Boundary shape identification in two-dimensional electrostatic problems using SQUIDs (Q2709824)

The authors consider the problem of determining structural flaws in materials used, e.g., in nuclear power plants by means of suitable measurements of magnetic flux densities. Assuming that the material has a cylindrical shape, the mathematical problem can be reduced to a 2D-one consisting in determining a function \(\varphi\), the electrical scalar potential, satisfying the elliptic boundary value problem NEWLINE\[NEWLINE \Delta \varphi = 0\quad \text{in} G_q,\qquad \varphi = v_i\quad \text{on } \partial G_i,\;i=1,3,\qquad {\partial \varphi\over \partial n} = 0 \quad \text{ on } \partial G_i,\;i=2,4. \tag{1}NEWLINE\]NEWLINE Here \(G_q=G\setminus C_q\), where \(G\) is a star-shaped bounded domain containing a star-shaped subdomain \(C_q\) depending ``continuously'' on a parameter \(q\) varying in a compact set \(Q \subset {\mathbb{R}}^m\). Moreover, the boundary \(\partial G\) is assumed to consist of four (disjoint) components of positive measures. NEWLINENEWLINENEWLINE\noindent The identification problem consists of recovering a pair \((\varphi,q)\), where \(\varphi\) satisfies equation (1), minimizing the following functional, the so-called additional information: NEWLINE\[NEWLINE\begin{multlined} \min_{q\in Q} \sum_{i=1}^m \Big|Y_p^i + {\sigma_0\mu_0l_p\over 4\pi |S^i_p|} \int_{S_p^i}\Big[ \int_G \Big( (x_2-x_2')D_{x_1}\varphi - (x_1-x_1')D_{x_2}\varphi\Big)\\ \times [(x_1-x_1')^2+(x_2-x_2')^2+h^2]^{-3/2} dx_1 dx_2\Big] dx_1' dx_2'\Big|^2. \end{multlined} \tag{2}NEWLINE\]NEWLINE The physical meaning of each integral in (2) is nothing but a measurement of the flux density by a superconducting quantum interference device through some surface \(S_p^i\subset \mathbb{R}^3\setminus {\overline G}\), the inner integral being an approximation of Biot-Savart's law. NEWLINENEWLINENEWLINEUsing well-known techniques in direct elliptic boundary-value problems the authors show that problem (1), (2) admits at least one solution \((\varphi,q)\) with \(q\in Q\) and \(\varphi\in H^1(G_q)\). Moreover, a finite Galerkin approach is used to construct a sequence of finite-dimensional approximating identification problems. These latter, in turn, easily allow to develop computational methods.