Laplace equation sample problem

Early predictions of corrosion rates and estimates of adequate CP have traditionally been based on case studies and sample exposure tests. Applying these techniques to real structures usually involve extrapolations, use of large safety factors and ongoing corrections and maintenance of the system. In the late 1960s the finite element method was applied to the problem by discretization of the electrolytically conductive environment into a mesh and solving numerically with Laplace equations to define the intersection points, or nodes of this mesh [17]. 

Equation (22) is particularly useful when a concentration gradient in depth exists. In this case, several spectra at different values of 9 are taken and the analysis is called angle resolved X-ray photoelectron spectroscopy. However, for a maximum efficiency, a flat surface (at an atomic level) is needed to avoid shade effects as shown by Fadley in his early works in the 1970s [44]. An additional problem exists the extraction of concentration profiles, cg(x), from Eq. (22) is an inverse problem the intensity as a function of the analysis angle is the Laplace transform of the composition depth profile of the sample [43] and does not have a unique solution. Several algorithms to solve the inversion problem were developed and tested [46]. They are all very unstable and sensitive to small statistical... 

The second is that propagation effects must be dealt with if sample dimensions are an appreciable fraction of a wavelength, and this situation is not readily avoided at frequencies for which the method is otherwise useful. Both problems are better handled by use of onesided Fourier (Laplace) transforms, rather than direct time domain solutions, as a result of the convolution theorem for the former, and solution of the field equations in the frequency domain for the latter. 

The renewal of liquid has been realized to avoid local chemistiy problems and to evacuate corrosion products in order to model the corrosion damage solving the Laplace s equation by M-BEM method (Moving Boundary Elements Method). After exposure to the corrosive solution for periods of 6-72 hours the heterogeneous electrode was cleaned in an ultrasonic bath to remove the corrosion products from the surface of the sample. 

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