Formulation of the Problem
As it is known that the principle of neutron conservation can be expressed in a simple form for a system of mono-energetic neutrons, one-group equations can be analyzed by considering the series of one-group equations.
The standard functional for the one-group diffusion equation may be written as
where
ф is the neutron flux D is the diffusion coefficient о is the absorption coefficient S is the source term
In the traditional FEM, the domain of the problem is divided into a number of subdomains and each is called an element. For each element, we may find the functional, and similarly, for the entire domain, the functional can be found by summing each functional element wise. This procedure may be written in the following way.
First, the domain R may be represented as
and the functional 1(ф) is defined as
where
n is the total number of elements
Т(ф) denotes the contribution of the element e to the functional 1(ф)
Now, Equation 6.1 for each elemental functional may be written as
For each element e, the scalar flux фе is approximated by a piece-wise interpolation polynomial. Depending on the interpolation polynomial, stiffness matrices are obtained by minimizing the elemental functional 1е(ф). The stiffness matrices are assembled, and finally, we get the algebraic form, which is represented as
where
[X] is the assembled stiffness matrix corresponding to the leakage and absorption terms {Q} is the assembled force vector for the source term
In general, when neutrons undergo scattering, the neutron transport equation involves uncertainty. The uncertainty occurs due to the imprecise value of the operating parameters, viz. geometry, diffusion and absorption coefficients. Considering uncertain parameters as fuzzy, we now investigate the uncertain spectrum of the neutron flux distribution. Accordingly, we now formulate the FFEM with the linear triangular fuzzy element discretization of the domain.
Let us write the coordinates of linear triangular elements in fuzzy form as
where Li (i = 1, 2, 3) are non-dimensionalized coordinates.
Equation 6.6 may be written in matrix form
111
~ 1
where the area of the fuzzy triangle is D = ^ X1 x2 x3.
y1 y2 y3
We now denote
If ф is the flux distribution, then it may be written as
The differentiation and integration formulae are then given by
Hence,
Similarly,
Using this formulation, one may get the leakage and absorption stiffness matrices. Accordingly, corresponding stiffness matrices of each element for the leakage and absorption terms are found to be
The source vector f} for each element may be written as
The limit method for fuzzy arithmetic in terms of а-cut has been used here for FEM. As such, using FFEM (Nayak and Chakraverty 2013), the uncertain fuzzy parameters are handled. The schematic diagram is presented in Figure 4.1, which gives an overall idea to encrypt the process of modified FFEM. It involves three steps such as the input, the output and the hidden layer. The uncertain (fuzzy) parameters are accumulated through a process, viz. FFEM, which plays the role of the hidden layer. Using the fuzzy input parameters, the hidden layer performs the fuzzy finite element procedure and gives fuzzy solutions in terms of fuzzy as output. In Figure 4.1, triangular fuzzy numbers have been considered as input parameters. The alpha (a)-level representation of two fuzzy sets X and Y with their triangular membership functions are operated through fuzzy arithmetic operation
(Chakraverty and Nayak 2012). The deterministic value is obtained for a4 level of fuzzy sets, whereas for a1, a2 and a3 levels we get interval values. The output can be generated by taking the union of all possible combinations of the alpha level.
