A solution technique for cathodic protection with dynamic boundary conditions by the boundary element method J.A.F. Santiago * , J.C.F. Telles COPPE / Universidade Federal do Rio de Janeiro, Programa de Engenharia Civil, Caixa Postal: 68506, 21945-970 Rio de Janeiro, RJ, Brazil Received 3 March 1997; accepted 2 August 1998 Abstract This article is concerned with the application of the Boundary Element Method to cathodic protection problems of submerged structures using polarization curves depending upon time and formation potential. These curves have been adjusted from potentiostatic data obtained from in-situ experiments, yielding a nonlinear functional representation. The solution technique adopts stepwise linearized polarization curves and is employed for sufficiently small time steps. The influence of varying formation potential is introduced into the analysis under two alternative hypotheses here designated fictitious time and fictitious potential.  1999 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. Keywords: Boundary elements; Cathodic protection; Dynamic polarization curves; Potential 1. Introduction The physical, chemical and biological phenomena, taking place on a cathodic surface in seawater is quite complex, being presently subject to study in many research centres around the world. The parameters which have major influ- ence in this phenomenon are potential, electric current, time, temperature, pressure, seawater chemistry, relative water velocity close to the cathodic surface and surface conditions [1]. For offshore structures, the behavior of cathodic surfaces is modeled by a time dependent polarization curve, different at each surface point, which describes the nonlinear relation between potential and current density. Dynamic polarization curves are influenced by scale deposition, which is strongly influenced by the current– potential history on the surface. This history may be quite complex, being impossible to cover all of the existing situa- tions by experiments. However, information about the polarization curve can be retrieved from potentiostatic polarization data (or galvanostatic polarization data), obtained from experiments in which a potential (or current density) is maintained constant in time. These curves can be used with adequate time marching schemes to simulate the time history of the varying potential and current density during the life of the cathodic protection system. In this work the Boundary Element Method is used in conjunction with two different time marching schemes to analyze cathodic protection systems of offshore structures [2–4]. These are, fictitious potential and fictitious time procedures and they are used with polarization curves deter- mined from potentiostatic data obtained from in-situ experi- ments. The solution routine adopted employs a new step by step linearization technique which reduces considerably the burden associated to the nonlinear polarization curve as boundary conditions. 2. Mathematical model Considering that the cathodic protection technique is developed within a homogeneous region V, surrounded by a boundary G and with electric conductivity k, the problem is governed by the Laplace equation: 7 2 f  0 1 subject to appropriate boundary conditions of the form: a:1 f   f on G 1 ; 2 a:2 i   ı on G 2 ; b i  vf + d on G 3 ; c i  F f f f; t on G 4 ; Advances in Engineering Software 30 (1999) 663–671 0965-9978/99/$ - see front matter  1999 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S0965-9978(98)00121-5 www.elsevier.com/locate/advengsoft * Corresponding author.