View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Ghent University Academic Bibliography On the analytic calculation of multiple integrals in electromagnetics L. Knockaert∗ Abstract — Making use of a powerful Gauss diver- If we could find a vector field G(x) such that gence theorem and a new Gauss bi-divergence the- orem, we transform a high dimensional moment in- div G(x)= (x) tegral into a lower dimensional boundary integral. This allows in most pertinent cases to replace the original singular multiple integral by a reduced non- then, by virtue of Gauss’ theorem, we can write singular multiple integral, which can be evaluated by ∫ means of standard analytic or numerical quadrature techniques. Some important electromagnetic cases ( )= nx ⋅ G(x) (1) are treated to indicate the strength and versatility ∂Ω of the proposed method. where nx is the outward unit normal to the sur- face ∂Ω. Note that a related Gauss theorem for the 1INTRODUCTION gradient is : The numerical evaluation of multiple integrals with ∫ singular kernels arises quite naturally in the method (∇ )= (x) nx (2) of moments solution [1] of electromagnetic scatter- ∂Ω ing problems. Most of the methods currently in use perform preliminary calculations to obtain analytic Let us first try to calculate the integral ( )for or quasi-analytic expressions for the integrand [2], (x) a monomial, i.e., most often by invoking a Gauss theorem tailored to the problem at hand, while afterwards employ- ∏ (x)= (x)= ing numerical quadrature techniques to calculate the remaining non-singular integrals. =1 Here we follow another approach. Starting with where the powers are natural numbers. Now it the kernel of the singular integrand, and making is easy to prove that use of a powerful Gauss divergence theorem [3] ( ) and a new Gauss bi-divergence approach, we trans- ∑ form the 2−dimensional moment integral into a div [x(x)] = + (x) (2 − 2)−dimensional boundary integral. This al- =1 lows in most pertinent cases to replace the original Hence for = we have singular multiple integral by a reduced non-singular ∫ multiple integral, which can be evaluated by means 1 of standard analytic or numerical quadrature tech- ()= ∑ nx ⋅ x (x) + niques [4]. Pertinent electromagnetic applications =1 ∂Ω include moment integrals for the Coulomb and re- Next, it is also easy to prove that tarded Coulomb interactions. ∫ 1 (x) −1 (x) = 2 GAUSS DIVERGENCE THEOREM ∑ 0 + =1 Let us start with the calculation of a multiple in- In other words we have proved that tegral over Ω ⊂ with Ω compact and simply connected, i.e., ∫ ∫ ∫ (x) = nx ⋅ G(x) (3) Ω ∂Ω ( )= (x) Ω where ∫ 1 ∗ −1 Dept. INTEC-IBCN-IBBT, Ghent University, G. G(x)=x (x) Crommenlaan 8, PB 201, B-9050 Gent, Belgium, e-mail: 0 [email protected], tel.: +3292643328, fax: +3292649969.This work was supported by a grant of the Since this is valid for all monomials (x), and Ω Research Foundation-Flanders (FWO-Vlaanderen) is compact, we can therefore generalize formula (3), in virtue of the Stone-Weierstrass theorem, to the 3 GAUSS BI-DIVERGENCE AP- space of all continuous functions over Ω, i.e., PROACH ∫ ∫ 1 Application of the method of moments [1] in , −1 ( )= nx ⋅ x (x) (4) in the presence of kernels which only depend on ∂Ω 0 the Euclidean distance between source point and Note that nx ⋅ x, considered as a weight function observation point, always leads to the evaluation of over ∂Ω, is also well-known in the study of the 2−fold multiple integrals. In general we need to Dirichlet eigenproblem for Ω, since the Dirichlet evaluate the integral eigenfunctions Φ(x) and eigenvalues are related ∫ ∫ by Rellich’s identity [5] : = (x)(∥x − x′∥)(x′) ′ Ω ℧ ∫ ∫ ( )2 2 ∂Φ where Ω, ℧ ⊂ are the compact, simply con- 2 Φ = nx ⋅ x Ω ∂Ω ∂ nected supports of the test function (x) and ba- sis function (x′), respectively. We further suppose Also, since the weight function nx ⋅ x is often pos- that (x)and(x′) vary linearly within their re- itive, formula (4) might provide useful quadrature spective supports with respective gradients p and formulas for Ω starting from quadrature formulas q. In other words for ∂Ω, in the same vein as the ones developed in [4] p. 43. (x)=(0) + p ⋅ x In [3] formula (4) was generalized to the transla- (x′)=(0) + q ⋅ x′ tionally invariant formula Consider the bi-divergence operator ∫ ∫ 1 −1 ′ def ′ ′ ( )= nx⋅(x−a) (x+(1−)a) ◇Ψ(x, x ) = ∇ ⋅∇Ψ(x, x ) ∂Ω 0 (5) It is easily proved, see [3, 6], that the displaymath with a any vector (translational change of center). ′ ′ Moreover, in formula (5), may be singular, with ◇ˆ (∥x − x ∥)=(∥x − x ∥) the proviso that is satisfied by the kernel ∫ ∫ lim (x +(1− )a)=0 1 →0+ ˆ ()=− ()−1 (8) 0 0 It is then easy to apply these formulas to functions Next consider the composite bi-divergence operator of the form ′ def ′ ′ ′ ◇Ψ(x, x ) = ∇ ⋅ [(x )∇ [(x)Ψ(x, x )]] 1(x)=(∥x − a∥) It is seen that and ′ ′ ′ ◇Ψ=(x )(x)◇Ψ+(x)q⋅∇Ψ+(x )p⋅∇ Ψ+p⋅qΨ F2(x)=(x − a)(∥x − a∥) ′ ˆ ′ with ∥⋅∥the Euclidean norm. Note that the func- Taking Ψ(x, x )=(∥x − x ∥), and defining tion ( + B ⋅ x)(∥x − a∥) can always be written ∫ ∫ = ◇ Ψ ′ as a superposition of 1(x)andF2(x). We have ∫Ω ∫℧ ′ ′ ′ = n ⋅ n ′ (x)(x )ˆ (∥x − x ∥) ∫ ∫ 1 x x −1 ∂Ω ∂℧ (1)= nx ⋅ (x − a) (∥x − a∥) ∂Ω 0 and since ◇Ψ=◇ˆ = , we obtain (6) and, since = − p ⋅ I1 − q ⋅ I2 − p ⋅ q 3 F (x)=∇(∥x − a∥) where 2 ∫ ∫ I = (x′)∇′ˆ (∥x − x′∥) ′ where ∫ 1 ∫Ω ∫℧ ()= () ˆ ′ ′ I2 = (x)∇(∥x − x ∥) ∫ ∫Ω ∫℧ (F )= (∥x − a∥) n (7) 2 x = ˆ (∥x − x′∥) ′ ∂Ω 3 Ω ℧ It is seen that Note that in the pulse-basis case we only need the ∫ ∫ kernel ˆ (). For =2wehaveˆ ()=−, as was ¯ ′ ′ 3 = nx ⋅ nx′ (∥x − x ∥) already indicated and discussed in [7]. ∂Ω ∂℧ Another, more general and difficult case is to deal where with retarded potentials, in which case the kernel ∫ ∫ 1 ¯ ˆ −1 is ()=− () (9) ′ −0∥x−x ∥ 0 0 (∥x − x′∥)= ′ since ◇¯ (∥x − x′∥)=ˆ (∥x − x′∥). The vectors ∥x − x ∥ I1 and I2 are more difficult to evaluate. After It is therefore important and potentially rewarding some algebraic manipulations, utilizing the formu- to calculate the ˆ () kernel associated with the las (1)-(2), (6), (7) and exploiting the fact that retarded Coulomb kernel ()=−0/. Appli- ′ ˆ ′ ˆ ′ ∇ (∥x − x ∥)=−∇(∥x − x ∥), we obtain, after cation of formula (8) yields some tedious calculations : ∫ ∫ ∫ ( ) 1 −0 ˜ ′ ′ − 1 −2 I1 = − (q ⋅ nx′ )nx(∥x − x ∥) ˆ ()= (16) ∂Ω ∂℧ ∫ ∫ 0 0 ′ ˘ ′ ′ − (x)((x − x) ⋅ nx′ )nx(∥x − x ∥) Although the kernel ˆ () defined by the integral ∂Ω ∂℧ and similarly (16) is an entire function with easily found series ∫ ∫ representation ′ ′ I = − (p ⋅ n )n ′ ˜ (∥x − x ∥) 2 x x ∞ ∂Ω ∂℧ ∑ (− ) ∫ ∫ ˆ ()=− 0 (17) ′ ′ ˘ ′ ′ ( + − 1)( +1)! − (x )((x − x ) ⋅ nx)nx′ (∥x − x ∥) =0 ∂Ω ∂℧ where it is seen that the delay information is completely ∫ ˜ ()= ˆ () (10) lost in the series representation (17). A potentially 0 better way to evaluate the kernel ˆ ()mightbe and ∫ to use an −point Gauss quadrature formula with 1 weight function −2 over the interval [0, 1]. Denot- ˘ ()= ˆ ()−1 (11) 0 ing the nodes and weights by and respectively, It is seen that , a2−fold multiple integral, has the integral (16) can then be approximated as been written as a sum of 2( − 1)−fold boundary ( ) integrals, which represents a serious computational ∑ −0 ˆ − 1 gain. Note that in the pulse-basis case, when p = () ≈ (18) 0 q =0, one simply has = . =1 4 SPECIAL CASES In contradistinction with formula (17), where the delay aspects are completely blurred, it is seen that An important special case is when the multiple mo- the delay information is still acutely present in for- ment integrals are generated by the Coulomb kernel mula (18). 1 (∥x − x′∥)= ∥x − x′∥ 5CONCLUSION Since ()=1/, the other pertinent kernels Starting with the singular kernel present in the in- ˆ ¯ ˜ ˘ (), (), ()and() are found by the an- tegrand, and making use of a powerful Gauss diver- alytical calculations of (8), (9), (10) and (11). We gence theorem and a new Gauss bi-divergence ap- obtain proach, we have transformed the 2−dimensional moment integral into a (2 − 2)−dimensional ˆ ()=− (12) − 1 boundary integral. This allows in most pertinent 3 cases to replace the original singular multiple in- ¯ ()= (13) tegral by a simpler non-singular multiple integral, 2 3( − 1) which can be evaluated by means of standard an- 3 alytic or numerical quadrature techniques. Some ˜ ()=− (14) 3( − 1) pertinent special cases, such as the Coulomb and retarded Coulomb kernels, were shown to fit into ˘ ()=− (15) 2 − 1 the general scope of the proposed approach. References [1] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968. [2] S. M. Rao, D. R. Wilton, and A. W. Glisson, ”Electromagnetic scattering by surfaces of arbi- trary shape,” IEEE Trans. Antennas Propagat., vol. AP-30, no. 3, pp. 409-418, May 1982. [3] L. Knockaert, ”A general Gauss theorem for evalu- ating singular integrals over polyhedral domains,” Electromagnetics, vol. 11, pp. 269-280, 1991. [4]A.H.Stroud,Approximate Calculation of Multi- ple Integrals. Englewood Cliffs, NJ: Prentice-Hall, 1971. [5] F. Rellich, ”Darstellung der Eigenwerte von Δ + = 0 durch ein Randintegral.,” Math. Z., vol. 46, pp. 635-636, 1940. [6] L. Knockaert, F. Olyslager, and D. Vande Gin- ste, ”On the evaluation of self-patch integrals in the method of moments,” Microwave Opt.