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FORMULATION OF FMBEM FOR 3D ELASTOSTATIC PROBLEMS

In this section, the FMBEM for 3D elastostatic boundary value problems is formulated.

The governing equation of elastostatic boundary value problems in the domain $D\in R^3$ is given by

$\displaystyle \Delta^*\mbox{\boldmath$u$ }=0~~\mbox{in}~D$     (7)

where $\Delta^*_{ik}$ is the Navier's operator defined by $\Delta^*_{ik}=C_{ijkl}\partial_j\partial_l$with elastic constant tensor $C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$, $(\lambda,\mu)$ are Lame's constants, $\delta_{ij}$ is Kronecker's delta and $\partial_i$ stands for the partial differentiation with respect to xi. For this problem, the integral equation represented by displacement $\mbox{\boldmath$u$ }$ is given by
 
$\displaystyle \frac{1}{2}u_i(x)= \int_S\Gamma_{ij}(x-y) t_j(y)dS_y
-\mbox{v.p.}\int_S T_{ij}(x-y) u_j(y)dS_y$     (8)

where S is the boundary of the domain D, $\mbox{\boldmath$t$ }$ is a traction vector defined by ti=Cijkluk,lnj, $\Gamma_{ij}$ and Tij is the first and second fundamental solutions, respectively given by
  
$\displaystyle \Gamma_{ij}(x-y)$ = $\displaystyle \frac{1}{8\pi\mu}\left(
\delta_{ij}\partial^y_l\partial^y_l
-\fra...
...\mu}{\lambda+2\mu}\partial^y_i\partial^y_j
\right)\vert\overrightarrow{xy}\vert$ (9)
Tij(x-y) = $\displaystyle C_{jknp}n_k(y)\partial^y_n\Gamma_{ip}(x-y)$ (10)

where $\partial^y_i$ denotes the partial differentiation by yi, and $\vert\overrightarrow{xy}\vert$ means the distance between two points x and y.

According to the formulation of FMBEM for 3D elastostatic crack problems developed by Nishimura et al.[2], the first fundamental solution given by Eq.(9) is expanded as

 
$\displaystyle \Gamma_{ij}(x-y)$
    $\displaystyle =\frac{1}{8\pi\mu}\sum_{N=0}^{\infty}\sum_{M=-N}^N
\left(F^S_{ij,...
...arrow{Ox})(\overrightarrow{Oy})_j\overline{R_{N,M}}(\overrightarrow{Oy})\right)$ (11)

where $\overline{(\cdot)}$ denotes the complex conjugate. And the functions FSij,N,M, GSi,N,M RN,M and SN,Mare defined by
    
$\displaystyle F^S_{ij,N,M}(\overrightarrow{Ox})$ = $\displaystyle \frac{\lambda+3\mu}{\lambda+2\mu}\delta_{ij}S_{N,M}(\overrightarr...
...{\lambda+2\mu}(\overrightarrow{Ox})_j
\partial^x_i S_{N,M}(\overrightarrow{Ox})$ (12)
$\displaystyle G^S_{i,N,M}(\overrightarrow{Ox})$ = $\displaystyle \frac{\lambda+\mu}{\lambda+2\mu}\partial^x_i S_{N,M}(\overrightarrow{Ox})$ (13)
$\displaystyle R_{N,M}(\overrightarrow{Ox})$ = $\displaystyle \frac1{(N+M)!}P_N^M(\cos \theta) e^{i M \phi} r^N$ (14)
$\displaystyle S_{N,M}(\overrightarrow{Ox})$ = $\displaystyle (N-M)! P_N^M(\cos \theta) e^{i M \phi} \frac1{r^{N+1}}$ (15)

where, in Eq.(14) and Eq.(15), $(r,\theta,\phi)$ denote the polar coordinates of point x with respect to the center O and PNM is the associated Legendre function.

Now, by using Eq.(11), the integration on $S_0\in S$ involved in Eq.(8) is evaluated as

 
$\displaystyle {\int_{S_0}\left( T_{ij}(x-y)u_j(y) - \Gamma_{ij}(x-y) t_j(y) \right)dS_y}$
    $\displaystyle =\frac{1}{8\pi\mu}\sum_{N=0}^{\infty}\sum_{M=-N}^N
\left(F^S_{ip,...
...ine{M_{p,N,M}}(O)
+G^S_{i,N,M}(\overrightarrow{Ox})\overline{M_{N,M}}(O)\right)$ (16)

assuming that x is such a far point from S0 as enough to be $\displaystyle\vert\overrightarrow{Ox}\vert > \max_{y \in S_0}\vert\overrightarrow{Oy}\vert $ . Here, terms Mp,N,M(O) and MN,M(O) involved in the above equation are the multipole moments with respect to O and defined as
Mp,N,M(O)
    $\displaystyle =\int_{S_0}\left(
u_j(y)C_{jknp}n_k(y)\partial^y_n R_{N,M}(\overrightarrow{Oy})
-t_p(y)R_{N,M}(\overrightarrow{Oy})
\right)dS_y$ (17)
$\displaystyle {M_{N,M}(O)}$
    $\displaystyle =\int_{S_0}\left(
u_j(y)C_{jknp}n_k(y)\partial^y_n(\overrightarro...
...y})dS_y
-t_j(y)(\overrightarrow{Oy})_j R_{N,M}(\overrightarrow{Oy})
\right)dS_y$ (18)

Eq.(16) is the multipole expansion formula for 3D elastostatic problem, which is entirely similar to one for the 3D elastostatic crack problem formulated by Nishimura et al.[2]. Therefore, some translation formulas are deduced similarly. In this paper, only the results are shown bellow without the proof.

When the center of the multipole moments are translated from O to O', the current multipole moments are translated as

$\displaystyle {M_{p,N,M}(O')=\sum_{N'=0}^{N}\sum_{M'=-N'}^{N'}
R_{N',M'}(\overrightarrow{O'O}) M_{p,N-N',M-M'}(O)}$
$\displaystyle {M_{N,M}(O')=}$
    $\displaystyle \sum_{N'=0}^{N}\sum_{M'=-N'}^{N'}
R_{N',M'}(\overrightarrow{O'O})\left(M_{N-N',M-M'}(O)+(\overrightarrow{O'O})_p M_{p,N-N',M-M'}(O)\right)$ (20)

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.

 
$\displaystyle {\int_{S_0}\left( T_{ij}(x-y)u_j(y) - \Gamma_{ij}(x-y) t_j(y) \right)dS_y}$
    $\displaystyle =\frac{1}{8\pi\mu}\sum_{N'=0}^{\infty}\sum_{M=-N}^{N}
\left(F^R_{...
...row{x_0x})b_{j,N,M}(x_0)
+G^R_{i,N,M}(\overrightarrow{x_0x})b_{N,M}(x_0)\right)$ (21)

where bp,N,M(x0) and bN,M(x0) are the coefficients of the local expansion with respect to the center x0 described as
$\displaystyle {b_{p,N,M}(x_0)=\sum_{N'=0}^{\infty}\sum_{M'=-N'}^{N'}(-1)^{N'+M'}
S_{N+N',M-M'}(\overrightarrow{Ox_0})\overline{M_{p,N,M}}(O)}$
$\displaystyle {b_{N,M}(x_0)=\sum_{N'=0}^{\infty}\sum_{M'=-N'}^{N'}(-1)^{N'+M'}
S_{N+N',M-M'}(\overrightarrow{Ox_0})}$
    $\displaystyle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times\left(\overline{M_{N,M}}(O)-(\overrightarrow{Ox_0})_p
\overline{M_{p,N,M}}(O)\right)$ (23)

Finally, with an assumption $\vert\overrightarrow{Ox_0}\vert > \vert\overrightarrow{x_0x}\vert$, when the center of the local expansion is translated from x0 to x1, the coefficients of the local expansion are translated as

  
$\displaystyle {b_{p,N',M'}(x_1)=
\sum_{N=N'}^{\infty}\sum_{M=-N}^{N}
R_{N-N',M-M'}(\overrightarrow{x_0x_1})b_{p,N,M}(x_0)}$
$\displaystyle {b_{N',M'}(x_1)}$
    $\displaystyle =\sum_{N=N'}^{\infty}\sum_{M=-N}^{N}
R_{N-N',M-M'}(\overrightarrow{x_0x_1})
\left(b_{N,M}(x_0)-(\overrightarrow{x_0x_1})_p b_{p,N,M}(x_0)\right)$ (25)

By using some formulas mentioned above, the FMBEM for 3D elastostatic problem can be carried out similarly described in Nishimura et al.[2]. 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[3] is used for solving the discretized linear equations, 3) the block diagonal scaling technique proposed by Nishida and Hayami[4] 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.

  
Fig.1 CPU Time per Iteration
\begin{figure}\epsfxsize=6cm
\epsffile{fig_BENCH.ps}
\end{figure}


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Next: SIMULATION Up: 無題 Previous: PRINCIPLE OF INITIAL STRESS
Toru Takahashi 平成11年10月13日