On the Mixed Discretization of the Time Domain Magnetic Field Integral Equation

Transcription

1 On the Mixed Discretization of the Tie Doain Magnetic Field Integral Equation H. A. Ülkü 1 I. Bogaert K. Cools 3 F. P. Andriulli 4 H. Bağ 1 Abstract Tie doain agnetic field integral equation (MFIE) is discretized using divergenceconforing Rao-Wilton-Glisson (RWG) and curl-conforing Buffa-Christiansen (BC) functions as spatial basis and testing functions, respectively. The resulting ixed discretization schee, unlike the classical schee which uses RWG functions as both basis and testing functions, is proper : Testing functions belong to dual space of the basis functions. Nuerical results deonstrate that the arching on-in-tie (MOT) solution of the ixed discretized MFIE yields ore accurate results than that of classically discretized MFIE. 1 INTRODUCTION Magnetic field integral equation (MFIE) is a second kind integral equation, i.e., it is constructed as a suation of an identity and a linear operator. Because of the identity operator, the doain and range of the MFIE operator are identical and a consistent discretization schee should use basis and testing functions, which belong to dual spaces of each other [1]. In particular, the testing function should be in the dual space of the MFIE operator s range (and doain) to obtain a proper discretization schee. Classical arching on-intie (MOT) based MFIE solvers expand the unknown surface current density using divergence-conforing Rao-Wilton-Glisson (RWG) functions [] in space and polynoial functions in tie. To obtain a proper discretization, spatial testing should be done using curl-conforing ˆn RWG functions, which belong to dual space of the divergence conforing RWG basis functions. However, resulting MOT atrix becoes singular and cannot be inverted accurately at every tie step as required by the MOT schee. Therefore, classical ipleentations use RWG functions (but not their duals) for spatial testing and violates the requireent of the proper discretization described above. Even though the solution of the MOT atrix syste resulting fro this discretization schee converges fast, it yields inaccurate results. In this work, tie doain MFIE is discretized using the ixed discretization schee, which is originally proposed for discretizing the frequency doain MFIE [3]. Mixed discretization schee akes use of recently proposed Buffa-Christiansen (BC) functions [4]-[5] to produce well-conditioned MOT atrices without violating the requireent of the proper discretization described above. Current density is expanded using divergence-conforing RWG functions in space and spatial testing is carried out using curl-conforing ˆn BC functions, which belong to dual space of the divergence-conforing RWG functions. 1 Division of Physical Science and Engineering, King Abdullah University of Science and Technology, Thuwal, KSA e-ail: huseyin.ulku@kaust.edu.sa, hakan.bagci@kaust.edu.sa tel.: , fax: Departent of Inforation Technology, Ghent University, Ghent, Belgiu. e-ail: Ignace.Bogaert@intec.ugent.be 3 Departent of Electrical and Electronic Engineering, University of Nottingha, Nottingha, UK. e-ail: Kristof.Cools@nottingha.ac.uk 4 Microwave Departent, TELECOM Bretagne, Brest, France. e-ail: francesco.andriulli@teleco-bretagne.eu

2 Nuerical results deonstrate that the MOT solution of the ixed discretized MFIE yields ore accurate results than that of the classically discretized MFIE, as expected. MAGNETIC FIELD INTEGRAL EQUATION Tie doain MFIE, which is obtained by enforcing the agnetic field boundary condition on perfect electrically conductor (PEC) scatterer surfaces, reads ˆn(r) H inc (r,t) = 1 J(r,t), r S. (1) δ (t R / c) ˆn(r) J( r,t) d r S 4π R Here, S denotes the scatter surface, H inc (r,t) is the (essentially) band-liited incident agnetic field, J(r,t) is unknown current density, R = r r is the distance between the source point r and the observation point r, ˆn(r) is the outward-pointing unit noral vector defined at r, c is the speed of light in the ediu, where S resides, denotes the teporal convolution operation, and δ (.) is the Dirac delta function. To nuerically solve (1) for the unknown current density J(r,t), MOT schee expands J(r,t) in ters of teporal and spatial basis functions: N J(r,t) = t I n,i (t)b n (r) () N n=1 In (), f n (r) denotes the n th spatial basis function, which is chosen as the RWG basis function, (t) is the i th teporal basis function, which is chosen as the shifted polynoial Lagrange interpolation function, and I n,i is the unknown coefficient associated with the n th spatial and i th teporal basis function. Here, N and N t are nubers of spatial basis functions and tie steps, respectively. Substituting () in (1) and testing the resulting equation with testing functions t (r), = 1,.., N, in space and δ (t t j ), j = 1,.., N t in tie, yield the linear MOT syste: t (r) ˆn(r) H inc (r,t j )dr = N j 1 n=1 I n,i t (r) )f n (r)dr t S (r) ˆn(r) ) H n (r,t j ) dr }. (3) Here, t j = jδt and Δt is the tie step size and H n (r,t) is the agnetic field due to ipulsively excited RWG basis function. It should be noted here that H n (r,t) is evaluated analytically as described in [6]-[8]; H n (r,t) does not have any spatial singularities, therefore there is no need for a singularity treatent schee to enhance accuracy of the resulting atrix eleents. Additionally, for polynoial (t), teporal convolution in (3) can be evaluated in analytically [6]-[8].

3 The choice of testing basis function t (r) deterines type of the discretization strategy as described next in Section.1 and., and one of factors that deterine the accuracy of the solution as deonstrated by the nuerical results presented in Section 3..1 Classical Discretization Strategy Classical Galerkin discretization strategy uses divergence-conforing RWG functions for spatial testing, i.e., t (r) = f (r) in (3), where f (r) denotes the th RWG function. Even though the Gra atrix [first ter on the right hand side of (3)] resulting fro this type of discretization is well-conditioned, this discretization schee is not proper: Testing and basis functions belong to the sae function space; see Section 1 for a brief explanation and see [3] for ore details.. Mixed Discretization Strategy Mixed discretization strategy uses curl-conforing rotated BC functions for spatial testing, i.e., t (r) = ˆn(r) g (r) in (3), where g (r) denotes the th BC function. After several atheatical anipulations, (3) can be rewritten as g (r) H inc (r,t j )dr = N j 1 n=1 I n,i ˆn(r) g (r) )f n (r)dr g S (r) ) H n (r,t j ) dr }. (4) It should be noted here that ˆn(r) g (r) belong to the dual space of f n (r). g (r) are defined on the barycentric refineent of the initial esh, where f n (r) are constructed; additionally they are linear cobinations of RWG functions constructed on the barycentric esh [9]-[10]. MOT syste (4) can easily be ipleented using existing codes that can account for (3) with RWG functions defined on the barycentric esh. Mixed discretization schee satisfies the proper discretization condition (see Section 1 for a brief explanation and see [3] for ore details) and generates a ore accurate MOT syste as shown by the nuerical results presented in the next section. 3 NUMERICAL RESULTS To deonstrate that the ixed discretization schee produces ore accurate MOT systes than the classical schee, transient scattering fro a unit sphere residing in free space is investigated. The sphere is discretized with 70 triangular patches. The excitation is chosen as odulated Gaussian plane wave propagating in ẑ direction: (t t p +r ẑ/c) H inc (r,t) = ŷcos π f 0 (t t p + r ẑ / c) e σ (5)

4 where f 0 = 75 MHz is the odulation frequency, f bw = 30 MHz is the effective bandwidth, σ = 7 / (π f bw ) is a easure of pulse duration, and t p = 3.5σ is the delay. Tie step size is chosen as Δt = 0.1/ ( f 0 + f bw ). Third order polynoial Lagrange interpolation function is used as (t). Both MOT systes in (3) and (4) are solved for N t = 000 tie steps. Figure 1 plots the coefficients of the 1st RWG basis function, which are obtained by solving (3) and (4). Figure plots the relative nor error in the radar cross section (RCS) with respect to Mie series solution. The error is coputed using err RCS ) = N θ =1 σ tie,φ) σ Mie,φ) N θ =1. (6) σ Mie,φ) Here, N θ = 181, θ = ( 1), φ = 0, f j = j180 khz, and σ tie,φ) and σ Mie,φ) are RCS saples obtained at these points by Fourier transforing the tie-doain MOT results and the Mie series solution, respectively. Figure clearly deonstrates the superior accuracy of the ixed discretization schee. References [1] P. Ylä-Oijala, S. P. Kiinki, K. Cools, F.P. Andriulli and S. Järvenpää, Mixed discretization schees for electroagnetic surface integral equations, International Journal of Nuerical Modelling: Electronic Networks, Devices and Fields, 01 (DOI: /jn.844). [] S. M. Rao, D. R. Wilton, and A. W. Glisson, Electroagnetic scattering by surfaces of arbitrary shape, IEEE Transactions on Antennas and Propagation, vol. 30, no. 3, pp , May 198. [3] K. Cools, F. P. Andriulli, D. De Zutter, and R. Michielssen, Accurate and conforing ixed discretization of the MFIE, IEEE Antennas and Wireless Propagation Letters, vol. 10, pp , 011. [4] A. Buffa and S. H. Christiansen, A dual finite eleent coplex on the barycentric refineent, Matheatics of Coputation, vol. 60, pp , 007. [5] A. Buffa and S.H. Christiansen, A dual finite eleent coplex on the barycentric refineent, Tech. Rep. PV-18 IMATI-CNR, 005. [6] H. A. Ülkü and A. A. Ergin, Analytical evaluation of transient agnetic fields due to RWG current bases, IEEE Transactions on Antennas and Propagation, vol. 55, no. 1, pp , Dec [7] H. A. Ülkü and A. A. Ergin, Application of analytical retarded-tie potential expressions to the solution of tie doain integral equations, IEEE Transactions on Antennas and Propagation, vol. 59, no. 11, pp , Nov [8] H. A. Ülkü and A. A. Ergin, Application of analytical expressions of transient potentials to the MOT solution of integral equations, in Proc. IEEE AP-S Int. Syp., San Diego, CA, 008.

5 [9] F. P. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, A ultiplicative Calderon preconditioner for the electric field integral equation, IEEE Transactions on Antennas and Propagation, vol. 56, no. 8, pp , Aug [10] H. Bagci, F.P. Andriulli, K. Cools, F. Olyslager, E. Michielssen, A Calderon ultiplicative preconditioner for the cobined field integral equation, IEEE Transactions on Antennas and Propagation, vol. 57, no. 10, pp , Oct Figure 1: Coefficients of the 1st RWG basis function obtained by solving the MOT systes (3) and (4). Figure : err RCS ( f ) obtained fro solutions of the MOT systes in (3) and (4).