Numerical solution of a boundary detection problem using genetic algorithms

Abstract

In this paper we consider the identification of the geometric structure of the boundary of the solution domain for the two-dimensional Laplace equation. We investigate the determination of the location, size and shape of an unknown portion γ⊂ ∂Ω of the boundary ∂Ω of a solution domain Ω⊂R 2 from Cauchy data on the remaining portion of the boundary ∂Ω\\γ or on a subset Γ⊂ ∂Ω\\γ of this part of the boundary. 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. The domain identification problem is considered as a variational problem to minimise a defect functional, which utilises some additional data on certain known parts of the boundary. A real coded genetic algorithm, combined with a function specification method, is used in order to minimise the objective functional. The Laplace equation is discretised using the boundary element method. Numerical results are presented and discussed for several test examples.