Advanced Numerical and Semi-Analytical Methods for Differential Equations

21Interval Finite Element Method

In Chapter 6, the crisp (certain) differential equations were solved using finite element method (FEM). In case of uncertainty, such differential equations are referred to as uncertain differential equations. As such, in this chapter we focus on solving uncertain (in terms of closed intervals) differential equation using Galerkin FEM viz. interval Galerkin FEM (IGFEM). Further, static and dynamic analyses of uncertain structural systems have also been discussed.

21.1 Introduction

As already mentioned earlier that in various engineering disciplines viz. structural mechanics, biomechanics, and electromagnetic field problems, the FEM is widely applicable for computing approximate solution of complex systems of which exact solutions may not be obtained. Readers interested in detailed study of FEM and its applications to various engineering fields are encouraged to refer Refs. [1–4]. In actual practice, the variables or parameters exhibit uncertainty due to measurement, observation, or truncation errors. Such uncertainties may be modeled through probabilistic approach, interval analysis, and fuzzy set theory. But probabilistic methods are not able to deliver reliable results without sufficient experimental data. Moreover, in fuzzy set theory a fuzzy number is approximately represented by a set of closed intervals using the α‐cut approach. As such, interval uncertainty is sufficient to understand since it forms a subset of fuzzy set. The application ...