An Optimization Technique for Inverse Crack Detection

Abstract

Any attempts to apply techniques that are based on indirect measurements of parameters that are believed to correlate to any material properties (or state) in an in-line situation must by necessity identify a mathematical model of this relationship. The most conventional approach is to use some empirically based model. If the analysis instead is based on an analytical model of a physical explanation, this trainee period can be minimized and the system is more dynamic and less sensitive to changes within the chain of production. A numerical solution to the inverse problem of ultrasonic crack detection is in this case investigated. This solution is achieved by applying optimization techniques to a realistic model of the ultrasonic defect detection situation. This model includes a general model of an ultrasonic contact probe working as transmitter and/or receiver and its interaction with the defect. The inverse problem is reduced to minimization of a nonlinear 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. The set of synthetic data the model is fitted with are generated both by numerical integration and with the two-dimensional stationary-phase method while the forward solver in the optimization procedure is based on the latter. In both these cases, the convergence, in terms of numbers of iterations, is sufficient when the initial guess is reasonably close.