Mathematical and Computer Modelling of Corrosion

Corrosion poses a significant threat to engineering infrastructure such as pipelines and offshore constructions, especially in regions that are difficult to access and cannot be monitored directly. To prevent failures in these areas, indirect detection methods based on inverse modelling are necessary.

This thesis develops the Boundary Element Method (BEM) to discretise both direct and inverse problems related to steady-state corrosion, modelled by the Laplace’s equation for the electric potential subjected to a Robin corrosion law on the corroded boundary that is not accessible to direct measurement. Inverse problems, such as reconstructing the corroded boundaries and coefficients from partial measurements, are highly ill-posed and require regularization techniques such as the Tikhonov or the truncated singular value decomposition methods to stabilize the solution in the presence of noise in the measured data.

The research addresses several specific cases of inverse problems, including Cauchy problems for data recovery with known geometry and coefficients, coefficient reconstruction, boundary identification, and joint reconstruction of both boundary and coefficients. These inverse problems are more difficult to solve than the corresponding direct problem since they are ill-posed i.e., either the existence, uniqueness or stability of solution is violated.

Numerical solutions are validated against analytical results and tested with various noise levels, demonstrating convergence and accuracy through stabilized systems solved by regularized least squares method.