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Chosen possibilities of ɛ-fuzzy boundary elements method application in the analysis of conductivity problems with uncertainties

Witek Halina, Witek Bernard

Computer Assisted Methods in Engineering and Science

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2023

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Vol. 30, no. 4

407--426

EN

The paper presents a methodology of 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. Fuzzy Boundary Element Method (FBEM) using ɛ-number will be called ɛ-Fuzzy Boundary Element Method (ɛ-FBEM). 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, power engineering 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, overexcitation or a fault) which cannot be tolerated due to the threat to the object and to the entire power system. Proposed method maybe used for electrical equipment diagnosis and in consequence as a power system failure prevention. In this paper calculation methodology is illustrated on 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. A dedicated computer program allows for the calculation of both temperature and temperature derivative for any number of boundary elements using ɛ-FBEM.

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Numerical modelling of cooling process using fuzzy boundary element method with alpha-cuts

Piasecka-Belkhayat A.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2011

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Vol. 10, nr 1

199-204

EN

In the paper, the description of an unsteady heat transfer for a two-dimensional problem is presented. It is assumed that all the thermophysical parameters appearing in the mathematical model of the problem analyzed are given as fuzzy numbers. The problem discussed has been solved by means of the 1st scheme of the fuzzy boundary element method using α-cuts. The application of α-cuts allows one to avoid complicated arithmetical operations in the fuzzy numbers set. The interval Gauss elimination method with the decomposition procedure has been applied to solve the obtained fuzzy system of equations. In the final part of the paper, the results of numerical computations are shown.

3

Metoda elementów brzegowych dla obszarów o kształcie rozmytym

Skrzypczyk J., Burczyński T.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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1998

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z. 6

317-322

PL

W pracy przedstawiono elementy nowej koncepcji metody rozmytych elementów brzegowych. Artykuł omawia odwzorowania o wartościach rozmytych, które są rozwiązaniami rozmytych brzegowych równań całkowych. Analiza przeprowadzona jest na przykładzie problemu brzegowego klasycznej teorii potencjału z uwzględnieniem rozmytych warunków brzegowych typu Dirichleta i Neumanna, rozmytej funkcji gęstości źródeł oraz rozmytego kształtu obszaru.

EN

In the paper basic concepts of a new methodology of the fuzzy boundary element method are presented. This article deals with fuzzy-set-valued mappings which are solutions of the fuzzy boundary integral equations. Exact fuzzy solutions of fuzzy boundary integral equations are defined as well as approximated ones. Computational fuzzy problems and applications are considered in details for boundary potential problems in a fuzzy domain with fuzzy Dirichlet and Neumann type boundary conditions and fuzzy density source functions.

4

Fuzzy boundary element method in the analysis of uncertain systems

Skrzypczyk J., Burczyński T.

Journal of Theoretical and Applied Mechanics

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1998

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nr 2

493-512

EN

In the paper basicconcepts of mew methodology of fuzzy boundary element method are presented. This article deals with fuzzy-set-valued mappings which are solutions of the fuzzy boundary integral equations. Extact fuzzy solutions of the fuzzy boundary integral equations are defined as well as conditional solutions. Computational fuzzy problems and applications are considered in details for boundary potential problems with fuzzy Dirichlet and Neumann type boundary conditions and fuzzy density source functions in a fuzzy domain.

PL

W pracy przedstawiono elementy nowej koncepcji metody rozmytych elementów brzegowych. Artykuł omawia odwzorowania o wartościach rozmytych, które są rozwiązaniami rozmytych brzegowych równań całkowych. Analiza przeprowadzona jest na przykładzie problemu brzegowego klasycznej teorii potencjału z uwzględnieniem rozmytych warunków brzegowych typu Dirichleta i Newmanna, rozmytej funkcji gęstości źródeł oraz rozmytego kształtu obszaru.