## 5 Computational Tests for Identification of PS  
On the basis of the projection scheme proposed in Section 4, a number of computational tests were carried out to identify the distribution function \(s_{rr}^{0}(\xi)\) of the flat PS for both plates. In the first two tests, the stresses arising during the initial radial loading of the plate by a normal load \(p_{0}\) applied on the outer radius in the case of a solid plate and on the inner radius in the case of an annular plate were considered as the PS field. In this case, under elastic deformation for the initial problems, it is possible to write out exact solutions in stresses. For a solid plate, the distribution of radial stresses is constant [29]:  
\[s_{rr}^{0}(\xi)=p_{0}. \tag{18}\]  
For an annular plate, it can be defined as a solution to the corresponding Lame problem for a disk [34]:  
\[s_{rr}^{0}(\xi)=\frac{p_{0}}{\xi_{0}^{-2}-1}\bigg{(}1-\frac{1}{\xi^{2}}\bigg{)}. \tag{19}\]  
In this case, for both formulations of inverse problems, the form of the function \(s_{rr}^{0}(\xi)\) is known, and it is necessary to find only the parameter of the initial load, which determines the level of PS. For this, the developed scheme can also be applied, while in expansion (16), it is necessary to use only one basis function of the form (18) or (19), respectively. Then, we can determine the desired parameter \(p_{0}\) from the solution of Equation (17) for one selected probe frequency. **Test 1**. The level of the uniform field of the PS was determined in the case of a solid plate for different values. The probing frequency was chosen near the first resonance \(\kappa=8.45\). In addition, tests were carried out with an additive noise (according to normal distribution law) of the input data by 0.5%, which by a significant margin exceeds the error of modern measuring instruments used in the implementation of the acoustic method. The obtained identification results are shown in Table 1; the identification error is indicated in parentheses. The above results show that the developed technique is applicable, among other things, to noisy input data. **Test 2**. Determination of the initial loading parameter, which caused a PS of the form (19) in an annular plate, was carried out for its various values. The results of the test are presented in Table 2.  
Note that with a decrease in the value of the PS level, its influence on the frequency response decreases, and therefore, the accuracy of its determination deteriorates, which is also evidenced by the results given in Section 3. In the following tests, situations are considered when the origin of PS is generally unknown; they could arise both in the case of elastic or inhomogeneous plastic deformation \begin{table}
\begin{tabular}{c c c c} \hline
**No** & **Exact \(p_{0}\)** & **Recovery (_error, \%_)** & **Noise 0.5\%, (_error, \%_)** \\ \hline
1 & **0.01** & 0.00999 (_0.10\%_) & 0.00986 (_1.40\%_) \\ \hline
2 & **0.005** & 0.00497 (_0.60\%_) & 0.00487 (_2.60\%_) \\ \hline
3 & **0.001** & 0.00098 (_2.00\%_) & 0.00096 (_4.00\%_) \\ \hline
4 & **0.0005** & 0.00048 (_4.00\%_) & 0.00045 (_10.0\%_) \\ \hline \end{tabular}
\end{table} 
Table 1: Identification of the prestress level for a solid plate and test with noise. \begin{table}
\begin{tabular}{c c c} \hline
**No** & **Exact \(p_{0}\)** & **Recovery (_error, \%_)** \\ \hline
1 & **0.01** & 0.01003 (_0.30\%_) \\ \hline
2 & **0.005** & 0.00497 (_0.60\%_) \\ \hline
3 & **0.001** & 0.00101 (_1.00\%_) \\ \hline
4 & **0.0005** & 0.00049 (2.00\%) \\ \hline \end{tabular}
\end{table} 
Table 2: Identification of the prestress level for an annular plate. and could be influenced by temperature or other influences. In such cases, an economical and easy-to-implement approach is needed to assess the level and nature of the PS distribution in the plate. Similar advantages are possessed by the projection technique proposed in this work, which makes it possible to determine the approximation of the sought functions in the class of given ones, for example, linear or quadratic. **Test 3.