FM-BIEM

Using Gegenbauer's addition theorem (See [2]), we can expand the fundamental solution in (3.117) into the following series

 

$$\frac{e^{ik R}}{4 \pi R} =\frac{i k}{4 \pi}\sum_{n=0}^{\infty}\su... ...ox{\boldmath$ x $ }}\vert > \vert\overrightarrow{O\mbox{\boldmath$ y $ }}\vert,$$ (3.120)

where $R=\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$, $\overline{・}$ denotes the complex conjugate and the functions ${\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$\space x $ }})$ and ${\cal I}_n^m(k,\overrightarrow{O\mbox{\boldmath$\space x $ }})$ are defined as

  

$${\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }}) h_n^{(1)}(k\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert) Y_n^m(\widehat{O\mbox{\boldmath$ x $ }}),$$ = (3.121)

$${\cal I}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }}) j_n(k\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert) Y_n^m(\widehat{O\mbox{\boldmath$ x $ }}).$$ = (3.122)

In these formulae $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ and $h_n^{(1)}(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ are the spherical Bessel function and the spherical Hankel function of the first kind, respectively, $\widehat{O\mbox{\boldmath$\space x $ }}$ indicates the unit vector given by $\overrightarrow{O\mbox{\boldmath$\space x $ }}/\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert$ and the spherical harmonics $Y_n^m(\widehat{O\mbox{\boldmath$\space x $ }})$ is defined as

$$\begin{eqnarray*}Y_n^m(\widehat{O\mbox{\boldmath$ x $ }})=\sqrt{\frac{(n-m)!}{(n+m)!}}P_n^m(\cos\theta) e^{i m \phi}. \end{eqnarray*}$$

The functions ${\cal O}_n^m$ and ${\cal I}_n^m$ have the following properties (See Epton and Dembart[17]):

 

$$\overline{{\cal I}}_n^m(k,\overrightarrow{O'\mbox{\boldmath$ y $ ... ...htarrow{O\mbox{\boldmath$ y $ }}) \ {\cal I}_l^{-m-m'}(k,\overrightarrow{O'O}),$$ (3.123)

 

$${\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }})= \sum_{n... ...ath$ x $ }_0}) {\cal O}_l^{m+m'}(k,\overrightarrow{O\mbox{\boldmath$ x $ }_0}),$$

$$\vert\overrightarrow{O\mbox{\boldmath$ x $ }_0}\vert > \vert\overrightarrow{\mbox{\boldmath$ x $ }_0\mbox{\boldmath$ x $ }}\vert,~$$ (3.124)

where W_n,n',m,m',l is given by

 

$$W_{n,n',m,m',l}=(2l+1)i^{n'-n+l}\left( { n \atop 0 }{ n' \atop 0 } { l \atop 0}\right) \left( { n \atop m }{ n' \atop m' } { l \atop -m-m'}\right).$$ (3.125)

In (3.125) $\left( { ・ \atop ・ }{ ・ \atop ・ } { ・ \atop ・}\right)$ stands for the Wigner-3j symbol (See Messiah[59]). The explicit form of the Wigner-3j symbol is expressed as

 

$${\left( { j_1 \atop m_1 }{ j_2 \atop m_2 } { j_3 \atop m_3}\right... ...c{(j_1+j_2-j_3)!(j_1-j_2+j_3)! (-j_1+j_2+j_3)!}{(j_1+j_2+j_3+1)!}\right]^{1/2}}$$

$$\times\left[(j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)! (j_3+m_3)!(j_3-m_3)!\right]^{1/2}$$ (3.126)

$$\times\sum_{i}\frac{(-1)^{i+j_1-j_2-m_3}} {i!(j_1+j_2-j_3-i)!(j_1-m_1-i)!(j_2+m_2-i)! (j_3-j_2+m_1+i)!(j_3-j_1-m_2+i)!}.$$

The summation is over such i that the numbers in the parentheses in the denominator are all non-negative. The recurrence formula for the Wigner-3j symbol are found in Schulten and Gordon[74]. Their implementation is available from Netlib (http://www.netlilb.org/).

In this section we use the hypersingular integral equation (3.118) for the formulation. We now compute the integral on the right-hand side of (3.118) over a subset of S denoted by S_y for $\mbox{\boldmath$\space x $ }$ which is away from S_y. Using (3.120), we obtain

 

$$\int_{S_y} \frac{\partial^2 }{\partial n_x \partial n_y} \frac{e^... ...rtial n_x }{\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }}) M_n^m(k,O),$$ (3.127)

where $R=\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ and M_n^m(k,O) is the multipole moment centred at Odefined as

 

$$M_n^m(k,O) = \int_{S_y} \frac{\partial }{\partial n_y } \overline... ...(k,\overrightarrow{O\mbox{\boldmath$ y $ }}) \phi(\mbox{\boldmath$ y $ }) dS_y.$$ (3.128)

In (3.127) we have assumed that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert > \vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert$ ( $\mbox{\boldmath$\space y $ }\in S_y$) holds.

The multipole moments are translated according to the following formulae as the centre of the multipole expansion is shifted from O to O':

 

$$M_n^m(k,O')= \sum_{n'=0}^{\infty}\sum_{m'=-n'}^{n'} \sum_{\script... ...m'} W_{n,n',m,m',l}{\cal I}_l^{-m-m'}(k,\overrightarrow{O'O}) M_{n'}^{-m'}(k,O)$$ (3.129)

where we have used (3.123) (See Appendix I.1). In the evaluation of the integral on the right-hand side of (3.118) one can use not only the multipole moments but also the coefficients of local expansion in the following manner:

 

$$\int_{S_y} \frac{\partial^2 }{\partial n_x \partial n_y} \frac{e^... ...oldmath$ x $ }_0\mbox{\boldmath$ x $ }}) L_{n}^{m}(k,\mbox{\boldmath$ x $ }_0),$$

where $R=\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$, $L_n^m(k,\mbox{\boldmath$\space x $ }_0)$ is expressed with M_n^m(k,O) by (See Appendix I.2)

 

$$L_n^m(k,\mbox{\boldmath$ x $ }_0)=\sum_{n'=0}^{\infty}\sum_{m'=-n... ...al O}}_l^{-m-m'}(k,\overrightarrow{O\mbox{\boldmath$ x $ }_0})M_{n'}^{m'}(k,O),$$ (3.130)

and $\widetilde{\cal O}_n^m$ is defined as

$$\begin{eqnarray*}\widetilde{\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ ... ... }}\vert) \overline{Y}_n^m(\widehat{O\mbox{\boldmath$ x $ }}). \end{eqnarray*}$$

In the derivation of (3.131) we have used (3.124) and have assumed that the inequality $\vert\overrightarrow{O\mbox{\boldmath$\space x $ }_0}\vert > \vert\overrightarrow{\mbox{\boldmath$\space x $ }_0\mbox{\boldmath$ x $ }}\vert$ holds.

The coefficients of the local expansion are translated according to the following formulae as the centre of the local expansion is shifted from $\mbox{\boldmath$\space x $ }_0$ to $\mbox{\boldmath$\space x $ }_1$:

 

$$L_n^m(k,\mbox{\boldmath$ x $ }_1)=\sum_{n'=0}^{\infty}\sum_{m'=-n... ...th$ x $ }_0 \mbox{\boldmath$ x $ }_1}) L_{n'}^{m'}(k,\mbox{\boldmath$ x $ }_0),$$ (3.131)

where we have used (3.123) (See Appendix I.3).

Notice that in this formulation the multipole moment and the coefficient do not have the properties like those of (3.23) and (3.27) in Laplace's equation because $\phi$ is complex-valued. Therefore, we can not save the memory with the use of symmetries.

We have thus prepared all the formulae needed for FM-BIEM. Using formulae presented above and the algorithm described in chapter 2, one can implement FM-BIEM.