An Efficient Numerical Method for Determination of Shapes, Sizes and Orientations of Flaws for Nondestructive Evaluation

Abstract

<p>The generalized pulse-spectrum technique (GPST)1 is a versatile and effcient iterative numerical algorithm for solving inverse problems (to determine the unknown coefficients, initial-boundary values, sources, and geometries of the space domain from the additionally measured data in the space-time domain or the space-complex frequency domain) of a system of nonlinear partial differential equations. Mathematically, inverse problems of partial differential equations can be formulated as ill-posed nonlinear operator equations. It is important to point out that the GPST is not a single narrowly defined iterative numerical algorithm but a broad class of iterative numerical algorithms based on the concept that either the nonlinear operator equation is first linearized by any one of the Newton-like iteration methods and then each iterate is solved by using a stabilizing method, e.g., the Tikhonov’s regularization method2, or the stabilizing method is first applied to the nonlinear operator equation and then the regularized nonlinear problem is solved by using a Newton-like iteration method. Hence different choices of various Newton-like iteration methods and stabilizing methods lead to different special forms of GPST, and the efficiency of GPST will then depend upon the particular choices of them and how efficiently one can treat every minute step in the numerical algorithm.</p>