This paper presents a new algorithm for PWA approximation of nonlinear systems. Such an approximation is very important to enable a reduction in the complexity of models of nonlinear systems while keeping the global validity of the models. The paper builds on previous work on piecewise affine (PWA) approximation methods, in particular on the work done by Casselman and Rodrigues, known as the Set of Linearization Points (SLP) PWA approximation. The proposed extension method can be used to approximate any continuous function of one variable by a PWA function. The algorithm is based on the points at which the linearization lines intersect with each other. The method assumes that a desired approximation error and one linearization point are given. The algorithm, then performs several linearizations. It is shown that the new linearization points are optimal in the sense of decreasing the error between the exact function and the approximation. The main advantages of this methodology compared to previous approaches are the reduction of the number of pieces of the PWA function, the guarantee that the approximation is continuous, and that the derivative of the approximation and the derivative of the exact function are equal at all linearization points. A detailed collection of examples from different fields of study highlight the effectiveness and the flexibility of the proposed method. It is shown that the proposed method compares favorably with other methods.

An electrolyte for electrochemical hydrogen charging of corrosion-susceptible alloys is developed, which preserves the surface integrity at nano-scale by minimizing the surface roughness alternation. To assure the formation and adsorption of the hydrogen from the electrolyte, permeation tests were performed on Fe 3wt.%Si ferritic steel.X-ray photoelectron spectroscopy method was used to check the effect of the glycerol-based solution on the chemical composition of the sample surface. The surface analysis revealed minimal chemical and topography alteration on the surface after different electrochemical treatments. Various types of in situ small-scale mechanical tests such as nano-indentation, micro-pillar compression, and micro-cantilever bending tests were performed inside this electrolyte while the samples being charged with hydrogen under cathodic potential. These small-scale mechanical tests showed that the solution facilitates studying hydrogen embrittlement in nano-or micro-scale.

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