Scattering at Open Perfectly Electrically Conducting ObjectsUsing Calderon Preconditioning and Fast Multipole Methods

Joris Peeters* and Femke OlyslagerDepartment of Information Technology, Ghent University,

Sint-Pietersnieuwstraat 41, 9000 Ghent, BelgiumE-mail: joris.peeters@intec.ugent.be

Introduction

When employing the Method of Moments (MoM) to solve scattering problems atPerfectly Electric Conducting (PEC) objects, two independent Boundary IntegralEquations (BIE) can be used: the Electric Field Integral Equation (EFIE) andthe Magnetic Field Integral Equation (MFIE) [1]. The EFIE is very accurate butill-posed while the solution of the MFIE converges rapidly when solved iterativelyat the expense of less accuracy. For closed objects, they can be combined in atrade-off between accuracy and conditioning. However, the MFIE is not valid atopen surfaces and hence in this case only the ill-posed EFIE remains. This meansthat preconditioning is required when scattering at open surfaces, in particularwhen the frequency drops, leading to an increase of the condition number of theMoM matrix. Currently, a variety of algebraic preconditioners are used, basedon approximate inverses of the impedance matrix. The most notable examplesare ILU and Block-Jacobi preconditioning. In high frequency simulations theyperform to some degree, but at lower frequencies they generally make the situa­tion even worse. Recently, a new type of preconditioners was discovered, whichworks on the level of the integral equation itself. These Calderon preconditionersessentially remove the hypersingular contribution in the integral kernel, which isresponsible for the conditioning problems. Using Buffa-Christiaensen (BC) [2]expansion functions for the preconditioner and Rao-Wilton-Glisson (RWG) [3]expansion functions for the impedance matrix, the cancelling of the hypersingularcontribution is maintained after discretisation. In this paper we will consider abroadband preconditioned EFIE accelerated by a suitable MLFMA to reduce theoverall complexity to 0 (N log N).

Calderon Preconditioning

In frequency domain (an ejwt time dependence is assumed and suppressed) theEFIE on a PEC scatterer is defined as -'1}T[J](r) = Un X Ei(r), with E i theincident electric field, J the unknown induced surface current density on the scat-

terer, E the permittivity, fJ, the permeability, fJ = vI¥ the characteristic impedance

and Un the unit normal on the scatterer surface. The electric-electric operator Tis given by

T[J](r) !Un x ( G(1' - 1") · J(1")dS' (1)'1} is-jkun x isg(R)J(1")ds' + ;~ x V' is V'g(R) · J(1")dS' (2)

Ts+Th (3)

978-1-4244-3647-7/09/$25.00 ©2009 IEEE

-jkRwith k = wyEii the wavenumber, R = Ir - r/l, g(R) = e41rR the homogeneousspace scalar Green function and G(r-r' ) = -jwJ.l(I+VV)g(R) the homogeneousspace Green dyadic with I the unit dyadic. The singular value spectrum of thisoperator has two branches, one going to infinity and one going to zero. Whenthe discretisation becomes finer, more singular functions can be resolved and asa result the condition number increases. The Calderon identity [4]' [5] used forpreconditioning this operator is

with

K[J](r) = -Un X is (\7 X g(R)I) · J(r')dS'

(4)

(5)

the magnetic-electric operator. The operator - i + K 2 is second kind and has abounded spectrum, which means that the integral equation

(6)

is well-posed. The reason for this is that T~ = O. When discretising both op­erators with RWG functions, however, this property is not maintained [5]. Arecently discovered solution [4] is to use BC functions for the discretisation of thepreconditioning T operator.

Fast Multipole Methods

The bottleneck of the iterative solution of MoM systems is the matrix-vectorproduct, which has 0 (N2 ) complexity, both in terms of memory and CPU-time.Using Fast Multipole Methods allows for a more efficient treatment of the distantinteractions and reduces the complexity to 0 (N log N). Because our interest isin both high frequency and low frequency simulations, a broadband approach isrequired. The Multi-Level Fast Multipole Algorithm (MLFMA) [6] breaks downat low frequencies, due to numerical instabilities. The spectral methods [7] andthe Nondirectional Stable Plane Wave MLFMA (NSPWMLFMA) [8], however,are stable over the entire frequency range. In this paper we will employ the NSP­WMLFMA, but it must be noted that the remainder of this paper is essentiallyindependent of this particular choice. Normally the vectorial formulation of NSP­WMLFMA (or any other) is used, leading to two radiation pattern componentsthat are orthogonal to the sampling vector. However, the BC functions lead toa finite contribution normal to the edge of the object [9]. In the calculation ofthe near interactions this leads to equivalent line charges that are non-integrable.These line charges require special handling beyond the scope of this contribution.As a consequence the scalar formulation needs to be employed, with four compo­nents (three Carthesian components for the vectorial potential and one componentfor the scalar potential).

Scattering at Open PEes

For surfaces that are closed and smooth, it can be shown that the operator - i +K 2 is second kind. However, when scattering at open surfaces is considered, theunknown current density is actually the sum of the contributions from both sides.In this case, the operator T can not be linked to K and it is not possible toprove that T 2 is second kind. As an experiment, a high frequency simulation at a20m x 20m plate was executed, at a frequency of 4.77 · 107 Hz. In Fig. 1 (left) theeigenvalue spectrum of the discretised operator T 2 is shown. While the conditionnumber is relatively low, a large scattering of the eigenvalues is observed as aconsequence of the edges of the open plate. Because iterative solvers require aclustering of the eigenvalues in certain areas of the complex plane, this will leadto a fairly slow convergence. The solution is to use a localised preconditioner TL.There are various methods to obtain a localised version of the preconditioner, buthere we will omit distant interactions beyond a certain distance, for instance .\(the wavelength). This will also make the scheme slightly more efficient. Theresulting eigenvalue spectrum is displayed in Fig. 1 (right). The eigenvalues aresignificantly better clustered and the iterative process converges much faster.

X 104

X 104

4OUo

40

2 0 2> ~ 2

0 000 ~ 0

~ff-2&0

-2

-40

-40 o~o

0

-6 -6-5 0 5 -5 0 5

x 104

X 104

Figure 1: Eigenvalue spectra (in the complex plane) of the matrices discretising-2 - -T (left) and T L T (right) for a square plate

To prove the stability of the iterative process, the number of iterations requiredto reach a 10-3 accuracy is plotted in Fig. 2, using different operators. The objectis a square plate discretised with mesh-size 0.1.\ (high frequency problem). Asthe size of the plate grows, the number of unknowns N increases. Fig. 2 clearly

800---B--- T

en~TT

.§ 600~

._._.TLTQ)

c::= 4000Q).0

§ 200c::

00 2 4 6 8

NX 10

4

Figure 2: Number of iterations to reach a 10-3 accuracy for operators T, T 2

and T LT for a high frequency simulation of a plate with variable size and Nunknowns.

demonstrates that the only way to obtain a stable iterative process for openstructures (at high frequencies) is to use a localised version of the preconditioner.At low frequencies, the scattering of the eigenvalues is much more limited andthese tend to cause less trouble. However, if the edge to surface ratio is veryhigh, then even a local preconditioner as described above may not be sufficient toguarantee an efficient iterative process.

Conclusion

Calderon preconditioning is very succesful in stabilising the EFIE and the use ofBC functions makes the formalism valid on open surfaces. Through applying abroadband fast multipole method, the complexity can be reduced from 0 (N2 )

to 0 (N log N), allowing the simulation of very large structures. In the highfrequency case a localised version of the preconditioner must be used to avoidexcessive scattering of the eigenvalues of the combined operator.

References

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