Loop-Star Decomposition for any Order Basis Functions with processing of Weak and Nearly Singularities for the Surface Integral Equation José Mª Gil (1), Rafael Gómez (2), Miguel A. González (1), Jesús García (1)
[email protected],
[email protected],
[email protected],
[email protected]. (1)
Departamento de Señales, Sistemas y Radiocomunicaciones, Universidad Politécnica de Madrid, Spain (2)
Departamento de Tecnología de los Computadores y de las Comunicaciones, Universidad de Extremadura, Spain
Abstract—A procedure to obtain div-conforming elements for the Surface Integral Equation, by rotating some popular curlconforming bases used in the Finite Elements Method (FEM) is presented. The method maintains the Helmholtz decomposition for surface currents, for any order of bases functions and any curvature order of the domain. Thanks to the current decomposition, a frequency scaling avoids the frequency breakdown. For addressing the problem of the singular term in the integral equation, we use a coordinate transformation to cancel out the singular term. The singularity cancellation is carried out directly on a plane local domain. The procedure is compatible with any order of curvature of the elements and invariant with the order of the bases functions. The coordinate transformation adds a parameter suitable to treat the near-singularities. Index Terms— Div-conforming higher-order elements, Helmholtz decomposition, singular cancelation.
I. INTRODUCCION Nowadays, the complexity of the structures used in electrical engineering designs requires to improve the efficiency and accuracy of the bases functions employed in codes based on the Integral Equation as the Method of Moments (MoM). This basis functions should increase its order and to be capable to handle curve domains. Furthermore, they must cope with the uncoupling of both solenoidal and non-solenoidal currents in the low frequency regime. A Huge decomposition of the currents is essential in the developing of the bases functions for any order of the bases, by simulating properly the physical character of the currents and making easy to apply pre-conditioners to avoid the frequency breakdown in the low frequency regime. In [1], a simple method to obtain higher-order div-conforming basis was presented; the method is based on rotating curlconforming basis with a Helmholtz decomposition by means of the theorem of the trace. A frequency scaling may then be applied in order to avoid the ill-conditioning of the overall matrix when the frequency is very low [2]. A similar procedure consists of scaling the basis functions following a normalization process as described in [3]. To reduce the integration errors due to the singular kernel in the Integral Equation a coordinate transformation as described in [4] is employed. The sub-triangles in local space are transformed into unity squares diluting the singularity in one point along an edge in the unity square. The Jacobian cancels out the weak singularity produced in the Electric Field Integral
Equation (EFIE). Because this method is applied in the local space, is compatible with any curvature order and invariant with the order of the bases functions. This transformation introduces, in addition, a parameter that constitutes a degree of freedom to tune the cancellation of the singularity. The called “nearsingularities” occurs when source and field elements are very close but they don’t overlap. This parameter can be used for the treatment of the near-singularities contributing to the reduction of the integration errors. II. DIV-CONFORMING ELEMENTS These elements are obtained by rotating the surface trace of curl-conforming basis functions, maintaining the advantages of the bases functions used within the FEM. A curl-conforming function is given by the following
(1) T ( x , y , z ) Tp ( p, q )p Tq ( p, q )q Tp (p, q), Tq (p, q) are the covariant components and p, q are the reciprocal base vectors. If we rotate 90º
nS T ( x , y , z) Tp ( p, q )[nS p] Tq ( p, q )[nS q ] (2) The div-conforming functions are given by N⃗(x, y, z) = T (p, q)τ + T (p, q)τ
With
p nS p
(3)
q nS q
(4)
Being ns, the unit external normal vector to the surface S. Furthermore, the Helmholtz decomposition for the curlconforming space is converted into a Hodge decomposition for the surface currents. Moreover, curl-conforming elements of orders (m, n) share the same components of the bases function at local space, Tp (p, q) and Tq (p, q), with the div-conforming ones. The space of functions Vmn is divided into a Loop space Lm formed by solenoidal (zero div) functions of degree m, and a non-solenoidal space Sn, formed by functions of degree n with nonzero div:
Vmn Lm Sn
(5)
We can also scale directly the basis functions trying to get the elements in the main diagonal of the matrix equal to 1. We multiply the basis functions by some weighting coefficients Ck as described in [3]
Lm { l Vmn / l 0}
Sn {s Vmn / s 0} The subspace Lm constitutes the null space of the divergence operator, N (div). The space Sn is made up of nonzero div functions to complete the vector space. Each subspace can be further decomposed into three subspaces associated with the edges of the triangle and one subspace of functions associated with the face of the triangle. These elements are hierarchical and an element can be obtained by adding new functions to the lower order one.
(8) This technique is added to our formulation with negligible computational cost. In Fig. 2 the condition number of the overall [Z] matrix, with and without frequency scaling and normalization of basis matrix versus the frequency shows the performance of these techniques.
III. NUMERICAL PERFORMANCE OF THE ELEMENTS Considering the Galerkin method applied to the electric field integral equation (EFIE); an algebraic system is obtained
[ Z ji ]Ii V j
(6)
From the calculated surface current density we can compute the far field and the Radar Cross Section (RCS). To check the elements, we analyse a metallic sphere with a diameter of 1 cm excited by a wave plane. The frequency is f=7.5 GHz. We compute the bistatic RCS along some samples and we calculate the root mean square (RMS) error described in (7), =
∑
[
( ) − ( )]
(7)
σref are the analytical samples obtained by the Mie series. Fig. 1 shows the monotonic convergence with an h-refinement for (1, 2) and (2, 3) elements.
Fig. 1. Monotonic h-convergence for (1, 2) and (2, 3) elements.
IV. NORMALIZATION OF THE MOM EQUATION In the EFIE, the contribution of the vector potential is negligible compared with the scalar potential at very low frequencies. The frequency breakdown inherent to the EFIE might be compensated by using the surface current decomposition into two components, loop and star. The overall [Z] matrix, when 0, is made up of blocks with different frequency dependence [2]. A frequency normalization makes the final matrix diagonally dominant and, then, stable at low frequencies.
Fig. 2. Conditioning of the [Z] matrix vs frequency with/ without scaling and basis functions normalization.
V. TREATMENT OF THE WEAK AND NEARLY SINGULARITY OF EFIE In the EFIE, a singular kernel of integrals arises when source and field elements coincide. If both elements are very close to each other, but they do not overlap, the integrals are nearly singular. We use a cancellation method based on a geometrical transformation whose Jacobean cancels out the singular term. The transformation is a conversion to triangular polar coordinates of the transformation due to Wait [5]. The cancellation is carried out directly in the local space. This is an advantage when curvilinear domains are involved, because the equations governing the geometrical transformation are linear and independent of the order of curvature. The domain is divided, in local space, into three rectilinear subtriangles, sharing the singular point. For each Gauss integration point for the outer integral (po, qo), a singularity occurs within the inner integral. Each of the three p-q local sub-triangle is transformed into the unity square in the (ρ, σ) space (Fig. 3). The vertex 1, which coincides with the point where the singularity is located, is transformed into the edge ρ = 0. The three sub-integrals are integrated on the unity square by means of a Gauss quadrature. q
3
3 1
1 2
2
Fig. 3. Mapping to unity square
p
The coordinate transformation is
t t1 [(t2 t1 ) (t3 t2 ) ] t , m
(9)
ti = (pi, qi) with i = 1, 2, 3 being the vertex of the element. The Jacobean of the transformation and the distance R are
J 2 m 1
∝
(10)
The singularity1/R is easily cancelled out just doing m = 1. This “m” constitutes an additional degree of freedom that allows to model singularities with different orders, useful to nearly singularities. In Table I, the RMS error in the RCS computation of a PEC sphere by using a 9 points Gauss formula and the current method is shown. Results are for both medium and high frequency regime and a relevant improvement is achieved.
Actually, the former condition is modulated with a factor taking into consideration the frequency regime, this factor is usually chosen as 3 for low frequency, 2 for medium frequency and 0.5 for high frequency. The “m” parameter must be chosen to suit the level of near singularity; the election of the “m” parameter requires a previous study. Results for a PEC sphere, are shown in Fig. 5. We can see an additional improvement in the RMS error for some values of the m parameter. This technique seems potentially effective although it requires a relevant computational cost in terms of CPU time, for which reason it is not worth doing for all the cases.
Table I. RMS for the RCS of a PEC sphere with and without the current singularity integration method.
RMS (Gauss) RMS (Current)
f=30 GHz 3.82 0.086
f=75 GHz 0.96 0.48
A. Treatment of near-singularities We have applied this coordinate transformation to elements that they are very close but they do not overlap. The size “S” of each element is the radio of a circumference equal to the distance between the centroid and the furthest vertex of the triangle (Fig. 4). If the distance “D” between two elements is lesser than or equal to S, the near processing is applied.
D S
Fig.9 Performance of the near-singularity treatment for the PEC sphere case in a medium frequency (MF) regime.
VI. CONCLUSIONS A method to develop any order curved hierarchical divconforming elements has been presented. The bases functions add a Helmholtz decomposition by making possible a frequency scaling for the low frequency regime. Another pre-conditioning based on the scaling of the bases functions is also checked. A coordinate transformation to cope with the weak and near singularity of the Electric Field Integral Equation is proposed. ACKNOWLEDGEMENTS This work was supported by The Spanish Ministry of Education under the reference TEC2013-46282-C2-1-P. REFERENCES
Fig. 4. Size “S” of the source element and distance “D” between the source and field element [1] J. M. Gil, "An Efficient Framework for using Higher-Order Curved DivConforming Elements with Cancellation of Weak Singularities in Local Space for Surface Integral Equation", IEEE, Trans. Antennas Propagat., Vol.60, NO. 8, pp. 3736-3743, August 2012. [2] J. S. Zhao and W. C. Chew, “Integral Equation Solution of Maxwell’s Equations from Zero Frequency to Microwave Frequencies”, IEEE Trans. Antennas Propagat., Vol. 48,No. 10, pp. 1635-1645, October 2000. [3] Milan M. Kostić and Branko M. Kolundžija, “Maximally Orthogonalized Higher Order Bases Over Generalized Wires, Quadrilaterals, and Hexahedra”, IEEE Trans. Antennas Propagat, Vol. 61, No. 6, pp. 3135-3147, June 2013.
[4] J. M. Gil and J. P. Webb, “A New Edge Element for the Modelling of Field Singularities in Transmission Lines and Waveguides”, IEEE Trans. Microwave Theory Tech., Vol. 45, No. 12, pp. 2125-2130, December 1997. [5] R Wait, “Finite Element Methods for Elliptic Problems with Singularities”, Computer Methods in Applied Mechanics and Engineering, Vol. 13, pp. 141150, 1978.
