# zbMATH — the first resource for mathematics

Using the Gaussian function to simulate constant potential anodes in multiobjective optimization of cathodic protection systems. (English) Zbl 1403.78035
Summary: The purpose of this work is to numerically find the optimum location of constant potential anodes to ensure complete structure surface protection using a cathodic protection technique. The existence of sacrificial anodes is originally introduced through the boundary conditions of the corresponding boundary value problem (BVP). However, if constant potential galvanic regions are introduced through its boundaries, then finding their optimal location is not an easy task due to the necessity of redefining boundary geometric nodes and the arrangement of virtual sources for the standard method of fundamental solutions (MFS) formulation. Therefore, in this work, the galvanic anodes are introduced as source terms using a Gaussian function. Hence, the boundary remains the same for different anode positions. The optimization process includes the identification of the following parameters characterizing the Gaussian function: the optimum coordinates of the centre of the anode, a factor that involves the inherent potential of the electrode and a proportionality factor for the electrode diameter. The MFS methodology coupled with a genetic algorithm presented good results for this multiobjective optimization procedure. This fact can be seen in the several results of applications that are discussed in this paper, considering numerical simulations in finite regions in $$\mathbb R^2$$.

##### MSC:
 78M25 Numerical methods in optics (MSC2010) 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs 65K10 Numerical optimization and variational techniques 78A25 Electromagnetic theory, general 90C29 Multi-objective and goal programming 90C90 Applications of mathematical programming
##### Software:
Genocop; HYBRJ; minpack
Full Text:
##### References:
 [1] Fontana, M. G.; Greene, N. D., Corrosion engineering, (1967), McGraw-Hill New York [2] Roberge, P. R., Handbook of corrosion engineering, (1999), McGraw-Hill New York [3] 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) [4] Santos, W. J.; Santiago, J. A.F.; Telles, J. C.F., Optimal positioning of anodes and virtual sources in the design of cathodic protection systems using the method of fundamental solutions, Eng Anal Bound Elem, 46, 67-74, (2014) · Zbl 1297.65197 [5] Montoya, R.; Rendon, O.; Genesca, J., Mathematical simulation of cathodic protection system by finite element method, Mater Corros, 56, 404-411, (2005) [6] 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) [7] Montoya, R.; Galvan, J. C.; Genesca, J., Using the right side of Poisson’s equation to save on numerical calculations in FEM simulation of electrochemical systems, Corros Sci, 53, 1806-1812, (2011) [8] Miyasaka, M.; Hashimoto, K.; Aoki, S., A boundary element analysis on galvanic corrosion problems - computational accuracy on galvanic fields with screen plates, Corros Sci, 30, 299-311, (1990) [9] Abootalebi, O.; Kermanpur, A.; Shishesaz, M. R.; Golozar, M. A., Optimizing the electrode position in sacrificial anode cathodic protection systems using boundary element method, Corros Sci, 52, 678-687, (2010) [10] 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) [11] 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 [12] BEASY User Guide Beasy. Computational mechanics. Ashurst, Southampton, UK: BEASY Ltd.; 2000. [13] 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 [14] Aoki, S.; Amaya, K., Optimization of cathodic protection system by BEM, Eng Anal Bound Elem, 19, 147-156, (1997) [15] Wrobel, L. C.; Miltiadou, P., Genetic algorithms for inverse cathodic protection problems, Eng Anal Bound Elem, 28, 267-277, (2004) · Zbl 1051.78019 [16] Kupradze, V. D.; Aleksidze, M. A., Aproximate method of solving certain boundary-value problems, Soobshch Akad Nauk Gruz SSR, 30, 529-536, (1963) [17] 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 [18] 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 [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] Garbow, B. S.; Hillstrom, K. E.; Mor, J. J., MINPACK project, (1980), Argonne National Laboratory, Argonne, Illinois, USA [21] More L. Levenberg-Marquardt algorithm: implementation and theory. In: Proceedings of the biennial conference held at Dundee, June 28-July 1. Argonne, IL, USA: Argonne National Laboratory; 1977 · Zbl 0372.65022 [22] Partridge, P.; Brebbia, C. A.; Wrobel, L. C., The dual reciprocity boundary element method, (1992), Computational Mechanics Publications Southampton · Zbl 0758.65071 [23] 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) [24] Atkinson, K. E., The numerical evaluation of particular solution for Poisson’s equation, IMA J Numer Anal, 5, 319-338, (1985) · Zbl 0576.65114 [25] Golberg, M. A., The numerical evaluation of particular solutions in the BEM - a review, Bound Elem Commun, 6, 99-106, (1995) [26] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv Comput Math, 4, 389-396, (1995) · Zbl 0838.41014 [27] Golberg, M. A., The method of fundamental solutions for Poisson’s equation, Eng Anal Bound Elem, 16, 205-213, (1995) [28] Chen CS, Golberg MA, Schaback RA. Recent developments of the dual reciprocity method using compactly supported radial basis functions. In: Transformation of domain effects to the boundary. Southampton, Boston: WIT Press; 2003. p. 183-225. · Zbl 1043.65133 [29] Fletcher, R., Practical methods of optimisation, (1987), John Wiley & Sons Chichester, UK · Zbl 0905.65002 [30] Srinivas, N.; Deb, K., Multiobjective optimization using nondominated sorting in genetic algorithms, Evol Comput, 2, 221-248, (1994) [31] Michalewicz, Z., Genetic algorithms + data structures=evolution programs, (1996), Springer-Verlag, New York: Springer-Verlag, Berlin Heidelberg · Zbl 0841.68047 [32] Mitchell, M., An introduction to genetic algorithms, (1997), The MIT Press Cambridge, MA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.