On the Analytic Calculation of Multiple Integrals in Electromagnetics

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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 theorem, we transform a high dimensional moment in- div G(x)=𝐹 (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 𝐼(𝐹 )= nx ⋅ G(x) 𝑑𝑆 (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 𝐼(∇𝐹 )= 𝐹 (x) nx 𝑑𝑆 (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 𝐼(𝐹 )for or quasi-analytic expressions for the integrand [2], 𝐹 (x) a monomial, i.e., most often by invoking a Gauss theorem tailored to the problem at hand, while afterwards employ- ∏𝑑 𝐹 (x)=𝐹 (x)= 𝑥𝛼𝑘 ing numerical quadrature techniques to calculate 𝛼 𝑘 the remaining non-singular integrals. 𝑘=1 Here we follow another approach. Starting with where the powers 𝛼𝑘 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- ∑𝑑 form the 2𝑑−dimensional moment integral into a div [x𝐹𝛼(x)] = 𝑑 + 𝛼𝑘 𝐹𝛼(x) (2𝑑 − 2)−dimensional boundary integral. This al- 𝑘=1 lows in most pertinent cases to replace the original Hence for 𝐹 = 𝐹 we have singular multiple integral by a reduced non-singular 𝛼 ∫ multiple integral, which can be evaluated by means 1 of standard analytic or numerical quadrature tech- 𝐼(𝐹𝛼)= ∑ nx ⋅ x 𝐹𝛼(x) 𝑑𝑆 𝑑 + 𝑑 𝛼 niques [4]. Pertinent electromagnetic applications 𝑘=1 𝑘 ∂Ω include moment integrals for the Coulomb and re- Next, it is also easy to prove that tarded Coulomb interactions. ∫ 1 𝐹 (x) 𝑡𝑑−1 𝐹 (𝑡x) 𝑑𝑡 = 𝛼 2 GAUSS DIVERGENCE THEOREM 𝛼 ∑𝑑 0 𝑑 + 𝑘=1 𝛼𝑘 Let us start with the calculation of a multiple in- In other words we have proved that tegral over Ω ⊂ 𝑅𝑑 with Ω compact and simply connected, i.e., ∫ ∫ ∫ 𝐹𝛼(x) 𝑑𝑉 = nx ⋅ G𝛼(x)𝑑𝑆 (3) Ω ∂Ω 𝐼(𝐹 )= 𝐹 (x) 𝑑𝑉 Ω where ∫ 1 ∗ 𝑑−1 Dept. INTEC-IBCN-IBBT, Ghent University, G. G𝛼(x)=x 𝑡 𝐹𝛼(𝑡x) 𝑑𝑡 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 𝐹𝛼(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 𝑅𝑑, 𝑑−1 𝐼(𝐹 )= nx ⋅ x 𝑑𝑆 𝑡 𝐹 (𝑡x) 𝑑𝑡 (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 2𝑑−fold multiple integrals. In general we need to Dirichlet eigenproblem for Ω, since the Dirichlet evaluate the integral eigenfunctions Φ𝑘(x) and eigenvalues 𝜆𝑘 are related ∫ ∫ by Rellich’s identity [5] : 𝐼 = 𝑡(x)𝐾(∥x − x′∥)𝑏(x′) 𝑑𝑉 𝑑𝑉 ′ Ω ℧ ∫ ∫ ( )2 𝑑 2 ∂Φ𝑘 where Ω, ℧ ⊂ 𝑅 are the compact, simply con- 2𝜆𝑘 Φ𝑘𝑑𝑉 = nx ⋅ x 𝑑𝑆 Ω ∂Ω ∂𝑛 nected supports of the test function 𝑡(x) and basis function 𝑏(x′), respectively. We further suppose Also, since the weight function nx ⋅ x is often pos- that 𝑡(x)and𝑏(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. 𝑡(x)=𝑡(0) + p ⋅ x In [3] formula (4) was generalized to the transla- 𝑏(x′)=𝑏(0) + q ⋅ x′ tionally invariant formula Consider the bi-divergence operator ∫ ∫ 1 𝑑−1 ′ def ′ ′ 𝐼(𝐹 )= nx⋅(x−a) 𝑑𝑆 𝑡 𝐹 (𝑡x+(1−𝑡)a) 𝑑𝑡 ◇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), 𝐹 may be singular, with ◇𝐾ˆ (∥x − x ∥)=𝐾(∥x − x ∥) the proviso that is satisfied by the kernel ∫ ∫ lim 𝑡𝑑𝐹 (𝑡x +(1− 𝑡)a)=0 𝑟 1 𝑡→0+ 𝐾ˆ (𝑟)=− 𝑢𝑑𝑢 𝐾(𝑢𝑡)𝑡𝑑−1𝑑𝑡 (8) 0 0 It is then easy to apply these formulas to functions Next consider the composite bi-divergence operator of the form ′ def ′ ′ ′ 𝑏◇𝑡Ψ(x, x ) = ∇ ⋅ [𝑏(x )∇ [𝑡(x)Ψ(x, x )]] 𝐹1(x)=𝐾(∥x − a∥) It is seen that and ′ ′ ′ 𝑏◇𝑡Ψ=𝑏(x )𝑡(x)◇Ψ+𝑡(x)q⋅∇Ψ+𝑏(x )p⋅∇ Ψ+p⋅qΨ F2(x)=(x − a)𝐾(∥x − a∥) ′ ˆ ′ with ∥⋅∥the Euclidean norm. Note that the func- Taking Ψ(x, x )=𝐾(∥x − x ∥), and defining tion (𝐴 + B ⋅ x)𝐾(∥x − a∥) can always be written ∫ ∫ 𝐽 = ◇ Ψ 𝑑𝑉 𝑑𝑉 ′ as a superposition of 𝐹1(x)andF2(x). 𝑏 𝑡 We have ∫Ω ∫℧ ′ ′ ′ = n ⋅ n ′ 𝑡(x)𝑏(x )𝐾ˆ (∥x − x ∥) 𝑑𝑆 𝑑𝑆 ∫ ∫ 1 x x 𝑑−1 ∂Ω ∂℧ 𝐼(𝐹1)= nx ⋅ (x − a) 𝑑𝑆 𝑡 𝐾(𝑡∥x − a∥) 𝑑𝑡 ∂Ω 0 and since ◇Ψ=◇𝐾ˆ = 𝐾, we obtain (6)

and, since 𝐼 = 𝐽 − p ⋅ I1 − q ⋅ I2 − p ⋅ q 𝐼3 F (x)=∇𝐿(∥x − a∥) where 2 ∫ ∫ I = 𝑏(x′)∇′𝐾ˆ (∥x − x′∥) 𝑑𝑉 𝑑𝑉 ′ where ∫ 1 𝑢 ∫Ω ∫℧ 𝐿(𝑢)= 𝑡𝐾(𝑡) 𝑑𝑡 ˆ ′ ′ I2 = 𝑡(x)∇𝐾(∥x − x ∥) 𝑑𝑉 𝑑𝑉 ∫ ∫Ω ∫℧ 𝐼(F )= 𝐿(∥x − a∥) n 𝑑𝑆 (7) 2 x 𝐼 = 𝐾ˆ (∥x − x′∥) 𝑑𝑉 𝑑𝑉 ′ ∂Ω 3 Ω ℧ It is seen that Note that in the pulse-basis case we only need the ∫ ∫ kernel 𝐾ˆ (𝑟). For 𝑑 =2wehave𝐾ˆ (𝑟)=−𝑟, as was ¯ ′ ′ 𝐼3 = nx ⋅ nx′ 𝐾(∥x − x ∥) 𝑑𝑆 𝑑𝑆 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 ∫ 𝑟 ∫ 1 ¯ ˆ 𝑑−1 is 𝐾(𝑟)=− 𝑢𝑑𝑢 𝐾(𝑢𝑡)𝑡 𝑑𝑡 (9) ′ 𝑒−𝑗𝑘0∥x−x ∥ 0 0 𝐾(∥x − x′∥)= ′ since ◇𝐾¯ (∥x − x′∥)=𝐾ˆ (∥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 𝐾ˆ (𝑟) kernel associated with the las (1)-(2), (6), (7) and exploiting the fact that retarded Coulomb kernel 𝐾(𝑟)=𝑒−𝑗𝑘0𝑟/𝑟. Appli- ′ ˆ ′ ˆ ′ ∇ 𝐾(∥x − x ∥)=−∇𝐾(∥x − x ∥), we obtain, after cation of formula (8) yields some tedious calculations : ∫ ∫ ∫ ( ) 1 −𝑗𝑘0𝑟𝑡 ˜ ′ ′ 𝑒 − 1 𝑑−2 I1 = − (q ⋅ nx′ )nx𝐾(∥x − x ∥) 𝑑𝑆 𝑑𝑆 𝐾ˆ (𝑟)= 𝑡 𝑑𝑡 (16) ∂Ω ∂℧ 𝑗𝑘 𝑡 ∫ ∫ 0 0 ′ ˘ ′ ′ − 𝑏(x)((x − x) ⋅ nx′ )nx𝐾(∥x − x ∥) 𝑑𝑆 𝑑𝑆 Although the kernel 𝐾ˆ (𝑟) defined by the integral ∂Ω ∂℧ and similarly (16) is an entire function with easily found series ∫ ∫ representation ′ ′ I = − (p ⋅ n )n ′ 𝐾˜ (∥x − x ∥) 𝑑𝑆 𝑑𝑆 2 x x ∞ ∂Ω ∂℧ ∑ (−𝑗𝑘 𝑟)𝑚 ∫ ∫ 𝐾ˆ (𝑟)=−𝑟 0 (17) ′ ′ ˘ ′ ′ (𝑚 + 𝑑 − 1)(𝑚 +1)! − 𝑡(x )((x − x ) ⋅ nx)nx′ 𝐾(∥x − x ∥) 𝑑𝑆 𝑑𝑆 𝑚=0 ∂Ω ∂℧ where it is seen that the delay information is completely ∫ 𝑟 𝐾˜ (𝑟)= 𝐾ˆ (𝑢)𝑢𝑑𝑢 (10) lost in the series representation (17). A potentially 0 better way to evaluate the kernel 𝐾ˆ (𝑟)mightbe and ∫ to use an 𝑁−point Gauss quadrature formula with 1 weight function 𝑡𝑑−2 over the interval [0, 1]. Denot- 𝐾˘ (𝑟)= 𝐾ˆ (𝑡𝑟)𝑡𝑑−1𝑑𝑡 (11) 0 ing the nodes and weights by 𝑡𝑘 and 𝑤𝑘 respectively, It is seen that 𝐼, a2𝑑−fold multiple integral, has the integral (16) can then be approximated as been written as a sum of 2(𝑑 − 1)−fold boundary 𝑁 ( ) integrals, which represents a serious computational ∑ −𝑗𝑘0𝑟𝑡𝑘 ˆ 𝑒 − 1 gain. Note that in the pulse-basis case, when p = 𝐾(𝑟) ≈ 𝑤𝑘 (18) 𝑗𝑘0𝑡𝑘 q =0, one simply has 𝐼 = 𝐽. 𝑘=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 𝐾(∥x − x′∥)= ∥x − x′∥ 5CONCLUSION

Since 𝐾(𝑟)=1/𝑟, the other pertinent kernels Starting with the singular kernel present in the in- ˆ ¯ ˜ ˘ 𝐾(𝑟), 𝐾(𝑟), 𝐾(𝑟)and𝐾(𝑟) 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 2𝑑−dimensional 𝑟 moment integral into a (2𝑑 − 2)−dimensional 𝐾ˆ (𝑟)=− (12) 𝑑 − 1 boundary integral. This allows in most pertinent 𝑟3 cases to replace the original singular multiple in- 𝐾¯ (𝑟)= (13) tegral by a simpler non-singular multiple integral, 2 3(𝑑 − 1) which can be evaluated by means of standard an- 𝑟3 alytic or numerical quadrature techniques. Some 𝐾˜ (𝑟)=− (14) 3(𝑑 − 1) pertinent special cases, such as the Coulomb and 𝑟 retarded Coulomb kernels, were shown to ﬁt into 𝐾˘ (𝑟)=− (15) 𝑑2 − 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 Δ𝑢 + 𝜆𝑢 = 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. Tech- nol. Lett., vol. 47, no. 1, pp. 22-26, Oct. 2005. [7] G. Antonini, A. Orlandi, and A. E. Ruehli, ”An- alytical integration of quasi-static potential integrals on nonorthogonal coplanar quadrilaterals for the PEEC Method,” IEEE Trans. Electromagnetic Compatibility, vol. 44, no. 2, pp. 399-403, May 2002.