Generalised bisection method for optimum ultrasonic ray tracing and focusing in multi-layered structures

Ultrasonic testing has been used for many decades, proving itself very efficient for detecting defects in many industrial sectors. The desire to apply ultrasonic testing to geometrically complex structures, and to anisotropic, inhomogeneous materials, together with the advent of more powerful electronics and software, is constantly pushing the applicability of ultrasonic waves to their limits. General ray tracing models, suitable for calculating the proper incident angle of single element probes and the proper time delay of phased array, are currently required. They can support the development of new imaging techniques, as Full Matrix Capture and Total Focusing Method, and the execution of very challenging ultrasonic inspections. This paper introduces a generalized iterative method for the computation of ultrasonic ray paths, when ultrasonic source and target are separated by multiple complex material interfaces in the two dimensional and three dimensional domains. The manuscript starts with a review of the well-known bisection method, and extends the applicability of the method to cases with increasing complexity. An application example, in the field of in-process weld inspection, shows that the introduced generalised bisection method can enable the computation of optimum incidence angles and focal delays for accurate ultrasonic focusing. There is no restriction on the analytical interfaces to be surjective. Interface folding is permitted. It is not necessary to know, a priori, with what sequence the interfaces are crossed by the rays. The presented implementation of the method completes each iteration of the bisection method in 4 ms, for a case with a single interface, and in 960 ms for the case with 52 interfaces.