Optimal positioning of anodes and virtual sources in the design of cathodic protection systems using the method of fundamental solutions

The method of fundamental solutions (MFS) is used for the solution of Laplace׳s equation, with nonlinear boundary conditions, aiming at analyzing cathodic protection systems. In the MFS procedure, it is necessary to determine the intensities and the distribution of the virtual sources so that the boundary conditions of the problem are satisfied. The metallic surfaces, in contact with the electrolyte, to be protected, are characterized by a nonlinear relationship between the electrochemical potential and current density, called cathodic polarization curve. Thus, the calculation of the intensities of the virtual sources entails a nonlinear least squares problem. Here, the MINPACK routine LMDIF is adopted to minimize the resulting nonlinear objective function whose design variables are the coefficients of the linear superposition of fundamental solutions and the positions of the virtual sources outside the problem domain. First, examples are presented to validate the standard MFS formulation as applied in the simulation of cathodic protection systems, comparing the results with a direct boundary element (BEM) solution procedure. Second, a MFS methodology is presented, coupled with a genetic algorithm (GA), for the optimization of anode positioning and their respective current intensity values. All simulations are performed considering finite regions in R2.