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Truncation of the infinite series

In FM-BIEM, we follow the techniques proposed by Song et al.[76] to truncate the infinite series taking P terms, where
 
$\displaystyle P(kR) = k R + 5 \ {\rm ln}(\pi + k R ),$     (3.132)

k is the wave number and R is the diameter of a sphere which circumscribes a cell. One has $R=\sqrt{3}D$, where D is the edge length of a cell. This means that we must change P as the level changes. Also, P increases as the size of a cell or the wave number becomes large. This property leads to a lot of computational cost when one deals with high frequency problems or large-scale problems. In order to overcome this problem some techniques are proposed by several authors (e.g. Rokhlin et al.[71] or Greengard et al.[34]). However, we do not use these techniques in this thesis since we are interested in low frequency problems.
  
Figure 3.39: kR vs P
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Ken-ichi Yoshida
2001-07-28