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

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

where $G (\mathbf{r}) = e^{- i k |\mathbf{r}|} / (4 \pi |\mathbf{r}|)$, $\mathrm{div}_S$ 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?

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?

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 $N_1 > 1$ straight line segments $[t_{1k}^- , t_{1 , k+1}^-]$ (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

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

where $G (\mathbf{r}) = e^{- i k |\mathbf{r}|} / (4 \pi |\mathbf{r}|)$, $\mathrm{div}_S$ 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

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

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?

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

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

where $G (\mathbf{r}) = e^{- i k |\mathbf{r}|} / (4 \pi |\mathbf{r}|)$, $\mathrm{div}_S$ 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?