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# Series expansion of the fundamental solution of three-dimensional elastostatics

We introduce a new coordinate system which has following base vectors:

where are base vectors in the Cartesian coordinates. In the new system, we have

where . The derivatives with respect to these variables satisfy

with which is written as

 (G.1)

The functions and (See (F.9)) satisfy relations given by

and the functions and (See (F.9)) satisfy relations given by

Using above relations, (D.1), (D.2), (D.3), (D.7), (D.8) and (D.11), we differentiate (F.8) using (G.1) to obtain the derivatives of as

 = = (G.2)

Another way to obtain this result is to use an expansion of given by[88]

 (G.3)

With (G.3) we can derive (G.2) as follows:

From (G.2) we obtain the second order derivatives of as

 = = (G.4)

where is the second order unit tensor. The fundamental solution of the equation of elastostatics is expressed as

 = = (G.5)

Using (G.3), (G.4) and (G.5), we obtain a series expansion of given by

 (G.6)

where FSij,n,m and GSi,n,m are functions defined as

Also, we can derive (G.6) in another way. We first rewrite then fundamental solution (G.5) as

 (G.7)

Substituting (G.3) into (G.7), we obtain
 = = = = =

Next: M2M, M2L, L2L in Up: Applications of Fast Multipole Previous: Series expansion of |x-y|
Ken-ichi Yoshida
2001-07-28