Case Study 1

The governing differential equation in crisp form for the bare homogeneous reactor (Glasstone and Sesonke 2004) is as follows:

The boundary condition is ф(х, ±1.5) = 0 = ф(±1.5, y), and it is solved first by the classical (traditional) FEM for the sake of completeness, and then the FFEM is presented. Here, the square homogeneous region has been divided into 18 and 72 elements as shown in Figures 6.1a and b and 6.2a and b. Two different types of discretizations are considered and the results are compared.

Initially, the eigenvalues and corresponding effective multiplication factors are investigated when the involved parameters, viz. diffusion coefficient (D) and absorption coefficient (c), are crisp. The different values of these parameters are given in Table 6.1. The observed results for two different types of discretizations are presented in Tables 6.2 through 6.7.

When neutrons undergo diffusion, they suffer scattering collisions with the nuclei assumed to be initially stationary, and as a result, a typical neutron trajectory consists of a number of short-path elements. These are scattering free paths which are uncertain in nature. As a result, the diffusion coefficient will lie in an uncertain region and may become fuzzy. Similarly, the absorption coefficient may also be taken as fuzzy. Here, we have taken two different types of fuzzy numbers (TFN and TRFN) to handle these uncertainties. The uncertain values along with the crisp values are given in Table 6.1.

FIGURE 6.1

Domain is discretized (type 1) into (a) 18 elements and (b) 72 elements.

FIGURE 6.2

Domain is discretized (type 2) into (a) 18 elements and (b) 72 elements.

TABLE 6.1

Crisp and Fuzzy Values of the Involved Parameters

Parameters	Crisp Value	TFN	TRFN
Diffusion coefficient	1	[0.5 + 0.5a, 1.5 - 0.5a]	[0.5 + 0.3a, 1.5 - 0.3a]
Absorption coefficient	1	[0.5 + 0.5a, 1.5 - 0.5a]	[0.5 + 0.3a, 1.5 - 0.3a]

TABLE 6.2

Comparison of Eigenvalues When a = 1 and D = [0.5 + 0.5a, 1.5 - 0.5a]

Number of Elements	Classical FEM (Figure 6.1a and b)	Proposed Fuzzy FEM (Figure 6.1a and b)	Classical FEM (Figure 6.2a and b)	Proposed Fuzzy FEM (Figure 6.2a and b)
18	2.6667	[1.3333 + 1.3334a, 4 - 1.3333a]	2.8195	[1.4097 + 1.4098a, 4.2293 - 1.4098a]
72	2.3426	[1.1713 + 1.1713a, 3.5140 - 1.1714a]	2.3454	[1.1727 +1.1727a, 3.5181 - 1.1727a]

TABLE 6.3

Comparison of Eigenvalues When D = 1 and a = [0.5 + 0.5a, 1.5 - 0.5a]


Number of Elements	Classical FEM (Figure 6.1a and b)	Proposed Fuzzy FEM (Figure 6.1a and b)	Classical FEM (Figure 6.2a and b)	Proposed Fuzzy FEM (Figure 6.2a and b)
18	2.6667	[1.7778 + 0.48887a, 5.3333 - 2.6666a]	2.8195	[1.8797 + 0.9398a, 5.6391 - 2.8196a]
72	2.3426	[1.5618 + 0.7808a, 4.6853 - 2.3427a]	2.3454	[1.5636 + 0.7818a, 4.6908 - 2.3454a]

TABLE 6.4

Comparison of Eigenvalues When D = [0.5 + 0.5a, 1.5 - 0.5a] and a = [0.5 + 0.5a, 1.5 - 0.5a]

Number of Elements	Classical FEM (Figure 6.1a and b)	Proposed Fuzzy FEM (Figure 6.1a and b)	Classical FEM (Figure 6.2a and b)	Proposed Fuzzy FEM (Figure 6.2a and b)
18	2.6667	[0.889 + 1.7777a, 8 - 5.3333a]	2.8195	[0.9398 + 1.8797a, 8.4587 - 5.6392a]
72	2.3426	[0.7809 + 1.5617a, 7.0279 - 4.6853a]	2.3454	[0.7818 + 1.5636a, 7.0363 - 4.6909a]

TABLE 6.5

Comparison of Eigenvalues When a = 1 and D = [0.5 + 0.3a, 1.5 - 0.3a]


Number of Elements	Classical FEM (Figure 6.1a and b)	Proposed Fuzzy FEM (Figure 6.1a and b)	Classical FEM (Figure 6.2a and b)	Proposed Fuzzy FEM (Figure 6.2a and b)
18	2.6667	[1.3333 + 0.8a, 4 - 0.8a]	2.8195	[1.4097 + 0.8459a, 4.2293 - 0.8458a]
72	2.3426	[1.1713 + 0.7028a, 3.5140 - 0.7028a]	2.3454	[1.1727 + 0.7036a, 3.5181 - 0.7036a]

TABLE 6.6

Comparison of Eigenvalues When D = 1 and a = [0.5 + 0.3a, 1.5 - 0.3a]

Number of Elements	Classical FEM (Figure 6.1a and b)	Proposed Fuzzy FEM (Figure 6.1a and b)	Classical FEM (Figure 6.2a and b)	Proposed Fuzzy FEM (Figure 6.2a and b)
18	2.6667	[1.7778 + 0.4444a, 5.3333 - 2a]	2.8195	[1.8797 + 0.4699a, 5.6391 - 2.1147a]
72	2.3426	[1.5618 + 0.3904a, 4.6853 - 1.757a]	2.3454	[1.5636 + 0.3909a, 4.6908 - 1.759a]

TABLE 6.7

Comparison of Eigenvalues When D = [0.5 + 0.3a, 1.5 - 0.3a] and a = [0.5 + 0.3a, 1.5 - 0.3a]

Number of Elements	Classical FEM (Figure 6.1a and b)	Proposed Fuzzy FEM (Figure 6.1a and b)	Classical FEM (Figure 6.2a and b)	Proposed Fuzzy FEM (Figure 6.2a and b)
18	2.6667	[0.889 + 0.8888a, 8 - 4a]	2.8195	[0.9398 + 0.9399a, 8.4587 - 4.2293a]
72	2.3426	[0.7809 + 0.7809a, 7.0279 - 3.5139a]	2.3454	[0.7818 + 0.7818a, 7.0363 - 3.5181a]

FIGURE 6.3

TFN for Figure 6.1a.

The uncertain values are used in the FFEM and eigenvalues are obtained. The uncertain fuzzy eigenvalues under different considerations are shown in Tables 6.2 through 6.7.

In view of these tabulated eigenvalues, the corresponding effective multiplication factors (keff) are plotted. These are given pictorially in Figures 6.3 through 6.10.