Chosen Possibilities of ε-Fuzzy Boundary Elements Method Application in the Analysis of Conductivity Problems with Uncertainties

This paper presents a methodology for solving boundary problems with uncertainty parameters based on the use of interval perturbation numbers. This methodology allows for the analysis of very complex problems with different uncertain parameters. The fuzzy boundary element method (FBEM) using ε̄-number will be called the ε-fuzzy boundary element method (ε-FBEM). A detailed discussion of the problems of computing and applications will be presented on the example of the fuzzy boundary integral equation arising from the boundary problem for the potential problems with heterogeneous, fuzzy boundary conditions of Dirichlet and Neumann type, fuzzy internal sources, fuzzy boundary, and fuzzy fundamental solution. The presented methodology can be used to solve various engineering problems (e.g., in civil engineering [9], power engineering [7, 15] and others) – e.g., to analyze the temperature distribution in structural elements or elements located in the vicinity of objects or devices. In the latter case, the increased temperature may be a symptom of a severe failure (e.g., power transformer overload, core overexcitation, or an internal fault), which cannot be tolerated due to the threat to the object itself as well as the entire power system operation. The proposed method may be used for electrical equipment diagnosis and, consequently, as a power system failure prevention [5, 6, 14]. In this paper, calculation methodology is illustrated with the example of an area bounded by a square, on the left boundary of which a certain temperature is set, while on the rest of the boundaries the conditions are equal to zero. The authors’ dedicated computer program written in the Fortran programming environment [13] allows the calculation of the temperature and temperature derivative for any number of boundary elements using ε̄-FBEM.