The governing equation of elastostatic boundary value problems in
is given by
According to the formulation of FMBEM for 3D elastostatic crack problems
developed by Nishimura et al.,
the first fundamental solution given by Eq.(9) is expanded as
Now, by using Eq.(11), the integration
involved in Eq.(8) is evaluated as
When the center of the multipole moments are translated from
O to O', the current multipole moments are translated as
Evaluating the multipole expansion formula given by Eq.(16),
for a point x with respect to its neighboring point x0,
the following local expansion is used.
Finally, with an assumption
when the center of the local expansion is translated from x0 to x1,
the coefficients of the local expansion are translated as
By using some formulas mentioned above, the FMBEM for 3D elastostatic problem can be carried out similarly described in Nishimura et al.. Following remarks should be mentioned about numerical analysis applied in the present paper. 1) the constant boundary elements are employed for both displacements and tractions, 2) non-restart GMRES is used for solving the discretized linear equations, 3) the block diagonal scaling technique proposed by Nishida and Hayami is applied for preconditioning the matrix.
To verify the program based on the above theory,
a sample test problem is solved, i.e. an elastic rectangular block
subjected to the uniaxial tension. In this example,
the truncation of infinite series involved in
Eqs.(21)-(25) is made at 10 terms and
the maximum number of boundary elements included in a cell is 50.
Fig.1 shows the computation time (CPU time[sec]) per one iteration process are plotted
to the degrees of freedom(DOF). It reveals that the CPU time is proportional to the degrees of freedom.
This fact implies that the present FMBEM is well done and that
it is confirmed that the solutions obtained in this analysis
are in good agreement with the theoretical ones.