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

Abstract 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 R 2 .

[1]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[2]  P. Ramachandran Method of fundamental solutions: singular value decomposition analysis , 2002 .

[3]  O. D. Kellogg Foundations of potential theory , 1934 .

[4]  William H. Press,et al.  The Art of Scientific Computing Second Edition , 1998 .

[5]  J.A.F. Santiago,et al.  On a regularized method of fundamental solutions coupled with the numerical Green's function procedure to solve embedded crack problems , 2013 .

[6]  J.A.F. Santiago,et al.  An Application of Genetic Algorithms and the Method of Fundamental Solutions to Simulate Cathodic Protection Systems , 2012 .

[7]  J. Kolodziej,et al.  Application of genetic algorithms for optimal positions of source points in the method of fundamental solutions. , 2008 .

[8]  R. Mathon,et al.  The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions , 1977 .

[9]  Luiz C. Wrobel,et al.  NUMERICAL SIMULATION OF A CATHODICALLY PROTECTED SEMISUBMERSIBLE PLATFORM USING THE PROCAT SYSTEM , 1990 .

[10]  Clifford H. Thurber,et al.  Parameter estimation and inverse problems , 2005 .

[11]  Trung Van Nguyen,et al.  Mathematical Modeling of Cathodic Protection Using the Boundary Element Method with a Nonlinear Polarization Curve , 1992 .

[12]  Thomas C. Hales,et al.  The Jordan Curve Theorem, Formally and Informally , 2007, Am. Math. Mon..

[13]  Non‐linear heat conduction in composite bodies: A boundary element formulation , 1988 .

[14]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[15]  G. Fairweather,et al.  THE SIMPLE LAYER POTENTIAL METHOD OF FUNDAMENTAL SOLUTIONS FOR CERTAIN BIHARMONIC PROBLEMS , 1989 .

[16]  W. Aperador,et al.  Influence of conductivity on cathodic protection of reinforced alkali-activated slag mortar using the finite element method , 2009 .

[17]  K. Atkinson The Numerical Evaluation of Particular Solutions for Poisson's Equation , 1985 .

[18]  Shigeru Aoki,et al.  Optimization of cathodic protection system by BEM , 1997 .

[19]  Oswald Veblen,et al.  Theory on plane curves in non-metrical analysis situs , 1905 .

[20]  Mohammad Habibi Parsa,et al.  Simulation of cathodic protection potential distributions on oil well casings , 2010 .

[21]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1996, Springer Berlin Heidelberg.

[22]  J. Genesca,et al.  Mathematical simulation of a cathodic protection system by finite element method , 2005 .

[23]  A. Karageorghis,et al.  THE METHOD OF FUNDAMENTAL SOLUTIONS FOR HEAT CONDUCTION IN LAYERED MATERIALS , 1999 .

[24]  Cleberson Dors,et al.  Efficient numerical models for the prediction of acoustic wave propagation in the vicinity of a wedge coastal region , 2011 .

[25]  L. Wrobel,et al.  Genetic algorithms for inverse cathodic protection problems , 2004 .

[26]  G. Fairweather,et al.  The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems , 1989 .

[27]  Leevan Ling,et al.  Optimality of the method of fundamental solutions , 2011 .

[28]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[29]  ON BOUNDARY ELEMENTS FOR SIMULATION OF CATHODIC PROTECTION SYSTEMS WITH DYNAMIC POLARIZATION CURVES , 1997 .

[30]  Graeme Fairweather,et al.  The method of fundamental solutions for the numerical solution of the biharmonic equation , 1987 .