Time Domain Adaptive Integral Method for the Combined Field Integral Equation Ali E. Yılmaz*, Jian-Ming Jin, and Eric Michielssen Center for Computational Electromagnetics Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA E-mail:
[email protected]
1.
Introduction
Recently, several FFT-based algorithms for efficiently solving time domain electric field integral equations (EFIEs) have been proposed [1-5]. These solvers treat the spatial dimensions of the space-time integral equations in a manner analogous to their frequency-domain counterparts [6, 7]. In addition, by using multilevel FFTs in the time dimension, they reduce the temporal computational complexity of classical solvers without sacrificing their time-marching nature. Among these FFT-based algorithms, the time domain adaptive integral method (TDAIM) is the most practical one for analyzing transient scattering from arbitrarily shaped, free-standing structures. By creating an auxiliary uniform mesh that encloses the scatterer and using it to compute far-field interactions, TD-AIM reduces the O ( Nt N s2 ) computational complexity of a classical time domain EFIE solver to O( Nt N c log( N g N c )log N g ) . Here, the integral equation is discretized with N s basis functions describing currents on the scatterer surface, the auxiliary mesh has N c nodes, the simulation is performed for Nt time steps, and the longest transit time across the scatterer is N g time steps [1]. In this article, TD-AIM is used to accelerate the process of solving a combined field integral equation (CFIE). To evaluate the magnetic field contribution to the CFIE, the vector potential is computed on an auxiliary mesh and spatial finite differences [8] are used to evaluate its curl. Simulation results demonstrate the accuracy and efficiency of this approach.
2.
Time Domain Integral Equations
Let S denote the surface of a closed perfect electrically conducting body that resides in an unbounded lossless dielectric medium and that is illuminated by an electromagnetic field {Einc (r, t ), H inc (r , t )} . This incident field induces the surface current and charge densities J (r, t ) and ρ (r , t ) that are related through the continuity equation, ∂t ρ (r, t ) = −∇ ⋅ J (r, t ) . The scattered electromagnetic field is given by Escat (r, t ) = −∂ t A (r , t ) − ∇Φ (r, t ),
H scat (r, t ) = ∇ × A(r , t ) / µ ,
(1)
where the vector and scalar potentials are defined as
A(r , t ) = ∫∫ S
µ J (r ′, t − R / c) ds′, 4π R
Φ(r, t ) = ∫∫ S
ρ (r ′, t − R / c) ds′ . 4πε R
(2)
Here, R is the distance between source point r′ and observation point r , µ and ε are the permeability and permittivity of the dielectric medium, η = µ / ε is its intrinsic
0-7803-7846-6/03/$17.00 ©2003 IEEE
impedance, and c = 1/ µε is the speed of light. Enforcing (the temporal derivatives of) the fundamental boundary conditions on the electric and magnetic fields tangential to S yields the following time domain EFIE and MFIE:
nˆ (r ) × nˆ (r ) × ∂ t Einc (r, t ) = nˆ (r ) × nˆ (r ) × (∂ t2 A(r, t ) − ∇∂ t Φ (r, t )) nˆ (r ) × ∂ t
(EFIE)
= ∂ t J (r, t ) − nˆ (r ) × ∇ × ∂ t A(r, t ) / µ CFIE = η (1-α )MFIE+α EFIE.
Hinc (r, t )
(MFIE) (CFIE)
(3)
Here nˆ (r ) is an outward directed unit vector normal to S. A linear combination of the EFIE and the MFIE gives the CFIE, represented by the last line in (3) [9].
3.
Marching-on-in-Time (MOT) and TD-AIM
To numerically solve the time domain integral equations in (3) by MOT, J (r, t ) is discretized using N s spatial and Nt temporal basis functions as J (r , t ) ≅
N s Nt
∑ ∑ I k ′,l′S k ′ (r)T ( t − l ′∆t ) ,
(4)
k ′=1 l ′=1
where ∆t is the time-step size. After substituting (4) into (3), the system is tested with the spatial functions S k (r ) at times l ∆t , resulting in Nt N s equations for Nt N s unknowns. One of these equations is given below before discretizing J (r, t ) for ease of presentation:
∫∫η (1 − α )Sk (r) ⋅ nˆ (r) × ∂t Hinc (r, l ∆t ) + α Sk (r) ⋅ ∂t Einc (r, l ∆t )ds S
= η (1 − α ) ∫∫ S k (r ) ⋅ [∂ t J (r, l ∆t ) − nˆ (r ) × ∇ × ∂ t A(r , l ∆t ) / µ ]ds
(5)
S
+α ∫∫ S k (r ) ⋅ [∂ t2 A(r, l ∆t ) − ∇∂ t Φ (r, l ∆t )]ds.
(Tested CFIE)
S
Once the current density is discretized as in (4), the MOT matrix system l −1
Z0 I l = Vlinc − Vlscat = Vlinc − ∑ Zl −l ′I l ′ ,
for l = 1, 2,… , Nt
(6)
l ′=1
is obtained. The vectors Vlinc and Vlscat capture the effects, at time l ∆t , of the incident field and the fields radiated by currents active prior to time step l, respectively. The dominant cost of MOT is the computation of the matrix-vector multiplications on the right-hand side of (6), which will be accelerated using TD-AIM. To apply TD-AIM, each of the matrices Zl −l ′ is separated into two parts: near far Zl −l ′ = Zlfar −l ′ + Zl −l ′ . The entries of Zl −l ′ are approximated using equivalent point sources residing on an auxiliary uniform mesh, whereas the entries of Zlnear −l ′ are constructed classically and pre-corrected in anticipation of the near-field errors introduced through the use of global FFTs. Indeed, the multiplications of the far-field components by past current vectors in (6) are all efficiently computed by multilevel space-time FFTs. Details on how to approximate entries of Zlfar −l ′ using point sources were presented in [4] and [5] for the EFIE. Here we will focus on the MFIE component of the CFIE.
Substituting (4) into (5) and using the vector identity A ⋅ B × C = C ⋅ A × B , the MFIE contribution to the matrices Zl −l ′ is given by ′ ′ ′ ZlMFIE −l ′ ( k , k ) = ∫∫ S k (r ) ⋅ S k ′ (r )∂ t T ((l − l ) ∆t ) ds S
+ ∫∫ ( S k (r ) × nˆ (r ) ) ⋅ ∇ × ∫∫ µ S k ′ (r ′)∂ t T ((l − l ′)∆t − R / c) /(4π R )ds′ds. S
(7)
S
TD-AIM approximates these matrix elements as
(k , k ′) = ZlMFIE,AIM −l ′
M (k )
ˆ x (C (u , k ), k ) + yΓ ˆ y (C (u , k ), k ) + zΓ ˆ z (C (u , k ), k ) ∑ xΓ
u =1
⋅∇×
M ( k ′)
∑
ˆ x (C (u ′, k ′), k ′) + yΛ ˆ y (C (u ′, k ′), k ′) + zΛ ˆ z (C (u ′, k ′), k ′) xΛ
(8)
u′=1
⋅ µ ∂ t T ((l − l ′)∆t − RC (u ,k ),C (u′,k ′) / c) /(4π RC (u ,k ),C (u′,k ′) ). Equation (8) is obtained by approximating the spatial functions S k ′ (r ′) and S k (r ) × nˆ (r ) as linear combinations of M (k ′) and M (k ) Dirac delta functions located on the auxiliary mesh, respectively. The weighting coefficients for the delta functions are chosen by moment matching [4, 5] and are stored in the projection matrices Λ and Γ . These matrices are of dimension N c × N s and have O ( N s ) nonzero entries. The function C returns the global ids of the delta functions that approximate the spatial functions, and RC ( u ,k ),C ( u′,k ′) is the distance between two such delta functions. Finally, finite difference schemes are used to approximate the curl of the vector potential on the auxiliary mesh [8]. The matrix-vector multiplications in (6) can then be efficiently computed because (i) the projection matrices are sparse, (ii) computing the curl requires O( Nt N c ) operations, and (iii) the resulting TD-AIM matrices are block-Toeplitz.
4.
Numerical Results
To demonstrate the capabilities of the above TD-AIM-based CFIE solver, scattering from a conducting sphere of radius 4 m is simulated and the monostatic RCS is compared to the analytical Mie series solution in Fig. 1(a). The CFIE is discretized using N s = 44,595 RWG triangular basis functions on the surface of the sphere, with average patch area of 67.6 cm 2 ( λ 2 /148 at 300 MHz) and Nt = 750 time steps. The time dependence of the incident plane wave is a modulated Gaussian with carrier frequency of 200 MHz and 200 MHz bandwidth (99.997% of the power of the pulse is in the band 100 MHz to 300 MHz). The time-step size is set to ∆t = 5 / 24 ns, the auxiliary mesh spacing is 0.1 m ( N c = 903 , N g = 144 ), and the near-field radius is 0.35 m. The total run time for this simulation using an MPI-based parallel scheme [10] on 90 Pentium III processors with 1 GHz clock speed was 44.5 min. Figure 1(b) plots the VV-polarized bistatic RCS for the NASA almond simulated by TD-AIM and compared to an independent frequencydomain method of moments (MOM) solver at various frequencies. There are N s = 19, 467 RWG triangular basis functions, with average patch area of 3.06 mm 2 ( λ 2 / 372 at 8.88 GHz). The time-step size is ∆t = 6 / 25 ps and Nt = 500 . TDAIM parameters are N c = 96 × 48 × 20 , N g = 149 , and the near-field radius is 1.2 cm. The total run time on 32 Pentium IIIs was 25 min.
5.
Conclusions
This article presented a time domain AIM for accelerating the solutions of CFIE. Sample numerical results validate the accuracy of the method for large-scale scattering analysis. 40 MOM TD−AIM
30 0.5
8.88 GHz
20 2
RCS (dB/λ )
Normalized Monostatic RCS (σ/π a2) (dB)
1
0
10
z
0
Einc kˆ
−10
−0.5
θ
−20 TD−AIM Mie series
−1 −1.2
1
1.5
2 2.5 3 Spere Radius in Wavelengths (a/λ)
(a)
3.5
−30 −35 4
2.95 GHz x
0
90
180 θ
270
360
(b)
Fig. 1. Broadband scattering analysis with CFIE accelerated by TD-AIM ( α = 0.5 ). (a) Monostatic RCS for a sphere of radius 4 m. (b) Bistatic RCS for the NASA almond at 2.95 GHz and 8.88 GHz compared to frequency domain simulations.
References [1] A. E. Yılmaz, D. S. Weile, J. M. Jin, and E. Michielssen, “A hierachical FFT algorithm for the fast analysis of transient electromagnetic scattering phenomena,” IEEE Trans. Antennas Propagat., vol. 50, no. 7, pp. 971-982, 2002. [2] A. E. Yılmaz, D. S. Weile, B. Shanker, J. M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propagat. Lett., vol.1, no.1, pp. 14-17, 2002. [3] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “A new fast time domain integral equation solution algorithm,” IEEE APS Symp. Digest, vol. 4, pp. 176-179, 2001. [4] A. E. Yılmaz, K. Aygün, J. M. Jin, and E. Michielssen, “Matching criteria and the accuracy of time domain adaptive integral method,” IEEE APS Symp. Digest, vol. 2, pp. 166-169, 2002. [5] A. E. Yılmaz, K. Aygün, J. M. Jin, and E. Michielssen, “Fast analysis of heatsink emissions with time domain AIM,” submitted to IEEE Int. Symp. EMC, 2003. [6] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Science, vol. 31, no. 5, 1996, pp. 1225-51. [7] M. F. Catedra, R. F. Torres, J. Basterrechea, and E. Gago, The CG-FFT Method: Application of Signal Processing Techniques to Electromagnetics. Norwood, MA: Artech House, 1995. [8] X. C. Nie, L. W. Li, and N. Yuan, “Precorrected-FFT algorithm for solving combined field integral equations in electromagnetic scattering,” IEEE APS Symp. Digest, vol. 3, pp. 574-577, 2002. [9] B. Shanker, A. A. Ergin, K. Aygün, and E. Michielssen, “Analysis of transient electromagnetic scattering from closed surfaces using a combined field integral equation,” IEEE Trans. Antennas Propagat., vol. 48, no. 7, pp. 1064-1074, 2000. [10] A. E. Yılmaz, S. Q. Li, J. M. Jin, and E. Michielssen, “A parallel framework for FFT-accelerated time-marching algorithms,” URSI Digest, p. 319, 2002.
