Boundary reconstruction in two-dimensional steady-state anisotropic heat conduction

We study the reconstruction of an unknown/inaccessible smooth inner boundary from the knowledge of the Dirichlet condition (temperature) on the entire boundary of a doubly connected domain occupied by a two-dimensional homogeneous anisotropic solid and an additional Neumann condition (normal heat flux) on the known, accessible, and smooth outer boundary in the framework of steady-state heat conduction with heat sources. This inverse geometric problem is approached through an operator that maps an admissible inner boundary belonging to the space of \(2\pi -\) $2\pi -$ periodic and twice continuously differentiable functions into the Neumann data on the outer boundary which is assumed to be continuous. We prove that this operator is differentiable, and hence, a gradient-based method that employs the anisotropic single layer representation of the solution to an appropriate Dirichlet problem for the two-dimensional anisotropic heat conduction is developed for approximating the unknown inner boundary. Numerical results are presented for both exact and perturbed Neumann data on the outer boundary and show the convergence, stability, and robustness of the proposed method.