# Ultrasonic NDT in Defect Detection - Mathematical modelling

Sammanfattning: This thesis is concerned with the mathematical modelling of the ultrasonic nondestructive testing situation.

The first part of the thesis treats a simplified model of an ultrasonic transducer and its implementation into a T-matrix method-based solution to a crack scattering problem. The source is an acoustic piston-like model of an unangled compressional transducer acting in pulse echo situation with its behaviour as a receiver treated with reciprocity. The considered crack is a freely oriented penny-shaped crack-like flaw, partly closed due to an external background pressure. In order to evaluate and verify the numerical results, comparisons with numerical calculations, provided by a GTD method-based program package, are performed.

The elastodynamic equivalence to the above is then considered. A more general probe model is proposed and modelled as boundary conditions on an elastic half-space. The action of the probe as a receiver is treated with a reciprocity argument. The electric signal is then obtained essentially as a product of the spherical expansion coefficients of the transmitting probe, the transition matrix of the defect and the spherical expansion coefficients of the receiving probe. The ultrasonic contact probe and its options as a transmitter are thoroughly investigated and several numerical examples are provided in the middle of the thesis. Numerically computed contour plots of the displacement field beneath the probe are given as snapshots, with varying contact conditions, frequency, bandwidth, angles and types. In order to enable experimental verification, a model for the ultrasonic probe evaluation situation, including the transmitting probe, an elastic plate and the receiving probe, has been developed. The reflection matrix representing reflection from the backwall is included and a model of the backwall measurement situation is deduced. Numerical examples are provided with varied couplants, shapes of effective area, traction distributions and different input signals. The final part of the thesis treats the ultrasonic inverse problem of determining the crack given the input and output electric signals. The numerical solution is achieved by applying optimisation techniques to the mathematical model of the ultrasonic NDT situation. The inverse problem is reduced to minimisation of a non-linear least squares problem and is performed with a quasi-Newton algorithm consisting of a locally convergent SVD-Newton method combined with a backtracking line search algorithm. Numerical examples for three specified realistic NDT situations are presented.

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