A Boundary Element Regularization Method for the Boundary Determination in Potential Corrosion Damage

In this paper, we consider the inverse problem for the Laplace equation in two-dimensions which requires the determination of the location, size and shape of an unknown, or partially unknown, portion n z of the boundary z of a solution domain z R 2 from additional Cauchy data on the remaining portion of the boundary o = z m n . This problem arises in the study of quantitative non-destructive evaluation of corrosion in materials in which boundary measurements of currents and voltages are used to determine the material loss caused by corrosion. This inverse problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularization procedure. The choice of the regularization parameter is based on an L-curve type criterion although, alternatively one may use the discrepancy principle. Several examples which involve noisy Cauchy input data are thoroughly investigated showing that the numerical method produces a stable approximate solution which is also convergent to the exact solution as the data errors tend to zero.