Finite Element Method with the Interval Set Parameters and its Applications in Computational Science ANDRZEJ POWNUK The University of Texas at El Paso Department of Mathematical Sciences 500 West University Avenue, El Paso, Texas USA andrzej@pownuk.com, http://andrzej.pownuk.com BEHZAD DJAFARI-ROUHANI The University of Texas at El Paso Department of Mathematical Sciences 500 West University Avenue, El Paso, Texas USA behzad@math.utep.edu NAVEEN KUMAR GOUD RAMUNIGARI The University of Texas at El Paso Department of Civil Engineering 500 West University Avenue, El Paso, Texas USA r.naveengoud@gmail.com Abstract: The Finite Element Method (FEM) is one of the most popular approach to describe engineering structures today. In order to apply this method efficiently, it is necessary to know the exact values of all parameters. In the case of uncertain shapes, the FEM method leads to a parameter dependent system of algebraic equations with the interval set parameters. In this paper the solutions for such equation will be presented. The method use the concept of topological derivative and monotonicity. Numerical examples will be presented. Key–Words: Interval sets, uncertainty, interval functional parameters, finite element method. 1 Engineering problems with the un- certain shape Almost all engineering problems require a very pre- cise information about the geometry (eg. height, thickness, curvature, coordinate of the characteristic points of the structure etc.) of the problem. Unfor- tunately, due to many reasons (unavoidable inaccu- racy in the construction process, bad materials, etc.) the real dimension of the engineering structure are not know exactly [5, 6, 12]. Civil engineering projects are usually very unique. Because of that it is very hard to get reliable probabilistic characteristics of the structures. One of the simplest methods for modeling uncertainty is based on the intervals. If Ω denotes the domain of the structure, then, due to uncertainty, we can assume that Ω ∈ [Ω , Ω] (1) where Ω , Ω denotes the extreme value of the shapes. If u = u(x, Ω) is a characteristic of the structure, e.g. displacement, then in the case of the uncertainty, in- stead of one number we have the whole interval [u (x), u(x)] = {u(x, Ω) : Ω ∈ [Ω , Ω]} (2) In this paper, some procedures for calculations [u (x), u(x)] will be presented. Problems with uncer- tain parameters were considered in the paper [7]. 2 Examples of set dependent func- tions 2.1 Center of gravity x coordinate of the center of gravity is a good example of a set dependent function x C (Ω) =  Ω xdμ  Ω dμ . (3) 2.2 Moment of inertia Different kinds of moment of inertia are also defined by using integrals, and they are also set dependent. I y (Ω) =  Ω x 2 dμ, I 0 (Ω) =  Ω (x 2 + y 2 )dμ (4)