Fuzzy and interval finite element method for heat conduction problem

TL;DR

This paper extends the finite element method to handle heat conduction problems with uncertain material properties by incorporating fuzzy and interval arithmetic, providing a more realistic analysis of heat transfer in composite materials.

Contribution

It introduces a fuzzy finite element approach for heat conduction problems with uncertain parameters and proposes new methods to solve the resulting fuzzy linear systems.

Findings

Fuzzy finite element method effectively models uncertainty in material properties.

New solution techniques for fuzzy linear systems are developed.

Results demonstrate the impact of uncertainty on heat conduction analysis.

Abstract

Traditional finite element method is a well-established method to solve various problems of science and engineering. Different authors have used various methods to solve governing differential equation of heat conduction problem. In this study, heat conduction in a circular rod has been considered which is made up of two different materials viz. aluminum and copper. In earlier studies parameters in the differential equation have been taken as fixed (crisp) numbers which actually may not. Those parameters are found in general by some measurements or experiments. So the material properties are actually uncertain and may be considered to vary in an interval or as fuzzy and in that case complex interval arithmetic or fuzzy arithmetic has to be considered in the analysis. As such the problem is discretized into finite number of elements which depend on interval/fuzzy parameters. Representation of interval/fuzzy numbers may give the clear picture of uncertainty. Hence interval/fuzzy arithmetic is applied in the finite element method to solve a steady state heat conduction problem. Application of fuzzy finite element method in the said problem gives fuzzy system of linear equations in general. Here new methods have also been proposed to handle such type of fuzzy system of linear equations. Corresponding results are computed and has been reported here.