Abstract
We consider the corrosion detection problem in terms of the Laplace equation and study a simply connected bounded domain with Wentzell-type GIBC boundary condition. We derive the systems of integral equations and establish the equivalence to the inverse shape problem in a Sobolev space setting. For the direct problem, we use potential theory to simulate the Neumann data from Dirichlet data on Dirichlet boundary. Then we propose a Newton iterative approach based on the boundary integral equations derived from Green’s representation theorem. After describing the linearization and the iteration scheme for the inverse shape, we compute the Fréchet derivatives with respect to the unknowns. We conclude by presenting several numerical examples for shape reconstructions to show the validity of the proposed method.
Highlights
The Laplace equation is a special partial differential equation
For instance, thermal imaging, electrostatic imaging, and corrosion detection, are mathematically regarded as inverse boundary value problems of the Laplace equation. Such inverse problems can be interpreted as reconstructing the boundary shape or the impedance coefficients on the boundary from the measurements
Much work has been done on these problems [7, 10, 11, 13, 14, 21, 22, 28], where the authors used the nonlinear integral equation method proposed by Kress and Rundell [24]
Summary
The Laplace equation is a special partial differential equation. Solving the boundary value problem of the Laplace equation is an important mathematical problem often encountered in the fields of electromagnetics, astronomy, thermodynamics, and hydrodynamics, because such equations describe the properties of physical objects such as electric, gravitational, and flow fields in the form of potential function. In [10, 13, 14] the authors applied the method of [24] to reconstruct the impedance boundary shape of the corrosion problems. We use a set of Cauchy data obtained from the direct problem to reconstruct the corrosion boundary shape with generalized impedance condition based on Green’s formula.
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