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Multipole moments and coefficients of the local expansion

The first step in FM-BIEM is to rewrite the fundamental solution in Eq. (4) into the following form:
$\displaystyle \Gamma_{ip}(x,y) =
\frac{1}{4\pi\mu k_T^2}
\left( e_{rqi}e_{rsp}\...
...l }{\partial x_i}\frac{\partial }{\partial y_p}
\frac{e^{i k_L r}}{r}\right). $     (6)

Also, we use the following result (Epton and Dembart[6]):
$\displaystyle \frac{e^{ikr}}{4\pi r} \approx \frac{ik}{16\pi^2} \int_{S^0}
e^{i...
...bf k $}}\cdot\overrightarrow{\scriptstyle y_0y}}dS_{\hat{\mbox{$\mathbf k $}}},$     (7)

where the inequality $ \vert$$ \mbox{$\mathbf x $}$$ -$$ \mbox{$\mathbf x $}$$ _0+$$ \mbox{$\mathbf y $}$$ _0-$$ \mbox{$\mathbf y $}$$ \vert < \vert$$ \mbox{$\mathbf x $}$$ _0-$$ \mbox{$\mathbf y $}$$ _0\vert$ is assumed to hold. In Eq. (8) $ \hat{\mbox{$\mathbf k $}}$ stands for the unit vector given by

$\displaystyle (\cos\phi \sin\theta, \sin\phi \sin\theta, \cos\theta),\quad
(0\le\theta\le\pi, 0\le\phi\le2\pi),
$

the integration is taken on the unit sphere $ S^0$, and $ \mathcal{T}(k,\hat{\mbox{$\mathbf k $}},\overrightarrow{y_0x_0})$ is given in terms of a large number $ p$ by
$\displaystyle \mathcal{T}(k,\hat{\mbox{$\mathbf k $}},\overrightarrow{y_0x_0})=...
..._0} }{\vert\overrightarrow{y_0x_0}\vert}\cdot \hat{\mbox{$\mathbf k $}}\right),$      

where $ h_{n}^{(1)}$ and $ P_n$ stand for the spherical Hankel function and the Legendre function, respectively.

We now compute the integral on the RHS of Eq. (5) over a subset of $ S$ denoted by $ S_0$ for an $ x$ which is away from $ S_0$. Using Eqs. (7) and (8) we obtain

$\displaystyle {-\int_{S_0} C_{jknp}\frac{\partial^2}{\partial x_{l}\partial
y_{n}} \Gamma_{mp}(x,y) n_{k}(y)\phi_{j}(y) dS_{y}}$
    $\displaystyle =\frac{i}{16\pi^2k_T^2\mu}\left((k_T)^5
\int_{S^0} \hat {k}_l \ha...
...0}}
L^T_r(\hat{\mbox{$\mathbf k $}};x_0) dS_{\hat{\mbox{$\mathbf k $}}} \right.$  
    $\displaystyle \left. - (k_L)^5 \int_{S^0} \hat {k}_l \hat {k}_m
e^{ik_L\hat{\mb...
..._0}} L^L(\hat{\mbox{$\mathbf k $}};x_0) dS_{\hat{\mbox{$\mathbf k $}}}\right) ,$ (8)

where $ L^T_r$ and $ L^L$ are the coefficients of local expansions centred at $ x_0$ expressed in terms of the multipole moments $ M^T_r$ and $ M^L$ centred at $ y_0$ as follows:
$\displaystyle L^T_r(\hat{\mbox{$\mathbf k $}};x_0) = {\mathcal T}(k_T,\hat{\mbox{$\mathbf k $}},\overrightarrow{y_0x_0})
M^T_r(\hat{\mbox{$\mathbf k $}};y_0),$     (9)
$\displaystyle L^L(\hat{\mbox{$\mathbf k $}};x_0) = {\mathcal T}(k_L,\hat{\mbox{$\mathbf k $}},\overrightarrow{y_0x_0})
M^L(\hat{\mbox{$\mathbf k $}};y_0).$     (10)

The multipole moments $ M^T_r$ and $ M^L$ are given by
$\displaystyle M^T_{r}(\hat{\mbox{$\mathbf k $}};y_0)$ $\displaystyle =$ $\displaystyle \int_{S_y}\hat k_s \hat k_n
C_{jknp}e_{rsp}\phi_{j}(y) n_{k}(y)
e^{ik_T\hat{\mbox{$\mathbf k $}}\cdot\overrightarrow{\scriptstyle y_0y}}dS_{y},$  
$\displaystyle M^L(\hat{\mbox{$\mathbf k $}};y_0)$ $\displaystyle =$ $\displaystyle \int_{S_y}\hat k_n \hat k_p
C_{jknp} \phi_{j}(y) n_{k}(y)
e^{ik_L\hat{\mbox{$\mathbf k $}}\cdot \overrightarrow{\scriptstyle y_0y}}dS_{y}.$  

Eqs. (10) and (11) are the M2L translation formulae which dominate the performance of FM-BIEM. Note that Eqs. (10) and (11), written for many $ \hat{\mbox{$\mathbf k $}}$s, have the form of the product of a diagonal matrix and a vector, in contrast to the M2L translation formulae in FM-BIEM with the Wigner-3j symbols which have the form of the product of a full matrix and a vector. Also, we notice that this formulation is in terms of 4 multipole moments (3 for $ M^T_{r}$ and 1 for $ M^L$) and 4 coefficients of the local expansion (3 for $ L^T_{r}$ and 1 for $ L^L$).


next up previous
Next: Numerical integration on Up: FM-BIEM Previous: FM-BIEM
2001-12-14