# A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms

Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit.

I am studying the paper by Putnam & Medgyesi on the CFIE method for BoR targets with junctions. In Section V (second paragraph) of the paper, the authors claim that by equating coefficients associated with half-triangle functions at the junction leads to a cancellation of boundary terms that one would get from integration-by-parts in the computation of the EFIE $$L$$ operator (see eqn. (5)) when the testing surface is open.

My issue is that I can't seem to reproduce this claim. I outline my approach below. Let us take eqn. (25) from the paper with $$\beta = 0$$. This formula is for the geometry in Fig (2). If I understand the discussion in Section V correctly, the boundary terms that arise in obtaining formulas for $$\mathcal{L} (S_1^c , S_1^c ; R_1)$$ and $$\mathcal{L} (S_1^c , S_1^d ; R_1)$$ should cancel if the expansion coefficients in the currents associated with the half-triangle functions are equated. After discretizing the generating curve of the BoR into $$N_1 - 1 > 1$$ straight line segments described by the interval $$[t_{1k}^- , t_{1 , k+1}^-]$$, $$k \in \{ 1 , \cdots , N_1 - 1 \}$$, (the subscript $$1$$ here indicates we are on the "outer" surface of the BoR) and parametrizing the surface by arclength along the generating curve $$t$$ and the polar angle $$\varphi$$, we expand the currents $$\mathbf{J}^c_1$$ according to

$$\mathbf{J}_1^c \simeq \sum_{n \in \mathbb{Z}} \sum_{k = 1}^{N_1^c} \left( c_{1nk}^t \mathbf{J}_{1nk}^t + c_{1nk}^{\varphi} \mathbf{J}_{1nk}^{\varphi} \right) ,$$

and likewise for $$\mathbf{J}^d_1$$. Above, $$\mathbf{J}_{1nk}^t (t , \varphi)$$ and $$\mathbf{J}_{1nk}^{\varphi} (t , \varphi)$$ are $$\mathbf{u}_t$$-directed and $$\mathbf{u}_{\varphi}$$-directed expansion functions which are both harmonic in $$\varphi$$ and use triangle functions in the $$t$$ variable, and $$N_1^c$$ is the number of segments making up the conducting part of the BoR outer surface (likewise, $$N_1^d$$). Above, $$\mathbf{u}_t$$ is the unit tangent vector along the BoR generating curve and $$\mathbf{u}_{\varphi}$$ is the usual polar unit vector. Equating the coefficients associated with half-triangle functions at the junctions means to me that $$c_{1 n , N_1^c}^t = d_{1 n , 1}^t$$ and $$c_{1 n , N_1^c}^{\varphi} = d_{1 n , 1}^{\varphi}$$. (This assumes the generating curve for the conducting part of the surface ends at the junction, and the dielectric part begins at the junction.)

The general formula for a matrix element is

$$\mathcal{L}^{\alpha \beta}_{\ell k} (S_1^c, S_1^d ; R) = i k_0 \eta_0 \int_{S_1^c} ( \mathbf{J}_{1 n \ell}^{\alpha} )^* (\mathbf{r}) \cdot \int_{S_1^d} \{ \mathbf{J}_{1nk}^{\beta} (\mathbf{r}') G (\mathbf{r} - \mathbf{r}') + k_0^{-2} (\nabla G) (\mathbf{r} - \mathbf{r}') \mathrm{div}_S~{\mathbf{J}_{1nk}^{\beta}} (\mathbf{r}') \} d S' d S ,$$

where $$G (\mathbf{r}) = e^{- i k_0 |\mathbf{r}|} / (4 \pi |\mathbf{r}|)$$, $$\mathrm{div}_S$$ is the surface divergence, and $$\alpha , \beta \in \{ t , \varphi \}$$. In the second term in the expression above, we integrate-by-parts to put the gradient on $$G$$ onto the testing function $$(\mathbf{J}_{i n \ell}^{\alpha} )^* (\mathbf{r})$$. Since $$S_1^c$$ is open, this will produce the boundary term

$$\oint_{\partial S_1^c} [ ( \mathbf{J}_{1 n \ell}^{\alpha} )^* \cdot \mathbf{u}_t ] (\mathbf{r}) \left( \int_{S_1^d} G (\mathbf{r} - \mathbf{r}') \mathrm{div}_S~{\mathbf{J}_{1nk}^{\beta}} (\mathbf{r}') d S' \right) d s .$$

This boundary term is nonzero only when $$\alpha = t$$ and $$\ell = N_1^c$$. Based on the above formula, my understanding is that the authors are claiming the term

$$\sum_{\beta \in \{t, \varphi\}} \oint_{\partial S_1^c} ( J_{1 n , N_1^c}^t )^* (\mathbf{r}) \left\lbrace \sum_{k = 1}^{N_1^c} c^{\beta}_{1nk} \left( \int_{S_1^c} G (\mathbf{r} - \mathbf{r}') \mathrm{div}_S~{\mathbf{J}_{1nk}^{\beta} (\mathbf{r}')} d S' \right) + \sum_{k = 1}^{N_1^d} d^{\beta}_{1nk} \left( \int_{S_1^d} G (\mathbf{r} - \mathbf{r}') \mathrm{div}_S~{\mathbf{J}_{1nk}^{\beta} (\mathbf{r}')} d S' \right) \right\rbrace d s$$

that would appear in equation (25) of the paper vanishes because $$c_{1 n , N_1^c}^t = d_{1 n , 1}^t$$ and $$c_{1 n , N_1^c}^{\varphi} = d_{1 n , 1}^{\varphi}$$. I am somewhat struggling to see this. The authors claim this is true because effectively $$S_1^c \cup S_1^d$$ form a closed surface. This suggests to me to try integrating-by-parts on the term

$$\int_{S_1^c} G (\mathbf{r} - \mathbf{r}') \mathrm{div}_S~{\mathbf{J}_{1nk}^{\beta} (\mathbf{r}')} d S' = \oint_{\partial S_1^c} G (\mathbf{r} - \mathbf{r}') [ \mathbf{J}_{1 n k}^{\beta} \cdot \mathbf{u}_t ] (\mathbf{r}') d S' - \int_{S_1^c} \nabla' G (\mathbf{r} - \mathbf{r}') \cdot \mathbf{J}_{1nk}^{\beta} (\mathbf{r}') d S' .$$

I can see how the boundary term for $$\beta = t$$ above would cancel out with a similar term in the similar formula for $$S_1^d$$ because $$c_{1 n , N_1^c}^t = d_{1 n , 1}^t$$. However, it is the lagging term

$$\int_{S_1^c} \nabla' G (\mathbf{r} - \mathbf{r}') \cdot \mathbf{J}_{1nk}^{\beta} (\mathbf{r}') d S'$$

that doesn't seem to disappear. This is making me quite confused about the authors' claim.

My question: Why do these boundary terms cancel? What am I missing? Is there anyone on this site who has worked on Method of Moments CEM codes or the EFIE and can shed light on the junction problem in general?

• I would be one, but I am not familiar with this particular paper. I might try looking into that on Christmas break, but there are also others who can potentially answer this. Dec 17, 2020 at 23:28
• The paper is behind a paywall, could you write down the relevant equations here? Also, could you write down the full methods names and not just the acronyms? I think I get most of them, except for BoR. In general, it helps the reader since some acronyms are used for several methods. Dec 24, 2020 at 16:27
• BoR stands for Body of Revolution. That is, the surface of the scattering body is a surface of revolution obtained by revolving a curve (called the generating curve above) about a fixed axis. The BoR assumption leads to modal decoupling and, hence, only a single mode $n$ appears in the matrix element for $\mathcal{L}$. The standard reference for this material is link.springer.com/article/10.1007/BF00382412. EFIE, MFIE, and CFIE stand for electric field, magnetic field, and combined field integral equation, respectively. Dec 24, 2020 at 16:48