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The inverse operational problem for the frequency-impedance model of the nonuniform acoustic medium: numerical and experimental implementations


B.K. Temyanov - Assistant, Department of «Radioelectronics and Quantum Devices», Kazan National Research Technical University named after A.N. Tupolev (KNRTU-KAI). E-mail: Yu.K. Evdokimov - Dr.Sc. (Eng.), Professor, Head of Department of «Radioelectronics and Quantum Devices» Kazan National Research Technical University named after A.N. Tupolev (KNRTU-KAI). E-mail:

Existing methods of ultrasonic inspection reveal defects in the studied objects in the form of sharp violations of homogeneity of the mechanical properties of sensed medium. In this paper, a numerical simulation and experimental implementation of the algorithm of continuously varying parameters reconstruction using the input complex conductivity (admittance) in the one-dimensional approximation were considered. The most difficult stage of numerical simulation was the solving Fredholm integral equation of the first kind with the use of regularization on the basis of the generalized discrepancy principle that determines the optimal value of the regularization parameter in accordance with a right side error of the integral equation. The error of the right side was simulated by random variable with known root – mean –square value. The values of the regularization parameter chosen in accordance with the generalized discrepancy principle using simulated right part error do not coincide with the optimum values. Despite this, the iterative structure of the algorithm with additional conditions, allows us to operate even with large differences between the chosen and the optimal values of the regularization parameter. An additional condition is the condition of not exceeding 100% accuracy of the solution at each iteration. In this case the obtained approximation will tend to the exact solution. Test regularization modeling showed that if the matched load is connected to the end of line error rate on the iteration does not exceed 100% in a wide range of values of the regularization parameter that determines high accuracy of the algorithm in this case. If a short circuit or idle running at the end of a line error value at iteration acquire much higher values, which leads to large errors in the solution. Given this fact, further reasoning were only the case of the connection of the matched load in the end of medium. Two functions of heterogeneity in the form of quadratic polynomials defining the distribution of continuously varying parameters of the sensed medium were considered using which the input admittance was calculated with the measurement error of 0.1% and 1%. The reconstruction errors for the case monotonically increasing function heterogeneity reached the values of the 0, 2% and 0.5%, respectively, at different levels of the input admittance measurement error. For the case of a monotonically decreasing function of heterogeneity error solutions were somewhat higher: 0.2% and 0.7%, respectively. The optimal values of frequency ranges of measurements have also been identified. Experimental realization of the algorithm was carried out on the basis of hardware-software complex of ultrasonography TOMOSCAN FOCUS LT. The main operating element of this complex is the ultrasonic phased array (UPA) is a set of piezoelectric elements in a polymeric insulating matrix. UPA is contacted with a test sample through a special wedge. One of the main stages of the experimental implementation was the measuring of the input admittance. The estimation of the input admittance was carried out on the results of measurements of the reflection coefficient of the border prism - the measurement object. For a more precise operation of the algorithm, all subsequent pulses after the first were cut off, so the connection of the matched load by the end of the line was simulated. Measurement error admittances were high enough; the standard deviation of the measured values relative to the average is 9%. The maximum standard deviation of the reconstructed functions heterogeneity relative to the average value does not exceed 7%. According to the results of the reconstruction we can conclude that the method restores the structure of the sample with high values of the errors introduced by the measurement of the input admittance.


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