# Fuzzy Finite Element Method

If the parameters as well as initial and boundary conditions are uncertain, then the governing differential equations become uncertain. Accordingly, the uncertainties are considered as intervals/fuzzy (Hanss 2005), and interval/FFEM is developed. Considering field variables and the involved parameters as interval/fuzzy, we get interval/fuzzy element properties in

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terms of interval/fuzzy matrices. Then, those matrices are assembled and the global matrix is obtained. Furthermore, boundary conditions are imposed, which may also be interval/fuzzy. From the global matrices and boundary conditions, we get a system of algebraic equations that are either a system of simultaneous equations or eigenvalue problems. It may be noted that due to the uncertainties (i.e. interval/fuzzy), we now have the interval/fuzzy system of simultaneous equations or eigenvalue problems. Here, a system of interval/fuzzy algebraic equations are investigated by using the interval/fuzzy arithmetic (limit method).

Figure 4.1 shows a schematic diagram of the fuzzy finite element procedure, which gives the basic idea to encrypt the process of the FFEM. We have modified the usual interval/ fuzzy arithmetic in the FEM. It involves three steps: input, hidden layer and output. In the input step, we have considered uncertain parameters and field variables. These uncertain parameters are taken as fuzzy. In the hidden layer step, element properties are obtained by using various fuzzy parameters. The element properties and fuzzy stiffness matrices are assembled and finally the global stiffness matrices are developed. Further, initial and boundary conditions are imposed and the transformed fuzzy system of equations is solved through the limit technique (Nayak and Chakraverty 2013), and various sub-steps are executed inside the hidden layer. Finally, we get uncertain fuzzy solutions as the output, which may be different in type and nature corresponding to the input fuzzy parameters. Here, in Figure 4.1, we have considered triangular fuzzy numbers as input parameters. The alpha (a)-level representation of two fuzzy sets *X* and Y with their triangular membership functions for the fuzzy arithmetic operation (Nayak and Chakraverty 2013) is shown in Figure 4.1. The deterministic value is obtained for a_{4} level of fuzzy sets, whereas for a_{1}, a_{2 }and a_{3} levels, we get different interval values. If we consider the value of a to be zero, then the deterministic interval lies on the x-axis. The output may be generated by considering all possible combinations of the alpha (a) levels.

FIGURE 4.1

Model diagram of the fuzzy finite element procedure.