NODAL- AND EDGE-BASED VECTOR BASIS FUNCTIONS ON ...

NODAL- AND EDGE-BASED VECTOR BASIS FUNCTIONS ON HIGHER ORDER AND RATIONAL GEOMETRIES IN THE BEM Andrew Hellicar, John Kot CSIRO ICT Centre Cnr Vimiera & Pembroke Rds, Marsfield, NSW 2122, Australia. [email protected] Abstract The problem of plane wave scattering by a PEC sphere is solved to investigate the performance of different geometric and surface vector bases. The vector bases are compared using an exact representation of the sphere to eliminate errors introduced by geometries with creases. Higher order and rational geometries are implemented to examine the effect of the geometry on solution accuracy.

I. INTRODUCTION The boundary element method (BEM) is a numerical technique which is often used in solving electromagnetic scattering problems. A boundary integral equation (BIE) for an equivalent surface current (J
s ) over the surface (σ) of the scatterer is set up. The unknown surface currents are represented by a finite-element basis over a set of surface patches, and a solvable linear system is produced by projection, using Galerkin’s method or similar technique. Here the BIE used is the well-known MFIE
inc incident upon surface σ with surface normal
n(x
), G(x
, x) is the (1) with known magnetic field H free space scalar Greens function.

inc (x
) = 1 J
s (x
) +
n(x
) × (1) J
s (x) × ∇G(x
, x)dσ(x)
n(x
) × H 2 σ(x) In this paper we compare different representations used to describe the equivalent currents. This includes the geometric basis functions describing the surface supporting the current, and the vector basis functions describing the flow of the current on that surface. Earlier work, [1], described the solution of the MFIE using just a new interpolatory nodal-based vector basis (the so-called
n ×
a basis), and just the one geometric basis (σ4r , see below for details). Here results for the
n ×
a basis are given alongside results for the popular Graglia, Wilton and Peterson (GWP) and the adaptive Wang and Webb (WW) vector bases published in [2] and [3] respectively. Results are also given using geometric basis functions of varying polynomial order in both rational and non-rational forms.

II. GEOMETRIC BASES All the implemented computational geometries are of the form  C [I] (ζ)P[I]
r(ζ) =

(2)

[I]

where the position vector
r is expressed as a summation over geometric basis functions C [I] (ζ), based upon the B´ezier [4] representation, and scaled by control points P[I] with index [I]. The control points are selected to generate the desired geometry. Higher order polynomial geometries were implemented that interpolate the surface at a discrete set of nodes, dependent upon the order of the geometry. These polynomial interpolatory representations

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are referred to as σpi where p is the order of geometric approximation p ∈ {1 . . . 4}. Two rational representations σpr , p ∈ {2, 4}, were implemented where, for spherical approximations, σ2r exactly represents the boundary of each patch in the tessellation, and σ4r exactly represents the spherical surface everywhere. The σ4r geometric basis is of particular interest because it exactly models the quadric class of surfaces, including ellipsoids, paraboloids and hyperboloids.

III. VECTOR BASES Three different surface vector bases were implemented to represent the surface currents: two based upon edge elements, and the third using the nodal elements introduced in [1]. Although both the WW and the
n ×
a elements have (p + 1)(p + 2) degrees of freedom (DOF) per patch, globally they contain a different number of DOF. The GWP elements are complete to polynomial order p in both current and divergence and thus contain more DOF. The expression for the
n ×
a representation is: 
L[J] (ζ)
a[J] (3) J(ζ) =
n(ζ) × [J]

where
a are nodal surface tangent vectors scaled by Lagrange interpolating polynomials L[J] . RMS % Error in

J
s Against Evaluated Integrals

100

+  10 % error

♦

GW P WW
n ×
a + 

♦ + 

♦ + 

1

0.1 1000

♦ + 

♦ + ♦ 

10000 integrals

100000

Figure 1: Convergence of the solution surface currents for the three vector bases as the polynomial order is increased, against the integrals evaluated to fill the BEM matrix.

IV. CONCLUSIONS Figure 1 shows that the nodal
n ×
a elements perform well alongside the conventional edge elements. Figure 2 illustrates the benefit of using higher order representations for the geometries. The abscissae used in both figures (FLOPS and integrals respectively) are proportional to the computational work required to generate the solution result. It is well known that large patches with a higher order vector bases perform better than smaller patches with lower order vector bases. Figures 1 and 2 show in combination, that (as was the case) if large patches are used to approximate a curved structure, then more benefit is derived by increasing the order of the geometric basis set, than by increasing the

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RMS % Error in J
s Against FLOPS to Evaluate a Point

♦  

100

♦ +  ×

10



♦

% error

 1

♦



 +



×  

tetpi ♦ tetpr + octpi  octpr × icopi  icopr 

0.1 10

100 FLOPS

1000

Figure 2: J
s RMS errors for higher order (σpi) and rational (σpr ) geometries based upon the tetrahedron,

octahedron and icosahedron tessellations (σ = tet, oct, ico) with 4, 8 and 20 patches calculated using n × a vector bases of orders 6, 6 and 4 respectively.

order of the vector basis set. Obviously for tessellations containing lower levels of curvature than the implemented 4, 8 and 20 patch tessellations, less benefit would be derived from increasing the order of the geometric basis. Examining the rational represententations, although the 2nd order rational approximation (σ2r ) produces more accurate results than the interpolatory method (σ2i), the FLOPS are better spent increasing the geometric basis function order to 3 (σ3i), rather than changing to a rational representation. σ4r allows the contribution of the geometric error to the BEM error to be seen and is useful for examining the convergence of the vector basis functions in Figure 1; however, standard higher order polynomial geometric basis functions proved to be quite adequate for producing accurate results.

References [1] A.D. Hellicar, G.K. Cambrell, J.S. Kot, and G.C. James, “Continuous vector elements on rational B´ezier surfaces in the BEM for scattering problems,” Proceedings, IEEE 2000 AP-S International Symposium, Salt Lake City, USA, Jul 2000. [2] R. D. Graglia, D. R. Wilton, A. F. Peterson “Higher Order Interpolatory Vector Basis for Computational Electromagnetics,” IEEE Trans. Antennas Propagat., vol 45, pp 329-342, Mar 1997. [3] J. Wang and J. P. Webb, “Hierarchal Vector Boundary Elements and p-Adaption for 3-D Electromagnetic Scattering,” IEEE Trans. Antennas Propagat., vol. 45, pp. 1869-1879, Dec 1997. [4] G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. Academic Press, 1990.

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