**2.2 Fuzzy logic implementation**

A first attempt in applying artificial intelligence techniques to reduce the computational time needed by FEM to evaluate the MVP values for different problem geometries have been made in (Satsios et al. 1999a, 1999b). The presented Fuzzy Logic Block had as input values the separation distance, *d*, between EPL and MP, the soil resistivity, *ρ,* and the coordinates *(x,y)* of the point where MVP is wanted to be evaluated:

> 0 The rule from the presented Fuzzy Logic Block's rule base : : , , and belong to the membership function, , and correspondingly *th j th j j j d x y j j j j jjj d x y j R dxy j A dxy* **IF THEN** (4)

The proposed Fuzzy Logic Block showed relatively good results for MVP's amplitude and phase evaluation according to the training database created by calculating MVP with FEM for a set of known problem geometries (figure 5):

Fig. 5. Absolute evaluation error provided by the presented Fuzzy Logic Block.

To identify the proper rule base and the optimal parameters for each rule an iterative technique has been applied using gradient based relations like:

$$\nabla \frac{\sigma\_{\alpha^{\bar{j}}}}{\alpha^{\bar{j}}} I^{\bar{p}} = \frac{\mu^{\bar{j}}}{\sum\_{j} \mu^{\bar{j}}} \Big[ \mathcal{A}^{\mathcal{P}} \{d, \mathbf{x}, \mathbf{y}, \boldsymbol{\rho}\} - \mathcal{A}^{\mathcal{P}}\_{\text{FEM}} \{d, \mathbf{x}, \mathbf{y}, \boldsymbol{\rho}\} \Big] \cdot \Big[ \mathcal{A}^{\bar{j}} - \mathcal{A}^{\mathcal{P}} \{d, \mathbf{x}, \mathbf{y}, \boldsymbol{\rho}\} \Big] \cdot \frac{\Delta^{\mathcal{P}} - \alpha^{\bar{j}}\_{\Lambda}}{\left(\sigma^{\bar{j}}\_{\Lambda}\right)^{2}} \tag{5}$$

and

258 Recurrent Neural Networks and Soft Computing

A first attempt in applying artificial intelligence techniques to reduce the computational time needed by FEM to evaluate the MVP values for different problem geometries have been made in (Satsios et al. 1999a, 1999b). The presented Fuzzy Logic Block had as input values the separation distance, *d*, between EPL and MP, the soil resistivity, *ρ,* and the

: , , and belong to the membership function, , and

 

The proposed Fuzzy Logic Block showed relatively good results for MVP's amplitude and phase evaluation according to the training database created by calculating MVP with FEM

*d x y*

(4)

 

*j th j j j*

Fig. 4. Cross section of the studied EPL-MP interference problem.

coordinates *(x,y)* of the point where MVP is wanted to be evaluated:

*j j j jjj d x y*

 

*A dxy*

The rule from the presented Fuzzy Logic Block's rule base :

 

Fig. 5. Absolute evaluation error provided by the presented Fuzzy Logic Block.

0

 

for a set of known problem geometries (figure 5):

*R dxy j*

correspondingly

**2.2 Fuzzy logic implementation** 

*th*

**IF**

*j*

*j*

**THEN**

$$\nabla \frac{\nabla \overline{\rho^{\text{I}}}}{\sigma^{\text{I}}\_{\text{A}}} = \frac{\mu^{\text{j}}}{\sum\_{\text{j}} \mu^{\text{j}}} \Big[ \mathbf{A}^{\text{p}} \{ \mathbf{d}, \mathbf{x}, \mathbf{y}, \rho \} - \mathbf{A}^{\text{p}}\_{\text{FEM}} \{ \mathbf{d}, \mathbf{x}, \mathbf{y}, \rho \} \Big] \cdot \Big[ \mathbf{A}^{\text{j}} - \mathbf{A}^{\text{p}} \{ \mathbf{d}, \mathbf{x}, \mathbf{y}, \rho \} \Big] \cdot \frac{\Delta^{\text{p}} - \alpha^{\text{j}}\_{\text{A}}}{\left( \sigma^{\text{I}}\_{\text{A}} \right)^{3}} \tag{6}$$

where *dxy* ,,, and ,,, , ,,, *<sup>p</sup> <sup>p</sup> A dxy A dxy FEM* are the MVP values obtained with FEM and respectively.

In the following, in order to improve the accuracy of the obtained results and to simplify the implementation process of the applied artificial intelligence technique, the authors propose an alternative by using a Neural Network solution instead of the presented Fuzzy Logic Block presented in (Satsios et al. 1999a, 1999b).
