On applying fuzzy arithmetic to finite element problems

Fuzzy arithmetic, based on Zadeh´s (1973) extension principle, is applied to solve finite element problems with uncertain parameters. As an example, a rather simple, one-dimensional static problem consisting of a two-component massless rod under tensile load is considered. Application of fuzzy arithmetic directly to the traditional techniques for the numerical solution of finite elements, i.e. primarily on the algorithms for solving systems of linear equations, however turns out to be impracticable in all circumstances. In contrast to the use of exclusively crisp numbers, the results for the calculations including fuzzy numbers usually differ to a large extent depending on the solution technique applied. The uncertainties expressed in the different calculation results are then basically twofold. On one hand, uncertainty is caused by the presence of parameters with fuzzy value, whilst on the other, an additional undesirable uncertainty is artificially created by the solution technique itself. For this reason, an overview of the most common techniques for solving finite element problems is offered, rating them with respect to minimizing the occurence of artificial uncertainties. Moreover a special technique is outlined which leads to modified solution procedures with reduced artificial uncertainties