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On the Analytic Calculation of Multiple Integrals in Electromagnetics

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On the Analytic Calculation of Multiple Integrals in Electromagnetics 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.
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