Identification of the electrical conductivity (Problem A1.2)

Since the aim of the present work is the identification of the material parameters we will consider the following: as described previously in section 6.1.2 we will use the identified heat source for the determination of the spatially distributed electrical conductivity as described in problem A1.2. For the optimization the gradient method is applied to the objective functional (cf. (A1.2’))

F(φ, σ) = 1 2S

p σ|∇φ|²−u_g

²_L2(Ω)+β

2∇σ²_L2(Ω).

For the results presented in figure 6.22 the previously identified heat source is used as measurement data u_g for the optimal control problem described in 5.2.2.3. The initial value is chosen as above asσ₀ = 3.0 S m⁻¹ with a scaling factorp= 350.Since the used generator power is much lower than in the example above, the scaling factor has to be smaller. With a tolerance of tol = 10⁻⁸ and a regularization parameter β = 10⁻⁸ the optimization with the gradient method needs 7 iterations for the presented results. The progression of the objective values is presented in figure 6.23.

The electrical conductivity is identified only accurate to a multiplication factor, which is set to one at the presented plot. Even if the conductivity does not match the given

6.2 Data from Agar-model

Figure 6.22: At the top left the calculated electrical conductivity (green asterisks) in comparison with the assumed conductivity (red circles) with 1.5% ature dependency is displayed. At the top right the corresponding temper-atures are depicted. The corresponding power density (green asterisks) is shown at the bottom left, compared with the previously identified power density (blue triangles). All displayed values are taken along the red line, depicted in the picture of the calculated power density at the bottom right.

5600 5700 5800 5900 6000 6100 6200 6300

1 2 3 4 5 6 7

objective values

number of iterations

Figure 6.23: The progression of the objective values for the identification of the electrical conductivity from temperature measurements of an RFA in Agar-gel via a previously identified heat source.

σ(T) = 0.56 + 0.015(T −T_body)/K, the tendency is visible. Moreover the correct elec-trical conductivity is not known and the used approximation with a linear temperature dependent conductivity is only an estimation [32]. Since the heat sourceu_g used in the objective functional is an approximation as well the results may be not as good as in the artificial setting. However, even for the artificial setting with the previously identified heat source the exact identification of the electrical conductivity was impossible at the tissue types which differ from the main liver tissue. Moreover, the initial temperature distribution is not the best data base since by rotating the two dimensional data addi-tional artifacts will appear and debase the identification.

In the figures 6.24 and 6.25 the identified electrical conductivity for a modified setting is presented. Instead of using c= 4181.3 J kg⁻¹K⁻¹ for the heat capacity, comparable with the heat capacity of water, we use the same capacity as in the artificial setting c= 3455 J kg⁻¹K⁻¹.The same for the density, which is no longer adapted to the density of water but to the human tissueρ= 1080 kg m⁻³.The scaling factor is set to p= 700.

The presented results are from two different settings. The first one is the identification of the electrical conductivity from a previously identified heat sourceug.The second one is slightly modified such that the heat sourceu_g is additionally smoothed. That means the heat source is convolved with a Gaussian function with meanκ= 0 and a standard deviationς= 1.5.For the first setting a regularization parameter ofβ = 2·10⁻⁸ is used whereas in the second case the regularization parameter is set toβ = 10⁻⁸.The resulting identified electrical conductivity for the second setting is considerably smoother than the one from the first setting without smoothed input data. These results illustrate that if the input data, i.e.ug, are smooth the resulting electrical conductivity will be smoother too and the calculated power density will fit the given one in a better way.

Further, the quality of the results depends on the spatial position in the three dimen-sional data setting. Especially the results for the non-smoothed input data are spatially varying a lot. Comparing the results for the temperature in figure 6.24 with those in figure 6.25 it is noticeable that the match of the temperature to the given one depends on the spatial distribution. In figure 6.25 the temperature calculated from the identified electrical conductivity fits the given values much better than the temperature calcu-lated corresponding to the electrical conductivity identified from the smoothed right heat source. On the other hand in figure 6.24 it is vice versa. This points out that a previously smoothing of the input data simplifies the identification of the electrical conductivity which additionally becomes smoother. However, if we do not know how smooth the original conductivity was, this may lead to more errors than an identification without smoothing. Furthermore even if the whole output is smoother the non-smooth output may fit in a better way to the given data, at least partially as seen above.

Since the scaling term ^P_P^eff

Ω is replaced by a scalarp= 700 (orp= 350 in the first setting) the identified values reflect the conductivity except a multiplication factor that is not known and cannot be calculated. However the calculated power density and the resulting temperature distribution are not affected by this unknown factor. The only limitation is that we cannot really compare the values with the electrical conductivity we assume to

6.2 Data from Agar-model be a good approximation. The results in figure 6.26 illustrate that we have to be careful in choosing the scaling factor due to the spatial position where we compare the results.

In the figure the results are multiplied with 0.8 and 1.2, which are chosen arbitrarily. The displayed diagrams suggest a multiplication factor larger than 1 will be a good choice for the present problem. However, the choice of the multiplication factors will always pose a challenge in real applications since the exact parameters are not known and even the approximations are not that good, otherwise we would not need an identification of the parameters. On the other hand the temperature distribution according to the identified parameters will be the same whatever is chosen as multiplication factor.

However, even if the presented results for the identification of the electrical conductivity from the measurements have not been proved satisfactory, the results are an improvement compared to the assumption of a constant value. In figure 6.27 temperature distributions calculated under different assumptions are compared. The original given temperature as well as the temperature corresponding to the identified heat source, the temperature according to the identified electrical conductivity and the temperature calculated with a constant electrical conductivity of σ = 0.56 Sm⁻¹ are displayed at different positions in the three dimensional object. The figures illustrate that the identified electrical con-ductivity accounts more for diverse spatial informations. Comparing the corresponding temperature distributions for the identified electrical conductivity and the constant con-ductivity, the results approve that the temperature for the identified conductivity fits better to the original data. Whereas for some spatial positions the temperature for a constant conductivity is superior.