Three-dimensional ray tracing of laser ultrasound for weld penetration sensing

Abstract

The time-of-flight (TOF) method is an ultrasonic nondestructive testing (NDT) technique. The TOF of an ultrasonic wave can be correlated to weld penetration depth, and hence weld quality. Changes in material properties due to temperature gradients will cause ultrasonic speed to vary during welding, which causes a curved propagation path. A ray tracing algorithm is required in order to study how ultrasound propagates within a weld sample. In this paper, a three-dimensional (3-D) ray tracing algorithm based on Fermat’s principle is presented. First, ray equations are derived using the calculus of variation. Then, a numerical algorithm is developed to solve the derived ray equations and obtain the curved propagation path. This algorithm includes finite element analysis (FEA) to obtain the transient temperature distribution during the welding and shooting method to solve the boundary value problem. After the curved ray path is obtained, the TOF can be found by integrating the time variable along the ray path. An analytical relationship between the TOF and penetration depth can be established by repeating the ray tracing algorithm for different penetration depths. Experimental measurements of TOF have been performed, and this data is to be used to validate the numerical results.