Variational Boundary Integral Equation

Multiplying (3.46) by a test function $\mbox{\boldmath$\space \psi $ }(\mbox{\boldmath$\space x $ })$ which satisfies the following condition:

$$\begin{eqnarray*}\mbox{\boldmath$ \psi $ }(\mbox{\boldmath$ x $ }) = 0 \quad \mbox{on} \ \partial S \end{eqnarray*}$$

and then integrating the resulting expressions on S, one obtains

 

$$\int_{S} \psi_a(\mbox{\boldmath$ x $ }) t^{\infty}_{a}(\mbox{\bol... ...cd\/jl} n_c(\mbox{\boldmath$ y $ }) \phi_d(\mbox{\boldmath$ y $ }) dS_{y} dS_x.$$ (3.100)

The non-regularised variational equation in (3.100) is used as the basis for the Galerkin BIEM.

Note that an arbitrary equilibrated stress field possesses an expression in terms of a stress function. Specifically, the ``stress'' computed from the fundamental solution can be expressed in terms of the stress function $\Phi_{pqrs}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ as (See Nishimura and Kobayashi [63])

 

$$C_{abik} \frac{\partial}{\partial x_k} \frac{\partial}{\partial y... ...tial}{\partial y_l} \Phi_{pqrs}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }),$$ (3.101)

where e_ijk is the permutation symbol and the function $\Phi_{pqrs}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is given by

$$\Phi_{pqrs}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = \fra... ...} \delta_{qr})\bigr) \ \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert.$$ (3.102)

With (3.100) and (3.101), one derives the following variational equation given by (See Appendix L.3)

 

$${\int_{S} \psi_a(\mbox{\boldmath$ x $ }) t^{\infty}_{a}(\mbox{\boldmath$ x $ }) dS_{x} }$$

$$= - \int_{S} n_b(\mbox{\boldmath$ x $ }) e_{bjq} \frac{\partial \... ...ckr} \frac{\partial \phi_d(\mbox{\boldmath$ y $ })}{\partial y_k} dS_y \ dS_x .$$ (3.103)

The variational equation in (3.103) has been considered useful in the standard Galerkin BIEM since the kernel function is regularised to an integrable one. In the present investigation with FMM, however, the variational equation in (3.100) is more useful. It is because we are interested in applying FMM in evaluating contributions to the integrals in (3.100) or (3.103) from distant $\mbox{\boldmath$\space x $ }$ and $\mbox{\boldmath$\space y $ }$, and because the kernel function in (3.100) decays more quickly than that in (3.103) as $\vert\mbox{\boldmath$\space x $ }- \mbox{\boldmath$\space y $ }\vert \to \infty$. Actually, FMM formulations with regularised integral equations have been tested in section 3.3 with collocation and were shown to introduce more moments than formulations without regularisation. Also, the numerical experiments in section 3.3 showed that the FMM with regularised integral equations is slower than that without regularisation. Notice, however, that the following identity holds for arbitrary smooth functions $\mbox{\boldmath$\space \psi $ }$and $\mbox{\boldmath$\space \phi $ }$ having supports within S:

 

$${\int_{S} \psi_a(\mbox{\boldmath$ x $ }) \ \pfint_{S} n_{b}(\mbox... ...cd\/jl} n_c(\mbox{\boldmath$ y $ }) \phi_d(\mbox{\boldmath$ y $ }) dS_{y} dS_x}$$

$$\int_{S} n_b(\mbox{\boldmath$ x $ }) e_{bjq} \frac{\partial \psi_... ...{ckr} \frac{\partial \phi_d(\mbox{\boldmath$ y $ })}{\partial y_k} dS_y \ dS_x.$$ = (3.104)

We shall use the regularised expression on the right-hand side of (3.104) in an auxiliary manner to compute the integral on the left-hand side of (3.104) with $\psi_a$ and $\phi_a$, respectively, taken as one of the basis functions (sometimes referred to as global shape functions) for the crack opening displacement. This calculation will be needed in the 5th step of the FMM algorithm in 3.4.3 when the supports of $\psi_a$ and $\phi_a$ are not separated sufficiently.