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Optimal positioning of anodes and virtual sources in the design of cathodic protection systems using the method of fundamental solutions. (English) Zbl 1297.65197
Summary: 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 $$\mathbb R^2$$.

##### MSC:
 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
##### Software:
Genocop; HYBRJ; Jordan; minpack
Full Text:
##### References:
 [1] Fontana, M. G.; Greene, N. D., Corrosion engineering, (1967), McGraw-Hill New York [2] Montoya, R.; Rendon, O.; Genesca, J., Genesca mathematical simulation of cathodic protection system by finite element method, Mater Corros, 56, 404-411, (2005) [3] Montoya, R.; Aperador, W.; Bastidas, D. M., Influence of conductivity on cathodic protection of reinforced alkali-activated slag mortar using the finite element method, Corros Sci, 51, 2857-2862, (2009) [4] Parsa, M. H.; Allahkaram, S. R.; Ghobadi, A. H., Simulation of cathodic protection potential distributions on oil well casings, J Pet Sci Eng, 72, 215-219, (2010) [5] Telles, J. C.F.; Mansur, W. J.; Wrobel, L. C.; Marinho, M. G., Numerical simulation of a cathodically protected semisubmersible platform using PROCAT system, Corrosion, 46, 513-518, (1990) [6] Santiago, J. A.F.; Telles, J. C.F., On boundary elements for simulation of cathodic protection systems with dynamic polarization curves, Int J Numer Methods Eng, 40, 2611-2622, (1997) · Zbl 0887.65126 [7] BEASY User Guide Beasy. Computational mechanics BEASY Ltd 2000. Ashurst, Southampton, UK. [8] Azevedo, J. P.S.; Wrobel, L. C., Nonlinear heat conduction in composite bodies: a boundary element formulation, Int J Numer Methods Eng, 26, 19-38, (1988) · Zbl 0633.65117 [9] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for numerical solution of the biharmonic equation, J Comput Phys, 69, 434-459, (1987) · Zbl 0618.65108 [10] Fairweather, G.; Karageorghis, A., The almansi of fundamental solutions for solving biharmonic problems, Int J Numer Methods Eng, 26, 1668-1682, (1988) · Zbl 0639.65066 [11] Fairweather, G.; Karageorghis, A., The simple layer potential method of fundamental solutions for certain biharmonic equation, Int J Numer Methods Fluids, 9, 1221-1234, (1989) · Zbl 0687.76028 [12] Ramachandran, P. A., Method of fundamental solutionssingular value decomposition analysis, Commun Numer Methods Eng, 18, 789-891, (2002) [13] Jopek, H.; Kolodziej, J. A., Application of genetic algorithms for optimal positions of source points in the method of fundamental solutions, Comput Assist Mech Eng Sci, 15, 215-224, (2008) · Zbl 1420.65130 [14] Wong, K. Y.; Ling, L., Optimality of the method of fundamental solutions, Eng Anal Bound Elem, 35, 42-46, (2011) · Zbl 1259.65198 [15] Fontes, E. F.; Santiago, J. A.F.; Telles, J. C.F., On a regularized method of fundamental solutions coupled with the numerical green׳s function procedure to solve embedded crack problems, Eng Anal Bound Elem, 37, 1-7, (2013) · Zbl 1351.74089 [16] Kupradze, V. D.; Aleksidze, M. A., Approximate method of solving certain boundary-value problems, Soobshch Akad Nauk Gruz SSR, 30, 529-536, (1963) [17] Mathon, R.; Johnson, R. L., The approximate solution of elliptic boundary-value problems by fundamental solution, SIAM J Numer Anal, 14, 638-650, (1977) · Zbl 0368.65058 [18] Berger, J. R.; Karageorghis, A., The method of fundamental solutions for heat conduction in layered materials, Int J Numer Methods Eng, 45, 1681-1694, (1999) · Zbl 0972.80014 [19] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for the solution of nonlinear plane potential problems, IMA J Numer Anal, 9, 231-242, (1989) · Zbl 0676.65110 [20] Costa, E. G.A.; Godinho, L.; Santiago, J. A.F.; Pereira, A.; Dors, C., Efficient numerical models for the prediction of acoustic wave propagation in the vicinity of a wedge coastal region, Eng Anal Bound Elem, 35, 855-867, (2011) · Zbl 1259.76024 [21] Santos, W. J.; Santiago, J. A.F.; Telles, J. C.F., An application of genetic algorithms and the method of fundamental solutions to simulate cathodic protection systems, Comput Model Eng Sci, 87, 23-40, (2012) [22] Aoki, S.; Amaya, K., Optimization of cathodic protection system by BEM, Eng Anal Bound Elem, 19, 147-156, (1997) [23] Wrobel, L. C.; Miltiadou, P., Genetic algorithms for inverse cathodic protection problems, Eng Anal Bound Elem, 28, 267-277, (2004) · Zbl 1051.78019 [24] Santos WJ, Santiago JAF, Telles JCF. Optimisation of the cathodic protection systems using genetic algorithms and boundary element method. In: Iberian Latin American Congress on Computational Methods in Engineering; 2011. [25] Garbow BS, Hillstrom KE, Moré JJ. MINPACK project 1980. Argonne National Laboratory, Argonne, Illinois, USA. [26] Kellog, O. D., Foundations of potential theory, (1954), Dover New York [27] Atkinson, K. E., The numerical evaluation of particular solution for poisson׳s equation, IMA J Numer Anal, 5, 319-338, (1985) · Zbl 0576.65114 [28] Yan, J. F.; Pakalapati, S. N.R.; Nguyen, T. V.; White, R. E., Mathematical modelling of cathodic protection using the boundary element method with nonlinear polarisation curves, J Electrochem Soc, 139, 1932-1936, (1992) [29] Aster, R. C.; Borchers, B.; Thurber, C. H., Parameter estimation and inverse problems, (2005), Elsevier Academic Press USA · Zbl 1088.35081 [30] Nocedal, J.; Wright, S. J., Numerical optimization, (2006), Springer New York · Zbl 1104.65059 [31] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flanner, B., Numerical recipes in C. the art of scientific computing, (1992), Cambridge University Press Cambridge · Zbl 0845.65001 [32] Fletcher, R., Practical methods of optimisation, (1987), John Wiley & Sons Chichester, U.K · Zbl 0905.65002 [33] Holland, J., Adaptation in natural and artificial systems, (1975), University of Michigan Press Ann Arbor [34] Goldberg, D. E., Genetic algorithms in search, optimisation and machine learning, (1989), Addison-Wesley Reading, MA [35] Michalewicz, Z., Genetic algorithms + data structures = evolution programs, (1996), Springer-Verlag, Berlin Heidelberg New York · Zbl 0841.68047 [36] Brebbia, C. A.; Telles, J. C.F.; Wrobel, L. C., Boundary elements techniques: theory and applications in engineering, (1984), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0556.73086 [37] Veblen, O., Theory on plane in non-metrical analysis situs, Trans Am Math Soc, 6, 83-98, (1905) · JFM 36.0530.02 [38] Hales, T. C., The Jordan curve theorem, formally and informally, Am Math Mon, 114, 882-894, (2007) · Zbl 1137.03305
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