A technique is developed for constructing the solution of ${\Delta }^{2}u=F$ in $R=\left\{\right(x,y):|x|\<a,|y|\<b\}$, subject to boundary conditions $u=\phi$, $\frac{\partial u}{\partial n}=\psi$ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2\left(R\right)$ onto the subspace $\mathbf{H}$ of square integrable functions harmonic in $\mathbf{R}$. This problem is solved by decomposition $\mathbf{H}$ into the closed direct (not orthogonal) sum of two subspaces ${\mathbf{H}}^{\left(1\right)},{\mathbf{H}}^{\left(2\right)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^{\left(1\right)}$, $P^{\left(2\right)}$ of $L^2\left(R\right)$ onto ${\mathbf{H}}^{\left(1\right)}$, ${\mathbf{H}}^{\left(2\right)}$ respectively. The resulting construction...

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical implementation...

In this paper we extend recent work on the detection of inclusions using electrostatic measurements to the problem of crack detection in a two-dimensional object. As in the inclusion case our method is based on a factorization of the difference between two Neumann-Dirichlet operators. The factorization possible in the case of cracks is much simpler than that for inclusions and the analysis is greatly simplified. However, the directional information carried by the crack makes the practical...