Case Study 2
Here, we have considered a triangular (equilateral) bare reactor, having each side of 4 units, and it is discretized into a triangular element, as shown in Figure 6.11.
Fuzzy parameters are taken for the diffusion and absorption coefficients, which are presented in Table 6.8.
Initially, the governing onegroup neutron diffusion equation is solved by considering only crisp parameters, and then the preceding method is used to solve the problem. Eigenvalues for both the crisp and fuzzy parameters are obtained, and the values are depicted in Table 6.9 for a different number of elements in the FEM and FFEM discretization.
For better visualization of the obtained results, eigenvalues for different numbers of discretizations of the domain are plotted and shown in Figures 6.12 through 6.19.
FIGURE 6.11
Triangular element discretization of triangular plate.
TABLE 6.8
Triangular Fuzzy Numbers for Uncertain Parameters
Parameters 
Crisp Value 
TFN 

Diffusion coefficient 
1 
[0.5 + 0.5a, 1.5  
 0.5a] 
Absorption coefficient 
1 
[0.5 + 0.5a, 1.5  
 0.5a] 
TABLE 6.9
Crisp and Triangular Fuzzy Eigenvalues for Triangular Plate
Number of Elements 
Crisp Eigenvalues 
Triangular Fuzzy Eigenvalues 
6 
0.6425 
[0.6377, 0.6425, 0.647] 
12 
0.6264 
[0.6236, 0.6264, 0.6297] 
24 
0.526 
[0.5251, 0.526, 0.527] 
48 
0.5083 
[0.508, 0.5083, 0.5087] 
96 
0.5034 
[0.5032, 0.5034, 0.5036] 
192 
0.5015 
[0.5015, 0.5015, 0.5016] 
384 
0.5007 
[0.5007, 0.5007, 0.5008] 
1536 
0.5002 
[0.5002, 0.5002, 0.5002] 
FIGURE 6.12
Six elements discretization of the domain.
The variation of both the crisp and fuzzy eigenvalues may be studied from Figure 6.20. Here, a set of eigenvalues are given and the convergence is studied.