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1 Uku, H. Arda and Bogaert, Ignace and Coos, Kristof and Andriui, Francesco P. and Bagci, Hakan (2017) of the time-domain MFIE at ow frequencies. IEEE Antennas and Wireess Propagation Letters, 16. pp ISSN Access from the University of Nottingham repository: Copyright and reuse: The Nottingham eprints service makes this work by researchers of the University of Nottingham avaiabe open access under the foowing conditions. This artice is made avaiabe under the University of Nottingham End User icence and may be reused according to the conditions of the icence. For more detais see: A note on versions: The version presented here may differ from the pubished version or from the version of record. If you wish to cite this item you are advised to consut the pubisher s version. Pease see the repository ur above for detais on accessing the pubished version and note that access may require a subscription. For more information, pease contact eprints@nottingham.ac.uk

2 IEEE ANTENNAS WIRELESS PROPAG. LETT., VOL. XX, NO. X, MONTH YEAR 1 Mixed Discretization of the Time Domain MFIE at Low Frequencies H. Arda Ükü, Member, IEEE, Ignace Bogaert, Kristof Coos, Francesco P. Andriui, Senior Member, IEEE, and Hakan Bağcı, Senior Member, IEEE Abstract Soution of the magnetic fied integra equation (MFIE), which is obtained by the cassica marching on-in-time (MOT) scheme, becomes inaccurate when the time step is arge, i.e., under ow-frequency excitation. It is shown here that the inaccuracy stems from the cassica MOT scheme s faiure to predict the correct scaing of the current s Hemhotz components for arge time steps. A recenty proposed mixed discretization strategy is used to aeviate the inaccuracy probem by restoring the correct scaing of the current s Hemhotz components under ow-frequency excitation. Index Terms Marching on-in-time (MOT) method, magnetic fied integra equation (MFIE), transient anaysis, ow-frequency anaysis, Buffa-Christiansen functions, mixed discretization. I. INTRODUCTION THE cassica marching on-in-time (MOT) scheme deveoped for soving time-domain eectric and magnetic fied integra equations (TD-EFIE and TD-MFIE) use Rao-Witon- Gisson (RWG) and poynomia basis functions to expand the current in space and time, respectivey. This expansion is inserted into the TD-EFIE and TD-MFIE and the resuting equations are tested with RWG functions at discrete times. The conditioning and accuracy of this cassica MOT-TD- EFIE sover at ow frequencies, i.e., when the scatterer size is comparabe to c t, have been studied extensivey [1 [4. Here, c is the speed of ight and t is the time step. Under this condition, the MOT-TD-EFIE matrix system becomes iconditioned and cannot be soved using iterative schemes. This probem, which is known as the ow-frequency breakdown of the EFIE, stems from the fact that the Hemhotz decomposed MOT-TD-EFIE matrix is not baanced in scaing with t as t and can be remedied with oop/star [1 and hierarchica preconditioning techniques [2 [4. On the other hand, behavior of the MOT-TD-MFIE matrix system as t has never been investigated. This work, for the first time, studies this behavior. Its contribution is twofod: (i) It rigorousy shows that the soution of the cassica H. A. Ükü was with the Division of Computer, Eectrica, and Mathematica Science and Engineering, KAUST, Thuwa, , Saudi Arabia. He is now with the Department of Eectronics Engineering, Gebze Technica University, Kocaei, Turkey. I. Bogaert was with the Department of Information Technoogy, Ghent University, Ghent, Begium. He is now with the ArceorMitta, Ghent. Begium. K. Coos is with the Department of Eectrica and Eectronic Engineering, University of Nottingham, Nottingham, UK. F. P. Andriui is with the Microwave Department, TELECOM Bretagne, Brest, France. H. Bağcı is with the Division of Computer, Eectrica, and Mathematica Science and Engineering, KAUST, Thuwa, , Saudi Arabia (emai: hakan.bagci@kaust.edu.sa). Manuscript received June 22, MOT-TD-MFIE matrix system does not scae correcty in t as t. The non-soenoida component of the current scaes as, which does not yied a finite vaue for the charge when integrated in time. Consequenty, the accuracy of the soution deteriorates regardess of the integration rue used for computing the MOT matrix entries. (ii) It shows that the mixed discretization scheme, which has been originay deveoped in [5 for soving the frequency domain MFIE, restores the correct scaing, i.e., the non-soenoida component of the current scaes as O( t 1 ). Consequenty, the soution of the mixed-discretized TD-MFIE maintains its accuracy for arge t. This is aso shown by numerica resuts. [5 uses divergence conforming RWG and cur conforming Buffa-Christiansen (BC) functions [6, i.e., ˆn BC, as basis and testing functions. Unike the cassica discretization, this scheme conforms with respect to the function spaces of the MFIE operator s both domain and range [5 [8 and preserves the correct frequency scaing of the soution s Hemhotz components [9, [10. As a resut, soution of the mixed-discretized MFIE is more accurate than the cassicaydiscretized MFIE especiay at ow frequencies [5 [10. II. FORMULATION A. TD-MFIE and MOT Scheme Let S represent the surface of a perfect eectric conductor residing in an unbounded homogeneous background medium. A magnetic fied H i (r, t) bandimited to f max is incident on the conductor. Enforcing the boundary condition on the tota magnetic fied on S yieds the TD-MFIE [11, [12: ˆn(r) H i (r, t) = 1 2 J(r, t) ˆn(r) S J(r, t R/c) dr. 4πR (1) Here, R = r r is the distance between observer and source points, r and r, and ˆn(r) is the outward pointing unit norma vector on S. To numericay sove (1), J(r, t) is approximated using spatia and tempora basis functions, f n (r) and T i (t): J(r, t) = N i=1 I i,nt i (t)f n (r). (2) Here, f n (r) are divergence conforming RWG functions, T i (t) = T (t i t), where T (t) is first order piecewise poynomia Lagrange interpoation function [11, [12, and I i,n are the unknown current coefficients. Inserting (2) into (1) and testing the resuting equation by t m (r), m = 1 : N, at times t = t yied the MOT matrix system [11, [12: Z 0 I = V 1 i=0 Z ii i, = 1 : N t. (3)

3 IEEE ANTENNAS WIRELESS PROPAG. LETT., VOL. XX, NO. X, MONTH YEAR 2 Here, {I i } n = I i,n, {V } m = V {t m }, and {Z k } m,n = Z k {t m, f n }, where the operators V {t m }, G k {t m, f n }, K k {t m, f n }, Z k {t m, f n } are defined as V {t m } = t m (r) ˆn H i (r, t)dr (4) Z k {t m, f n } = 1 2 G k{t m, f n } 1 4π K k{t m, f n } (5) G k {t m, f n } = t m (r) f n (r)t (k t)dr (6) K k {t m, f n } = (7) [ T (k t R/c) t m (r) ˆn(r) f n (r )dr dr. R S n The choice of t m (r) determines the spatia discretization schemes termed cassica and mixed as detaied in Sections II-B and II-C, respectivey. t depends ony on f max : t = 1/(αf max ), where α is the over-samping factor and 5 α 20. For a given spatia discretization, choosing a high vaue for α increases N t unnecessariy without any additiona gain in accuracy. When f max is sma, t shoud ideay be chosen arge. B. Cassica Discretization It has been shown in [5 and [8 that to obtain accurate resuts, the discretization of an integra equation shoud be conforming with respect to the function spaces, where the range and domain of the integra operator reside, and the resuting matrix system shoud be we-conditioned. For the MFIE, conforming discretization means that the testing functions t m (r) shoud reside in the dua space of the divergence conforming RWG basis functions f n (r). Cur conforming RWG testing functions t m (r) = ˆn(r) f m (r) satisfy this condition. However, the resuting Gram matrix with entries G 0 {ˆn f m, f n } is singuar, which makes the soution of (3) impossibe. Therefore, in the iterature, the choice t m (r) = f m (r) is adopted. In this work the scheme resuting from this choice of testing function is termed the cassica discretization scheme. In what foows here, it is rigorousy shown that the current obtained by soving the cassicay-discretized TD- MFIE has incorrect scaing in t under pane-wave excitation. It shoud be noted here that the resuts of the anaysis carried out in Sections II-B and II-C are vaid for any excitation that can be represented by a pane-wave expansion. The behavior of the Hemhotz components of the current as t can be anayzed by decomposing the RWG space into two components spanned by oop and star functions. Assume that J(r, t) is approximated as J(r, t) = N + N s i=1 I i,nt i (t)f n(r) (8) i=1 Is i,nt i (t)f s n(r). Here, N +N s = N and f n(r) and f s n(r) are oop and star basis functions, which are constructed from inear combinations of RWG functions [13. Inserting (8) into (2) and testing the resuting equation with f m(r), m = 1 : N and f s m(r), m = 1 : N s, at times t = t yied Z s 0 I s = V s 1 i=0 Zs ii s i, = 1 : N t. (9) Here, the entries of the bocks of the matrix Z s k and the vectors Vi s and I s i are given by Z k {f m, fn}, m = 1 : N, n = 1 : N {Z s Z k } m,n = k {fm N s, fn}, m = N + 1 : N, n = 1 : N Z k {f m, fn N s }, m = 1 : N, n = N + 1 : N Z k {fm N s, fn N s }, N + 1 : N, n = N + 1 : N { {V s V {f } m = m}, m = 1 : N V {fm N s (10) }, m = N + 1 : N { {I {I s i } n = i } n = Ii,n, n = 1 : N {I s i } n N = Ii,n N s, n = N + 1 : N. The scaing of I s as t can be obtained from the scaing of Z s 0 and V s. It is obvious that G 0{fm, p fn} q, p, q {, s}. Consider the Tayor series expansion of the gradient term in (7) with k = 0: [ [ T ( R/c) ( 1) p T (p) (0)R p 1 = R p=0 p!c p (11) = T (0) R 1 + ˆR ( 1) p T (p) (0)R p 2 p=2 p!c p. Here, T (p) (0) = p t T (t) t=0 and ˆR = (r r )/R. The first term in the right hand side (RHS) of (11) scaes as. The terms in the summation scae as O( t p ), p 2, due to the derivatives T (p) (0). By inserting (11) into the expression of K 0 {fm, p fn}, q p, q {, s}, one can show that, its dominant terms a scae as. From expressions of V {fm} and V {fm}, s it can be concuded that V {fm} and V {fm} s scae as. Consequenty, one obtains Z s 0 [, V s [, I s i [. (12) Inserting (8) into the equation of continuity ρ(r, t) = t 0 J(r, τ)dτ and using the fact that f n(r) = 0, one can obtain ρ(r, t) = N s i=1 Is i,n f s n(r) t 0 T i (τ)dτ. (13) Here, ρ(r, t) is the charge density. The integra in (13) scaes as O( t). Therefore, I s i has to scae as O( t 1 ) to obtain a finite vaue for ρ(r, t). This immediatey shows that I s i obtained by soving (9) does not scae correcty as t. C. Mixed Discretization scheme uses the rotated BC functions as testing functions, i.e., t m (r) = ˆn(r) g m (r), m = 1 : N, where g m (r) denote the divergence conforming BC functions [6. Since ˆn(r) g m (r) are cur conforming, the mixed discretization is conforming with respect to the function space of the MFIE operator s range. Additionay, the resuting Gram matrix with entries G 0 {ˆn g m, f n } is we-conditioned [14. In what foows here, it is shown that the current obtained by

4 IEEE ANTENNAS WIRELESS PROPAG. LETT., VOL. XX, NO. X, MONTH YEAR Cassica discretization (ev=1) Cassica discretization (ev=2) (ev=1) 10-6 (ev=2) Fig. 1. L 2 -norm of the star coefficients at the first time step, t = t. soving the mixed-discretized TD-MFIE has the correct scaing in t under pane-wave excitation. The behavior of I s i as t is anayzed by decomposing the RWG space into two subspaces spanned fn(r) and fn(r), s respectivey. Expansion (8) is inserted into (1) and the resuting equation is tested with oop and star functions generated from inear combinations of BC functions, ˆn(r) gm(r), m = 1 : N and ˆn(r) gm(r), s m = 1 : N s, at times t = t. The resuting inear MOT system reads Z s 0 I s = Ṽ s 1 Z s ii s i, = 1 : N t. (14) i=0 The entries of the matrix Z s k and the vector Ṽs are given by the expressions obtained by repacing fm and fm s with ˆn as t can be obtained from the scaing of Z s 0 and Ṽ s g m and ˆn g s m, in (10) respectivey. The scaing of I s The dominant terms in Z 0 {ˆn g m, f s n}, Z 0 {ˆn g s m, f n}, and Z 0 {ˆn g s m, f s n} come from G 0 {ˆn g m, f s n}, G 0 {ˆn g s m, f n}, and G 0 {ˆn g s m, f s n}, respectivey. Therefore, they a scae as. On the other hand, anaysis of Z 0 {ˆn g m, f n} s scaing requires a more detaied investigation as described next. The RWG and BC oop functions can be represented as the surface cur of the pyramid shaped functions: f n(r) = ˆn(r) S ϕ n (r) (15) g m(r) = ˆn(r) S ϑ m (r) (16) where ϕ n (r), r Sn [10, and ϑ m (r), r [6, are the pyramid shape functions, Sn and, represent the union of trianguar patches that support ϕ n (r) and ϑ m (r), and S is the surface gradient operator. It shoud be noted that ϕ n (r) and ϑ m (r) are continuous on trianguar patches and vanish at the boundaries of Sn and. Inserting (15) and (16) into G 0 {ˆn gm, fn} and appying surface divergence theorem to the resuting equation yied: G 0 {ˆn gm, fn} = ϑ m (r) ˆn(r) [ S S ϕ n (r) dr ϑ m (r) ˆm(r) [ˆn(r) S ϕ n (r) dr (17) Here, ˆm(r) is the outward pointing unit norma vector of, the boundary of. Since ϑ m (r) = 0 for r and. S S ϕ n (r) = 0, then G 0 {ˆn gm, fn} = 0 and Z 0 {ˆn gm, fn} = K 0 {ˆn gm, fn}/(4π). Inserting (10) into the expression of K 0 {ˆn gm, fn} yieds: K 0 {ˆn gm, fn} = + gm(r) S p=2 n gm(r) R 1 fn(r )dr dr (18) S n ( 1) p T (p) (0)R p 2 ˆR p!c p fn(r )dr dr. The first term of the RHS in (18) represents the static magnetic fied due to a oop source tested by a oop function and is zero [15. Terms of the summation in the RHS scae as O( t p ), p 2, due to derivatives T (p) (0). Therefore, the dominant term in K 0 {ˆn gm, fn} and hence Z 0 {ˆn gm, fn} scaes as O( t 2 ). As a resut, Z s 0 [ O( t 2 ). (19) The scaing of Ṽ s can be determined using V {ˆn gm} and V {ˆn gm}. s It can be easiy seen that V {ˆn gm} s. Inserting (16) into the expression of V {ˆn g m} and appying the chain rue and severa vector manipuations to the resuting equation yied: V {ˆn gm} = S [ϑ m (r) ˆn(r) H i (r, t ) dr ϑ m (r) ˆn(r) [ H i (r, t ) dr. (20) Surface divergence theorem is appied to the first term of the RHS. The second term is simpified assuming H i (r, t) is the magnetic fied of a pane wave. These operations yied: V {ˆn gm} = ˆm(r) [ϑ m (r) ˆn(r) H i (r, t ) dr + c 1 (ˆn(r) ˆk) ϑ m (r) t H i (r, t)dr (21) where ˆk is the pane wave s direction of propagation. First integra in (21) is zero since ϑ m (r) = 0 for r. Derivative in the second term indicates that V {ˆn gm} O( t 1 ). Consequenty, one can obtain Ṽ s [ O( t 1 ), I s [ O( t 1. (22) ) Indeed, the scaing of I s obtained by soving (14) matches that predicted by the continuity equation as t. III. NUMERICAL RESULTS In this section, the MOT-TD-MFIE sover, which uses cassica and mixed discretization schemes, is appied to the characterization of transient scattering from a unit sphere that resides in free space and is centered at the origin. Retarded-time source integras and the test integras in the MOT matrix entries in (7) are computed using the semianaytica integration scheme described in [11, [12 and the Gauss-Legendre quadrature rue, respectivey. Two eves of numerica integration are used: (i) ev=1 uses seven quadrature points. (ii) ev=2 first divides the trianges into four and uses seven quadrature points in each sub-triange. The

5 IEEE ANTENNAS WIRELESS PROPAG. LETT., VOL. XX, NO. X, MONTH YEAR Cassica discretization (a) Cassica discretization (b) Cassica discretization (c) Fig. 2. The x-component of the range-corrected scattered eectric fied, r ˆx E scat (r) for (a) t = 1.33 ms, (b) t = 2 ms, and (c) t = 4 ms. excitation is a pane wave: H i (r, t) = η 1 ŷg(t + r ẑ/c) where G(t) = cos(2πf 0 [t t 0 )e (t t0)2 /(2σ 2) is a Gaussian puse with moduation frequency f 0 = 0.66f max, duration σ = 3.34/f max, and deay t 0 = 21.7/f max. To investigate the scaing of I s as t, f max is swept in the interva [24 khz-192 MHz, and t = 1/(10f max ). J(r, t) induced on the sphere is discretized using N = 750 RWG basis functions. Fig. 1 pots the L 2 norm of the star current coefficients at t = t, i.e., I s Ns 1 = I s 2 1,n, vs. t as fmax and t are swept. Note that I s 1 are computed via oop/star decomposition after the I 1 are computed by the MOT sover using the cassica and mixed discretization schemes. They are not obtained by soving the MOT systems in (9) or (14). Fig. 1 ceary demonstrates that I s 1 obtained by soving the cassicay-discretized TD-MFIE saturates and scaes as as t gets arger. On the other hand, I s 1 obtained by soving the mixed-discretized TD-MFIE scaes as O( t 1 ). These resuts verify the anaysis carried out in Sections II-B and II-C. Fig. 1 aso shows that the cassica discretization produces the wrong scaing regardess of the integration accuracy (due to non-conforming testing). On the other hand, higher integration accuracy heps the mixed discretization scheme achieve the correct scaing especiay as t gets arger. Figs. 2(a)-(c) pot the x-component of the (range-corrected) scattered eectric fied obtained from the three sets of MOT soutions with f 0 = 0 Hz and f max {75, 50, 25} Hz ( t {1.33, 2, 4} ms). The figures ceary show that, as t gets arger, the accuracy of the cassicay-discretized TD- MFIE s soution deteriorates whie the soution of the mixeddiscretized TD-MFIE maintains its accuracy. IV. CONCLUSION The TD-MFIE discretized using RWG basis and testing functions produces inaccurate resuts when the t is arge because this discretization scheme can not predict the correct scaing of the current s Hemhotz components as t. This can be avoided by using the mixed discretization scheme with RWG basis and BC testing functions that are conforming with respect to the function spaces of the MFIE operator. REFERENCES [1 N. W. Chen, K. Aygun, and E. Michiessen, Integra-equation-based anaysis of transient scattering and radiation from conducting bodies at very ow frequencies, IET Microwaves Antennas Propag., vo. 148, no. 6, pp , Dec [2 F. P. Andriui, H. Bagci, F. Vipiana, G. Vecchi, and E. Michiessen, A marching-on-in-time hierarchica scheme for the soution of the time domain eectric fied integra equation, IEEE Trans. Antennas. Propag., vo. 55, no. 12, pp , Dec [3, Anaysis and reguarization of the TD-EFIE ow-frequency breakdown, IEEE Trans. Antennas. Propag., vo. 57, no. 7, pp , Juy [4 H. Bagci, F. P. Andriui, F. Vipiana, G. Vecchi, and E. Michiessen, A we-conditioned integra-equation formuation for efficient transient anaysis of eectricay sma microeectronic devices, IEEE Trans. Adv. Packag., vo. 33, no. 2, pp , May [5 K. Coos, F. P. Andriui, D. D. Zutter, and E. Michiessen, Accurate and conforming mixed discretization of the MFIE, IEEE Antennas Wireess Propag. Lett., vo. 10, pp , [6 A. Buffa and S. H. Christiansen, A dua finite eement compex on the barycentric refinement, Math. Comput., vo. 76, pp , Dec [7 Y. Beghein, K. Coos, H. Bagci, and D. D. Zutter, A space-time mixed Gaerkin marching-on-in-time scheme for the time-domain combined fied integra equation, IEEE Trans. Antennas. Propag., vo. 61, no. 3, pp , Mar [8 S. H. Christiansen and J.-C. Nedeec, A preconditioner for the eectric fied integra equation based on Caderon formuas, SIAM Journa on Numerica Anaysis, vo. 40, no. 3, pp , [9 I. Bogaert, K. Coos, F. P. Andriui, and H. Bagci, Low-frequency scaing of the standard and mixed magnetic fied and Müer integra equations, IEEE Trans. Antennas. Propag., vo. 62, no. 2, pp , Feb [10 Y. Zhang, T. J. Cui, W. C. Chew, and J.-S. Zhao, Magnetic fied integra equation at very ow frequencies, IEEE Trans. Antennas. Propag., vo. 51, no. 8, pp , Aug [11 H. A. Uku and A. A. Ergin, Anaytica evauation of transient magnetic fieds due to RWG current bases, IEEE Trans. Antennas. Propag., vo. 55, no. 12, pp , Dec [12, Appication of anaytica retarded-time potentia expressions to the soution of time domain integra equations, IEEE Trans. Antennas. Propag., vo. 59, no. 11, pp , Nov [13 G. Vecchi, Loop-star decomposition of basis functions in the discretization of the EFIE, IEEE Trans. Antennas. Propag., vo. 47, no. 2, pp , Feb [14 F. P. Andriui, K. Coos, H. Bagci, F. Oysager, A. Buffa, S. Christiansen, and E. Michiessen, A mutipicative Caderon preconditioner for the eectric fied integra equation, IEEE Trans. Antennas. Propag., vo. 56, no. 8, pp , Aug [15 S. Y. Chen, W. C. Chew, J. M. Song, and J.-S. Zhao, Anaysis of ow frequency scattering from penetrabe scatterers, IEEE Trans. Geosci. Remote Sens., vo. 39, no. 4, pp , Apr