

A well-conditioned combined field integral equation based on quasi-Helmholtz projectors

https://biblio.ugent.be/publication/4248759/file/4248765

A well-conditioned combined field integral equation based on quasi-Helmholtz projectors

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography